From EQ to Reverb & Distortion: DSP Audio Effects in Matlab

Transcription

1 From EQ o Reverb & Disorion: DSP Audio Effecs in Malab By Henry D. Pfiser Overview These noes are designed for a wo-hour summer enrichmen workshop for high-school sudens. The aciviy inroduces he sudens o digial signal processing (DSP) for audio signals. Sudens will record and play sound on a compuer using he Malab scriping language and hen apply audio effecs (e.g., equalizaion, reverberaion, and disorion) o ha sound. Sound is ransmied hrough he air as a pressure wave ha varies boh in ime and space (e.g., see Figure ). An audio signal is a funcion ha describes he pressure, measured a a fixed poin in space (e.g., your ear), as a funcion of ime. Figure : A speaker generaes a pressure wave in space by vibraing. This generaes a pressure profile ha varies in ime and across space. High pressure regions in space have more gas molecules and low pressure regions have fewer gas molecules. The sinusoid shows he pressure as a funcion of space for a fixed insance in ime. Figure 2: For a pressure wave, he disance beween peaks in space is called he wavelengh while he inerval beween peaks in ime is called he period.

2 . Signals An audio signal is periodic wih period T seconds if i is formed by repeaing a single segmen of duraion T seconds. The smalles possible T for which his is rue is called he fundamenal period of he signal. A signal wih fundamenal period T repeas one cycle every T seconds and has a fundamenal frequency of F = /T cycles per second. The erms cycles per second is also known as Herz and is abbreviaed Hz. Example. Since he funcion sin(2π) is periodic wih a fundamenal frequency of Hz, he funcion = sin(2πf ) is periodic wih fundamenal frequency f Hz and fundamenal period /f sec. Example 2. For each signal below, deermine if he signal is periodic and, if so, wha is is fundamenal period and fundamenal frequency. = sin(4π) = e Square Wave Sawooh Wave Digial Represenaions Signals relaed o physical quaniies, such as pressure, are ypically considered o ake coninuous values ha lie in some real inerval and o be coninuous ime (i.e., he signal value is defined for all real valued imes). On he oher hand, digial compuers can only represen a finie number of signal values and ime insans. A discree-ime signal is defined only for ineger ime insans. One can conver a coninuous-ime signal ino a discree-ime signal by sampling a uniformly-spaced imes. For example, y[n] = y(nt s ) is a discree-ime sampled version of wih a sampling period of T s. The sampling frequency F s = /T s is defined o be he number of samples aken per second. Figure 3 shows some example discree-ime signals. A quanized signal is a signal where he ampliude values are limied o finie se. If he se is large enough, hen his deail can ypically be ignored. For example, he compac disc (CD) audio sandard 2

3 y[n] = sin(πn/6) y[n] = cos(.5n) y[n] y[n] 2 2 n 5 5 n Figure 3: Examples of discree-ime signals uses a sampling rae of 44, Hz and quanizes he signal o one of differen values. These values were chosen o so ha, for ypical liseners, he perceived loss in qualiy is negligible. Also, he number = 2 6 can be represened by 6 bis..3 Malab Malab is an inerpreed language ha makes i easy o manipulae marices and vecors. Now, we will sar o explore he use of MATLAB o generae and plo signals. To learn more abou any funcion is Malab, you can ype help [funcion name] in he MATLAB command window. If you don know he exac funcion name, you may also ype lookfor [keyword]. This will lis all funcions relaed o he keyword. Type in he following commands o see a few Malab commands in acion: *2 3/2 pi cos(pi) a = [ ] a2 = :5 max(a2) b = [ ] b2 = :2:8 lengh(b2) min(b2) 5*a a+b b-a a.* b help.* a./ b help./ ones(,) zeros(,) 3

4 Type (or cu and pase) he following commands o see how Malab can be used o define, plo, and analyze signals: = -3:3 y = cos(2*pi*/7) plo(,y) xlabel(''); ylabel(''); ile('cos(2*pi*/7)') % Seup ime indices % Discree-ime signal y % Plo signal y % Label plo % Label plo % Label plo Fs = 225; Ts = /Fs; = :Ts:3; y = cos(2*pi**8) soundsc(y,fs) % Seup sample rae % Seup sample period % Generae ime indices in seconds % Generae 8 Hz signal % Scale and play sound hrough speaker Exercise 3. Change he previous example o play a 4 Hz one. Exercise 4. Consider he following signal: = 2 sin(2πf) + 4 cos(πf + p), where f = 5 Hz, p = π/4, F s = Hz, and = : /F s : 2. Use MATLAB o: (a) Plo y (b) Find he lengh of y (c) Find he maximum of y (d) Find he minimum of y 2 Audio Processing For his secion, you will need o download he accompanying files guiar4.wav and guiar6.wav. 2. Imporing WAV files Malab makes i fairly easy o impor audio files. Type help audioread o find ou more deails. Use he following command o load and play a WAV file guiar riff: [y,fs] = audioread('guiar4.wav'); soundsc(y,fs); % Read WAV file % Play WAV file Exercise 5. Repea he previous seps wih he file guiar6.wav. Nex, we will exrac a porion of he WAV file and deermine he fundamenal frequency of one noe index = 33:34985; yy = y(index); soundsc(yy,fs) plo(index,yy) % Index samples for second noe % Exrac indexed samples ino yy % Play exraced waveform % Plo exraced waveform From he picure, we see ha his porion of he signal appears o be periodic. One can esimae he fundamenal frequency by couning he number of samples S in one cycle. Analyzing he unis, one finds ha Hz = cycles second = samples sec cycles sample = F s/s. Exercise 6. Can you hink of a more accurae way o esimae he fundamenal period and frequency from his plo. Describe your mehod and use i o esimae he frequency in Herz. 4

5 2.2 Echo and Reverberaion An echo occurs when a sound reflecs off a disance surface. Due o he longer pah, he refleced sound arrives laer han he original sound and can be heard as a disinc second copy of he original sound. delay =.2; index = round(delay*fs); yy = [zeros(index,); y]; yy = yy(:lengh(y)); yy = yy+y; soundsc(yy,fs); % Delay in seconds % Delay in samples % Zero pad he beginning o add delay % Cu vecor o correc lengh % Add original sound o echo % Play i Exercise 7. Play wih he value of delay o see he effec. Reverberaion (or reverb) is he effec generaed by many copies of a sound wih differen delays summing ogeher due o reflecions off he walls of a room. In conras o echo, he individual copies are no disinguishable bu overall effec ypically makes he sound richer or fuller. delay = round(fs*.8); % FIR Delay delay2 = round(fs*.25); % IIR Delay coef =.7; % IIR Decay rae yy = filer([ zeros(,delay) coef],[ zeros(,delay2) -coef],y); soundsc(yy,fs); % Play i Exercise 8. Adjus he values of delay, delay2, and coef o make a reverb ha sounds beer o you. 2.3 Disorion An ideal amplifier simply akes he inpu signal and muliplies i by some consan larger han. Some amplifier designs are nearly ideal when he oupu level is no oo large. Bu, heir characerisics change for larger oupu values. In paricular, he signal is disored when he amplifier is driven ino sauraion. For vacuum ube amplifiers, his disorion has a sof edge and many rock musicians enjoy he resuling sound. hard-limier sof-limier 2 oupu oupu inpu inpu For our experimen, we will use a sof-limier based on he inverse angen funcion aan(x). Anoher good choice is he hyperbolic angen anh(x). yy = aan(8*y); soundsc(yy,fs); Exercise 9. Adjus he consan 8 in he previous example o find your favorie disorion. 5

6 2.4 Recording Sound in Malab Now, we will ry some of hese effecs on sounds ha we have recorded. The following code excerp shows how o record sound in Malab. r = audiorecorder(fs,6,); record(r); sop(r); mywav = geaudiodaa(r,'double'); soundsc(mywav,fs); % Ge recorder wih Fs samples/sec, 6 bi, mono % Sar speaking or singing afer his line % Execue his line when you are done % Ge sound % Replay sound Exercise. Now ry recording your singing or humming. Then, plo he signal o find a nice periodic secion. Based on he earlier example, ry o esimae he fundamenal frequency of ha periodic secion. Exercise. Now ry downloading a WAV or MP3 file from he inerne. This file can be loaded ino Malab and processed in a similar. Ideally, he sound file should use a sampling rae of 44 HZ. Also, if he file is sereo (i.e., i has lef/righ channels), hen you will need o ake he lef channel by wriing y=y(,:);. 3 Filers and Equalizaion A filer is a device ha amplifies differen frequencies by differen amouns. Many of you have probably seen a picure of he graphical equalizers ha were popular in he 98s. Figure 4: Graphical equalizer The heory behind filering a bi oo mahemaical for his workshop bu i is covered in a juniorlevel course called Signals and Sysems. For now, we will ake a cookbook approach and use Malab o design 3 filers for us: a low-pass filer, a high-pass filer, and a band-pass filer. lowpass = fir(2,2/225); % low-pass filer wih cuoff 2 Hz yylow = filer(lowpass,,y); % filer signal soundsc(yylow,fs);pause; % play filered signal, wai for keypress bandpass = fir(2,[2/225 /225],'DC-'); % band-pass filer ha passes 2-2 Hz yyband = filer(bandpass,,y); % filer signal soundsc(yyband,fs);pause; % play filered signal, wai for keypress highpass = fir(2,2/225,'dc-'); % high-pass filer wih cuoff 2 Hz yyhigh = filer(highpass,,y); % filer signal soundsc(yyhigh,fs);pause; % play filered signal, wai for keypress soundsc(yylow+yyband+yyhigh,fs);pause; % play he sum off all bands, awai keypress yyeq = yylow+4*yyband+yyhigh; % 3-band EQ for mid-range boos soundsc(yyeq,fs); % Play EQ wih mid-range boos 6

7 4 Muliple Effecs Now ha we have consruced 3 differen effecs in Malab, we can pu hem ogeher in an arbirary fashion. yyeqd = aan(8*yyeq); yyfinal = filer([ zeros(,delay) coef],[ zeros(,delay2) -coef],yyeqd); soundsc(yyfinal,fs); Exercise 2. There are many oher applicaions of DSP for audio. Try o lis a few of hem. 7