ADSP-ASSNMNT 2

8/6/2019 ADSP-ASSNMNT 2

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I. TIME DOMAIN CHARACTERIZATION

UP-SAMPLING OPERATION

The program shown below can be used to study up-sampling of a sinusoidal input sequences, the

up-sampling factor, and the frequency of the sinusoid in Hz. It then plots the input sequence and

its up sampled version.

PROGRAM

% Illustration of Up-Sampling by an Integer Factor

clf;

N = input('Input length = ');

L = input('Up-sampling Factor = ');

fo = input('Input signal frequency = ');

% Generate the input sinusoidal sequence

n = 0:N-1;

x = sin(2*pi*fo*n);

% Generate the up-sampled sequence

y = zeros(1, L*length(x));

y([1: L: length(y)]) = x;

% Plot the input and the output sequences

subplot(2,1,1)

stem(n,x);

title('Input Sequence');

xlabel('Time index n');ylabel('Amplitude');

subplot(2,1,2)

stem(n,y(1:length(x)));

title(['Output sequence up-sampled by', num2str(L)]);


INPUT DATA

Length of the Sequence-50

Frequency:0.12

Up-sampling factor:3


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INPUT /OUTPUT WAVEFORM

DOWN

SAMPLING OPERATION

The program shown below can be used to study down sampling of a sinusoidal

input sequences, the down sampling factor, and the frequency of the sinusoid in

Hz. It then plots the input sequence and its down sampled sampled version.

PROGRAM

% Illustration of Down-Sampling by an Integer Factor

%

clf;

N = input('Output length = ');

M = input('Down-sampling Factor = ');

fo = input('Input signal frequency = ');

% Generate the input sinusoidal sequence

n = 0 : N-1;

m = 0 : N*M-1;

0 5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5

1Input Sequence

Time index n

Amplitude

0 5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5

1Output sequence up-sampled by3

Time index n

Amplitude


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x = sin(2*pi*fo*m);

% Generate the down-sampled sequence

y = x([1 : M : length(x)]);


subplot(2,1,1)

stem(n, x(1:N));

title('Input Sequence');


subplot(2,1,2)

stem(n, y);

title(['Output sequence down-sampled by ',num2str(M)]);

ylabel('Amplitude');xlabel('Time index n');

INPUT DATA

Length of the Sequence-50

Frequency: 0.042 Hz

Down sampling factor: 3


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INPUT/OUTPUT WAVEFORM

0 5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5

1Input Sequence

Timeindex n

Amplitude

0 5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5

1Output sequencedown-sampledby 3

Amplitude

Timeindex n

II. FREQUENCY DOMAIN CHARACTERIZATION

FREQUENCY-DOMAIN PROPERTIES OF THE UP-SAMPLER USING MATLAB

The program determines the output of the up-sampler and then plots the input and

output spectrums for L=5. The output spectrum consist of a factor of 5 compressed

version of the input spectrum followed by L-1=4 images.

PROGRAM

% Effect of Up-Sampling in the Frequency Domain

% Use fir2 to create a bandlimited input sequence

freq = [0 0.45 0.5 1];

mag = [0 1 0 0];

x = fir2(99, freq, mag);

% Evaluate and plot the input spectrum

[Xz, w] = freqz(x, 1, 512);


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plot(w/pi, abs(Xz)); grid

xlabel('\omega/\pi'); ylabel('Magnitude');

title('Input spectrum');

pause

% Generate the up-sampled sequence

L = input('Type in the up-sampling factor = ');

y = zeros(1, L*length(x));

y([1: L: length(y)]) = x;

% Evaluate and plot the output spectrum

[Yz, w] = freqz(y, 1, 512);

plot(w/pi, abs(Yz)); grid


title('Output spectrum');


0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1

0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1

/

M

agnitude

In p u t s p e c tru m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

/

Magnitude

Output spectrum

Fig:MATLAB input and output spectrum of a factor of 5-up-sampler

FREQUENCY-DOMAIN PROPERTIES OF THE DOWN SAMPLER USING MATLAB

1. Down sampling factor of M=3,

2. Down sampling factor of M=2

PROGRAM

% Effect of Down-Sampling in the Frequency Domain

% Use fir2 to create a bandlimited input sequence


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freq = [0 0.42 0.48 1];

mag = [0 1 0 0];

x = fir2(101, freq, mag);

% Evaluate and plot the input spectrum

[Xz, w] = freqz(x, 1, 512);

plot(w/pi, abs(Xz)); grid


title('Input spectrum');

pause

% Generate the down-sampled sequence

M = input('Type in the down-sampling factor = ');

y = x([1: M: length(x)]);

% Evaluate and plot the output spectrum

[Yz, w] = freqz(y, 1, 512);

plot(w/pi, abs(Yz)); grid


title('Output spectrum');


0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 10

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1

/

M

agnitude

In p u t s p e c t ru m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

/

Magnitude

Output spectrum


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

/

M

agnitude

Output spectrum

Fig: input spectrum,output spectrum for down sampling factor of M=3,output spectrum for a

down sampling factor of M=2

III. DECIMATIONDECIMATION OPERATION USING MATLAB

We consider the decimation of a sum of two sinusoidal sequences of normalized

frequencies f1 and f2 by an arbitrary down sampling factor M using program

shown below. The input data to the program are the length N of the input sequence

x[n],the down sampling factor M ,and two normalized frequencies in Hz.The

program uses a 30-tap FIR low pass decimation filter designed to have a stop band

edge at /M power5 .It then plots the input and the output sequences,for

N=100,M=2,f1=0.043,f2=0.031.

PROGRAM:

% Illustration of Decimation Process%

clf;

N = input('Length of input signal = ');

M = input('Down-sampling factor = ');

f1 = input('Frequency of first sinusoid = ');

f2 = input('Frequency of second sinusoid = ');

n = 0:N-1;

% Generate the input sequence

x = sin(2*pi*f1*n) + sin(2*pi*f2*n);


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% Generate the decimated output sequence

y = decimate(x,M,'fir');


subplot(2,1,1)

stem(n,x(1:N));

title('Input sequence');


subplot(2,1,2)

m=0:N/M-1;

stem(m,y(1:N/M));

title('Output sequence');

xlabel('Time index n'); ylabel('Amplitude');


0 10 20 30 40 50 60 70 80 90 100-2

-1

0

1

2Input sequence

Time index n

Am

plitude

0 5 10 15 20 25 30 35 40 45 50-2

-1

0

1

2Output sequence

Time index n

Am

plitude


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IV. INTERPOLATION

INTERPOLATION OPERATION USING MATLAB

To illustrate interpolation process,we consider an input x that is given by a sum of two sinusoidal

sequences. Program is used to find its interpolated output y for an upsampling factor of L. The

input data to th program are the length N of the input vector x,the upsamler factor L,anKd two

normalized frequencies in hz.The frequencies of the sinusoids considered here are the f1=0.043

and f2=0.031,while the interpolation factor is 2.The input length is 50.

PROGRAM:

% Illustration of Interpolation Process

%

clf;


L = input('Up-sampling factor = ');




n = 0:N-1;


% Generate the interpolated output sequence

y = interp(x,L);


subplot(2,1,1)

stem(n,x(1:N));



subplot(2,1,2)

m=0:N*L-1;

stem(m,y(1:N*L));




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INPUT/OUTPUT WAVEFORM:

0 5 10 15 20 25 30 35 40 45 50-2

-1

0

1

2Input sequence

Timeindexn

Amplitude

0 10 20 30 40 50 60 70 80 90 100-2

-1

0

1

2Output sequence

Timeindexn

Amplitude

V. SAMPLE RATE ALTERATION

SAMPLE RATE ALTERATION BY A RATIO OF TWO INTEGERS

In this program, we consider the sampling rate increase by a ratio of two positive integers of a

signal composed of two sinusoids. The input data to the program are the length N of the input

vector x, the upsampling factor L, the down sampling factor M, and the two frequencies.

PROGRAM

% Program

% Illustration of Sampling Rate Alteration by

% a Ratio of Two Integers

%

clf;


L = input('Up-sampling factor = ');

M = input('Down-sampling factor = ');





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n = 0:N-1;


% Generate the resampled output sequence

y = resample(x,L,M);


subplot(2,1,1)

stem(n,x(1:N));



subplot(2,1,2)

m=0:N*L/M-1;

stem(m,y(1:N*L/M));



INPUT

Length of input signal = 25

Up-sampling factor = 5

Down-sampling factor = 3

Frequency of first sinusoid = 0.043

Frequency of second sinusoid = 0.031



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0 5 10 15 20 25-2

-1

0

1

2Input sequence

Time index n

Amplitude

0 5 10 15 20 25 30 35 40-2

-1

0

1

2Output sequence

Time index n

Amp

litude

Fig: Illustration Of Sample Rate Increases By A Rational Number 5/3

VI. NYQUIST FILTERS

DESIGN OF Lth BAND FIR FILTER USING THE WINDOWED FORIER SERIES APPROACH

Design a length-23 half-band (L=2) linear phase low pass filter using the

windowed forier series approach with a Hamming window.

PROGRAM

% Program

% Design of Lth Band FIR Filter Using the

% Windowed Fourier Series Approach

%


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N = input('Type in the filter length = ');

L = input('Type in the value of L = ');

K = (N-1)/2;

n = -K:K;

% Generate the truncated impulse response of the ideal

% lowpass filter

b = sinc(n/L)/L;

% Generate the window sequence

win = hamming(N);

% Generate the coefficients of the windowed filter

c = b.*win';

% Plot the gain response of the windowed filter

[h,w] = freqz(c,1,256);

g = 20*log10(abs(h));

plot(w/pi,g);grid

xlabel('\omega/\pi');ylabel('Gain, dB');

INPUT

Type in the filter length = 23

Type in the value of L = 2



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VII. QMF FILTER BANK

FREQUENCY RESPONSES OF JOHNSTON'S QMF FILTERS

The impulse response coefficients of the length -12 linear phase low pass filter 12B

of Johnston are given by

Filter coefficients = -0.006443977 -0.00758164 +0.09808522 +0.02745539 -0.0913825

+0.4807962

Matlab program is used to verify the performance of the above filter H0(z).

PROGRAM

% Program

% Frequency Responses of Johnston's QMF Filters

% Type in prototype lowpass filter coefficients

clf;

B1 = input('Filter coefficients = ');

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-80

-60

-40

-20

0

20

/

Gain,

dB


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B1 = [B1 fliplr(B1)];

% Generate the complementary highpass filter

L = length(B1);

for k = 1:L

B2(k) = ((-1)^k)*B1(k);

end

% Compute the gain responses of the two filters

[H1z, w] = freqz(B1, 1, 256);

h1 = abs(H1z); g1 = 20*log10(h1);

[H2z, w] = freqz(B2, 1, 256);

h2 = abs(H2z); g2 = 20*log10(h2);

% Plot the gain responses of the two filters

plot(w/pi, g1,'-',w/pi, g2,'--');axis([0 1 -80 5]);grid

xlabel('Normalized frequency');ylabel('Gain in dB')

pause

% Compute the sum of the square magnitude responses

for i = 1:256,

sum(i) = (h1(i)*h2(i)) + (h2(i)*h1(i));

end

d =10*log10(sum);

% Plot the sum of the square magnitude responses

plot(w/pi,sum);grid;

xlabel('Normalized frequency')

ylabel('Amplitude distortion in dB')

INPUT


+0.4807962


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INPUTOUTPUT WAVEFORM

Fig: Johnstons 12 B filter: Gain respons

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Normalizedfrequency

AmplitudedistortionindB

Fig: Johnstons 12 B filter: Amplitude distortion

FREQUENCY RESPONSES OF TREE-STRUCTURED QMF FILTERS

Design of a four channel QMF bank by iterating the two-channel QMF bank based on the filter

12B of Johnston

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80

-70

-60

-50

-40

-30

-20

-10

0

Normalizedfrequency

GainindB


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PROGRAM

% Program

% Frequency Responses of Tree-Structured QMF Filters

%

clf;

% Type in prototype lowpass filter coefficients

B = input('Filter coefficients = ');

B1 = [B fliplr(B)];

% Generate the complementary highpass filter

L = length(B1);

for k = 1:LB2(k) = ((-1)^k)*B1(k);

end

% Determine the coefficients of the four filters

B10 = zeros(1, 2*length(B1));

B10([1: 2: length(B10)]) = B1;

B11 = zeros(1, 2*length(B2));

B11([1: 2: length(B11)]) = B2;

C0 = conv(B1, B10);C1 = conv(B1, B11);

C2 = conv(B2, B10);C3 = conv(B2, B11);

% Determine the frequency responses

[H0z, w] = freqz(C0, 1, 256);

h0 = abs(H0z);

M0 = 20*log10(h0);

[H1z, w] = freqz(C1, 1, 256);

h1 = abs(H1z);

M1 = 20*log10(h1);

[H2z, w] = freqz(C2, 1, 256);

h2 = abs(H2z);

M2 = 20*log10(h2);


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[H3z, w] = freqz(C3, 1, 256);

h3 = abs(H3z);

M3 = 20*log10(h3);

plot(w/pi, M0,'r-',w/pi, M1,'b--',w/pi, M3,'g-.',w/pi, M2,'m:');grid

axis([0 1 -100 20]);

xlabel('\omega /\pi');ylabel('Gain, dB')

legend('H_{O}(z)','H_{1}(z)', 'H_{2}(z)', 'H_{3}(z)')

INPUT


+0.4807962

INPUTOUTPUT WAVEFORM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-80

-60

-40

-20

0

20

/

Gain

,dB

HO(z)

H1(z)

H2(z)

H3(z)

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ADSP-ASSNMNT 2