Moving Media Alta Frequenza Di Taglio

qualsiasi disturbo che viaggia attraverso un mezzo elastico come l'aria, terra, acqua o di essere ascoltato dall'orecchio umano. Quando un corpo vibra, o si muove avanti e indietro (vedi vibrazioni vibrazioni, in fisica, comunemente un movimento motionmdasha oscillatorio prima in un senso e poi di nuovo nella direzione opposta. Si è esposta, ad esempio, da un pendolo oscillante, dal rebbi di un diapason che è stato colpito, o dalla stringa di un musical. clicca sul link per ulteriori informazioni.), l'oscillazione provoca un disturbo periodico dell'aria circostante o altro mezzo che si irradia verso l'esterno in linea retta in forma di onda onda di pressione, in fisica, il trasferimento di energia dalla vibrazione regolare, o moto oscillatorio, o di qualche mezzo materiale o dalla variazione di grandezza dei vettori di campo di un campo elettromagnetico (vedi radiazione elettromagnetica). . Fare clic sul link per maggiori informazioni. L'effetto queste onde producono sull'orecchio è percepito come suono. Dal punto di vista della fisica, il suono è considerato onde del moto vibratorio stessi, anche se non sono ascoltati dall'orecchio umano. Generazione di onde sonore Le onde sonore sono generate da qualsiasi corpo vibrante. Ad esempio, quando una corda di violino vibra dopo essere piegato o strappato, il suo movimento in una direzione spinge le molecole dell'aria prima, li affollamento insieme nel suo percorso. Quando si muove indietro oltre la sua posizione originale e dall'altra parte, si lascia dietro uno spazio quasi vuoto, cioè uno spazio relativamente poche molecole in esso. Nel frattempo, tuttavia, le molecole che erano inizialmente ammassati hanno trasmesso parte della loro energia di movimento per altre molecole ancora più avanti e sono ritornare per riempire di nuovo lo spazio originariamente occupato e ora lasciato vuoto dalla stringa violino ritirata. In altre parole, il movimento vibratorio istituito dalla corda di violino provoca alternativamente in un dato spazio un affollamento insieme delle molecole d'aria (a condensazione) e un diradamento delle molecole (una rarefazione). Presi insieme una condensazione e rarefazione formano un'onda sonora una tale onda è chiamato longitudinale o compressivo, perché il moto vibratorio è avanti e indietro lungo la direzione che l'onda segue. Poiché un tale onda viaggia disturbando le particelle di un mezzo materiale, le onde sonore non possono viaggiare attraverso il vuoto. Caratteristiche delle onde sonore I suoni sono generalmente udibile dall'orecchio umano, se la loro frequenza (numero di vibrazioni al secondo) compreso tra 20 e 20.000 vibrazioni al secondo, ma la gamma varia notevolmente con l'individuo. Le onde sonore con frequenze inferiori a quelle delle onde acustiche sono chiamati subsonici quelli con frequenze superiori della gamma udibile sono chiamati ad ultrasuoni (vedi ultrasuoni ultrasuoni, studio e applicazione dell'energia delle onde sonore vibrante a frequenze superiori a 20.000 cicli al secondo, cioè oltre la gamma dell'udito umano. l'applicazione di energia sonora nella gamma udibile è limitata quasi esclusivamente alle comunicazioni, da allora. Fare clic sul link per maggiori informazioni.). Un'onda sonora è solitamente rappresentato graficamente uno ondulato, linea orizzontale la parte superiore dell'onda (cresta) indica una condensazione e la parte inferiore (trogolo) indica una rarefazione. Questo grafico, tuttavia, è solo una rappresentazione e non è un'immagine reale di un'onda. La lunghezza di un'onda sonora, o la lunghezza d'onda, è misurata come la distanza da un punto di maggiore condensazione al successivo o da qualsiasi punto su un'onda al punto corrispondente sulla successiva in un treno di onde. La lunghezza d'onda dipende dalla velocità del suono in un dato mezzo ad una data temperatura e dalla frequenza di vibrazione. La lunghezza d'onda di un suono può essere determinato dividendo il valore numerico per la velocità del suono nel mezzo data alla temperatura in frequenza di vibrazione. Ad esempio, se la velocità del suono in aria è 1.130 ft al secondo e la frequenza di vibrazione è 256, allora la lunghezza d'onda è di circa 4.4 ft. La velocità del suono non è costante, tuttavia, esso varia in diversi media e lo stesso mezzo a diverse temperature. Ad esempio, in aria a 0degC. è di circa 1.089 piedi al secondo, ma a 20degC. è aumentata a circa 1.130 ft al secondo, o un aumento di circa 2 piedi al secondo per ogni aumento di grado centigrado della temperatura. Il suono viaggia più lentamente nei gas rispetto a liquidi, e più lentamente nei liquidi che nei solidi. Poiché la capacità di condurre il suono dipende dalla densità del mezzo, i solidi sono conduttori migliori di liquidi, liquidi sono conduttori migliori di gas. Le onde sonore può essere riflessa, rifratti (o piegato), e assorbiti come le onde luminose possono essere. La riflessione delle onde sonore può provocare un eco eco, la riflessione di un'onda sonora alla sua fonte in forza sufficiente e con un tempo di ritardo sufficiente per essere separatamente distinta. Se un'onda sonora ritorna entro 1-10 secondi, l'orecchio umano non è in grado di distinguere da quello originale. . Fare clic sul link per maggiori informazioni. mdashan fattore importante nelle acustica acustica Gr., i fatti circa l'udito, la scienza del suono, compresi i suoi produzione, la propagazione e gli effetti. Vari rami di acustica che si occupano di diversi aspetti del suono e dell'udito comprendono bioacustica, fisica acustica, ultrasuoni, e architettonico. Fare clic sul link per maggiori informazioni. di teatri e auditorium. Un'onda sonora può essere rinforzata con onde da un corpo avente la stessa frequenza di vibrazione, ma la combinazione di onde di diverse frequenze di vibrazione può produrre battiti o pulsazioni o può provocare altre forme di interferenza interferenza, in fisica, l'effetto prodotto dalla combinazione o sovrapposizione di due sistemi di onde, in cui queste onde rafforzare, neutralizzano, o in altri modi interferire con l'altro. . Fare clic sul link per maggiori informazioni. Caratteristiche dei suoni musicali suoni musicali si distinguono dai rumori in quanto sono composti da regolari, vibrazioni uniformi, mentre i rumori sono vibrazioni irregolari e disordinati. Autori, tuttavia, usano spesso i rumori e suoni musicali. Un tono musicale si distingue da un altro sulla base di altezza, intensità o loudness e qualità o timbro. Pitch descrive come alto o basso tono è e dipende dalla rapidità con cui un corpo sonoro vibra, cioè dalla frequenza di vibrazione. Più alta è la frequenza di vibrazione, maggiore è il tono il passo di una sirena ottiene più in alto la frequenza di vibrazione aumenta. La variazione apparente nel campo di un suono come fonte avvicina o si allontana da un osservatore è descritto dall'effetto Doppler effetto Doppler, variazione della lunghezza d'onda (o frequenza) di energia sotto forma di onde, ad esempio suono o la luce, come risultato del movimento della sorgente o il ricevitore delle onde dell'effetto è chiamato per scienziato austriaco Christian Doppler, che ha dimostrato l'effetto. Fare clic sul link per maggiori informazioni. L'intensità o volume di un suono dipende dalla misura in cui il corpo sonoro vibra, cioè l'ampiezza della vibrazione. Un suono è più forte come l'ampiezza della vibrazione è maggiore, e l'intensità diminuisce all'aumentare della distanza dalla sorgente aumenta. Loudness è misurata in unità chiamate decibel decibel, abbr. dB, unità utilizzata per misurare il volume del suono. È un decimo di bel (chiamato per A. G. Bell), ma l'unità più grande è raramente usato. Il decibel è una misura dell'intensità sonora in funzione del rapporto di potenza, con la differenza in decibel tra due suoni essere. Fare clic sul link per maggiori informazioni. Le onde sonore emesse dai vari organi vibranti differiscono in termini di qualità, o il timbro. Una nota da un sassofono, per esempio, si differenzia da una nota della stessa altezza e l'intensità prodotta da un violino o di uno xilofono simile vibrante canne, le colonne d'aria, e archi tutto differiscono. Qualità dipende dal numero e intensità relativa dei toni prodotti dal corpo vibrante (vedi armonica armonica. 1 termine fisica descrive la vibrazione in segmenti di un corpo sonoro produttrice (vedi suono). Una stringa vibra contemporaneamente in tutta la sua lunghezza e segmenti della metà, terzi, quarti, ecc. Fare clic sul link per maggiori informazioni.), e questi a loro volta dipendono dalla natura del corpo vibrante. Bibliografia Cfr G. Chedd, suono (1970). L'eccitazione meccanica di un mezzo elastico. In origine, il suono è stato considerato solo ciò che si sente. Questo domande ammesse, come se suono è stato generato da alberi che cadono dove nessuno poteva sentire. Un approccio più meccanicistico evita questi problemi e permette anche disturbi acustici troppo alta frequenza (ultrasuoni) di essere ascoltato o troppo bassa (infrasuoni) per essere classificato come estensioni di quegli eventi che possono essere sentite. Una fonte di suono subisce rapidi cambiamenti di forma, dimensione, o posizione che disturbano elementi adiacenti del mezzo circostante, causando loro di muoversi loro posizione di equilibrio. Questi disturbi a loro volta vengono trasmessi elasticamente elementi vicini. Questa catena di eventi propaga a distanze più grandi, costituendo un'onda viaggiante attraverso il mezzo. Se l'onda contiene la gamma appropriata di frequenze e incide su un orecchio, che genera gli impulsi nervosi che sono percepite come l'udito. Pressione acustica Un'onda sonora comprime e dilata gli elementi materiali che passa attraverso, generando oscillazioni di pressione associati. Un apposito sensore (un microfono, per esempio) posta nel campo sonoro registrerà una deviazione variabile nel tempo dalla pressione di equilibrio trovato in quel punto nel fluido. La pressione totale cambiando P misurata varierà sulla pressione di equilibrio P 0 da una piccola quantità chiamata pressione acustica, p P - P 0. L'unità di misura della pressione è il pascal (Pa), pari a 1 newton per metro quadrato (Nm 2). Standard pressione atmosferica (14,7 lbin. 2) è di circa 1 bar 10 6 dynecm 2 10 5 Pa. Per un tipico suono in aria, l'ampiezza della pressione acustica può essere di circa 0,1 Pa (un milionesimo di atmosfera) più suoni causa relativamente lievi perturbazioni della pressione totale. Vedere pressione. misurazione della pressione. Trasduttori di pressione. Onde piane di pressione sonora Una delle onde sonore più fondamentali è l'onda piana in viaggio. Questo è un'onda di pressione progredire attraverso il mezzo in una direzione, dire la direzione x, con estensione infinita nelle yez direzioni. Un analogo bidimensionale è Ocean Surf avanzando verso un tempo molto lungo, dritto, e anche spiaggia. Vedi Wave (fisica). Equazione delle onde. Una moto ondoso più importante un'onda piana, chiamato harmonie, è il fluido oscillante onda piana monofrequenza descritto da Eq. (1). L'ampiezza di questa onda è P. La fase (argomento del coseno) aumenta con il tempo, e ad un punto nello spazio coseno passerà attraverso un ciclo completo per ogni incremento in fase di 2amppgr. Il periodo T richiesto per ogni ciclo deve essere tale che 2amppgr fT 2amppgr o T 1 f. in modo che f 1 T può essere identificato come la frequenza di oscillazione dell'onda di pressione. Durante questo periodo T. ogni porzione della forma d'onda è avanzata attraverso una TAC a distanza x3bb. e questa distanza x3bb deve essere la lunghezza d'onda. Questo dà la relazione fondamentale (2) tra la frequenza, lunghezza d'onda, e la velocità del suono c con qualsiasi mezzo. Ad esempio, in aria a temperatura ambiente, la velocità del suono è 343 ms (1125 FTS). Un suono della frequenza di 1 kHz (1000 cicli al secondo) avrà una lunghezza d'onda di x3bb C f 3.431.000 m m (1,1 ft). Le frequenze più basse avranno lunghezze d'onda più lunghe: un suono di 100 Hz in aria ha una lunghezza d'onda di 3,4 m (11 ft). Per confronto, in acqua dolce a temperatura ambiente la velocità del suono è 1480 ms (4856 FTS), e la lunghezza d'onda del suono 1 kHz è quasi 1,5 m (5 piedi), quasi cinque volte maggiore della lunghezza d'onda per la stessa frequenza in aria. Descrizione del suono La caratterizzazione di un suono si basa principalmente sulle risposte psicologiche ad esso. A causa della natura delle percezioni umane, le correlazioni tra valutazioni soggettive fondamentalmente come il volume, il tono e il timbro e le qualità più fisici come l'energia, la frequenza e spettro di frequenza sono sottili e non necessariamente universale. La forza di un'onda sonora è descritta dalla sua intensità. Dai principi fisici di base, la velocità istantanea in cui l'energia viene trasmessa da un'onda sonora attraverso unità di area è data dal prodotto della pressione acustica e la componente della velocità delle particelle perpendicolare all'area. Il tempo medio di questa quantità è l'intensità acustica. Se tutte le quantità sono espresse in unità SI (ampiezza della pressione o ampiezza pressione effettiva in Pa, velocità del suono in ms, e la densità in kgm 3), allora l'intensità sarà in watt per metro quadrato (Wm 2). Vedere intensità sonora causa del modo la forza di un suono è percepito, è diventato convenzionale per specificare l'intensità del suono in termini di una scala logaritmica con il (adimensionale) unità di decibel (dB). Un individuo con udito intatta ha una soglia di percezione vicino 10 -12 Wm 2 tra circa 2 e 4 kHz, la gamma di frequenza di maggiore sensibilità. Aumentando l'intensità di un suono di frequenza fissa, la valutazione soggettiva di rumore aumenta anche, ma non proporzionalmente. Piuttosto, l'ascoltatore tende a giudicare che ogni raddoppio successivo della intensità acustica corrisponde lo stesso aumento di volume. Per i suoni che giace superiore a 4 kHz o inferiore a 500 Hz, la sensibilità dell'orecchio è sensibilmente diminuita. Suoni a questi estremi di frequenza deve avere elevati livelli di intensità soglia prima che possano essere percepite, e raddoppio del volume richiede piccoli cambiamenti nell'intensità con il risultato che a livelli più alti suoni di pari intensità tendono ad avere loudnesses simili. È a causa di questa caratteristica che la riduzione del volume della musica registrata induce a suonare sottile o metallico, privo sia alti e bassi di frequenza. Poiché più apparecchiature audio-misura rileva la pressione acustica piuttosto che di intensità, è conveniente definire una scala equivalente in termini di livello di pressione sonora. Il livello di intensità e il livello di pressione sonora sono generalmente considerate come identiche, ma questo non è sempre vero. Vedere Decibel Come suono x201chighx201d di una particolare frequenza sembra essere descritto dal senso di campo. A pochi minuti con un generatore di frequenza e uno spettacolo diffusore che passo è strettamente legata alla frequenza. campo superiore corrisponde alla frequenza più alta, con piccole influenze seconda loudness, la durata e la complessità della forma d'onda. Per i toni puri (monofrequenza sembra) incontrato principalmente in laboratorio, passo e frequenza non si trovano ad essere proporzionale. Raddoppiare la frequenza inferiore raddoppia il campo. Per le forme d'onda più complesse solito incontrano, tuttavia, la presenza di armoniche favorisce un rapporto proporzionale tra passo e frequenza. Propagazione di onde piane sonori sono una notevole semplificazione di un campo sonoro reale. Il suono irradiato da una sorgente (ad esempio un altoparlante, un battito di mani, o una voce) deve diffondersi verso l'esterno molto simile ai cerchi ampliamento da un ciottolo gettato in un lago. Un semplice modello di questo caso più realistica è una sorgente sferica vibrante uniformemente in tutte le direzioni con una singola frequenza di movimento. Il campo sonoro deve essere simmetria sferica con una ampiezza che diminuisce all'aumentare della distanza dalla sorgente, e gli elementi fluidi deve avere velocità di particelle che sono diretti radialmente. Non tutte le sorgenti irradiano il loro suono in modo uniforme in tutte le direzioni. Quando qualcuno sta parlando in uno spazio confinato, ad esempio un campo aperto, un ascoltatore che circonda l'altoparlante ascolta la voce più ben definita quando l'altoparlante si trova di fronte l'ascoltatore. La voce perde definizione quando l'altoparlante è rivolto verso l'ascoltatore. frequenze più elevate tendono ad essere più pronunciati di fronte al diffusore, mentre le frequenze più basse sono percepiti più o meno uniformemente attorno all'altoparlante. La diffrazione E 'possibile sentire ma non vedere dietro l'angolo di un edificio alto. Tuttavia, audio-frequenza più alta (con lunghezza d'onda più corta) tende a piegarsi o x201cspillx201d meno intorno ai bordi e gli angoli di quanto non faccia rumore di frequenza inferiore. La capacità di un'onda a diffondersi dopo viaggiando attraverso un'apertura e piegare intorno agli ostacoli è chiamato diffrazione. Questo è il motivo per cui è spesso difficile da proteggere un ascoltatore da una fonte indesiderato di rumore, come il blocco aereo o rumore del traffico dalle residenze vicine. Semplicemente erigere un mattone o muro di cemento tra sorgente e ricevitore è spesso un rimedio insufficiente, perché i suoni possono diffract intorno alla parte superiore del muro e raggiungere gli ascoltatori con intensità sufficiente a distrarre o fastidioso. Vedere il rumore acustico. Diffrazione Poiché la velocità del suono varia con la temperatura locale (e pressione, in diversa gas perfetti), la velocità di un'onda sonora può essere una funzione della posizione. Diverse porzioni di un'onda sonora può viaggiare con diverse velocità del suono. Ogni piccolo elemento di una superficie di fase costante traccia una linea nello spazio, definendo un raggio lungo il quale viaggia energia acustica. Il fascio suono può quindi essere vista come un fascio di raggi, come un fascio di grano, con i raggi distribuiti su tutta l'area della sezione trasversale della superficie di fase costante. Come la maggior lobo si diffonde con la distanza, questa zona aumenta e i raggi sono meno densamente concentrate. Il numero di raggi per unità di area trasversale al percorso di propagazione misura la densità di energia del suono in quel punto. E 'possibile utilizzare il concetto di raggi per studiare la propagazione di un campo sonoro. I percorsi ray definiscono le traiettorie su cui l'energia acustica viene trasportato dal dell'onda viaggiante, e la densità di flusso dei raggi misura l'intensità si trovano in ogni punto dello spazio. Questo approccio, un modo alternativo per studiare la propagazione del suono, è approssimativa in natura, ma ha il vantaggio di essere molto facile da visualizzare. Riflessione e trasmissione Se un'onda sonora viaggia in un fluido colpisce un confine tra il primo fluido e un secondo, poi ci possono essere riflessione e trasmissione del suono. Per la maggior parte dei casi, è sufficiente considerare le onde essere planare. Il primo fluido contiene l'onda incidente di intensità I ie onda riflessa di intensità I r secondo fluido, da cui si riflette il suono, contiene l'onda trasmessa di intensità I t. Le direzioni dell'incidente, riflesso, e le onde sonore piane trasmesse possono essere specificati dal pascolo angoli x398 i. x398 r. e x398 t (misurata tra le rispettive direzioni di propagazione e il piano della superficie riflettente). Vedere Riflessione di assorbimento acustico Quando il suono si propaga attraverso un mezzo, ci sono una serie di meccanismi per cui l'energia acustica viene convertita in calore e l'onda sonora indebolito finché non è completamente dissipata. Questo assorbimento di energia acustica è caratterizzata da un coefficiente di assorbimento spaziale per viaggiare onde. Vedere assorbimento acustico La sensazione stimolata negli organi uditivi da un disturbo vibratorio. in senso lato, il movimento vibratorio delle particelle in media anelastico, moltiplicazione, come onde in gassoso, liquido o solido medium in senso stretto, un fenomeno che è percepito da un apposito organo sensoriale nell'uomo e negli animali. Gli esseri umani sentire il suono con una frequenza compresa tra 16 e 20.000 hertz (Hz). Il concetto fisico del suono comprende sia suono udibile e non udibile. Suono con una frequenza inferiore a 16 Hz è chiamato infrasuoni con una frequenza superiore a 20.000 Hz, ultrasuoni. ad alta frequenza molto onde elastiche nell'intervallo dal 10 al 10 9 12 ndash10 13 Hz sono chiamati iper-suono. La regione di frequenza infrasuoni ha incontrato sono praticamente limite non inferiore vibrazioni infrasoniche in natura a frequenze di decimi e centesimi di hertz. La gamma di frequenza delle onde ipersonico è limitata superiormente da fattori fisici che caratterizzano la struttura atomica e molecolare del mezzo: la lunghezza di un'onda elastica deve essere sostanzialmente maggiore della lunghezza percorso libero delle molecole nei gas e superiore alle distanze interatomiche nei fluidi e solidi. Come risultato, Hypersound non può propagarsi in aria ad una frequenza di 10 Hz 9 o superiore o in solidi ad una frequenza maggiore di 10 12 ndash10 13 rdquo Hz. Caratteristiche di base. Una caratteristica importante del suono è suo spettro, che è prodotto espandendo il suono in semplici vibrazioni armoniche (cosiddetta analisi suono di frequenza). Lo spettro è continuo se l'energia delle vibrazioni sonore e distribuita su una gamma di frequenze piuttosto ampio è uno spettro lineare se ha un insieme di componenti di frequenza discreti. Suono avere uno spettro continuo è percepito come un rumore come il fruscio delle foglie al vento o suoni di meccanismi di funzionamento. Un suono musicale ha uno spettro lineare con frequenze multiple della frequenza fondamentale determina l'altezza del suono percepita ascoltando, e l'insieme delle componenti armoniche contraddistingue il suo timbro. Lo spettro dei suoni del linguaggio contiene formanti, che sono gruppi stabili di componenti di frequenza corrispondenti ad alcuni elementi fonetici. Il parametro di energia delle vibrazioni sonore è l'energia sonora intensitymdashthe trasportata dall'onda suono attraverso una unità di superficie, perpendicolare alla direzione di propagazione, per unità di tempo. intensità del suono è funzione dell'ampiezza di pressione sonora, nonché delle proprietà del mezzo e la forma dell'onda. Il parametro personale, che è associato con l'intensità del suono, è loudness, che dipende dalla frequenza. L'orecchio umano è più sensibile nella gamma di frequenza da 1 a 5 kHz. In questa regione la soglia di audibilitymdashthat è, l'intensità dei deboli soundsmdashis acustici dell'ordine di 10 -12 watt per mq (Wm 2) e la corrispondente pressione sonora è 10 -5 nuovi a tonnellata per mq (Nm 2 ). Il limite superiore di intensità per percezione del suono dall'orecchio umano è detta soglia del dolore è leggermente dipende dalla frequenza della gamma udibile ed è approssimativamente uguale a 1 Wm 2. molto maggiore intensità (fino a 104 kWm 2) vengono raggiunti in tecnologia a ultrasuoni. fonti sonore. sorgenti sonore sono fenomeni che producono una variazione locale di pressione o stress meccanici. Vibrazione corpi solidi, come i coni altoparlanti, i diaframmi in telefoni e le corde e tavole sonda in strumenti musicali, sono le sorgenti sonore più comuni nel campo di frequenza ultrasonica le fonti possono essere piastre e barre realizzati piezoelettrici o materiali magnetostrittivi . Le vibrazioni di volumi limitati del mezzo itselfmdashfor esempio, in canne d'organo, strumenti a fiato, e whistlesmdashmay anche essere sorgenti sonore. L'apparato vocale dell'uomo e degli animali è un sistema vibrante complesso. La vibrazione di sorgenti sonore possono essere eccitata da un colpo (nel caso di una campana) o pizzicando (nel caso di una stringa) una modalità di vibrazione autoeccitato può essere mantenuta in tali oggetti da una corrente d'aria (in vento strumenti). trasduttori elettroacustici, in cui le vibrazioni meccaniche vengono prodotte convertendo oscillazioni correnti elettriche della stessa frequenza, sono una vasta classe di sorgenti sonore. In natura, il suono è prodotto quando l'aria fluisce attorno corpi solidi a causa della creazione e distacco dei vortici (ad esempio, quando il vento soffia oltre fili, tubi, e le creste delle onde del mare). Bassa frequenza e suoni infrasuoni sono prodotti da esplosioni e valanghe. Le fonti di rumore acustico comprendono macchine e meccanismi utilizzati nella tecnologia, nonché getti di gas e acqua. Lo studio di industriale, dei trasporti, e le fonti di rumore aerodinamico sta ricevendo una grande quantità di attenzione in vista dei loro effetti dannosi sul corpo umano e attrezzature industriali. ricevitori audio. ricevitori audio raccolgono energia sonora e la convertono in altre forme. L'apparato uditivo degli esseri umani e animali è in questa categoria. Nella tecnologia, trasduttori elettroacustici sono generalmente utilizzati per la ricezione del suono: microfoni in aria, idrofoni in acqua, e geofoni nella crosta earthrsquos. In aggiunta a tali trasduttori, che riproducono la dipendenza dal tempo del segnale sonoro, sono ricevitori che misurano i parametri di onde sonore media rispetto al timemdashfor esempio, il disco Raleigh e radiometro acustica. Propagazione di onde sonore. La propagazione delle onde sonore è caratterizzata principalmente dalla velocità del suono. onde longitudinali propagano in mezzi gassosi e liquidi (la direzione del moto vibratorio particlesrsquo coincide con la direzione di propagazione dell'onda) ad una velocità determinata dalla compressibilità e densità del mezzo. La velocità del suono in aria secca a temperatura di 0degC è di 330 msec, e in acqua dolce al 17degC è 1,430 msec. Oltre alle onde longitudinali, onde trasversali, per cui la direzione delle vibrazioni è perpendicolare alla direzione di propagazione dell'onda, così come onde di superficie (onde di Rayleigh), possono propagare nei solidi. Per la maggior parte dei metalli velocità delle onde longitudinali varia da 4.000 a 7.000 msec la velocità delle onde trasversali, tra 2.000 e 3.500 msec. Durante la propagazione delle onde di grande ampiezza, la fase di compressione si propaga ad una velocità superiore a quella della fase di rarefazione, in modo che la forma sinusoidale dell'onda diventa gradualmente distorto e l'onda sonora viene convertita in un'onda d'urto. In molti casi si osserva dispersione del suono, cioè, la velocità di propagazione è una funzione della frequenza. dispersione del suono porta ad un cambiamento nella forma di segnali acustici complessi, compresi alcuni componenti armoniche, e, in particolare, alla distorsione di impulsi sonori. I fenomeni di interferenza e di diffrazione, tipici di tutti i tipi di onde, possono verificarsi durante la propagazione delle onde sonore. Quando la dimensione di ostacoli e disomogeneità del prodotto sono grandi rispetto alla lunghezza d'onda, la propagazione del suono obbedisce usuali leggi della riflessione e rifrazione per onde e può essere affrontato dal punto di vista dell'acustica geometrica. Durante la propagazione di un'onda sonora in una data direzione, graduale attenuazione occursmdashthat è, la sua intensità e diminuzione dell'ampiezza. Conoscenza delle leggi di attenuazione è di importanza pratica nel determinare la distanza massima propagazione di un segnale acustico. Attenuazione dipende da una serie di fattori, che si manifestano in misura maggiore o minore a seconda delle caratteristiche del suono stesso (soprattutto la sua frequenza) e le proprietà del mezzo. Tutti questi fattori possono essere classificati in due grandi gruppi. Il primo gruppo comprende i fattori associati con le leggi della propagazione delle onde nel mezzo. Così, nel caso di propagazione in un mezzo infinito, l'intensità di un suono da una sorgente di dimensioni finite diminuisce inversamente proporzionale al quadrato della distanza. Disomogeneità delle proprietà mediumrsquos provoca dispersione dell'onda in varie direzioni, indebolendo così nella direzione originale, come è il caso con il suono dispersa dalle bolle in acqua, dalla superficie agitata di un oceano, e dalla turbolenza atmosferica ultrasuoni ad alta frequenza è sparsi in metalli policristallini e dislocazioni nei cristalli. La propagazione del suono nell'atmosfera e nell'oceano è influenzata dalla temperatura e pressione e dalla forza e velocità del vento. Questi fattori determinano la piegatura del raysmdashthat suono è, refractionmdashwhich conti particolare per il fatto che il suono udibile più lontano con il vento che contro di esso. La distribuzione della velocità del suono con la profondità nell'oceano spiega l'esistenza del cosiddetto canale subacqueo suono, in cui si osserva estremamente lungo raggio propagazione del suono: per esempio, il suono di un'esplosione propaga per più di 5.000 chilometri. Il secondo gruppo di fattori determinanti attenuazione acustica è associato con i processi fisici in una sostanza, compresa la trasformazione irreversibile di energia sonora in altre forme, principalmente heatmdashthat a dire, l'assorbimento del suono, che è causato dalla viscosità e la conducibilità termica del mezzo (absorptionrdquo ldquoclassical) mdashand la trasformazione di energia sonora in energia dei processi intramolecolari (molecolari o di assorbimento di rilassamento). di assorbimento acustico aumenta notevolmente con la frequenza. Pertanto, ultrasuoni ad alta frequenza e Hypersound solito propagano solo su distanze molto brevi, spesso non più di alcuni centimetri. infrasuoni, che si distinguono per un basso assorbimento e dispersione debole, si propagano più lontano nell'atmosfera, in acqua, e nella crosta earthrsquos. Alle alte frequenze ultrasoniche e ipersonico assorbimento ulteriore verifica nei solidi come risultato dell'interazione delle onde con le vibrazioni termiche del reticolo cristallino, con gli elettroni, e con onde luminose. In determinate condizioni tale interazione può produrre assorbimento ldquonegative, rdquo o l'amplificazione delle onde sonore. L'importanza delle onde sonore, e di conseguenza il loro studio (in acustica), è estremamente grande. Sin dai tempi antichi, il suono è servito come un mezzo di comunicazione e segnalazione. Lo studio di tutte le sue caratteristiche ha reso possibile lo sviluppo di sistemi di trasmissione dati più avanzati, un aumento della gamma di sistemi di segnalazione, e la creazione di migliori strumenti musicali. Le onde sonore sono praticamente l'unica forma di segnali che si propagano in acqua, dove vengono utilizzati per le comunicazioni sottomarine, la navigazione, e ecolocalizzazione. sonoro a bassa frequenza è uno strumento per lo studio della crosta earthrsquos. L'applicazione pratica degli ultrasuoni ha ultrasuoni creati, un intero ramo della tecnologia moderna. Gli ultrasuoni sono utilizzati per il monitoraggio e la misura (in particolare in difetto di rilevamento), nonché per operazioni su sostanze (lavaggio ad ultrasuoni, trattamento meccanico, e saldatura). onde sonore ad alta frequenza, in particolare Hypersound, sono un mezzo importante per la ricerca in fisica dello stato solido. RIFERIMENTI Strutt, J. (Lord Rayleigh). Teoriia zvuka. 2a ed. voll. 1ndash2. Mosca, 1955. (tradotto dall'inglese.) Krasilrsquo Nikov, V. A. Zvukovye i ultrazvukovye Volny v vozdukhe, vode i tverdykh telakh. 3a ed. Mosca, 1960. Rozenberg, L. D. Rasskaz O neslyshimom zvuke. Mosca, 1961.Created da Olli Niemitalo su 2003/01/21, modificato l'ultima volta 2012-08-04 Nel 1998, ho avuto un po 'di tempo in più, mentre altri leggevano per gli esami finali del liceo, e ottenuto in elaborazione del segnale digitale. Ho scritto come ho imparato, ed ecco il risultato. It is not entirely accurate in places but may serve as a nice tutorial into the world of audio DSP. Previously this document was called Yehars digital sound processing tutorial for the braindead, but I have kinda grown out of my scene identity over the years. Enjoy the ASCII art This is written for the audio digital signal processing enthusiasts (as the title suggests ) and others who need practical information on the subject. If you dont have this as a linear reading experience and encounter difficulties, check if theres something to help you out in the previous chapters. In filter frequency response plots, linear frequency and magnitude scales are used. Page changes are designed for 60 linespage printers. Chapter Shuffling IIR equations is written by my big brother Kalle. And, thanks to Timo Tossavainen for sharing his DSP knowledge Copy and use this text freely. by Olli Niemitalo, oiki. fi Note that sample can mean (1) a sampled sound or (2) a samplepoint Sampled sound data is a pile of samples, amplitude values taken from the actual sound wave. Sampling rate is the frequency of the shots. For example, if the frequency is 44100, 44100 samples have been taken in one second. Heres an example of sampling: The original sound is the curve, and 0s are the sampled points. The horizontal straight line is the zero level. A sampled sound can only represent frequencies up to half the samplerate. This is called the Nyquist frequency. An easy proof: You need to have stored at least two samplepoints per wave cycle, the top and the bottom of the wave to be able to reconstruct it later on: If you try to include above Nyquist frequencies in your sampled sound, all you get is extra distortion as they appear as lower frequencies. A Sound consists of frequency components. They all look exactly like sine waves, but they have different frequencies, phases and amplitudes. Lets look at a single frequency: Now, we take the same frequency from another sound and notice that it has the same amplitude, but the opposite (rotated 180 degrees) phase. Merging two signals is done simply by adding them together. If we do the same with these two sine waves, the result will be: It gets silent. If we think of other cases, where the phase difference is less than 180 degrees, we get sine waves that all have different amplitudes and phases, but the same frequency. Heres the way to calculate the phase and the amplitude of the resulting sinewave. Convert the amplitude and phase into one complex number, where angle is the phase, and absolute value the amplitude. If you do this to both of the sinewaves, you can add them together as complex numbers. As you see, the phase of the new sine wave is 45 degrees and the amplitude sqrt(1212) sqrt(2) about 1.4 It is very important that you understand this, because in many cases, it is more practical to present the amplitude and the phase of a frequency as a complex number. When adding two sampled sounds together, you may actually wipe out some frequencies, those that had opposite phases and equal amplitudes. The average amplitude of the resulting sound is (for independent originals) sqrt(a2b2) where a and b are the amplitudes of the original signals. The main use of a filter is to scale the amplitudes of the frequency components in a sound. For example, a lowpass filter mutes all frequency components above the cutoff frequency, in other words, multiplies the amplitudes by 0. It lets through all the frequencies below the cutoff frequency unattenuated. If you investigate the behaviour of a lowpass filter by driving various sinewaves of different frequencies through it, and measure the amplifications, you get the magnitude frequency response. Heres a plot of the magnitude frequency response curve of a lowpass filter: Frequency is on the - axis and amplification on the axis. As you see, the amplification ( scaling) of the frequencies below the cutoff frequency is 1. So, their amplitudes are not affected in any way. But the amplitudes of frequencies above the cutoff frequency get multiplied by zero so they vanish. Filters never add any new frequency components to the sound. They can only scale the amplitudes of already existing frequencies. For example, if you have a completely quiet sample, you cant get any sound out of it by filtering. Also, if you have a sine wave sample and filter it, the result will still be the same sine wave, only maybe with different amplitude and phase - no other frequencies can appear. Professionals never get tired of reminding us how important it is not to forget the phase. The frequency components in a sound have their amplitudes and. phases. If we take a sine wave and a cosine wave, we see that they look alike, but they have a phase difference of pi2, one fourth of a full cycle. Also, when you play them, they sound alike. But, try wearing a headset and play the sinewave on the left channel and the cosine wave on the right channel. Now you hear the difference Phase itself doesnt contain important information for us so its not heard, but the phase difference, of a frequency, between the two ears can be used in estimating the position of the origin of the sound so its heard. Filters have a magnitude frequency response, but they also have a phase frequency response. Heres an example curve that could be from a lowpass filter: If you filter a sound, the values from the phase frequency response are added to the phases of the frequencies of the original sound. Linear (straight line) phase is the same thing as a plain delay, although it may look wild in the plot if it goes around several times. If your, for example, lowpass filter doesnt have a linear phase frequency response, you cant turn it into a highpass filter by simply subtracting its output from the original with equal delay. Complex math with filters The response of a filter for a single frequency can be expressed as a complex number, where the angle is the phase response of the filter and the absolute value the magnitude response. When you apply the filter to a sound, you actually do a complex multiplication of all the frequency components in the sound by the corresponding filter response values. (Read chapter Adding two sinewaves together if you find this hard to understand.) Example: The response of a filter is (0,1) at 1000Hz. You filter a sine wave, with the phase amp amplitude information presented as the complex number (0,1), of the same frequency with it: The phase of the sine wave got rotated 90 degrees. No change in the amplitude. Combining filters The combined response of these two filters put in serial is the response of A multiplied by the response of B (Complex numbers as always). If you only need to know the magnitude response, you could as well multiply the absolute values. In the figure, both filters get their inputs from the same source. Their outputs are then added back together, forming the final output. Now you need to use addition in solving the combined response. FIR filter is more straightforward, and easier to understand. Finite impulse response means that when the filter input has remained zero for a certain time, the filter output also becomes zero. An Infinite impulse response filter never fully settles down after turning off the input, but it does get quieter and quieter though. A basic FIR filter could be: where input means the sample values fed to the filter. In this case, people would speak of a 3 tap filter. Its up to the coefficients (a0, a1, a2) what this filter will do to the sound. Choosing the coefficient values is the hardest part, and well get to that later. To design your own filters, you need to understand some of the math behind and know the right methods. In the above filter example, only past input values are used. In realtime filters, this is a requirement, because you dont know the future inputs. In sample editors and such, you dont have this limitation, because you have the whole input data ready when you begin. If your filter is: and you need a realtime version of it, just convert it to: The only difference is the one sample delay in the realtime filter. Unlike FIR filters, IIR filters also use their previous output values in creating their present output. Heres a simple example: This could be called 3 input, 3 output tap filter. IIR filters can never use future output values, because such dont yet exist There can be several ways of implementing the same IIR filter. Some may be faster than the usual input-output-and-coefficients way. Anyhow, every IIR filter can be written in this form, and it must be used in filter design and examining calculations. An impulse response ( What the filter will do to a one samplepoint impulse) of an IIR filter often looks more or less like this in the sampledata: Some badly designed IIR filters are unstable. This results in ouput getting louder and louder instead of quieter and quieter. A simple example of this is: output(t) input(t) 2output(t-1). As soon as it gets input data, it gets crazy. The above described filter types process the data sample by sample. Not so, if you implement your filter using FFT, Fast Fourier Transformation. FFT usually operates on chunks of length 2n. First, you should have your planned filter impulse response ready. Then convert it, using FFT, to spectral information - complex numbers representing the phases and amplitudes of the frequency components. These components are called bins, because their frequencies are fixed and evenly distributed, and if the original data contained any in-between frequencies, then most of the energy of such a frequency will be distributed amongst the nearby bins. Now, you FFT also the sample data you want to filter, and multiply the resulting frequency bins with those from the filter. Then IFFT (Inverse FFT) is used to convert the information into a chunk of filtered sample data. So, multiplication of the two frequency domain data resulted in convolution of the two time domain data. However, theres a catch: FFT operates on periodic signals, that is, if you have a filter impulse response as long as the FFT chunk then any non-zero sample data in the middle of the FFT chunk will result with the convolution wrapping the tail of the filter around the FFT boundary. In order to avoid this problem, you can use FFT twice as long as the filter impulse response, and when doing FFT on the sample data, only fill up the FFT input buffer to half way and set rest of the input to zero. For longer inputs, you would process the data in chunks like that and then add the resulting filtered chunks together. This is called the overlap-add method. Another option is overlap-save (look it up if you like). FFT can also be used to analyze the frequency content of sample data, for whatever reason. If you just take a chunk of sample data, it has sharp edges, which is bad for FFT. Windowing functions are used to smoothen these edges. Raised cosine, cos(x pi2)2, is one possible windowing function. Here you see what happens when you apply that windowing function to a chunk of sample data: Sometimes (resampling, precisely defined delay) you need to get samplevalues from between the known samplepoints. Thats when you need interpolation. If you dont interpolate, and just throw away the fractional part of your sampleoffset, you get a lot of high frequency distortion: In the example, the original samplepoints try to represent a sine wave. The closer the interpolated curve is to a sine wave, the better the interpolation algorithm is. The simpliest interpolation method is linear interpolation. Straight lines are drawn between two adjacent samplepoints: Still looks quite edgy to be a sine wave. However, the improvement to uninterpolated is significant. Theres also a drawback - the frequencies just below the Nyquist frequency get attenuated, even more than without interpolation. Heres the formula for linear interpolation: new old(int)(old(int1)-old(int))fract, where int means the integer part of sample offset and fract the fractional part. Next step could be Hermite curve, which gives in every way better quality than linear interpolation: With linear interpolation, you needed to know 2 samplepoints at time to be able to draw the line. With Hermite curve, the number is 4. The interpolation curve goes through the two middle points, and the points 1 and 4 are used in shaping the curve. The formula is a cubic: And this one here is where a, b,c, d were solved from: A perfect interpolation also exists. By replacing all the sample points with correctly scaled sinc curves, sin(pi x)(pi x), and by adding them together, you get exact, perfect interpolation. Here is one of the samplepoints replaced with a scaled sinc curve: Sinc curve is endlessly long, so youd have to use all the samplepoints in calculation of one interpolated value. A practical solution would be to limit the number of samples to say 1000. It will still be too slow for a realtime application, but itll give great accuracy. If you insist to use sinc in a realtime interpolation algorithm, try using a windowing function and a low number (at least 6) of sinc curves. Downsampling If you want to downsample (decrease the samplerate), you must first filter away the above Nyquist frequencies, or they will appear as distortion in the downsampled sample. In the process of filter design, you often need to make compromises. To have sharp edges or steep slopes in the magnitude response, you will need a big, and therefore slow filter. In other words, filters with low number of taps practically always have gently sloping magnitude responses. In the case of IIR filters, sharp edges in magnitude often mean an ugly (very nonlinear) phase frequency response, and close-to-linear phase response a gently sloping magnitude response. With FIR filters, an attempt to create very sharp edges may cause waving in the magnitudes of nearby frequencies. IIR filters are great for a realtime routine, because they are fast, their properties (for example cutoff frequency) can be quickly changed in the middle of action, and, they sound like real analog filters. ) The nonlinear phase response of IIR filters usually doesnt matter. FIR filters could be used where the quality and linear phase are important, for example, in a sample editor. People who filter other signals than sound, often desire linear phase frequency response. With stereo signal, it is important to have identical phase changes on left and right channels. Some filters and their stylized magnitude frequency responses: If you have a symbolic calculation program, i strongly recommend you to use it in the mechanical calculations, just to make your life easier. Derive is an old DOS program, but still very useful. White noise White noise means the sort of noise that has flat spectrum. You can easily create it by using random numbers as samplevalues. If you want to know the magnitude frequency response of a filter, apply it on a long sample of white noise and then run a spectrum analysis on the output. What you see is the magnitude frequency response of the filter. Another way is to send a one-sample impulse, which originally has a flat spectrum. An impulse looks like this in the sampledata: 0, 0, 0, 0, 1, 0, 0, 0, 0 - where the impulse is the 1 in the middle. From the two, the impulse thingy is faster, but using white noise can give cleaner-looking results, because errors will be less visible. For much the same reasons, when you are watching videos, a still picture will look more snowy than the running picture. Taking a spectrum analysis on a long sample is usually done by dividing it to smaller pieces, analyzing them separately and then taking the average of all the analyses. My personal choice here would be the program Cool Edit 96, which is for Windows. Pole-zero method is the easiest way of designing fast and simple IIR filters. When you have learned it, you will be able to design filters by yourself. Heres the complex Z-plane, the one used in the pole-zero method: Imagine the frequencies to be wrapped around the unit circle. At angle 0 we have 0Hz, at pi2 we have samplerate4, at pi we have samplerate2, the Nyquist frequency. You shouldnt care about higher frequencies, since they will never appear in the signal, but anyway, at 2pi (full cycle) we have the sampling frequency. So if you used sampling frequency 44100 Hz, 0 Hz would be at (1,0), 11025 Hz at (0,1) and 22050 Hz at (-1,0). What are poles and zeros then They are cute little things you can place on the Z-plane, like this: There are some rules you have to remember. Poles must always be inside the unit circle, never outside or on it. Zeros can be put anywhere. You can use any number of poles and zeros, but they must all have conjugate pairs, if they are not positioned on the - axis. Conjugate pairs means that if you put for example a zero to (0.6, 0.3), you must put another zero to the conjugate coordinate, (0.6,-0.3). And the same thing with poles. But hey What do poles and zeros DO Poles amplify frequencies, zeros attenuate. The closer a pole is to a frequency (on the unit circle, remember), the more it gets amplified. The closer a zero is to a frequency, the more it gets attenuated. A zero on the unit circle completely mutes the frequency it is sitting on. Now it could be the right time to try this out yourself. There are free filter design programs around that allow you to play with poles and zeros. One candidate could be: QEDesign 1000 demo for Windows. Its somewhere on the Internet, youll find it. Designing a bandpass filter The simpliest filter designed using pole-zero is the following bandpass filter: Poles amplify frequencies, so you could draw the conclusion that the most amplified frequency is the one at the same angle as the pole. And you are almost right The only problem comes from the conjugate pole, which also gives its own amplification. The effect is strongest at angles close to 0 and pi, where the distance between the two poles is the smallest. But dont let this confuse you, well get back to it later. So the angle of the pole determines the passband frequency. Whats the effect of the absolute value ( radius) then As stated, poles amplify frequencies, and the amplification is stronger when the pole is closer to a frequency. In our bandpass filter, increasing the radius of the pole causes the magnitude response to become steeper and passband narrower, as you see here: Positions of poles: Corresponding magnitude frequency response plots (normalized): Lets call the radius r from now on. (Some of you might remember the letter q from analog, resonant filters. This is much the same.) In this case we have the limitation: 0 r lt 1, since poles must be inside the unit circle. So changing r changes steepness, resonance. This resonance - its not a magic thing, just one frequency being amplified more than others. From poles and zeros to filter coefficients There is a transfer function: where z is frequency, in the (complex) wrapped-around-the-unit-circle coordinate form. H(z) gives the response (complex) of the filter at the frequency z. P1, p2, p3 and so on are positions of poles and z1, z2, z3 and so on positions of zeros. A0 is the first input coefficient of the filter. Heres the IIR filter formula again, in case you have forgotten: Our bandpass filter only has one pole, and its conjugate pair, so we can simplify the transfer function: and replace p1 and p2 with the coordinates of the conjugate poles: Lets give the divisor a closer look. Say: Powers of z here are actually indexes to the output of the filter: So we know how to calculate the output side coefficients from the position of the pole: OK Lets say the passband frequency is at the Z-plane at position ph: The pole is at the same angle as the frequency on the unit circle, but has radius r. Therefore: Now that we know how the position of the pole depends on the frequency, we can rewrite the output side coefficients: But we mustnt forget the dividend (of the transfer function), where powers of z are indexes to the input of the filter: This must be added to what we already have solved from the output side: Next we have to decide what to put to a0. This is called normalization. The purpose of a0 is just to scale the output of the filter. In our bandpass filter, we want the amplification at the passband frequency to be 1. So we can write the equation: There it is Now the filter is ready: Improving the simple bandpass filter We could compensate the effect of the conjugate pole by adding a zero onto the - axis, between the poles. For example, if we had poles at coordinates (0.6, 0.5) and (0.6,-0.5), wed put a zero at (0.6, 0): The transfer function for this is: The output side coefficients are exactly the same as before. Input side coefficients can be solved like this: In case you want to use this filter, you should be able to do the normalization yourself. I wont do it here. Words of wisdom It is easy to make a filter more efficient: Double all poles and zeros. The frequency response of the new filter is the square of the old. There are better ways, but this is the easiest. If you put a zero on a pole, you neutralize both. A pole outside the unit circle causes the filter to become unstable. A pole on the unit circle may turn the filter into an oscillator. Large number of poles and zeros means large number of taps. Zeros affect the input coefficients, poles output. Poles and zeros must have conjugate pairs, because otherwise youd get complex filter coefficients and, consequently, complex output signal. With low r values, the most amplified frequency is not always at the same angle with the pole, because of the effect of the conjugate pole. Try differentiating the magnitude response if you want exact precision. An IIR filter with no poles is a FIR filter. 0 r lt 1 always applies. Bandpass with r Read chapter IIR filter design using pole-zero method. Notch with r The higher the r, the narrower the stopband. Lowpass with r This can be done in several ways: The higher the r, the stronger the resonation. Resonant lowpass filter is surely the most used filter type in synthesizers. Allpass with r Highpass with r Impulse, sinc If you read about sinc interpolation in the chapter Interpolation of sampled sound, you know that you can replace a single sample peak ( impulse) in the sampledata with a correctly stretched sinc function. Correctly stretched means amplitudesinc(t). When you run a spectrum analysis on an impulse, you get a flat spectrum with upper limit at samplerate2, the Nyquist frequency. Because impulse sinc, this is also the spectrum of sinc: You could draw the conclusion that you get the sinc function if you sum together all the frequencies from 0 to SR2, and divide the sum by the number of frequencies, to fulfil the equation sinc(0) 1. And youd be right. From the spectrum analysis, you know that all the frequencies summed together have equal amplitudes. But whats their phase at the center of the impulse Sinc function is symmetrical around x0, so is cosine - so sinc must be made of cosines. If you test this with about 100 cosines, you get a pretty close approximation of sinc. The sum of all frequencies from 0 to 1 (comparable to SR2), divided with their number, can be written as: (Here oo means infinite) As done above, x must be replaced with pi t, because the cycle length of sin is 2 pi, which must be stretched to 2 (which is the wavelength of the Nyquist frequency in the sampledata). Phase shift What if we replaced the cosines with sines Lets try it Theres a universal formula (which, btw, i invented myself) we can use: Now, if we replace all the impulses in the sound with this new function, we actually perform a -90 degree phase shift This can be done by creating a FIR filter, where the coefficients are taken from this new function: (1-cos(pi t))(pi t), but in reverse order, by replacing t with - t, so it becomes: (cos(pi t)-1)(pi t). Heres an example that explains why it is necessary to use - t instead of t: Say you want to replace all the impulses in the signal with the sequence 1,2,3. If the input signal is 0,1,0, common sense says it should become 1,2,3. If you just use 1,2,3 as filter coefficients in that order, the filtered signal becomes: Which is not what you asked for But if you use coefficients 3,2,1, you get the right result, Ok, back to the -90 degree phase shift filter. When you are picking the filter coefficients from (cos(pi t)-1)(pi t), at t0 you unluckily get a division by zero. Avoid this by calculating the limit t-gt0, on paper, or with a math proggy. If you use your brains a little, you notice it is 0, because the filter formula is a sum of sines, and sin(0)0, so at t0 it is a sum of zeros. Like sinc, our new function has no ending, so a compromise must be made in the number of taps. This causes waves in the magnitude response, and attenuation of the very lowest and highest frequencies. By applying a windowing function to the coefficients, you can get rid of the waves, but i dont know anything that would help with the attenuation, except more taps. The windowing functions used with FFT work here also. The center of the windowing function must be at t0, and it must be stretched so that the edges are lay on the first and the last tap. You can also get a phase shift of any angle a: Note that reversing t has already been done here, so we can take the coefficients directly from this formula. The limit t-gt0 is naturally cos(a), because all the cosines added together had phase a at x0. In case you didnt yet realize it, the main idea in FIR filter making is to create a function that contains the frequencies you want to pass the filtering. The amplitudes of the frequencies in the function directly define the magnitude frequency response of the filter. The phases of the frequencies define the phase response. Reversing the coefficients is only necessary with phase shifting filters, because filters that do not introduce a phase shift of any kind are symmetrical around t0. Defining the frequency range included If you use sinc as your filter coefficient formula, you actually do no filtering, because all the frequencies from 0 to Nyquist are equally presented in sinc. Here youll see how you can select which frequencies will be present in your filter coefficient formula. Remember where we originally got sinc from: In the integral, the upper limit (1x) actually represents the highest frequency included (1), and the lower limit (0x) the lowest (0). So if you want a formula for a bandpass filter, you can write: where top and bottom are the cutoff frequencies in such way that 1 means the Nyquist frequency, and 0 means 0Hz. Now just put there whatever frequencies you want, calculate, and replace x with (pi t). Theres your filter coefficient formula ready For example, if you want to make a halfband lowpass filter (which naturally has cutoff frequency at samplerate4, same as Nyquist frequency 2): To create multi-band filters, you can combine several bandpass filter formulas by adding them together. The equalizer example If you want to make an equalizer (a filter that allows you to define the magnitudes for certain frequencies), you probably sum together a lot of bandpass filter formulas, scaled by the magnitudes you want for the frequency segments. This gives you a magnitude response that looks very much like as if it was made of bricks: Maybe youd want it to look more like this instead: There are three ways. The first way is to use smaller bricks, meaning that you divide the frequency into narrower-than-before segments and use interpolation to get the magnitude values for the new narrow bandpass filters you then combine. The second way is to define a polynomial (like ax3bx2cxd) that has the wanted characteristics (and where x1 represents freqSR2), and to make the magnitude response of your filter to follow it. This is possible. The third way is to add together several bandpass ramp filter formulas. In the magnitude response this solution looks like straight lines drawn between the adjacent defined frequencies. This is also possible, and, in my opinion, the best solution. Polynomial shaped magnitude frequency response In sinc, all the cosine waves added together have equal amplitudes, as you see here - all the frequencies are treated equally: You can change this by putting there a function g() that defines the amplitudes of the cosine waves of different frequencies: If the function g(x) is form axb, the calculations go like this: For a simple example, if we want the magnitude frequency response to be a straight line, starting from 0 at 0Hz, and ending at 1 at SR2, we define g(x) x: And the filter coefficient formula calculations for this: In other cases, to get the formula for a full polynomial, do the calculations for each of its terms (axb) separately and sum the results. Bandpass magnitude-ramp Heres an example of the magnitude frequency response of a ramp filter: To make a bandpass ramp, you must first define the polynomial g(x) that describes how the magnitude behaves inside the bandpass limits. The magnitude is linear inside the limits, so the polynomial g(x) must be form cxd. C and d can be solved from the equations: where x1 is the lower frequency limit, and x2 the higher. Y1 and y2 are the magnitudes of the limit frequencies. Remember that here x1 equals frequencySR2. OK, here are c and d solved: G(x)cxd is a polynomial, and you already know how to make the magnitude frequency response have the same shape (Section Polynomial shaped magnitude frequency response) as a polynomial. You also already know how to include only limited range of frequencies (Section Defining the frequency range included) in your coefficient formula. Combine this knowledge, and you can write the coefficient formula for the ramp bandpass filter: A note about implementing the equalizer. If the equalizer is to be adjustable realtime, recalculating the whole equalizer filter formula with all the trigonometric functions may turn out too heavy. It may be better to precalculate coefficients for several overlapping filters, for example these for a three-channel equalizer: When calculating the coefficients for the whole equalizer, just pick the corresponding coefficients from these, scale according to the equalizer sliders, and sum. If you take your FIR filter coefficients directly from your filter formula, you get a very wavy magnitude response. The reason is simple: The number of coefficients is limited, but the filter formula is not, it continues having nonzero values outside the range you are using for the coefficients. A windowing function helps. Not using a windowing function is the same thing as using a rectangular ( flat inside its limits) windowing function. Using a windowing function means that you multiply the values taken from your infinitely long filter formula by the corresponding values taken from your finitely long windowing function, and use the results as filter coefficients. Here are some windowing functions, and the produced magnitude responses of a FIR lowpass filter with a low number of taps, illustrated: As you see, the steeper the cutoff, the more waves you get. Also, if wed look at the magnitude responses in dB scale, wed notice that from the three, cos4 gives the best stopband ( the frequency range that should have 0 magnitude) attenuation. Mathematically, multiplication in the time domain is convolution in the frequency domain, and windowing is exactly that. (Also, multiplication in the frequency domain is convolution in the time domain.) I hope i didnt slam too many new words to your face. Time domain means the familiar time-amplitude world, where we do all the FIR and IIR filtering. The frequency domain means the frequency-amplitudeampphase world that you get into through Fourier transformation. And convolution In the time domain, FIR filtering is convolution of the input signal with the filter coefficients. Say you convolute 0,1,0,0,2,0,1,0 with 1,2,3 (where 2 is at the center): Youll get 1,2,3,2,4,7,2,3. If you understand this example, you surely understand convolution too. Ideally (impossible), there would be no windowing, just the constant value 1 infinitely in time. And a steady constant value in the time domain is same as 0Hz in the frequency domain, and if you (in the frequency domain) convolute with 0Hz, it is the same as no convolution. Convolution in the frequency domain equals to multiplication in the time domain, and convolution in the time domain equals to multiplication in the frequency domain. Sounds simple, eh But note that in this frequency domain, there are positive AND NEGATIVE frequencies. Youll learn about those in chapter Positive and negative frequencies. Words of wisdom You get flat (but not necessarily continuous) phase response if your filter (filter coefficients) is symmetrical or antisymmetrical (sides are symmetrical but have opposite signs, and the center crosses zero) around t0, even if you limit the number of coefs andor window them. Sometimes you can optimize your filter code a lot. Some coefficients may turn zero, so you can skip their multiplications. If your filter is symmetrical around t0, you can instead of input(t)ainput(-t)a write (input(t)input(-t))a). If your filter is antisymmetrical around t0, replace input(t)a-input(-t)a) with (input(t)-input(-t))a. Sinc(t) is 1 at t0, and 0 at other integer t values. Calculating the limit t-gt0 is very simple. If your filter formula was originally a sum cosines (meaning its not a phase shift filter), the limit t-gt0 is simply the area of the magnitude frequency response, in such way that the area of no filtering is 1. The actual filter implementation (after possible coefficient calculations) depends much on how the input data is fed to the filter. I can see three cases: You have the whole input data in front of you right when you start. A sample editor is a good example on this. This is the easiest case. With FIR filters, just take values from the input data, multiply with coefficients and sum, like this: output(t) a0input(t-2) a1input(t-1) a2input(t) a3input(t1) a4input(t2). The only problem is what to do at the start and at the end of the input table, because reading data from outside it would only cause problems and mispredictability. A lazy but well working solution is to pad the input data with zeroes, like this: This is how its mostly done with FFT filtering. With FIR filters, it isnt that hard to write a version of the routine that only uses a limited range of its taps, like this: and to use that version at the start and at the end. For this, it is easiest if you have a table of coefficients instead of hard-coding them into the routine. Data is fed to the filter in small chunks, but it is continuous over the chunk borders. This is the most common situation in programs handling realtime audio. One sample at a time. Case 2 can be treated as this, because the chunks can always be chopped into single samples. It is a fact that you cannot use future inputs in this case, so a FIR filter would have to be of form such as: output(t) a0input(t-4) a1input(t-3) a2input(t-2) a3input(t-1) a4input(t). Clearly this kind of a filter creates a delay, but thats just a thing you have to learn to live with. Also, you only get in one sample at a time, which is not enough for filtering, so you have to store the old input values somehow. This is done using a circular buffer. The buffer is circular, because otherwise youd soon run out of memory. Heres a set of pictures to explain the scheme: The buffer must be at least as long as the filter, but it is practical to set the length to an integer power of 2 (In the above example: 2532), because then you can use the binary AND operation to handle pointer wrapping always after increasing or decreasing one (In the above example, AND with 31). Even better, use byte or word instructions, and wrapping will be automatically handled in overunderflows caused by the natural limits of byte or word. Note that the buffer should be filled with zeroes before starting. A similar circular buffer scheme is also often the best solution for implementing the output part of an IIR filter, no matter how the input part was realized. There are both positive and negative frequencies. Until now we havent had to know this, because we have been able to do all the calculations by using sines as frequencies. Dont be fooled that positive frequencies would be sines, and negative ones something else, because that is not the case. In all real (meaning, not complex) signals, positive and negative frequencies are equal, whereas in a complex signal the positive and negative frequencies dont depend on each other. A single sinewave (real) consists of a positive and a negative frequency. So any sine frequency could be expressed as a sum of its positive and negative component. A single, positive or negative, frequency is: and could also be written as: As stated, a sinewave consists of a positive and a negative frequency component. Heres the proof: (The phase of the negative frequency must also be inverted, because it rotates to the other direction) As you see, the imaginary parts nullify each others, and all that remains is the real part, the sine wave. Amplitude of the sine wave is the sum of the amplitudes of the positive and the negative frequency component (which are the same). This also proves that in any real signal, positive and negative frequencies are equal, because a real signal can be constructed of sine waves. The complex Z-plane is a good place to look at positive and negative frequencies: Positive frequencies are on the upper half of the circle and negative frequencies on the lower half. They meet at angles 0 and the Nyquist frequency. Aliasing usually means that when you try to create a sine wave of a frequency greater than the Nyquist frequency, you get another frequency below the Nyquist frequency as result. The new frequency looks like as if the original frequency would have reflected around the Nyquist frequency. Heres an example: The cause of aliasing can be easily explained with positive and negative frequencies. The positive component of the sine wave actually gets over the Nyquist frequency, but as it follows the unit circle, it ends up on the side of negative frequencies And, for the same reasons, the negative component arrives on the side of positive frequencies: The result is a sine wave, of frequency SR-f. Analytic signal It is sometimes needed to first create a version of the original signal that only contains the positive frequencies. A signal like that is called an analytic signal, and it is complex. How does one get rid of the negative frequencies Through filtering It is possible to do the job with an IIR filter that doesnt follow the conjugate-pair-poles-and-zeros rule, but a FIR filter is significantly easier to create. Well use the old formula that we first used to create sinc: but this time, instead of cosines, only including the positive frequencies: As you see, the filter coefficients are complex. We should also halve the amplitude of the positive frequency (it should be half of the amplitude of the cosine, because the negative component is gone) but thats not necessary, because itd only scale the magnitude. To convert the complex analytic signal back to real, just throw away the imaginary parts and all the frequencies get a conjugate (on the z-plane) pair frequency. Here the amplitudes drop to half, but as we skipped the halving in the filtering phase, it is only welcome. The real to analytic signal conversion could also be a good spot for filtering the signal in other ways, because you can combine other filters with the negative frequency removal filter. Amplitude modulation Amplitude modulation means multiplying two signals. All samplepoints in the modulated signal are multiplied by the corresponding samplepoints in the modulator signal. Heres an example: What happens if we modulate a signal with a sinewave The original signal is (as we have learned) a sum of frequecy components, sinewaves of various frequencies, amplitudes and phases. Note that the signal we are talking about here is real, not complex. Say sNUMBER is one of the frequency components. So, we can write the original signal as: Now, if we multiply this signal with the modulator signal m, we get: This is good, because as you see, its the same as if the frequency components were processed separately, so we can also look at what happens to each frequency component separately. A frequency component can be written as: where amp is the amplitude, f the frequency and a the phase. The modulator sine can be written the same way (Only added the letter m): Multiply those and you get: If we discard the phase and amplitude information, we get: which is two frequencies instead of the origial one. Heres a graph that shows how the frequencies get shifted and copied. The original frequency is on the - axis and the resulting frequencyfrequencies on the axis: In the graph Modulated, the frequencies that would seem to go below zero, get aliased and therefore reflect back to above zero. In sampled signal, the Nyquist frequency also mirrors the frequencies. Frequency shifting With some tweaking and limitations, you could make a frequency shifter by using sinewave modulation, but theres a better way. Lets try modulating the signal with e(i mf x) instead of cos(mf x). Phases and amplitudes are irrelevant, so ive just ignored them. (I hope you dont mind) Lets see what happens to a single positivenegative frequency when it is modulated: The answer is very simple. The original frequency got shifted by the modulator frequency. Notice how the rule Multiplication in the time domain is convolution in the frequency domain. applies here also. Heres an example on the z-plane unit circle. p0, p1, p2 are the positive frequencies and n0, n1, n2 their negative conjugate frequencies. Say the modulator frequency rotates the frequencies 14 full cycle counterclockwise: In the modulated signal, the original pair frequencies (like p0 and n0) are no longer conjugate pairs. Thats bad. Another bad thing is that negative frequencies get on the side of positive frequencies and vice versa. But if we first filter all the negative, and those of the positive frequencies that would arrive on the wrong side of the cirle, and then modulate the filtered signal: (The filter formula is in the chapter A collection of FIR filters in section Combined negative frequency removal and bandpass) Now it looks better To make this filtered amp modulated complex signal back to real again, just discard the imaginary part and all the frequencies get a conjugate pair: For most sounds, frequency shifing doesnt do a very good job, because they consist of a fundamental frequency and its harmonics. Harmonic frequencies are integer multiples of the fundamental frequency. After you have shift all these frequencies by the same constant frequency, they no longer are harmonics of the fundamental frequency. There are ways to do scaling instead of shifting, but just scaling the frequencies would be same as resampling, and resampling also stretches the sound in time, so it has to be something smarter. The main idea is to divide the sound into narrow frequency bands and to shiftscale them separately. OK, so frequencies usually come with harmonics - Why Just think where sounds in nature originate from: vocal cords in our throat, quitar strings, air inside a flute. All vibrating objects, and you have probably learned at school that objects have several frequencies in which they like to vibrate, and those frequencies are harmonics of some frequency. What happens in those objects is that they get energy from somewhere (moving air, players fingers, air turbulence), which starts all kinds of vibrationsfrequencies to travel in them. When the frequencies get reflected, or say, go around a church bell, they meet other copies of themselves. If the copies are in the same phase when they meet, they amplify each other. In the opposite phases they attenuate each other. Soon, only few frequencies remain, and these frequencies are all harmonics of same frequency. Like so often in physics, this is just a simplified model. A note about notation. ) The fundamental frequency itself is called the 1st harmonic, fundamental2 the 2nd, fundamental3 the 3rd, and so on. Chromatic scale In music, harmonics play a very important role. The chromatic scale, used in most western music, is divided into octaves, and each octave is divided into 12 notes. The step between two adjanced notes is called a halftone. A halftone is divided into hundred cents. An octave up (12 halftones) means doubling the frequency, an octave down (-12 halftones) means halving it. If we look at all the notes defined in the chromatic scale on a logarithmic frequency scale, we note that they are evenly located. This means that the ratio between the frequencies of any two adjacent notes is a constant. The definition of octave causes that constant12 2, so constant 2(112) 1.059463. If you know the frequency of a note and want the frequency of the note n halftones up (Use negative n to go downwards) from it, the new frequency is 2(n12) times the old frequency. If you want to go n octaves up, multiply by 2n. But why 12 notes per octave As said, harmonics are important, so it would be a good thing to have a scale where you can form harmonics. Lets see how well the chromatic scale can represent harmonics. The first harmonic is at the note itself: 0 halfnotes 1. The second harmonic is at 1 octave 2. The third harmonic is very close to 1 octave 7 halftones 19 halftones 2(1912) 2.996614. E così via. Heres a table that shows how and how well harmonics can be constructed: Not bad at all The lowest harmonics are the most important, and as you see, the errors with them are tiny. I also tried this with other numbers than 12, but 12 was clearly the best of those below 30. So, the ancient Chinese did a very good choice The above table could also be used as reference when tuning an instrument, for example a piano (bad example - no digital tuning in pianos), to play some keys and chords more beautifully, by forcing some notes to be exact harmonics of some other notes. A common agreement is that one of the notes, middle-a, is defined to be at 440Hz. This is just to ensure that different instruments are in tune. Flanger is simply: where d is the length of the variable delay. D values have a lower limit, and the variation comes from sine: Because d is not integer, we must interpolate. Most probably, annoying high frequency hissing still appears. It can be reduced by lowpass filtering the delayed signal. Wavetable synthesis means that the instruments being played are constructed of sampled sound data. MOD music is a well-known example. Also most of the basic home synthesizers use wavetable synthesis. Say you have a sampled instrument, and want to play it at frequency f 440Hz, which is middle A in the chromatic scale. To be able to do this, you need to know A) the samplerate of the sample and the frequency of the sampled instrument, or B) the wavelength of the instrument expressed as number of samples (doesnt have to be integer). So you decide to precalculate the wavelength to speed up the realtime routines a little: The samplerate of your mixing system, SR, is 44100Hz. Now that you know this, you can calculate the new wavelength, the one you want (number of samples): In the mixer innerloop, a sample offset variable is used in pointing to the sampledata. Every time a value is read from the sampledata and output for further mixing, sample offset is advanced by adding variable A to it. Now we must define A so that ol (256) is stretched (here shortened) to nl (100.22727), in other words, so that for ol samplepoints in the sampledata, you produce nl output values: Everything on one line: Thats it By using A as the addvalue, you get the right tone. Click removal There are some situations when unwanted clicks appear in the output sound of a simple wavetable synthesizer: Abrupt volume (or panningbalance) changes. A sample starts to play and it doesnt start from zero amplitude. A sample is played to the end and it doesnt end at zero amplitude. (Biased sampledata or badly cut out sample) A sample is killed abruptly, mostly happens when new notes kill the old ones. Poor loops in a sample. And what does help Heres some advice: Volume changes must be smoothed, maybe ramped, so that itll always take a short time for the new volume to replace the old. Clicky sample starts can be muffled, meaning that the volume is first set to zero and then slided up. This could of course be done beforehand too, and some think muffling sample starts is wrong, because the click may be deliberate. Some drum sounds lose a lot of their power when the starts are muffled. Another case is when the playing of a sample is not started from its beginning. That will most probably cause a click, but muffling is not the only aid - starting to play from the nearest zero crossing also helps. Abrupt sample ends should also be faded down. This may require some sort of prediction, if you want to fade down the sound before its ran over by another sound. This prediction can be made by using a short information delay buffer. It may be easier to just use more channels, to allow the new sound to start while the other one is being faded out in the background, on another channel. When the sampledata ends at a value other than zero, the cause may be that the sampledata is not centered around the zero level, or that the creator of the sample has just cut the end of the sample away. The easiest way to fix this is to fade out the end of the sample beforehand. However, this is not always possible. Symmetric form Turning an IIR filter backwards Getting rid of output(tn) Getting rid of input(tn) FIR frequency response IIR frequency response. Olli wrote he tried to make his text as down-to-earth as possible. Well, heres a more mathematical approach. But Ive still tried to make this intuitive and FUN rather than boring myself with lengthy proofs. This also means that there may be errors, most probably in signs. Symmetric form Say you have this IIR filter: You can put its equation to this symmetric form: Now define a new function, middle(t): You can rewrite this as: Notice how the transition from input(t) to middle(t) is a FIR filter and the transition from output(t) to middle(t) is another. So the IIR filter in fact consists of two FIR filters facing each other. This gives a simple approach to frequency response calculations (see the section IIR frequency response). Turning an IIR filter backwards You can solve input(t) from the IIR equation: Now swap input and output and you have a filter that undoes what the original did. But if the frequency response of the original filter was ZERO for some frequency, the inverted one will amplify that frequency INFINITELY. This is just logical. The inverted filter will also have an opposite phase shift, so that if R(f) is the frequency response of the original filter as a complex number and r(f) is the frequency response of the inverted filter, R(f)r(f)1 for every f. Getting rid of output(tn) Say you have somehow found that you need an IIR filter like this: You need to know both output(t2) and output(t-2) to be able to compute output(t). Doesnt seem very practical. But you can shuffle the equation a little: Now define a new variable ut2 and use it instead of t: Then solve output(u): Now you can use the filter. Getting rid of input(tn) Notice how in the previous example, input(t) became input(u-2). Had there been input(t1), it would have become input(u-1) which can be used in real time filters. Generally, you can get rid of input(tn) this way if the equation also uses output(tm) where mn, because you can define utm which turns input(tn) to input(u-(m-n)) which you get in time. If mltn, this is not possible: Here m0 and n1, so you cant get rid of input(t1) and keep the filter mathematically equivalent to the original. However, you can delay the output by one time unit: Usually, this small delay doesnt matter. But it changes the phase frequency response of the filter and this DOES matter if you then mix the filtered signal with the original one or others derived from it in that case, youd better make sure that all of the signals have the same delay. (Except if you happen to like the extra effect.) (For example, if you have a filter output(t)input(t-1), it doesnt do much as such. But if you mix the filtered signal with the original one, the mixing becomes a filter in itself and you can compute its frequency response and all.) If you try to force the original filter through the utm trick by introducing a dummy 0output(t1) term: youll just get division by zero. FIR frequency response Treat a sine wave as a rotating phasor e(it2piffs) where: The real component of this phasor is the regular sine wave. The neat thing about this is that you can multiply it with various complex numbers to scale the magnitude and shift the phase at the same time. By defining ze(i2piffs), the phasor can be written as zt. This is the same z that is used in pole-zero calculations (see chapter IIR filter design using pole-zero method). Heres the general FIR equation: Now, lets look what the filter does to an infinitely long sine wave with frequency f. But this sine wave can be replaced with the rotating phasor if we then throw away the imaginary component of the output. m(k) is real so the real and imaginary components cant affect each other. Here the zt factor doesnt depend on k, so it can be moved outside the sum: z depends on f (ze(i2piffs), remember) but the value of the sum doesnt depend on t. Ill call it R(f): output(t) is a rotating phasor at the same frequency as input(t) it just has a different amplitude and phase as defined by R(f). This means that for an infinitely long sine wave of frequency f, R(f) shows how the filter affects its amplitude and phase. In other words, R(f) is the frequency response of the filter. Its a complex function. If you dont remember what this means, see section Complex math with filters in chapter Whats a filter in this file. IIR frequency response When two filters are concatenated so that one filters output is fed to the other filters input, the responses are multiplied at each frequency: A filter that just connects its input to its output and doesnt change the signal at all has a frequency response of 1 at all frequencies: Now assume that we have a filter with frequency response R(f) and we make another filter with frequency response Rinv(f) that UNDOES everything the first filter did to the signal when they are concatenated. So the inverse filter also has an inverse frequency response. Remember, an IIR filter consists of two FIR filters facing each other (see section Symmetric form). This setup can be treated as a normal FIR filter followed by an inverted FIR filter: This means that if you can calculate the frequency responses of the two FIR filters, you can calculate the IIR frequency response by dividing one with the other. An example. You have this IIR filter. Change the names of functions a little: Compute the frequency response of filter input1-gtoutput1 (originally input-gtmiddle). The general formulas: In this particular case: The input2-gtoutput2 (originally output-gtmiddle) filter: Now the whole IIR: To actually calculate the frequency response at some frequency, youd apply Eulers formula and the usual complex number rules: R in the filters means resonance, steepness and narrowness. Fastest and simplest lowpass ever Fast lowpass with resonance v1 19 Comments raquo Thanks for posting this. It8217s a nice collection of audio DSP nuggets. May I suggest that the URL at the top of the original text document (iki. fiodspdspstuff. txt ) be pointed directly to this page. Comment by ColdCold 8212 2009-11-16 16:06 Thanks Mate, Greatly appreciate this tutorial. DSP in simple terms is not easy to come by on the Web Comment by Don 8212 2010-05-10 04:29 Thanks a lot. Very useful concepts explained in a lucid manner. Comment by Ravi 8212 2010-08-30 14:59 Hi, About notch filter.. Why I can8217t get the frequency cut effect Sample rate:1600 freq 1950 q 0.1 z1x cos(2pifreqsamplerate) a0a2 ((1-q)(1-q))(2(fabs(z1x)1)) q a1 -2z1xa0a2 b1 2z1xq b2 -(qq) 8212821282128212- frequency: 1950.000000 q: 0.100000 z1x: 0.195090 a0a2: 0.438887 a1: -0.171245 b1: 0.039018 b2: -0.010000 Each sample calculation: 82128212821282128212821282128212821282128212821282128212 reg0 a0a2 ((double)samplecurrentsampleminus2) a1sampleminus1 b1reg1 b2reg2 reg2 reg1 reg1 reg0 82128212821282128212821282128212821282128212821282128212 Is it correct Output is clean voice, but 1950Hz carrier is still there. BR Comment by Alexander Vangelov 8212 2011-03-16 22:46 Freq should be between 0 and samplerate2. (Just a quick comment before I go to bed) Comment by Olli Niemitalo 8212 2011-03-17 00:53 Thank you, it works :) I missed a zerro digit in parametters (just before I go to bed) Sample rate: 16000 Freq: 1950.000000 q: 0.400000 z1x: 0.720854 a0a2: 0.504599 a1: -0.727484 b1: -0.576683 b2: 0.160000 Comment by Alexander Vangelov 8212 2011-03-17 10:43 Very good tutorial, thanks Comment by Vadim 8212 2011-10-11 19:42 man, this is the best introduction (covering all topics) into DSP I stumbled upon perhaps I do have a chance to pass the exam. D sorry, for a double post. but8230 can you attest everything is correct for example, 822082218221 You can use any number of poles and zeros, but they must all have 8220conjugate pairs8221, if they are not positioned on the 8220-8221 axis. 822082218221 is this true I8217m playing with applets that allow for poles without conjugate pairs and seemingly band-pass filters (with regard to the magnitude response) can be built this way. can you please explain ( laps. fri. uni-lj. sidpsarhivappletiisipsystemv4.0srcapplet. html ) Doug, it is true, IF you want the filter to have a real output, not complex. If you make a bandpass with just one pole, and have the pole so close to the unit circle that the filter output is pretty much a single frequency, then the output of the filter will be a complex phasor rotating in one direction on the complex plane. If you switch the sign of the imaginary part of the position of the pole then you get as output a phasor that rotates in the opposite direction. If you have poles in both of those positions, then the output must contain both of those complex phasors in equal parts, thus the imaginary parts of the phasors cancel each other. So you get as output a real sinusoid. Good luck with the exam Comment by Olli Niemitalo 8212 2011-12-27 13:41 This is the first cogent explanation of poles and zeros that I have ever received. I feel better and worse at the same time, if you know what I mean. In any case. THANK YOU Comment by Mark McConnell 8212 2012-05-09 01:12 8230 Yehar8217s Digital Sound Processing Tutorial for the Braindead 8230 Nice Job Men82308230. I found it very helpful. Grazie. Can you put implementation of audio effects in computer. Comment by Trnform3r 8212 2012-09-16 10:07 Sure, for example as a VST effect. Comment by Olli Niemitalo 8212 2012-09-16 22:14 This is fantastic nice work and a very well explanation of DSP. Thank you :D Comment by tor 8212 2013-02-16 01:42 Thank you so much for this informative writing on the subject which makes life much easier since no-where could I find any book on the subject which makes it as clear as you did here. Keep it going and thank you again. Comment by FJ Botha 8212 2015-02-21 10:14 Frickin delicious Seriously, i thank people like you for simply existing and count my blessings that i found this brilliant introduction you created. The note takingoutline is digestable in one bite and it will stick with me during my upcoming solo winter sound holiday to the pampa and magellians strait, the large uninhabited Falkland rock, and if im still alive - christmas island. Dec to Feb. I hope to capture enough sound to keep me glazed and deadeyed until black metal villians capture Oslo Comment by Mick Dkaye 8212 2016-10-18 19:13 And love that Black Deck. Masonna weeps Comment by Mick Dkaye 8212 2016-10-18 19:16 Leave a commentNyquist Effect Plug-ins Noise Gate Author: Steve Daulton. Noise Gates may be used to cut the level of noise between sections of a recording. While this is essentially a very simple effect, this Noise Gate has a number of features and settings that allow it to be both effective and unobtrusive and well suited to most types of audio. Select Function: Apply the Noise Gate effect Test the noise level View one of the Help screens. Stereo Linking: Link Stereo Tracks (gate audio when both channels fall below the gate threshold) Dont Link Stereo (gate channels independently) Apply Low-Cut filter: No (Do not apply filter) 10Hz 6dBoctave 20Hz 6dBoctave Removes sub-sonic frequencies including DC offset. Gate frequencies above: 0 kHz to 10 kHz Applies the gate only to frequencies above the set level which may be useful for reducing tape hiss, but will also introduce some phase shift. Setting this below 0.1 kHz will switch this feature off. Level reduction: -100 dB to 0 dB How much the gated sections are reduced in volume. Values below -96 dB shut the gate to produce absolute silence. Gate threshold: -96 dB to -6 dB When the audio level drops below this threshold the gate will close and the output level will be reduced. When the audio level rises above this threshold the gate will open and the output will return to the same level as the input. AttackDecay: 10 to 1000 milliseconds How quickly the gate opens and closes. At the minimum (10 ms) the gate will fully open and close almost instantly as the audio level crosses the threshold. At the maximum (1000 ms), the gate will begin to slowly open (fade-in) 1 second before the sound level exceeds the Threshold, and will gradually close (fade-out) after the sound level drops below the Threshold over a period of 1 second. Longer gate times (up to 10 seconds) may be achieved using text input rather than the slider. For more detailed information and usage tips, read the help file included in this ZIP package. or the help screens included in the plug-in. Author: Steve Daulton. The effect is like an upside-down Noise Gate. Whereas a Noise Gate attenuates sounds that are below a specified threshold level, Pop Mute attenuates sounds that are above a specified threshold level. The effect can be used to heavily attenuate loud sounds. It may be useful for rescuing recordings that suffer from loud clicks or pops. Sounds (such as pops) that have a peak level above the Threshold level will be lowered to a residual level set by the Mute Level. Be aware that ALL sounds above the threshold will be affected. Take care to avoid selecting loud sounds that should not be muted. The effect looks ahead for peaks so that it can begin to lower the level of the sound smoothly a short time before the peak occurs. This is set by the Look ahead time value. After the peak has passed, the level will smoothly return to normal over a period set by the Release time setting. To attenuate brief clicks, time values of around 5 ms are likely to work well. For larger pops, values of 10 ms or more may sound better. For reverberant sounds such as hand claps, the Release time may be increased so as to catch some of the reverberation. View Help: No Yes (default No) View the built-in help screen. Threshold: -24 dB to 0 dB (default -6 dB) This is the level above which sounds are acted on (reduced in level) Mute Level: -100 dB to 0 dB (default -24 dB) How much to reduce the peak level by. Look ahead: 1 to 100 milliseconds (default 10 millisecond) How far to look ahead for the next pop or crackle. Release time: 1 to 1000 milliseconds (default 10 millisecond) How rapidly to release the effect and return to normal volume after the pop has passed. Text Envelope Author: Steve Daulton. Provides an alternative to the Envelope Tool that is accessible for visually impaired and other users that do not use pointing devices. This effect provides a means to shape the volume level of a track or selection by fading from one control point level to the next. Control points are defined by a pair of numbers, the first of which sets the time position of the control point and the second defines the amplification level. Initial and final amplification settings may also be defined. Help screens are available in the Select Function control of this effect. Select function: choices: Apply Effect, View Quick Help, View Examples, View Tips. Default Apply Effect Time Units: choices: milliseconds, seconds, minutes, percent. Default seconds Amplification Units: choices: dB or Percent. Default dB Initial Amplification Numeric input. Default none Final Amplification Numeric input. Default none Intermediate Control Points as pairs of time and amplification Pairs of numbers. Default none Note: Decimal values must use a dot as the decimal separator. Band Stop Filter Author: Steve Daulton. A band-rejection filter that passes most frequencies unaltered, but stops those in a specific range. Set the Center Frequency slider, or type in a value for the center of the frequency band to block. Set the Stop-Band Width to determine how wide the cut frequency band will be. Smaller numbers will produce a narrower notch and larger numbers will cut a broader band of frequencies. This filter uses steep high pass and low pass filters to achieve the band stop effect. The filters iterate to improve the band stop efficiency for narrow band width and can thereby perform close to total blocking down to almost 14 octave. For even narrower notches a notch filter should be used. Chebyshev Type I Filter Author: Kai Fisher A Chebyshev filter with options for high-pass or low-pass operation. Type I Chebyshev filters can provide a steeper roll-off than Butterworth filters but at the expense of more ripple in the passband. The plug-in provides unity gain (except for ripple) in the passband. This plug-in is capable of providing an exceptionally steep cutoff transition by selecting a high order. Filter Type: choice: Lowpass Highpass (default Lowpass) Order: choice 2 to 30 in steps of 2 (default 6) The higher the order number, the steeper the cutoff transition from the passband to stopband. Cutoff Frequency: 1 to 48000 Hz (default 1000 Hz). The actual filter frequency is limited to half of the track sample rate (the Nyquist frequency ). For example, if the track sample rate is 44100 Hz, then setting the Cutoff frequency to any value greater than 22050 will produce the same result as setting the frequency to 22050 Hz. Ripple: 0.0 to 3.0 dB (default 0.05) Lower values will produce less ripple in the passband at the expense of a less steep cutoff. Higher values will produce a steeper cutoff but with more ripple in the passband. The difference in ripple and cutoff slope is likely to be most noticeable with low order filters and may be noticed as a slight boost or ringing in the passband just before the cutoff frequency. When Ripple is set to zero, the passband response is essentially flat and the filter has the characteristics of a Butterworth filter. The high-pass and low-pass filters may be used one after the other to produce a flat topped band-pass effect, in which the lower cutoff is provided by the high-pass filter and the upper cutoff provided by the low pass filter. The passband is the frequency band that passes between these two cutoff frequencies. Classic EQ Authors: Josu Etxeberria and David R. Sky. An Equaliser (EQ) that can modify more than one band at a time. You have 15 bands to choose from and can manipulate all of of them independently by moving their sliders. Example clips: clip 1 is a phrase spoken twice, first with no equalisation and then with the five lowest frequency bands raised 10 dB in clip 2, the five highest frequency bands are raised 10 dB. Comb Filter The name comb filter comes from how it acts on the audio spectrum its applied to: it looks like a comb with the teeth pointing up. For example, if you set the comb frequency at 1000 Hz, the comb filter emphasizes 1000 Hz as well as 2000, 3000, 4000 Hz and succeeding frequencies. Produces an airy effect, which is more pronounced the higher the comb decay value is set, and resonance is increasingly produced as well. A comb filter can be produced using flanger-like settings on a delay effect, but this filter does not use a delay to get the result, so it does sound somewhat different. Comb frequency: Hz, 20 - 5000, default 440 Comb decay: 0 - 0.1, default 0.025 Normalization level: 0.0 - 1.0, default 0.95 Customizable EQ Center frequency: Hz, 20 - 20000, default 440 Band width in octaves octaves, 0.1 - 5.0, default 1.0 Gain: dB, -48.0 - 48.0, default 0.0 Apply normalization Default no Normalization level: 0.0 - 1.0, default 0.95 Author: Steve Daulton This EQ is modelled on the EQ section of the Allen amp Heath(TM) GL series mixing desk. It is a 4-band EQ (equaliser) with two semi-parametric mids and provides independent control of four frequency bands plus a low frequency roll-off switch (HPF). Allen amp Heath (along with Soundcraft and Neve) are well known for their distinctive British EQ . The two mid filters are bell shaped peakdip filters which affect frequencies around a center point which can be swept from 500 Hz to 15 kHz, and 35 Hz to 1 kHz respectively. The width of the band is selected to provide effective control for both creative and corrective equalisation. 100 Hz HPF: (- 15 dB) attenuates frequencies below 100 Hz by 12 dB per octave. It may be used to reduce low frequency noise such as microphone popping, stage noise and tape transport rumble. HF Gain: sets the gain of the high frequency shelf filter which boosts or cuts high frequencies. Positive values will tend to make the sound brighter. Negative values will tend to make the sound less bright. High-Mid Frequency: (500 Hz to 15 kHz) sets the center frequency of the high-mid band filter. High-Mid Gain: (- 15 dB) sets the gain of the high-mid band filter. Low-Mid Frequency: (35 Hz to 1 kHz) sets the center frequency of the low-mid band filter. Low-Mid Gain: (- 15 dB) sets the gain of the low-mid band filter. LF Gain: (- 15 dB) sets the gain of the low frequency shelf filter. Positive values will tend to give the sound more bass and negative values will reduce the bass. High Pass Filter with q A high pass filter with q, or resonance. A high pass filter attenuates frequencies below a given cutoff point. The higher q is, the more the cutoff frequency will resonate (produce a tone). Applied to white noise, both this filter and the low pass filter with Q can be used to produce wind-like sounds at a constant frequency. See the high pass filter (LFO) and low pass filter (LFO) for ability to modulate a fixed resonance cutoff frequency. Cutoff frequency: 20 - 10000 Hz, default 1000 Filter q (resonance): 0 - 5, default 1 High Pass Filter (LFO) A high pass filter with a low frequency oscillator (LFO). A high pass filter attenuates frequencies below a given cutoff point. The LFO in this plug-in modulates the cutoff frequency up and down, like on an electronic synthesizer. LFO frequency: 0 - 20 Hz, default 0.2 - defines the speed of the oscillation, higher is faster Lower cutoff frequency: 20 - 20000 Hz, default 160 Upper cutoff frequency: 20 - 20000 Hz, default 2560 LFO starting phase: -180 to 180 degrees, default 0 Example clip 1: LFO frequency of 1.0 Hz, lower frequency 113 Hz, upper frequency 3620 Hz, applied to 110Hz square wave. Example clip 2: LFO frequency of 5.0 Hz, lower frequency 113 Hz, upper frequency 3620 Hz, applied three times to a voice. Alternative version Center cutoff frequency: 20 to 20000 Hz, default 640 LFO depth (radius): 0.0 to 10.0, default 1 - how far (in octaves) from center f the filter sweeps. LFO frequency: 0.0 to 20.0, default 0.2 LFO starting phase: -180 to 180 degrees, default 0 Hum Remover Author: Steve Daulton A filter for removing the sound of mains hum from recordings. The frequency of mains electricity is 60 Hz in the US, 50 Hz in Europe. This can create electrical interference in recordings with many harmonics. To remove the hum, this effect applies a series of notch filters based on the frequencies of mains electricity and the harmonics, which have frequencies that are at exact multiples of that frequency. To minimize loss of audio data, the number of harmonics may be adjusted so that only as many notches as required to eliminate the audible hum are applied. There are often more odd harmonics than even harmonics, so this effect allows the number of odd and even harmonic filters to be set independently. Unless the amount of hum is very bad, high level audio will often mask the hum, making removal unnecessary, but during quiet parts of the recording the hum may be unpleasantly obtrusive. This effect therefore has a threshold level control so that only quiet sounds (where the hum will be most noticeable) are filtered. Select Region: Europe (50Hz) or USA (60Hz), default 50Hz - Sets the fundamental hum frequency. Number of odd Harmonics: 0 to 200, default 1 - The first harmonic is 50 or 60 Hz depending on the region selected. Number of even Harmonics: 0 to 200, default 0 - The number of even harmonics to filter. Hum Threshold Level(0 to 100): 0 to 100, default 10 - The signal level, as a percentage of full scale below which the filters are applied. The Plot Spectrum effect can often provide a useful guide as to which frequencies need to be removed. First, select 50 or 60 Hz with the first control as appropriate, then set the other controls to maximum. Preview the effect frequently while reducing one control at a time to find the minimum settings required to remove the hum. Low Pass Filter (LFO) A low pass filter with a low frequency oscillator (LFO). A low pass filter attenuates frequencies above a given cutoff point. The LFO in this plug-in modulates the cutoff frequency up and down, like on an electronic synthesizer. LFO frequency: 0 - 20 Hz, default 0.2 - defines the speed of the oscillation, higher is faster Lower cutoff frequency: 20 - 20000 Hz, default 160 Upper cutoff frequency: 20 - 20000 Hz, default 2560 LFO starting phase: -180 to 180 degrees, default 0 Example clips 1 - 3: LFO frequency of 0.2 Hz, lower frequency 320 Hz, upper frequency 1280 Hz, applied to white noise once, twice and three times respectively. Example clip 4: LFO frequency of 1.0 Hz, lower frequency 320 Hz, upper frequency 1280 Hz, applied to 640 Hz square wave. Alternative version Center cutoff frequency: 20 20000 Hz, default 640 LFO depth (radius): 0.0 to 10.0, default 1 - how far (in octaves) from center f the filter sweeps. LFO frequency: 0.0 to 20.0, default 0.2 LFO starting phase: -180 to 180 degrees, default 0 Low Pass Filter with Q A low pass filter with q, or resonance. A low pass filter attenuates frequencies above a given cutoff point. The higher q is, the more the cutoff frequency will resonate (produce a tone). Applied to white noise, both this filter and the high pass filter with Q can be used to produce wind-like sounds at a constant frequency. See the low pass filter (LFO) and high pass filter (LFO) for ability to modulate a fixed resonance cutoff frequency. Cutoff frequency: 20 - 10000 Hz, default 1000 Filter q (resonance): 0 - 5, default 1 Multiband EQ Select total number of bands (T, from 2 to 30), band number (1 to 30, depending on how many total bands T you chose), and apply gain (-24 to 24 db). Determines width of band depending on total band number T you chose. Author: Steven Jones. Loosely based on the Mutron stomp box from the late 70s. Basically it is a filter controlled by an envelope follower. CenterCutoff: 0 - 10000 Hz, default 100 - sets the static filter frequency Depth: -10000 - 10000 Hz, default 5000 - sets the negative or positive filter modulation depth Band Width: 50 - 400 Hz, default 100 - controls the resonance, lower values being more resonant Mode: 0Low 1High 2Notch 3Band (default) - sets the filter mode: 0 Low pass, 1 High pass, 2 Band Reject (cut a notch at the filter frequency), 3 Band Pass Notch Filter Authors: Steve Daulton and Bill Wharrie. Like its name suggests, a notch filter cuts out a notch in the spectrum of your audio. The default frequency (60 Hz) can remove much of the hum that recordings can acquire from 60 Hz mains supply (as used in North and Central America and much of South America). You can set Frequency to 50 Hz to counteract mains hum in other countries. See chart of mains frequencies by country. Filter frequencies above 10000 Hz may be entered by typing the value but are only valid up to half of the sample rate of the audio being processed. Q values outside of the slider range can be entered by typing the values but must be greater than 0.01. Frequency: 0 - 10000 Hz, default 60 Hz Q: 0.1 - 20.00, default 1.00 - determines the width of the notch. Below 1 creates a wider notch, above 1 creates a narrower notch. Parametric EQ Author: Steve Daulton and Bill Wharrie A parametric equalizer is a variable equalizer effect which provides control of three parameters: amplitude, center frequency and bandwidth. This plug-in provides control of one frequency band that can be tuned to a user defined center frequency. The width of the affected frequency band may be adjusted with the Width control and the defined frequency band may be boosted or attenuated according to the Gain control. Frequency (Hz): 10 to 10000 Hz, default 1000 Hz - sets the center frequency of the filter Width: 0 to 10, default 5 - determines the width of the affected frequency band. Greater width settings affect a broader range of frequencies. Smaller width affects a narrower band of frequencies. Numerically the width setting is approximately the half gain width in half octaves, thus the default setting of 5 has a half gain width of approximately 2.5 octaves. Gain (dB): -15 to 15 dB, default 0 dB (no effect) - how much the filter center frequency is boosted or attenuated. Random Low Pass Filter Like someone is playing around with the cutoff frequency knob of your low pass filter. Because of the way the random signal is generated, the lower the maximum speed is, the higher the depth factor must be to produce a similar depth of filtering changes. If you generate white noise then apply this effect, you can to some extent simulate constant pitch wind noise. Max filter sweep speed: 0.01 - 10.0 Hz, default 0.2 - maximum speed of the random filter cutoff changes Filter depth factor: 1 - 300, default 20 - how extreme the random filter cutoff changes are Maximum cutoff frequency: 20 - 5000 H, default 2000 - the filters maximum cutoff frequency Resonant Filter Author: Steve Daulton A filter with low pass, high pass and band pass options with a resonance control. Audio filters are commonly designed to have a smooth frequency response that is essentially flat in the pass band then rolls off to a lower level in the stop band, but in some cases it is desirable to use a filter that has a peak and accentuates frequencies close to the defined filter frequency. Such filters are commonly used in sound synthesis to cause ringing at specified frequencies. This tends to be most effective with sounds that have complex frequency content, such as noise. Filter frequency: 1 to 20000 Hz (default: 1000 Hz) - The corner frequency of the filter. The frequency must be below the Nyquist Frequency (half the sample rate) or an error message will be displayed. Resonance (Q): 0.1 to 100 (default: 10) - The amount of resonance. Higher values will produce a more pronounced and narrower peak at the corner frequency. Lower values will produce a less prominant peak with values below 0.7 showing no peak at all. Filter type: choice: Low Pass, High Pass, Band Pass (default: Low Pass) - Low pass allows frequencies below the corner frequency to pass through the filter and reduces frequencies above the corner. High Pass allows frequencies above the corner to pass and reduces frequencies below the corner. Band Pass reduces frequencies that are below the corner and reduces frequencies that are above the corner, allowing only a band of frequencies around the corner frequency to pass. Output Gain: -60 to 0 dB (default -12 dB) - Because the resonance accentuates frequencies around the corner frequency it is often necessary to reduce the output level of this effect. Lower (more negative) values reduce the level more. Shelf Filter Author: Steve Daulton A shelf filter with options for high shelf, low shelf or mid-band. Low-shelf filter passes all frequencies, but increases or reduces frequencies below the shelf frequency by specified amount. High-shelf filter passes all frequencies, but increases or reduces frequencies above the shelf frequency by specified amount. Mid-band shelf filter passes all frequencies, but increases or reduces frequencies between the low and high cutoff frequencies by specified amount. Filter type: low-shelf high-shelf mid-band - specifies which type of filter Low frequency cutoff: 1 to 10000 Hz - The corner frequency for the low shelf filter, or the lower corner frequency for the mid-band filter. High frequency cutoff: 0.1 to 20 kHz - The corner frequency for the high shelf filter, or the upper corner frequency for the mid-band filter. The high frequency cutoff must be less than half the track sample rate. Filter gain: - 30 dB - how much to boost or cut the filtered audio. Positive values boot and negative values reduce the level. Ten Band EQ An Equaliser (EQ) that can modify one band at a time. Select the band number (1 to 10) and gain (-24 to 24 dB).