Dsp iit workshop

Digital Signal Processing Using Scilab

R.Senthilkumar,Assistant Professor,

Department of Electronics and Communication Engineering,

Institute of Road and Transport Technology,Erode – 638 316

Email Id: senthilkumar@irttech.ac.inrsenthil_signalprocess@in.com

Mobile no: 9940882605

Topics to be covered in signal processing

[1]. Generation of Discrete basic sequences:

unit impulse, unit step sequence, Ramp, sine, cosine and exponential sequences.

[2]. Linear Convolution

[3]. Circular Convolution

[4]. Linear Convolution using Circular Convolution

[5]. DFT and IDFT

[6]. DFT and IDFT using FFT inbuilt function

[7]. DCT and IDCT

[8]. Cross correlation and Auto correlation

[9]. Sampling and aliasing effect

[10]. FIR digital filter design

[11]. IIR digital filter design

[12]. Power Spectrum Estimation using DFT

[1]. Generation of basic sequences

//Caption:Unit Impulse Sequence [Program]

clear;

clc;

close;

L = 4; //Upperlimit

n = -L:L;

x = [zeros(1,L),1,zeros(1,L)];

b = gca();

b.y_location = "middle";

plot2d3('gnn',n,x)

a=gce();

a.children(1).thickness =4;

xtitle('Graphical Representation of Unit Impulse Sequence','n','x[n]');

//Caption: Unit Step Sequence [Program]

clear;

clc;

close;

L = 4; //Upperlimit

n = -L:L;

x = [zeros(1,L),ones(1,L+1)];

a=gca();

a.y_location = "middle";

plot2d3('gnn',n,x)

title('Graphical Representation of Unit Step Signal')

xlabel(' n');

ylabel(' x[n]');

//Caption: Discrete Ramp Sequence [Program]

clear;

clc;

close;

L = 4; //Upperlimit

n = -L:L;

x = [zeros(1,L),0:L];

b = gca();

b.y_location = 'middle';

plot2d3('gnn',n,x)

a=gce();

a.children(1).thickness =2;

xtitle('Graphical Representation of Discrete Unit Ramp Sequence','n','x[n]');

//Caption: Exponentially Decreasing Signal [Program]

clear;

clc;

close;

a =0.5;

n = 0:10;

x = (a)^n;

a=gca();

a.x_location = "origin";

a.y_location = "origin";

plot2d3('gnn',n,x)

a.thickness = 2;

xtitle('Graphical Representation of Exponentially Decreasing Signal','n','x[n]');

//Caption:Exponentially Increasing Signal [Program]

clear;

clc;

close;

a =1.5;

n =1:10;

x = (a)^n;

a=gca();

a.thickness = 2;

plot2d3('gnn',n,x)

xtitle('Graphical Representation of Exponentially Increasing Signal','n','x[n]');

//Caption: Sine wave and Cosine wave plot [Program]

clc;

clear;

close;

f = 3; //Frequency = 3 Hertz

t = 0:0.01:1;

X = sin(2*%pi*f*t);

Y = cos(2*%pi*f*t);

subplot(2,1,1)

plot2d3('gnn',t,X,5)

title("Sine waveform")

subplot(2,1,2)

plot2d3('gnn',t,Y,3)

title('Cosine Waveform')

[2]. Linear Convolution

//Caption:Program for Linear Convolution [Program]

clc;

clear all;

close ;

x = input('enter x seq');

h = input('enter h seq');

m = length(x);

n = length(h);

//Method 1 Using Direct Convolution Sum Formula

for i = 1:n+m-1

conv_sum = 0;

for j = 1:i

if (((i-j+1) <= n)&(j <= m))

conv_sum = conv_sum + x(j)*h(i-j+1);

end;

y(i) = conv_sum;

end;

end;

disp(y','Convolution Sum using Direct Formula Method =')

//Method 2 Using Inbuilt Function

f = convol(x,h)

disp(f,'Convolution Sum Result using Inbuilt Funtion =')

subplot(3,1,1);

plot2d3('gnn',x)

xtitle('Graphical Representation of Input signal x');

subplot(3,1,2);

plot2d3('gnn',h)

xtitle('Graphical Representation of Impulse signal h');

subplot(3,1,3);

plot2d3('gnn',y)

xtitle('Graphical Representation of Output signal y');

[3]. Circular Convolution

//Caption: Program to find the Cicrcular Convolution of given [Program]

//discrete sequences using Matrix method

clear;

clc;

x1 = [2,1,2,1]; //First sequence

x2 = [1,2,3,4]; //Second sequence

m = length(x1); //length of first sequence

n = length(x2); //length of second sequence

//To make length of x1 and x2 are Equal

if (m >n)

for i = n+1:m

x2(i) = 0;

end

elseif (n>m)

for i = m+1:n

x1(i) = 0;

end

end

N = length(x1);

x3 = zeros(1,N); //x3 = Circular convolution result

a(1) = x2(1);

for j = 2:N

a(j) = x2(N-j+2);

end

for i =1:N

x3(1) = x3(1)+x1(i)*a(i);

end

X(1,:)=a;

//Calculation of circular convolution

for k = 2:N

for j =2:N

x2(j) = a(j-1);

end

x2(1) = a(N);

X(k,:)= x2;

for i = 1:N

a(i) = x2(i);

x3(k) = x3(k)+x1(i)*a(i);

end

end

disp(X,'Circular Convolution Matrix x2[n]=')

disp(x3,'Circular Convolution Result x3[n] = ')

//Result

//Circular Convolution Matrix x2[n]=

// 1. 4. 3. 2.

// 2. 1. 4. 3.

// 3. 2. 1. 4.

// 4. 3. 2. 1.

// Circular Convolution Result x3[n] =

// 14. 16. 14. 16.

[4]. Linear Convolution using Circular Convolution

//Method 3 Linear Convolution using Circular Convolution

//Circular Convolution Using frequency Domain multiplication (DFT-IDFT method)[Program]

N = n+m-1;

x = [x zeros(1,N-m)];

h = [h zeros(1,N-n)];

f1 = fft(x)

f2 = fft(h)

f3 = f1.*f2; // freq domain multiplication

f4 = ifft(f3)

disp(f4,'Convolution Sum Result DFT - IDFT method =')

//Result

//enter x seq [1 1 1 1]

//enter h seq [1 2 3]

// Convolution Sum Result DFT - IDFT method =

// 1. 3. 6. 6. 5. 3.

[5]. DFT and IDFT using Formula

//Caption: Discrete Fourier Transform and its Inverse using formula

//Program to find the spectral information of discrete time signal [Program]

clc;

close;

clear;

xn = input('Enter the real input discrete sequence x[n]=');

N = length(xn);

XK = zeros(1,N);

IXK = zeros(1,N);

//Code block to find the DFT of the Sequence

for K = 0:N-1

for n = 0:N-1

XK(K+1) = XK(K+1)+xn(n+1)*exp(-%i*2*%pi*K*n/N);

end

end

[phase,db] = phasemag(XK)

disp(XK,'Discrete Fourier Transform X(k)=')

disp(abs(XK),'Magnitude Spectral Samples=')

disp(phase,'Phase Spectral Samples=')

n = 0:N-1;

K = 0:N-1;

subplot(2,2,1)

a = gca();

a.x_location = "origin";

a.y_location = "origin";

plot2d3('gnn',n,xn)

xlabel('Time Index n---->')

ylabel('Amplitude xn---->')

title('Discrete Input Sequence')

subplot(2,2,2)

a = gca();

a.x_location = "origin";

a.y_location = "origin";

plot2d3('gnn',K,abs(XK))

xlabel('Frequency Sample Index K---->')

ylabel('|X(K)|---->')

title('Magnitude Spectrum')

subplot(2,2,3)

a = gca();

a.x_location = "origin";

a.y_location = "origin";

plot2d3('gnn',K,phase)

xlabel('Frequency Sample Index K---->')

ylabel('<X(K) in radians---->')

title('Phase Spectrum')

//Code block to find the IDFT of the sequence

for n = 0:N-1

for K = 0:N-1

IXK(n+1) = IXK(n+1)+XK(K+1)*exp(%i*2*%pi*K*n/N);

end

end

IXK = IXK/N;

ixn = real(IXK);

subplot(2,2,4)

a = gca();

a.x_location = "origin";

a.y_location = "origin";

plot2d3('gnn',[0:N-1],ixn)

xlabel('Discrete Time Index n ---->')

ylabel('Amplitude x[n]---->')

title('IDFT sequence')

[6]. DFT and IDFT using FFT algorithm (inbuilt function)

//Caption:DFT and IDFT using FFT algorithm[Program]

clear;

clc;

close;

x = input("enter the input discrete sequence");

L = length(x); //Length of the Sequence

N = input('Enter the N-point value:='); // N -point DFT

//Computing DFT

X = fft(x,-1);

disp(X,'FFT of x[n] is X(k)=')

x =abs(fft(X,1))

disp(x,'IFFT of X(k) is x[n]=')

//Plotting the spectrum of Discrete Sequence

subplot(2,1,1)

a=gca();

a.data_bounds=[0,0;5,10];

plot2d3('gnn',0:length(x)-1,x)

b = gce();

b.children(1).thickness =3;

xtitle('Graphical Representation of Discrete Sequence','n','x[n]');

subplot(2,1,2)

a=gce();

a.data_bounds=[0,0;5,10];

plot2d3('gnn',0:length(X)-1,abs(X))

b = gce();

b.children(1).thickness =3;

xtitle('Graphical Representation of Discrete Spectrum','k','X(k)');

[7]. Discrete Cosine Transform and Inverse Discrete Cosine TransformProblem: If x[n] = {1,3,5,2,3,6,1,2} compute signal in frequency domain using DCT algorithm.

XDCT(K) = {8.131,-0.0342,-2.263,0.1573,-2.474,-1.7718,2.8508,-0.5784}

xIDCT[n] = {1,3,5,2,3,6,1,2}

Program

//Caption: Discrete Cosine Transform and Inverse Discrete Cosine Transform

clear;

clc;

close;

x = [1,3,5,2,3,6,1,2];

XDCT = dct(x,-1);//Discrete Cosine Transform

xIDCT = dct(XDCT,1);//Inverse Discrete Cosine Transform

disp(XDCT,'DCT of x[n]=')

disp(xIDCT,'Inverse DCT of x =')

//RESULT

//DCT of x[n]=

// 8.131728 - 0.0342533 - 2.2632715 0.1573526 - 2.4748737 - 1.7718504 2.8508949 - 0.5784576

// Inverse DCT of x =

// 1. 3. 5. 2. 3. 6. 1. 2.

[8]. Auto correlation and Cross correlation

a). Auto correlation [Program]

//Caption: Program to Compute the Autocorrelation of a Sequence And verfication of Autocorrelation property

clc;

clear;

close;

x = input('Enter the input Sequence:=');

m = length(x);

lx = input('Enter the lower index of input sequence:=')

hx = lx+m-1;

n = lx:1:hx;

x_fold = x($:-1:1);

nx = lx+lx;

nh = nx+m+m-2;

r = nx:nh;

Rxx = convol(x,x_fold);

disp(Rxx,'Auto Correlation Rxx[n]:=')

//Property 1: Autocorrelation of a sequence has even symmetry

//Rxx[n] = Rxx[-n]

Rxx_flip = Rxx([$:-1:1]);

if Rxx_flip==Rxx then

disp('Property 1:Auto Correlation has Even Symmetry');

disp(Rxx_flip,'Auto Correlation time reversed Rxx[-n]:=');

end

//Property 2: Center value Rxx[0]= total power of the sequence

Tot_Px = sum(x.^2);

Mid = ceil(length(Rxx)/2);

if Tot_Px == Rxx(Mid) then

disp('Property 2:Rxx[0]=center value=max. value=Total power of i/p sequence');

end

subplot(2,1,1)

plot2d3('gnn',n,x)

xlabel('n===>')

ylabel('Amplitude-->')

title('Input Sequence x[n]')

subplot(2,1,2)

plot2d3('gnn',r,Rxx)

xlabel('n===>')

ylabel('Amplitude-->')

title('Auto correlation Sequence Rxx[n]')

/Example

//Enter the input Sequence:=[2,-1,3,4,1]

//

//Enter the lower index of input sequence:=-2

//

// Auto Correlation Rxx[n]:=

//

// 2. 7. 5. 11. 31. 11. 5. 7. 2.

//

// Property 1:Auto Correlation has Even Symmetry

//

// Auto Correlation time reversed Rxx[-n]:=

//

// 2. 7. 5. 11. 31. 11. 5. 7. 2.

//

// Property 2:Rxx[0]=center value=max. value=Total power of i/p sequence

[b]. Cross Correlation [Program]

9. Sampling and Aliasing Effect//Caption: Verification of Sampling Theorem [Program]

//[1].Right Sampling [2]. Under Sampling [3]. Over Sampling

clc;

close;

clear;

fm=input('Enter the input signal frequency:');

k=input('Enter the number of Cycles of input signal:');

A=input('Enter the amplitude of input signal:');

tm=0:1/(fm*fm):k/fm;

x=A*cos(2*%pi*fm*tm);

figure(1);

a = gca();

a.x_location = "origin";

a.y_location = "origin";

plot(tm,x);

title('ORIGINAL SIGNAL');

xlabel('Time');

ylabel('Amplitude');

xgrid(1)

//Sampling Rate(Nyquist Rate)=2*fm

fnyq=2*fm;

// UNDER SAMPLING

fs=(3/4)*fnyq;

n=0:1/fs:k/fm;

xn=A*cos(2*%pi*fm*n);

figure(2);

a = gca();

a.x_location = "origin";

a.y_location = "origin";

plot2d3('gnn',n,xn);

plot(n,xn,'r');

title('Under Sampling');

xlabel('Time');

ylabel('Amplitude');

legend('Sampled Signal', 'Reconstructed Signal');

xgrid(1)

//NYQUIST SAMPLING

fs=fnyq;

n=0:1/fs:k/fm;

xn=A*cos(2*%pi*fm*n);

figure(3);

a = gca();

a.x_location = "origin";

a.y_location = "origin";

plot2d3('gnn',n,xn);

plot(n,xn,'r');

title('Nyquist Sampling');

xlabel('Time');

ylabel('Amplitude');

legend('Sampled Signal', 'Reconstructed Signal');

xgrid(1)

//OVER SAMPLING

fs=fnyq*10;

n=0:1/fs:k/fm;

xn=A*cos(2*%pi*fm*n);

figure(4);

a = gca();

a.x_location = "origin";

a.y_location = "origin";

plot2d3('gnn',n,xn);

plot(n,xn,'r');

title('Over Sampling');

xlabel('Time');

ylabel('Amplitude');

legend('Sampled Signal', 'Reconstructed Signal');

xgrid(1)

//Result

//Enter the input signal frequency:100

//Enter the number of Cycles of input signal:2

//Enter the amplitude of input signal:2

10. FIR Digital Filter Design

Low Pass Filter Design [Program]

//Caption: Program to Design FIR Low Pass Filter

//Reference: Digital Signal Processing A Practical Approach

//by Emmaanuel Ifeachor and Barrie Jervis, Second Edition, Pearson Education

clc;

close;

clear;

fp = input("Enter the passband edge frequency in Hz=")

delf = input("Enter the transition width in Hz =")

A = input("Enter the stop band attenuation in dB =")

Fs = input("Enter the sampling frequency in Hz=")

M = (A-7.95)/(14.36*delf/Fs) //Filter length

M = ceil(M);

Wc = ((fp+delf/2)/Fs)*%pi; //Digital Cutoff frequency

Tuo = (M-1)/2 //Center Value

for n = 1:M

if (n == Tuo+1)

hd(n) = Wc/%pi;

else

hd(n) = sin(Wc*((n-1)-Tuo))/(((n-1)-Tuo)*%pi);

end

end

//Rectangular Window

for n = 1:M

W(n) = 1;

end

//Windowing Fitler Coefficients

h = hd.*W;

disp(h,'Filter Coefficients are')

[hzm,fr]=frmag(h,256);

hzm_dB = 20*log10(hzm)./max(hzm);

subplot(2,1,1)

plot(2*fr,hzm)

xlabel('Normalized Digital Frequency W');

ylabel('Magnitude');

title('Frequency Response 0f FIR LPF using Rectangular window')

xgrid(1)

subplot(2,1,2)

plot(2*fr,hzm_dB)

xlabel('Normalized Digital Frequency W');

ylabel('Magnitude in dB');

title('Frequency Response 0f FIR LPF using Rectangular window')

xgrid(1)

//Result

///

//Enter the passband edge frequency in Hz=1500

//Enter the transition width in Hz =500

//Enter the stop band attenuation in dB =50

//Enter the sampling frequency in Hz=8000

High Pass Filter Design [Program]//Caption: Program to Design FIR High Pass Filterclear;clc;close;M = input('Enter the Odd Filter Length ='); //Filter length Wc = input('Enter the Digital Cutoff frequency ='); //Digital Cutoff frequencyTuo = (M-1)/2 //Center Valuefor n = 1:M if (n == Tuo+1) hd(n) = 1-Wc/%pi; else hd(n) = (sin(%pi*((n-1)-Tuo)) -sin(Wc*((n-1)-Tuo)))/(((n-1)-Tuo)*%pi); endend//Rectangular Windowfor n = 1:M W(n) = 1;end

//Windowing Fitler Coefficients

h = hd.*W;

disp(h,'Filter Coefficients are')

[hzm,fr]=frmag(h,256);

hzm_dB = 20*log10(hzm)./max(hzm);

subplot(2,1,1)

plot(2*fr,hzm)

xlabel('Normalized Digital Frequency W');

ylabel('Magnitude');

title('Frequency Response 0f FIR HPF using Rectangular window')

xgrid(1)

subplot(2,1,2)

plot(2*fr,hzm_dB)

xlabel('Normalized Digital Frequency W');

ylabel('Magnitude in dB');

title('Frequency Response 0f FIR HPF using Rectangular window')

xgrid(1)

//Result

//Enter the Odd Filter Length = 5

//Enter the Digital Cutoff frequency = %pi/4

//Filter Coefficients are

//

// - 0.1591549

// - 0.2250791

// 0.75

// - 0.2250791

// - 0.1591549

Band Pass Filter [Program]

//Caption: Program to Design FIR Band Pass Filter

clear;

clc;

close;

M = input('Enter the Odd Filter Length ='); //Filter length

//Digital Cutoff frequency [Lower Cutoff, Upper Cutoff]

Wc = input('Enter the Digital Cutoff frequency =');

Wc2 = Wc(2)

Wc1 = Wc(1)

Tuo = (M-1)/2 //Center Value

hd = zeros(1,M);

W = zeros(1,M);

for n = 1:11

if (n == Tuo+1)

hd(n) = (Wc2-Wc1)/%pi;

else

n

hd(n) = (sin(Wc2*((n-1)-Tuo)) -sin(Wc1*((n-1)-Tuo)))/(((n-1)-Tuo)*%pi);

end

if(abs(hd(n))<(0.00001))

hd(n)=0;

end

end

hd;

//Rectangular Window

for n = 1:M

W(n) = 1;

end

//Windowing Fitler Coefficients

h = hd.*W;

disp(h,'Filter Coefficients are')

[hzm,fr]=frmag(h,256);

hzm_dB = 20*log10(hzm)./max(hzm);

subplot(2,1,1)

plot(2*fr,hzm)

xlabel('Normalized Digital Frequency W');

ylabel('Magnitude');

title('Frequency Response 0f FIR BPF using Rectangular window')

xgrid(1)

subplot(2,1,2)

plot(2*fr,hzm_dB)

xlabel('Normalized Digital Frequency W');

ylabel('Magnitude in dB');

title('Frequency Response 0f FIR BPF using Rectangular window')

xgrid(1)

//Result

//Enter the Odd Filter Length = 11

//Enter the Digital Cutoff frequency = [%pi/4,3*%pi/4]

//Filter Coefficients are

// 0. 0. 0. - 0.3183099 0. 0.5 0. - 0.3183099 0. 0. 0.

FIR Band Stop Filter [Program]

//Caption: Program to Design FIR Band Reject Filter

clear ;

clc;

close;

M = input('Enter the Odd Filter Length ='); //Filter length

//Digital Cutoff frequency [Lower Cutoff, Upper Cutoff]

Wc = input('Enter the Digital Cutoff frequency =');

Wc2 = Wc(2)

Wc1 = Wc(1)

Tuo = (M-1)/2 //Center Value

hd = zeros(1,M);

W = zeros(1,M);

for n = 1:M

if (n == Tuo+1)

hd(n) = 1-((Wc2-Wc1)/%pi);

else

hd(n)=(sin(%pi*((n-1)-Tuo))-sin(Wc2*((n-1)-Tuo))+sin(Wc1*((n-1)-Tuo)))/(((n-1)-Tuo)*%pi);

end

if(abs(hd(n))<(0.00001))

hd(n)=0;

end

end

//Rectangular Window

for n = 1:M

W(n) = 1;

end

//Windowing Fitler Coefficients

h = hd.*W;

disp(h,'Filter Coefficients are')

[hzm,fr]=frmag(h,256);

hzm_dB = 20*log10(hzm)./max(hzm);

subplot(2,1,1)

plot(2*fr,hzm)

xlabel('Normalized Digital Frequency W');

ylabel('Magnitude');

title('Frequency Response 0f FIR BSF using Rectangular window')

xgrid(1)

subplot(2,1,2)

plot(2*fr,hzm_dB)

xlabel('Normalized Digital Frequency W');

ylabel('Magnitude in dB');

title('Frequency Response 0f FIR BSF using Rectangular window')

xgrid(1)

[11]. IIR Butterworth Filter Design(Digital Filter Design using Bilinear

Transform Method)a) .Digital IIR Low Pass Butterworth Filter

b). Digital IIR High Pass Butterworth Filter

c). Digital IIR Band Pass Butterworth Filter

d).Digital IIR Band Stop Butterworth Filter

[12]. IIR Chebyshev Digital Filter Design

(Using Bilinear Transformation)

Program

[13]. Power Spectrum Estimation (Periodogram)

Program