Post on 04-Oct-2021
transcript
1
LOGO
LAB 3
Sampling
and
Quantization
3
Sampling Theorem
4
( ) ( )a sx t x nT
5
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
6
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
7
8
Ex: x(t)=5 cos (2pi*2000* t)3+ cos (2pi *3000* t)
Fs=8000 Hz
Fs> 2Fm=2*3000=6 kHZ
9
Ex: x(t)=5 cos (2pi*2000* t)3+ cos (2pi *5000* t)
Fs=8000 Hz
Fs< 2Fm=2*5000=10 kHZ
10
k k 1, 2, 3,......S = aliased baseF F F
11
Anti-aliasing Filter
12
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
14
b) Use subplot to plot x(t) for Fo=300 Hz and 700 Hz
15
0 1 2 3 4 5
x 10-3
-1
0
1
0 1 2 3 4 5
x 10-3
-1
0
1
16
c) Use subplot to plot x(n) for Fo=300 Hz and 700 Hz
17
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
18
d) Use subplot to plot x(t) for Fo=9700 Hz and 9300 Hz
19
0 1 2 3 4 5
x 10-3
-1
0
1
0 1 2 3 4 5
x 10-3
-1
0
1
20
c) Use subplot to plot x(n) for Fo=9700 Hz and 9300 Hz
21
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
22
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
23
24
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
25
d) Use subplot to plot x(t) for Fo=10300 Hz and 10700 Hz
26
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
27
c) Use subplot to plot x(n) for Fo=10300 Hz and 10700 Hz
28
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
29
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
30
Part 2: Aliasing in Frequency Domain
31
32
33
34
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));
35
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
36
4
0
37
38
2
2
15
0
39
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
40
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
41
Image Quantization
Exercise 1
42
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
43
b=1
44
b=2
45
b=3
46
b=4
47
b=5
48
b=6
49
b=7
50
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
51
Exercise 2
Audio Quantization
52
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
53
Simulink model for sampling and quantization
54
Exercise 3
55
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
56
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
12
57
Signal to quantization noise ratio
2
12
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
58
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.
LOGO
59