Sampling اهمتسیس و اهلانگیس یرهطم لضفلاوبا دیس فیرش یتعنص هاگشناد رتویپماک هدکشناد

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1 Sampling سیگنالها و سیستمها سید ابوالفضل مطهری دانشکده کامپیوتر دانشگاه صنعتی شریف

2 Sampling Conversion of a continuous-time signal to discrete time. x(t) x[n] t n

3 Sampling Applications x(t) Sampling y n DT processing w n Interpolation v(t) Sampling allows the use of modern digital electronics to process, record, transmit, store, and retrieve CT signals. audio: MP3, CD, cell phone pictures: digital camera, printer video: DVD everything on the web Sampling allows reduction in amount/size of data without information loss or with some loss

4 Relating DT and CT Signals x[n] D/A y(t) x[n] y(t) T Time domain relation ÿ Œ y(t) = x[n] (t nt ) n= Œ Œÿ

5 Relating DT and CT Signals X e j! Y (j!) T T Frequency domain relation Y (j!) =X e j!t

6 Relating DT and CT Signals x[n] D/A y(t) LP x c (t) Y (j!) T T H(j!) T T X c (j!) T T

7 Relating DT and CT Signals x c (t) =h(t) y(t) 1X = x[n]h(t) (t nt ) = n= 1 1X n= 1 x[n]h(t nt ) T t h(t) = sin t = 1 T sinc T t x c (t) = 1 T 1X x[n]sinc T (t nt ) n= 1 Show that: x c (mt )=x[m]

8 Relating DT and CT Signals T x c (t) x c (mt )=x[m]

9 Relating DT and CT Signals Conclusion: Start from a continuous signal and sample it at rate T, then the signal can be recovered by using the following system: x[n] D/A y(t) LP x c (t) NO

10 Relating DT and CT Signals Both have the same sample points but reconstruction is unique Amongst all possible CT signals with the above sample points which one is reconstructed by the above procedure?

11 Nyquist Theorem If x(t) is bandlimited so that X(jÊ) =0 for Ê > Ê m then x(t) is uniquely determined by its samples x(nt ) if 2fi Ê s = > 2Ê m. T The minimum sampling frequency, 2Ê m, is called the Nyquist rate. X c (j!) T! m T

12 Analog to Digital x c (t) Sampler y(t) A/D x[n] 1X y(t) =x c (t) (t nt ) 1X = = n= 1 1X n= 1 n= 1 x c (nt ) (t nt ) x[n] (t nt )

13 Analog to Digital x c (t) y(t) x[n] Sampler A/D Y (j!) = 1 2 X c(j!) 2 T 1X n= 1 (! n! s )! s = 2 T Y (j!) = 1 T 1X X c (j(! n! s )) n= 1 Periodic X e j!t = Y (j!)

14 Analog to Digital X c (j!) T! m T Y (j!)! s T! m! s T X e j!t! s

15 Aliasing X c (j!) T! m T Y (j!)! s! m! s

16 Anti-Aliasing Filter To avoid aliasing, remove frequency components that alias before sampling. x(t) Anti-aliasing Filter 1 Ê Ê s Ês 2 2 x p (t) Reconstruction Filter T Ê Ê s Ês 2 2 x r (t) p(t)

17 Sampling We can hear sounds with frequency components between 20 Hz and 20 khz. What is the maximum sampling interval T that can be used to sample a signal without loss of audible information? µs µs µs fi µs 5. 50fi µs 6. 25fi µs Ê s 2fi 2fif m = Ê m < = 2 2T 1 1 T< = = 25 µs 2f m 2 20 khz

18 Sampling of DT Signals

19 Sampling in our Eyes Every image that we see is sampled by the retina, which contains 100 million rods and 6 million cones (average spacing 3µm) which act as discrete sensors. ourtesy of e ga o duardo ernande and a ph e son. sed ith permission.

20 Camera

21 Quantization We measure discrete amplitudes in bits. Bit rate = (# bits/sample) (# samples/sec) 8

22 Quantization We measure discrete amplitudes in bits. Example: audio CD 2 channels 16 bits sample 44, 100 samples sec 60 sec min 74 min 6.3 G bits 0.78 Gbytes 13

23 Quantizing Images Converting an image from a continuous representation to a discrete representation involves the same sort of issues. This image has pixels, with brightness quantized to 8 bits. 14

24 Quantizing Images 8 bit image 7 bit image 15

25 Quantizing Images 8 bit image 6 bit image 16

26 Quantizing Images 8 bit image 5 bit image 17

27 Quantizing Images 8 bit image 4 bit image 18

28 Quantizing Images 8 bit image 3 bit image 19

29 Quantizing Images 8 bit image 2 bit image 20

30 Quantizing Images 8 bit image 1 bit image 21

31 Check Yourself What is the most objectionable artifact of coarse quantization? 8 bit image 4 bit image 22

32 Dithering One very annoying artifact is banding caused by clustering of pixels that quantize to the same level. Banding can be reduced by dithering. Dithering: adding a small amount (± 1 2 the image before quantizing. quantum) of random noise to Since the noise is di erent for each pixel in the band, the noise causes some of the pixels to quantize to a higher value and some to a lower. But the average value of the brightness is preserved. 23

33 Quantizing Images with Dither 7 bit image 7 bits with dither 24

34 Quantizing Images with Dither 6 bit image 6 bits with dither 25

35 Quantizing Images with Dither 5 bit image 5 bits with dither 26

36 Quantizing Images with Dither 4 bit image 4 bits with dither 27

37 Quantizing Images with Dither 3 bit image 3 bits with dither 28

38 Quantizing Images with Dither 2 bit image 2 bits with dither 29

39 Quantizing Images with Dither 1 bit image 1 bit with dither 30

40 Check Yourself What is the most objectionable artifact of dithering? 3 bit image 3 bit dithered image 31

41 Check Yourself What is the most objectionable artifact of dithering? One annoying feature of dithering is that it adds noise. 32

42 Quantization Schemes Example: slowly changing backgrounds. Quantization: y = Q(x) Quantization with dither: y = Q(x + n) 33

43 Check Yourself What is the most objectionable artifact of dithering? One annoying feature of dithering is that it adds noise. Robert s technique: add a small amount (± 1 2 quantum) of random noise before quantizing, then subtract that same amount of random noise. 34

44 Quantization Schemes Example: slowly changing backgrounds. Quantization: y = Q(x) Quantization with dither: y = Q(x + n) Quantization with Robert s technique: y = Q(x + n) n 35

45 Quantizing Images with Robert s Method 7 bits with dither 7 bits with Robert s method 36

46 Quantizing Images with Robert s Method 6 bits with dither 6 bits with Robert s method 37

47 Quantizing Images with Robert s Method 5 bits with dither 5 bits with Robert s method 38

48 Quantizing Images with Robert s Method 4 bits with dither 4 bits with Robert s method 39

49 Quantizing Images with Robert s Method 3 bits with dither 3 bits with Robert s method 40

50 Quantizing Images with Robert s Method 2 bits with dither 2 bits with Robert s method 41

51 Quantizing Images with Robert s Method 1 bits with dither 1 bit with Robert s method 42

6.003: Signals and Systems. Sampling and Quantization

6.003: Signals and Systems. Sampling and Quantization 6.003: Signals and Systems Sampling and Quantization December 1, 2009 Last Time: Sampling and Reconstruction Uniform sampling (sampling interval T ): x[n] = x(nt ) t n Impulse reconstruction: x p (t) =