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LAB 3

Sampling

and

Quantization

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Sampling Theorem

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( ) ( )a sx t x nT

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1( )s

n

X (t) = t nTT

1

( )s

ns

X (f) = f nfT

( )SX (t) = X t X (t)1

( )* ( )s s

n

X (f) = X f X (f) X f nfT

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1( )

1 1 1 1 1....... ( 2 ) ( ) ( ) ( ) ( 2 ) .......

s s

n

s s s s s

X (f) = X f nfT

X f f X f f X f X f f X f fT T T T T

Sampled Signal

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Ex: x(t)=5 cos (2pi*2000* t)3+‏ cos (2pi *3000* t)

Fs=8000 Hz

Fs> 2Fm=2*3000=6 kHZ

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Ex: x(t)=5 cos (2pi*2000* t)3+‏ cos (2pi *5000* t)

Fs=8000 Hz

Fs< 2Fm=2*5000=10 kHZ

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k k 1, 2, 3,......S = aliased baseF F F

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Anti-aliasing Filter

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Practical Parts

Part 1: Aliasing in Time Domain

a) Let Fs=10 kHz and Fo=1 kHz. Compute and

plot x[n] using stem.

130 10 20 30 40 50 60

-1

-0.5

0

0.5

1

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b) Use subplot to plot x(t) for Fo=300 Hz and 700 Hz

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0 1 2 3 4 5

x 10-3

-1

0

1

0 1 2 3 4 5

x 10-3

-1

0

1

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c) Use subplot to plot x(n) for Fo=300 Hz and 700 Hz

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0 10 20 30 40 50 60-1

-0.5

0

0.5

1

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

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d) Use subplot to plot x(t) for Fo=9700 Hz and 9300 Hz

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0 1 2 3 4 5

x 10-3

-1

0

1

0 1 2 3 4 5

x 10-3

-1

0

1

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c) Use subplot to plot x(n) for Fo=9700 Hz and 9300 Hz

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0 10 20 30 40 50 60-1

-0.5

0

0.5

1

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

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k k 1, 2, 3,......S = aliased baseF F F

300 ( 1) 10000 =9700

700 ( 1) 10000 =9300

300 9700

700 9300

aliased

aliased

F

F

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

x 10-3

-1

-0.5

0

0.5

1

gives the same sample every one Ts

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d) Use subplot to plot x(t) for Fo=10300 Hz and 10700 Hz

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-1

-0.5

0

0.5

1

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c) Use subplot to plot x(n) for Fo=10300 Hz and 10700 Hz

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0 10 20 30 40 50 60-1

-0.5

0

0.5

1

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

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k k 1, 2, 3,......S = aliased baseF F F

300 ( 1) 10000 =10300

700 ( 1) 10000 =10700

300 9700 10300

700 9300 10700

aliased

aliased

F

F

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Part 2: Aliasing in Frequency Domain

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32

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Part 3: Quantization

function y=uquant(x,n)

del=((max(max(x))-(min(min(x)))))/(n-1);

r=(x-min(min(x)))/del;

r=round(r);

y=r*del+min(min(x));

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Example: Quantized x=2sin (2pi*t) using 16 levels.

max min 2 ( 2)4 /15

1 16 1

X Xdel

L

2

2

4

0

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4

0

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2

2

15

0

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t=0:.001:1;

y=2*sin(2*pi*t)

figure(1)

subplot(311)

plot(y)

q1=uquant(y,4)

subplot(312)

plot(q1)

q2=uquant(y,32)

subplot(313)

plot(q2)

Ps=mean(y.^2);

Pq1=mean(q1.^2);

Pq2=mean(q2.^2);

SQR1=Ps/Pq1

SQR2=Ps/Pq2

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0 200 400 600 800 1000 1200-2

0

2

0 200 400 600 800 1000 1200-2

0

2

0 200 400 600 800 1000 1200-2

0

2

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Image Quantization

Exercise 1

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clc

clear all

y1=imread('office_4.jpg');

y=rgb2gray(y1);

for i=1:7;

L=2^i;

Q=uquant(y,L);

i=i+1;

pause

L

figure(i)

imshow(Q)

end

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b=1

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b=2

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b=3

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b=4

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b=5

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b=6

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b=7

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clc

clear all

[y,fs]=wavread('speech_dft.wav');

sound(y,fs)

for b=1:7;

L=2.^b;

yQ=uquant(y,L);

pause

b

sound(yQ,fs);

end

Audio Quantization

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Exercise 2

Audio Quantization

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2 3 4 5 6 7 810

-2

10-1

100

101

102

103

104

plotting SNR

0 1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400plotting SNR

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Simulink model for sampling and quantization

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Exercise 3

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Q ua ntization error : ( ) ( ) ( )q qe n x n x n

2 2error

max minQuantization step =1

x x

L

0.1 0.1

2 2error

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Quantization of sinusoidal signal

2 2

2 2

0 0

2 2

2

2

2

1 1P ( ) ( sin )

2 2

( ) for ( T t T )2

1 1P ( ( )) ( )

2 2

1

2 2

2

2

sig

q

T T

q q

T T

T

T

S t dt A wt dt

e t tT

e t dt t dtT T T

t dtT T

A

Average power of sinusoidal signal :

Average power of quantized signal :

2

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Signal to quantization noise ratio

2

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2

max min

2

2

2

2

2

the signal to quantization noise ratio

P

P

( ) 2

P 2

P

2

12

4

12

3

2

sig

q

sig

q

SQNR

x x A A A

L L L

A

SQNR

A

LA

L

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Note:

Your report should include the following:

All Matlab program and its results with a short comment

on each result.

Answer any internal questions in practical parts.

Solve all lab exercises.

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