DSP Simulation Assignment using Octave

V. A. Amarnath

November 17, 2009

1

Contents

I Generate following sequences 4

1 Unit Impulse Function 4

2 Unit Step Function 5

3 Ramp Function 6

4 Exponential Sequence 7

5 Sinusoidal Sequence 8

6 given x[n] 9

II Linear Convolution 10

III Circular Convolution 13

IV FFT Algorithms 16

V Symmetry Property of DFT 20

VI Scaling Property of DFT 22

VII Overlap Save Method 24

VIII Overlap Add Method 27

IX Linear Convolution using DFT 30

X Analyse an audio �le 32

2

List of Figures

1 Unit Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . 42 Unit Step Function . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Ramp Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Exponential Sequence . . . . . . . . . . . . . . . . . . . . . . . . 75 Sinusoidal Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 86 Given x[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 x[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 y[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 z[n] = x[n]*y[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210 x[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1411 y[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1412 z[n] = x[n]~y[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513 x[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814 real{DFT (x[n])} . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815 imag{DFT (x[n])} . . . . . . . . . . . . . . . . . . . . . . . . . . 1916 Symmetry Property of DFT . . . . . . . . . . . . . . . . . . . . . 2117 Scaling Property of DFT . . . . . . . . . . . . . . . . . . . . . . . 2318 x[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2519 h[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2520 Result of Overlap Save Method using l = 4 . . . . . . . . . . . . 2621 x[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2822 h[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2823 Result of Overlap Add Method using l = 3 . . . . . . . . . . . . 2924 x[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3025 y[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3126 x[n] * y[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3127 Audio Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3228 Frequency Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 33

3

Part I

Generate following sequences

1 Unit Impulse Function

Code

clf;

y = [linspace(0,0,10),1,linspace(0,0,10)];

l = length(y);

x = [-(l-1)/2:1:(l-1)/2];

stem (x,y);

axis ([-9, 9, -2, 2]);

grid on;

hold on;

print impulse.jpg -djpg;

Figure

Figure 1: Unit Impulse Response

4

2 Unit Step Function

Code

clf;

y = [linspace(0,0,9),linspace(1,1,10)];

l = length(y);

x = [-(l-1)/2:1:(l-1)/2];

stem (x,y);

axis ([-9, 9, -2, 2]);

grid on;

hold on;

print step.jpg -djpg;

Figure

Figure 2: Unit Step Function

5

3 Ramp Function

Code

clf;

y = [linspace(0,0,10),linspace(0,10,11)];

l = length(y);

x = [-(l-1)/2:1:(l-1)/2];

stem (x,y);

axis ([-10, 10, -10, 10]);

grid on;

hold on;

print ramp.jpg -djpg;

Figure

Figure 3: Ramp Function

6

4 Exponential Sequence

Code

clf;

y = [linspace(0,5,101),exp(linspace(0,5,101))];

l = length(y);

x = 0.1*[-(l-1)/2:1:(l-1)/2];

stem (x,y);

axis ([0, 11, -1, 150]);

grid on;

hold on;

print expo.jpg -djpg;

Figure

Figure 4: Exponential Sequence

7

5 Sinusoidal Sequence

Code

clf;

y = [linspace(0,0,101),sin(linspace(0,10,101))];

l = length(y);

x = [-(l-1)/2:1:(l-1)/2];

stem (x,y);

axis ([-100, 100, -1, 1]);

grid on;

hold on;

print sine.jpg -djpg;

Figure

Figure 5: Sinusoidal Sequence

8

6 given x[n]

x[n] = 5 cos{ 32πn + 3

4π} + 4 cos{ 35πn} - sin{1

2πn}

Code

clf;

function p = z(n)

p = 5*cos(1.5*pi*n+.75*pi)+4*cos(.6*pi*n)-sin(.5*pi*n);

endfunction

y = [linspace(0,0,11),z(linspace(0,10,11))];

l = length(y);

x = [-(l-1)/2:1:(l-1)/2];

stem (x,y);

axis ([-10, 10, -10, 10]);

grid on;

hold on;

print xn.jpg -djpg;

Figure

Figure 6: Given x[n]

9

Part II

Linear Convolution

Code

function [res, num] = linconv(a,b,n1,n2)

r=zeros(length(a),length(b));

for m = 1 : length(a)

for n = 1 : length (b)

r(n,m) = a(m)*b(n);

end

end

for k = 2 : length(a)+length(b)

res(k-1) = 0;

for m = 1 : length(a)

for n = 1 : length(b)

if (m + n) == k

res(k-1) = r(n,m) + res(k-1);

end

end

end

end

num = n1+n2:1:length(a)+length(b)+n1+n2-2;

end

10

Figures

Figure 7: x[n]

Figure 8: y[n]

11

Figure 9: z[n] = x[n]*y[n]

12

Part III

Circular Convolution

Code

function all = circonv(a,b)

n = max(length(a),length(b));

if length(a) ~= n

for k = length(a)+1 : n

a(k) = 0;

end;

end;

if length(b) ~= n

for k = n - length(b) : n

b(k) = 0;

end;

end;

t = zeros(n);

temp = zeros(n,n);

for k = 1 : n

t = shift(b,k-1);

for l = 1 : n

temp(k,l)=t(l);

end;

end;

all = zeros(n);

all = (a*temp);

end;

13

Figures

Figure 10: x[n]

Figure 11: y[n]

14

Figure 12: z[n] = x[n]~y[n]

15

Part IV

FFT Algorithms

DIT FFT Algorithm

Code

function z = ditfft(x)

N = length(x);

n = log2(N);

clear t1;

z = zeros(1, N);

for p = 1 : N

t = dec2bin(p-1,n);

for q = 1 : n

t1(q) = t(n-q+1);

end;

t = t1;

z (p) = x(bin2dec(t)+1);

end;

for p = 1:n

m = 2^(p);

w= exp(-i*2*pi/m);

for l = 1: N/m

temp = z(1+(l-1)*m:l*m);

for k= 1:m/2

z(k+(l-1)*m) = temp(k)+temp(m/2+k)*w^(k-1);

z(m/2+k+(l-1)*m) = temp(k)-temp(m/2+k)*w^(k-1);

end;

end;

end;

end;

16

DIF FFT Algorithm

Code

function y = diffft(x)

len = length(x);

n = log2(len);

g = x;

for p = n:-1:1

lt = 2^(p);

Wn= exp(-i*2*pi/lt);

for l = 1: len/lt

temp = g(1+(l-1)*lt:l*lt);

for k= 1:lt/2

g(k+(l-1)*lt) = temp(k)+temp(lt/2+k);

g(lt/2+k+(l-1)*lt)=(temp(k)-temp(lt/2+k))*Wn^(k-1);

end;

end;

end;

tmp=g;

for p = n:-1:1

lt = 2^(p);

m=1;

for l = 1: len/lt

for k = 1:2:lt

y(m) = tmp(k+lt*(l-1));

y(lt/2+m)=tmp(k+lt*(l-1)+1);

m=m+1;

end;

m=l*lt+1;

end;

tmp=y;

end;

end;

17

Figures

Figure 13: x[n]

Figure 14: real{DFT (x[n])}

18

Figure 15: imag{DFT (x[n])}

19

Part V

Symmetry Property of DFT

Code

x = [1 2 3 4 4 3 2 1];

N = length(x);

n= 0:N-1;

y = fft(x);

z = [y y];

for k = 1:N

sym(k) = conj(z(N+1-(k-1)));

end;

subplot(2,2,1)

stem(n,abs(y));

grid on;

xlabel('time');

ylabel('amplitude');

title('magnitude of X(k)');

subplot(2,2,2)

stem(n,angle(y));

grid on;

xlabel('time');

ylabel('amplitude');

title('phase of X(k)');

subplot(2,2,3)

stem(n,abs(sym));

grid on;

xlabel('time');

ylabel('amplitude');

title('magnitude of X*(N-k)');

subplot(2,2,4)

stem(n,angle(sym));

grid on;

xlabel('time');

ylabel('amplitude');

title('phase of X*(N-k)');

20

Figure

Figure 16: Symmetry Property of DFT

21

Part VI

Scaling Property of DFT

Code

x=[8 7 6 5 4 3 2 1];

len = length(x);

y=0:1:len-1;

subplot(3,1,1)

stem(y,x,'fillled');

grid on;

axis([0 8 0 10]);

xlabel('Time ');

ylabel('Amplitude');

title('Input sequence');

z=fft(x,len);

subplot(3,1,2)

stem(y,abs(z),'fillled');

axis([0 8 0 40]);

grid on

xlabel('Time ');

ylabel('Amplitude');

title('8 point');

z=fft(x,4*len);

y=0:1:4*len-1;

subplot(3,1,3);

stem(y,abs(z),'fillled');

axis([0 32 0 40]);

grid on

xlabel('Time ');

ylabel('Amplitude');

title('32 point');

22

Figure

Figure 17: Scaling Property of DFT

23

Part VII

Overlap Save Method

Code

function res = oversave(x, h,l)

m = length (h);

L = length (x);

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

r = rem (length (y), l);

y = [y, zeros(1, r)]

if r == 0

y = [y, zeros(1, l/2)];

end;

xn = length (y);

h1 = [h, zeros(1, l - m)];

res = zeros (1, L + m -1);

k = 1;

pos = 0;

p = 0;

count = 0;

while k <= length(res)

temp = zeros (1,l);

for p = 0 : l-1

temp(p+1) = y(k+p);

end;

t = circonv (temp, h1);1

p = 1;

while p <= l - m + 1

res (p + pos) = t (p + m - 1);

p = p + 1;

end;

pos = pos + l - m + 1;

k = k + l - m + 1;

end;

end;

1circonv function used from Q. 3

24

Figures

Figure 18: x[n]

Figure 19: h[n]

25

Figure 20: Result of Overlap Save Method using l = 4

26

Part VIII

Overlap Add Method

Code

function res = overadd(x,h,l)

n = length(x);

m = length(h);

r = rem(n, l);

if r ~= 0

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

end;

xn = length(x);

k = 1;

h = [h, zeros(1, l - 1)];

res = zeros(1, n + m -1);

rp = 1;

while k <= xn

t = zeros(1, l + m - 1);

for p = 1 : l

t(p) = x(k + p - 1);

end;

t = circonv(t, h);2

k = k + l;

for p = 1 : l + m - 1

res(rp) = res(rp) + t(p);

rp = rp +1;

if rp > n + m - 1

break;

end;

end;

rp = rp - m + 1;

end;

end;

2circonv function used from Q. 3

27

Figures

Figure 21: x[n]

Figure 22: h[n]

28

Figure 23: Result of Overlap Add Method using l = 3

29

Part IX

Linear Convolution using DFT

Code

function res = lindft(x, y)

n1 = length(x);

n2 = length(y);

x1 = [x, zeros(1,n2-1)];

y1 = [y, zeros(1,n1-1)];

res = ifft(fft(x1).*fft(y1));

end;

Figures

Figure 24: x[n]

30

Figure 25: y[n]

Figure 26: x[n] * y[n]

31

Part X

Analyse an audio �le

Code

x = wavread(`mohanlal.wav');

stem (x);

y = fft(x);

N = length(x);

z = zeros(1,N/2);

for k = 1:N/2

z(k) = y(k);

end

c = linspace(0,pi,N/2);

c = c/(2*pi);

c = c*8000;

stem(c,abs(z));

grid on;

Figures

Figure 27: Audio Sequence

32

Figure 28: Frequency Spectrum

33