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CM0340 Tutorial 5: MATLAB BASIC DSPand Synthesis
In this tutorial we explain some of the DSP and MATLAB conceptsdiscussed in the lectures leading up to Digital Audio Synthesis.
We start by looking at basic signal waveforms.
The code that is used over the next few slides is waveforms.m
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Simple Waveforms
• Frequency is the number of cycles per second and is measured inHertz (Hz)
• Wavelength is inversely proportional to frequencyi.e. Wavelength varies as 1
frequency
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The Sine Wave and Sound
The general form of the sine wave we shall use (quite a lot of) is asfollows:
y = A.sin(2π.n.Fw/Fs)
where:
A is the amplitude of the wave,Fw is the frequency of the wave,Fs is the sample frequency,n is the sample index.
MATLAB function: sin() used — works in radians
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MATLAB Sine Wave Radian FrequencyPeriod
Basic 1 period Simple Sine wave — 1 period is 2π radians
% Basic 1 period Simple Sine wave
i=0:0.2:2*pi;y = sin(i);figure(1)plot(y);
% use stem(y) as alternative plot as in lecture notes to% see sample values
title(’Simple 1 Period Sine Wave’);
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MATLAB Sine Wave Amplitude
Sine Wave Amplitude is -1 to +1.
To change amplitude multiply by some gain (amp):
% Now Change amplitude
amp = 2.0;
y = amp*sin(i);
figure(2)plot(y);title(’Simple 1 Period Sine Wave Modified Amplitude’);
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MATLAB Sine Wave Frequency% Natural frequency is 2*pi radians% If sample rate is F_s HZ then 1 HZ is 2*pi/F_s% If wave frequency is F_w then freequency is F_w* (2*pi/F_s)% set n samples steps up to sum duration nsec*F_s where nsec is the% duration in seconds% So we get y = amp*sin(2*pi*n*F_w/F_s);
F_s = 11025;F_w = 440;nsec = 2;dur= nsec*F_s;
n = 0:dur;
y = amp*sin(2*pi*n*F_w/F_s);
figure(3)plot(y(1:500));title(’N second Duration Sine Wave’);
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MATLAB Sine Wave Plot of n cycles
% To plot n cycles of a waveform
ncyc = 2;
n=0:floor(ncyc*F_s/F_w);
y = amp*sin(2*pi*n*F_w/F_s);
figure(4)plot(y);title(’N Cycle Duration Sine Wave’);
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Cosine, Square and Sawtooth WaveformsMATLAB functions cos() (cosine), square() and sawtooth()
similar.
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MATLAB Cos v Sin Wave% Cosine is same as Sine (except 90 degrees out of phase)
yc = amp*cos(2*pi*n*F_w/F_s);
figure(5);hold onplot(yc,’b’);plot(y,’r’);title(’Cos (Blue)/Sin (Red) Plot (Note Phase Difference)’);hold off;
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Relationship Between Amplitude,Frequency and Phase
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Amplitudes of a Sine Wave% Simple Sin Amplitude Demo
samp_freq = 400;dur = 800; % 2 secondsamp = 1; phase = 0; freq = 1;s1 = mysin(amp,freq,phase,dur,samp_freq);
axisx = (1:dur)*360/samp_freq; % x axis in degreesplot(axisx,s1);set(gca,’XTick’,[0:90:axisx(end)]);
fprintf(’Initial Wave: \t Amplitude = ...\n’, amp, freq, phase,...);
% change amplitudeamp = input(’\nEnter Ampltude:\n\n’);
s2 = mysin(amp,freq,phase,dur,samp_freq);hold on;plot(axisx, s2,’r’);set(gca,’XTick’,[0:90:axisx(end)]);
The code is sinampdemo.m
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mysin MATLAB codeThe above call function mysin.m which a simple modified version
of previous MATLAB sin function to account for phase.function s = mysin(amp,freq,phase,dur, samp_freq)% example function to so show how amplitude,frequency and phase% are changed in a sin function% Inputs: amp - amplitude of the wave% freq - frequency of the wave% phase - phase of the wave in degree% dur - duration in number of samples% samp_freq - sample frequncy
x = 0:dur-1;phase = phase*pi/180;
s = amp*sin( 2*pi*x*freq/samp_freq + phase);
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Amplitudes of a Sine Wave: sinampdemooutput
0 90 180 270 360 450 540 630 720−1
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Frequencies of a Sine Wave% Simple Sin Frequency Demo
samp_freq = 400;dur = 800; % 2 secondsamp = 1; phase = 0; freq = 1;s1 = mysin(amp,freq,phase,dur,samp_freq);
axisx = (1:dur)*360/samp_freq; % x axis in degreesplot(axisx,s1);set(gca,’XTick’,[0:90:axisx(end)]);
fprintf(’Initial Wave: \t Amplitude = ...\n’, amp, freq, phase,...);
% change amplitudefreq = input(’\nEnter Frequency:\n\n’);
s2 = mysin(amp,freq,phase,dur,samp_freq);hold on;plot(axisx, s2,’r’);set(gca,’XTick’,[0:90:axisx(end)]);
The code is sinfreqdemo.m
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Amplitudes of a Sine Wave: sinfreqdemooutput
0 90 180 270 360 450 540 630 720−1
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Phases of a Sine Wave% Simple Sin Phase Demo
samp_freq = 400;dur = 800; % 2 secondsamp = 1; phase = 0; freq = 1;s1 = mysin(amp,freq,phase,dur,samp_freq);
axisx = (1:dur)*360/samp_freq; % x axis in degreesplot(axisx,s1);set(gca,’XTick’,[0:90:axisx(end)]);
fprintf(’Initial Wave: \t Amplitude = ...\n’, amp, freq, phase,...);
% change amplitudephase = input(’\nEnter Phase:\n\n’);
s2 = mysin(amp,freq,phase,dur,samp_freq);hold on;plot(axisx, s2,’r’);set(gca,’XTick’,[0:90:axisx(end)]);
The code is sinphasedemo.m
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Amplitudes of a Sine Wave: sinphasedemooutput
0 90 180 270 360 450 540 630−1
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MATLAB Square and Sawtooth Waveforms% Square and Sawtooth Waveforms created using Radians
ysq = amp*square(2*pi*n*F_w/F_s);ysaw = amp*sawtooth(2*pi*n*F_w/F_s);
figure(6);hold onplot(ysq,’b’);plot(ysaw,’r’);title(’Square (Blue)/Sawtooth (Red) Waveform Plots’);hold off;
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Triangular WaveformMATLAB function sawtooth(t,width = 0.5) can create a
triangular waveform, but its easy to build one ourselves (later wemake a smoother sounding sawtooth in similar fashion):% Half Frequencydelta = 2*F_w/F_s;% min and max values of simple waveformminf=0;maxf=1;% create triangle wave of centre frequency valuesfigure(7); hold onytri = [];% plot n cycleswhile(length(ytri) < floor(ncyc*F_s/F_w) )
ytri = [ ytri amp*(minf:delta:maxf) ]; %upslopedoplot = input(’\nPlot Figure? y/[n]:\n\n’, ’s’);if doplot == ’y’,
plot(ytri,’r’);figure(7);
endlasti = length(ytri);ytri = [ ytri amp*(maxf:-delta:minf) ]; %downslopedoplot = input(’\nPlot Figure? y/[n]:\n\n’, ’s’);if doplot == ’y’,
plot(ytri,’b’);figure(7);
endendtitle(’Triangular Waveform Plots’); hold off;
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Triangular Waveform Display
0 5 10 15 20 25 30 35 400
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Using these WaveformsAll above waveforms used (as seen in Lecture notes):
• Base waveforms for various forms of synthesis: Subtractive, FM,Additive
• Modulators and Carrier waveforms for various Digital Audioeffects.
– Low Frequency Oscillators (LFO) to vary filter cut-offfrequencies and delay times
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MATLAB filtersMatlab filter() function implements an IIR(or an FIR no A components).
Type help filter:
FILTER One-dimensional digital filter.Y = FILTER(B,A,X) filters the data in vector X with thefilter described by vectors A and B to create the filtereddata Y. The filter is a "Direct Form II Transposed"implementation of the standard difference equation:
a(1)*y(n) = b(1)*x(n) + b(2)*x(n-1) + ... + b(nb+1)*x(n-nb)- a(2)*y(n-1) - ... - a(na+1)*y(n-na)
If a(1) is not equal to 1, FILTER normalizes the filtercoefficients by a(1).
FILTER always operates along the first non-singleton dimension,namely dimension 1 for column vectors and non-trivial matrices,and dimension 2 for row vectors.
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A Complete IIR Systemx(n)
T T T
x(n ! 1) x(n ! 2) x(n ! N + 1)
" " " " "b0 b1 b2 bN!2 bN!1
+ + + + +y(n)
" " " "!aM !aM!1 !aM!2 !a1
T T T
y(n ! M) y(n ! 1)
1
Here we extend:The input delay line up to N − 1 elements andThe output delay line by M elements.
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Complete IIR System Algorithmx(n)
T T T
x(n ! 1) x(n ! 2) x(n ! N + 1)
" " " " "b0 b1 b2 bN!2 bN!1
+ + + + +y(n)
" " " "!aM !aM!1 !aM!2 !a1
T T T
y(n ! M) y(n ! 1)
1
We can represent the IIR system algorithm by the differenceequation:
y(n) = −M∑
k=1
ak y(n− k) +
N−1∑k=0
bk x(n− k)
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Using filter() in PracticeWe have two filter banks defined by vectors: A = {ak}, B = {bk}.
We have to specify some values for them.
• We can do this by hand — we could design our own filters
• MATLAB provides standard functions to set up A and B formany common filters.
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Using filter() in Practice:Hand CodingRecall from lecture notes:The MATLAB file IIRdemo.m sets up the filter banks as follows:
fg=4000;fa=48000;k=tan(pi*fg/fa);
b(1)=1/(1+sqrt(2)*k+kˆ2);b(2)=-2/(1+sqrt(2)*k+kˆ2);b(3)=1/(1+sqrt(2)*k+kˆ2);a(1)=1;a(2)=2*(kˆ2-1)/(1+sqrt(2)*k+kˆ2);a(3)=(1-sqrt(2)*k+kˆ2)/(1+sqrt(2)*k+kˆ2);
and then applies the difference equation:
for n=1:Ny(n)=b(1)*x(n) + b(2)*xh1 + b(3)*xh2 - a(2)*yh1 - a(3)*yh2;xh2=xh1;xh1=x(n);yh2=yh1;yh1=y(n);end;
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Filtering with IIR: Simple Example Output
This produces the following output:
0 2 4 6 8 10 12 14 16 18
−1
−0.5
0
0.5
1
n →
x(n)
→
0 2 4 6 8 10 12 14 16 18
−1
−0.5
0
0.5
1
n →
y(n)
→
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Using MATLAB to make filtersMATLAB provides a few built-in functions to create ready made
filter parameter A and B:E.g: butter, buttord, besself, cheby1, cheby2,
ellip, freqz, filter.For our purposes the Butterworth filter will create suitable filters,
help butter:
BUTTER Butterworth digital and analog filter design.[B,A] = BUTTER(N,Wn) designs an Nth order lowpass digitalButterworth filter and returns the filter coefficients in lengthN+1 vectors B (numerator) and A (denominator). The coefficientsare listed in descending powers of z. The cutoff frequencyWn must be 0.0 < Wn < 1.0, with 1.0 corresponding tohalf the sample rate.
If Wn is a two-element vector, Wn = [W1 W2], BUTTER returns anorder 2N bandpass filter with passband W1 < W < W2.[B,A] = BUTTER(N,Wn,’high’) designs a highpass filter.[B,A] = BUTTER(N,Wn,’low’) designs a lowpass filter.[B,A] = BUTTER(N,Wn,’stop’) is a bandstop filter if Wn = [W1 W2].
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Using MATLAB to make filtershelp buttord:BUTTORD Butterworth filter order selection.[N, Wn] = BUTTORD(Wp, Ws, Rp, Rs) returns the order N of the lowestorder digital Butterworth filter that loses no more than Rp dB inthe passband and has at least Rs dB of attenuation in the stopband.Wp and Ws are the passband and stopband edge frequencies, normalizedfrom 0 to 1 (where 1 corresponds to pi radians/sample). For example,
Lowpass: Wp = .1, Ws = .2Highpass: Wp = .2, Ws = .1Bandpass: Wp = [.2 .7], Ws = [.1 .8]Bandstop: Wp = [.1 .8], Ws = [.2 .7]
BUTTORD also returns Wn, the Butterworth natural frequency (or,the "3 dB frequency") to use with BUTTER to achieve thespecifications.
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Using MATLAB Filter Example: Subtractive Synthesis LectureExample
The example for studying subtractive synthesis, subtract synth.m,uses the butter and filter MATLAB functions:
% simple low pas filter example of subtractive synthesisFs = 22050;y = synth(440,2,0.9,22050,’saw’);
% play sawtooth e.g. waveformdoit = input(’\nPlay Raw Sawtooth? Y/[N]:\n\n’, ’s’);if doit == ’y’,
figure(1)plot(y(1:440));playsound(y,Fs);end
%make lowpass filter and filter y[B, A] = butter(1,0.04, ’low’);yf = filter(B,A,y);
[B, A] = butter(4,0.04, ’low’);yf2 = filter(B,A,y);
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% play filtererd sawtoothsdoit = ...
input(’\nPlay Low Pass Filtered (Low order) ? Y/[N]:\n\n’, ’s’);if doit == ’y’,figure(2)plot(yf(1:440));playsound(yf,Fs);end
doit = ...input(’\nPlay Low Pass Filtered (Higher order)? Y/[N]:\n\n’, ’s’);
if doit == ’y’,figure(3)
plot(yf2(1:440));playsound(yf2,Fs);end
%plot figuresdoit = input(’\Plot All Figures? Y/[N]:\n\n’, ’s’);if doit == ’y’,figure(4)plot(y(1:440));hold onplot(yf(1:440),’r+’);plot(yf2(1:440),’g-’);end
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synth.m
The supporting function, synth.m, generates waveforms as we haveseen earlier in this tutorial:function y=synth(freq,dur,amp,Fs,type)% y=synth(freq,dur,amp,Fs,type)%% Synthesize a single note%% Inputs:% freq - frequency in Hz% dur - duration in seconds% amp - Amplitude in range [0,1]% Fs - sampling frequency in Hz% type - string to select synthesis type% current options: ’fm’, ’sine’, or ’saw’
if nargin<5error(’Five arguments required for synth()’);
end
N = floor(dur*Fs);n=0:N-1;if (strcmp(type,’sine’))
y = amp.*sin(2*pi*n*freq/Fs);
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elseif (strcmp(type,’saw’))
T = (1/freq)*Fs; % period in fractional samplesramp = (0:(N-1))/T;y = ramp-fix(ramp);y = amp.*y;y = y - mean(y);
elseif (strcmp(type,’fm’))
t = 0:(1/Fs):dur;envel = interp1([0 dur/6 dur/3 dur/5 dur], [0 1 .75 .6 0], 0:(1/Fs):dur);I_env = 5.*envel;y = envel.*sin(2.*pi.*freq.*t + I_env.*sin(2.*pi.*freq.*t));
elseerror(’Unknown synthesis type’);
end
% smooth edges w/ 10ms rampif (dur > .02)
L = 2*fix(.01*Fs)+1; % L oddramp = bartlett(L)’; % odd lengthL = ceil(L/2);y(1:L) = y(1:L) .* ramp(1:L);y(end-L+1:end) = y(end-L+1:end) .* ramp(end-L+1:end);
end
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synth.m (Cont.)
Note the sawtooth waveform generated here has a non-linear up slope:
0 10 20 30 40 50 60 70 80 90 100−0.15
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This is created with:
ramp = (0:(N-1))/T;y = ramp-fix(ramp);
fix rounds the elements of X to the nearest integers towards zero.
This form of sawtooth sounds slightly less harsh and is more suitablefor audio synthesis purposes.
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Wavetable Synthesis ExamplesSimple form as discussed in Lectures is a simple creation one sine
wave and one saw and then some simple cross-fading between thewaves. The simple example , :
% simple example of Wavetable synthesis
f1 = 440; f2 = 500; f3 = 620;Fs = 22050;
%Create a single sine waves of frequencie f1y1 = synth(f1,1/f1,0.9,Fs,’sine’);
doit = input(’\nPlay/Plot Raw Sine y1 looped for 10 ...seconds? Y/[N]:\n\n’, ’s’);
if doit == ’y’,figure(1)plot(y1);loopsound(y1,Fs,10*Fs/f1);end
%Create a single Saw waves of frequencie f2y2 = synth(f2,1/f2,0.9,Fs,’saw’);
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doit = input(’\nPlay/Plot Raw saw y2 looped for 10 ...seconds? Y/[N]:\n\n’, ’s’);
if doit == ’y’,figure(2)plot(y2);loopsound(y2,Fs,10*Fs/f2);end
%concatenate waveywave = [y1 , y2];
% Create Cross fade half width of wave y1 for xfade windowxfadewidth = floor(Fs/(f1*2));ramp1 = (0:xfadewidth)/xfadewidth;ramp2 = 1 - ramp1;
doit = input(’\nShow Crossfade Y/[N]:\n\n’, ’s’);if doit == ’y’,figure(4)plot(ramp1);hold on;plot(ramp2,’r’);end;
%apply crossfade centered over the join of y1 and y2pad = (Fs/f1) + (Fs/f2) - 2.5*xfadewidth;
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xramp1 = [ones(1,1.5*xfadewidth) , ramp2, zeros(1,pad)];xramp2 = [zeros(1,1.5*xfadewidth) , ramp1, ones(1,pad)];
% Create two period waveforms to fade betweenywave2 = [y1 , zeros(1,Fs/f2)];ytemp = [zeros(1,Fs/f1), y2];
ywave = ywave2;% do xfade
% add two waves together over the period modulated by xfade ramps% (recall .* to multiply matrices element by element% NOT MATRIX mutliplication
ywave2 = xramp1.*ywave2 + xramp2.*ytemp;
doit = input(’\nPlay/Plot Additive Sines together? Y/[N]:\n\n’, ’s’);if doit == ’y’,figure(5)
subplot(4,1,1);plot(ywave);
hold off
set(gca,’fontsize’,18);ylabel(’Amplitude’);
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title(’Wave 1’);set(gca,’fontsize’,18);subplot(4,1,2);plot(ytemp);set(gca,’fontsize’,18);ylabel(’Amplitude’);title(’Wave 2’);set(gca,’fontsize’,18);subplot(4,1,3);plot(xramp1);hold onplot(xramp2,’r’)hold offset(gca,’fontsize’,18);ylabel(’Amplitude’);title(’Crossfade Masks’);set(gca,’fontsize’,18);subplot(4,1,4);plot(ywave2);set(gca,’fontsize’,18);ylabel(’Amplitude’);title(’WaveTable Synthesis’);set(gca,’fontsize’,18);loopsound(ywave2,Fs,10*Fs/(f1 + f2));end
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RECAP: Wavetable synthesis:Dynamic Waveshaping (1)
Simplest idea: Linear crossfading
• Crossfade from one wavetable to the next sequentially.
• Crossfade = apply some envelope to smoothly merge waveforms.
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Simple MATLAB Example: Linear Crossfading
% Create Cross fade half width of wave y1 for xfade windowxfadewidth = floor(Fs/(f1*2));ramp1 = (0:xfadewidth)/xfadewidth;ramp2 = 1 - ramp1;
%apply crossfade centered over the join of y1 and y2pad = (Fs/f1) + (Fs/f2) - 2.5*xfadewidth;xramp1 = [ones(1,1.5*xfadewidth) , ramp2, zeros(1,pad)];xramp2 = [zeros(1,1.5*xfadewidth) , ramp1, ones(1,pad)];% Create two period waveforms to fade betweenywave2 = [y1 , zeros(1,Fs/f2)];ytemp = [zeros(1,Fs/f1), y2];
% do xfade% add two waves together over the period modulated by xfade rampsywave2 = xramp1.*ywave2 + xramp2.*ytemp;
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Simple MATLAB Example: Linear Crossfading (Cont.)
0 20 40 60 80 100−1
0
1
Ampl
itude
Wave 1
0 20 40 60 80 100−0.5
0
0.5Am
plitu
deWave 2
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itude
Crossfade Masks
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itude
WaveTable Synthesis
Note: This sort of technique is useful to create an ADSR envelopein MATLAB
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