1 LOGO LAB 3 Sampling and Quantization Sampling Theorem 3 x a (t ) x (nT s ) 4 1 X (t) = T (t nT n s ) X S (t) = X (t )X (t) 1 X (f) = Ts (f n 1 X s (f) = X (f ) * X (f) T nf s ) X (f n nf s ) 5 Sampled Signal 1 X s (f) = X (f nf s ) T n 1 1 1 1 1 ....... X (f 2f s ) X (f f s ) X (f s ) X (f f s ) X (f 2f s ) ....... T T T T T 6 7 Ex: x(t)=5 cos (2pi*2000* t)+3 cos (2pi *3000* t) Fs=8000 Hz Fs> 2Fm=2*3000=6 kHZ 8 Ex: x(t)=5 cos (2pi*2000* t)+3 cos (2pi *5000* t) Fs=8000 Hz Fs< 2Fm=2*5000=10 kHZ 9 Faliased Fbase k FS k = 1, 2, 3,...... 10 Anti-aliasing Filter 11 Practical Parts Part 1: Aliasing in Time Domain a) Let Fs=10 kHz and Fo=1 kHz. Compute and plot x[n] using stem. 12 1 0.5 0 -0.5 -1 0 10 20 30 40 50 60 13 b) Use subplot to plot x(t) for Fo=300 Hz and 700 Hz 14 1 0 -1 0 1 2 3 4 5 -3 x 10 1 0 -1 0 1 2 3 4 5 -3 x 10 15 c) Use subplot to plot x(n) for Fo=300 Hz and 700 Hz 16 1 0.5 0 -0.5 -1 0 10 20 30 40 50 60 0 10 20 30 40 50 60 1 0.5 0 -0.5 -1 17 d) Use subplot to plot x(t) for Fo=9700 Hz and 9300 Hz 18 1 0 -1 0 1 2 3 4 5 -3 x 10 1 0 -1 0 1 2 3 4 5 -3 x 10 19 c) Use subplot to plot x(n) for Fo=9700 Hz and 9300 Hz 20 1 0.5 0 -0.5 -1 0 10 20 30 40 50 60 0 10 20 30 40 50 60 1 0.5 0 -0.5 -1 21 Faliased Fbase k FS k = 1, 2, 3,...... Faliased 300 (1) 10000 =9700 Faliased 700 (1) 10000 =9300 300 9700 700 9300 22 23 1 0.5 0 -0.5 -1 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 gives the same sample every one Ts -3 x 10 24 d) Use subplot to plot x(t) for Fo=10300 Hz and 10700 Hz 25 1 0.5 0 -0.5 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 x 10 1 0.5 0 -0.5 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 x 10 26 c) Use subplot to plot x(n) for Fo=10300 Hz and 10700 Hz 27 1 0.5 0 -0.5 -1 0 10 20 30 40 50 60 0 10 20 30 40 50 60 1 0.5 0 -0.5 -1 28 Faliased Fbase k FS k = 1, 2, 3,...... Faliased 300 (1) 10000 =10300 Faliased 700 (1) 10000 =10700 300 9700 10300 700 9300 10700 29 Part 2: Aliasing in Frequency Domain 30 31 32 33 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)); 34 Example: Quantized x=2sin (2pi*t) using 16 levels. 2 4 2 0 X max X min 2 (2) del 4 /15 L 1 16 1 35 4 0 36 37 2 15 2 0 38 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 39 2 0 -2 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 2 0 -2 2 0 -2 40 Image Quantization Exercise 1 41 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 42 b=1 43 b=2 44 b=3 45 b=4 46 b=5 47 b=6 48 b=7 49 Audio Quantization 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 50 Audio Quantization Exercise 2 51 plotting SNR plotting SNR 4 1400 10 1200 3 10 1000 2 10 800 1 10 600 0 10 400 -1 10 200 0 -2 0 1 2 3 4 5 6 7 8 10 2 3 4 5 6 7 8 52 Simulink model for sampling and quantization 53 Exercise 3 54 x max x min Quantization step = L 1 Quantization error : eq (n ) x q (n ) x (n ) error 2 2 0.1 0.1 error 2 2 55 Quantization of sinusoidal signal Average power of sinusoidal signal : Psig 1 2 2 1 2 0 S (t )dt 2 2 2 A 2 ( A sin wt ) dt 0 2 Average power of quantized signal : eq (t ) t for ( T t T ) 2T T T 1 1 2 2 Pq (eq (t )) dt ( t ) dt 2T T 2T T 2T 1 2T 2T 2 2 t T dt 12 2 T 56 Signal to quantization noise ratio the signal to quantization noise ratio A2 Psig 2 SQNR Pq 2 12 12 2 x max x min A ( A ) 2A L L L A2 Psig 2 3 2 SQNR L 2 Pq 4A 2 2 12L 57 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. 58 LOGO 59