6.003: Signals and Systems. Sampling

Transcription

1 6.003: Signals and Systems Sampling April 27, 200

2 Mid-term Examination #3 om orrow: W ednesday, A pril 2 8, 7 : : 3 0 pm. No recitations tomorrow. Coverage: Lectures 20 Recitations 20 Homeworks Homework will not collected or graded. Solutions are posted. Closed book: 3 pages of notes (8 2 inches; front and back). Designed as -hour exam; two hours to complete.

3 Sampling Conversion of a continuous-time signal to discrete time. x(t) x[n] t n We have used sampling a number of times before. oday: new insights from Fourier representations.

4 Sampling Sampling allows the use of modern digital electronics to process, record, transmit, store, and retrieve C signals. audio: MP3, CD, cell phone pictures: digital camera, printer video: DVD everything on the web

5 Sampling Sampling is pervasive. Example: digital cameras record sampled images. y I(x, y) n I[m, n] x m

6 Sampling Photographs in newsprint are half-tone images. black or white and the average conveys brightness. Each point is

7 Sampling Zoom in to see the binary pattern.

8 Sampling Even high-quality photographic paper records discrete images. When AgBr crystals (0.04.5μm) are exposed to light, some of the Ag is reduced to metal. During development the exposed grains are completely reduced to metal and unexposed grains are removed. Images of discrete grains in photographic paper removed due to copyright restrictions.

9 Sampling Every image that we see is sampled by the retina, which contains 00 million rods and 6 million cones (average spacing 3μm) which act as discrete sensors. Courtesy of Helga Kolb, Eduardo Fernandez, and Ralph Nelson. Used with permission.

10 Check Yourself Your retina is sampling this slide, which is composed of pixels. Is the spatial sampling done by your rods and cones adequate to resolve individual pixels in this slide?

11 Check Yourself he spacing of rods and cones limits the angular resolution of your retina to approximately θeye = rod/cone spacing diameter of eye m 0 4 radians 3 cm he angle between pixels viewed from the center of the classroom is approximately screen size / m/ θ pixels = radians distance to screen 0 m Light from a single pixel falls upon multiple rods and cones.

12 Sampling How does sampling affect the information contained in a signal?

13 Sampling We would like to sample in a way that preserves information, which may not seem possible. x(t) t Information between samples is lost. herefore, the same samples can represent multiple signals. cos 7 π 3 n? cos π 3 n? t

14 Sampling and Reconstruction o determine the effect of sampling, compare the original signal x(t) to the signal x p (t) that is reconstructed from the samples x[n]. Uniform sampling (sampling interval ). x[n] =x(n ) t n Impulse reconstruction. x p (t) = x[n]δ(t n ) n t n

15 Reconstruction Impulse reconstuction produces a signal x p (t) that is equal to the original signal x(t) multiplied by an impulse train. x p (t) = x[n]δ(t n ) n= = x(n )δ(t n ) n= = x(t)δ(t n ) n= = x(t) δ(t n ) n= }{{} p(t) x p (t) is motivated by impulse reconstruction (top line) can be understood entirely within C framework (bottom line)

16 Sampling Multiplication by an impulse train in time is equivalent to convolution by an impulse train in frequency. generates multiple copies of original frequency content. X(j) W W s P (j) s X p (j)= 2 π s (X(j ) P (j ))() s =

17 Check Yourself What is the relation between the DF of x[n] = x(n ) and the CF of x p (t) = x[n]δ(t n ) for X(j) below. X(j) W W. X p (j)=x(e jω ) Ω= 2. X p (j)=x(e jω ) Ω= 3. X p (j)=x(e jω ) Ω= 4. X p (j)=x(e jω ) Ω= 5. none of the above

18 Check Yourself DF X(e jω )= x[n]e jωn CF of x p (t) X p (j)= n= n= = x[n] δ(t n )e jt dt n= = x[n]e jn n= = X(e jω ) Ω= x[n]δ(t n )e jt dt

19 Check Yourself X p (j)=x(e jω ) Ω= X(j) W W X p (j)= 2 π s (X(j ) P (j ))() s = X(e jω )=X p (j) = Ω Ω

20 Check Yourself What is the relation between the DF of x[n] = x(n ) and the CF of x p (t) = x[n]δ(t n ) for X(j) below. X(j) W W. X p (j)=x(e jω ) Ω= 2. X p (j)=x(e jω ) Ω= 3. X p (j)=x(e jω ) Ω= 4. X p (j)=x(e jω ) Ω= 5. none of the above

21 Sampling he high frequency copies can be removed with a low-pass filter (also multiply by to undo the amplitude scaling). X p (j)= (X(j ) P (j ))() s 2 s 2 Impulse reconstruction followed by ideal low-pass filtering is called bandlimited reconstruction.

22 he Sampling heorem If signal is bandlimited sample without loosing information. If x(t) is bandlimited so that X(j)=0 for > m then x(t) is uniquely determined by its samples x(n ) if s = > 2 m. he minimum sampling frequency, 2 m, is called the Nyquist rate.

23 Summary hree important ideas. Sampling x(t) x[n] =x(n ) Bandlimited Reconstruction x[n] Impulse Reconstruction x p (t) = x[n]δ(t n ) LPF s s 2 2 x r (t) Sampling heorem: If X(j)=0 > s then xr (t) =x(t). 2

24 Check Yourself We can hear sounds with frequency components between 20 Hz and 20 khz. What is the maximum sampling interval that can be used to sample a signal without loss of audible information?. 00 μs μs μs 4. 00πμs 5. 50πμs 6. 25πμs

25 Check Yourself s f m = m < = 2 2 < = =25 μs 2f m 2 20 khz

26 Check Yourself We can hear sounds with frequency components between 20 Hz and 20 khz. What is the maximum sampling interval that can be used to sample a signal without loss of audible information?. 00 μs μs μs 4. 00πμs 5. 50πμs 6. 25πμs

27 C Model of Sampling and Reconstruction Sampling followed by bandlimited reconstruction is equivalent to multiplying by an impulse train and then low-pass filtering. x(t) x p (t) LPF s s 2 2 x r (t) p(t) p(t) = sampling function 0 t

28 Aliasing π What happens if X contains frequencies >? X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

29 Aliasing π What happens if X contains frequencies >? X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

30 Aliasing π What happens if X contains frequencies >? X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

31 Aliasing π What happens if X contains frequencies >? X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

32 Aliasing he effect of aliasing is to wrap frequencies. Output frequency s 2 X(j) s 2 Input frequency s s 2 2

33 Aliasing he effect of aliasing is to wrap frequencies. Output frequency s 2 X(j) s 2 Input frequency s s 2 2

34 Aliasing he effect of aliasing is to wrap frequencies. Output frequency s 2 X(j) s 2 Input frequency s s 2 2

35 Aliasing he effect of aliasing is to wrap frequencies. Output frequency s 2 X(j) s 2 Input frequency s s 2 2

36 Check Yourself A periodic signal with a period of 0. ms is sampled at 44 khz. o what frequency does the eighth harmonic alias?. 8 khz 2. 6 khz 3. 4 khz 4. 8 khz 5. 6 khz 6. none of the above

37 Check Yourself Output frequency (khz) Input frequency (khz)

38 Check Yourself Output frequency (khz) Input frequency (khz) Harmonic Alias 0 khz 0 khz 20 khz 20 khz 30 khz 44 khz-30 khz =4 khz 40 khz 44 khz-40 khz = 4 khz 50 khz 50 khz-44 khz = 6 khz 60 khz 60 khz-44 khz =6 khz 70 khz 88 khz-70 khz =8 khz 80 khz 88 khz-80 khz = 8 khz

39 Check Yourself A periodic signal with a period of 0. ms is sampled at 44 khz. o what frequency does the eighth harmonic alias?. 8 khz 2. 6 khz 3. 4 khz 4. 8 khz 5. 6 khz 6. none of the above

40 Aliasing High frequency components of complex signals also wrap. X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

41 Aliasing High frequency components of complex signals also wrap. X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

42 Aliasing High frequency components of complex signals also wrap. X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

43 Aliasing High frequency components of complex signals also wrap. X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

44 Aliasing Aliasing increases as the sampling rate decreases. X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

45 Aliasing Aliasing increases as the sampling rate decreases. X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

46 Aliasing Aliasing increases as the sampling rate decreases. X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

47 Aliasing Aliasing increases as the sampling rate decreases. X(j) X p (j)= s P (j) s (X(j ) P (j ))() s s 2 2

48 Aliasing Demonstration Sampling Music s = =f s f s =44. khz f s =22 khz f s = khz f s =5.5 khz f s =2.8 khz J.S. Bach, Sonata No. in G minor Mvmt. IV. Presto Nathan Milstein, violin

49 Aliasing Aliasing increases as the sampling rate decreases. X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

50 Aliasing Aliasing increases as the sampling rate decreases. X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

51 Aliasing Aliasing increases as the sampling rate decreases. X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

52 Aliasing Aliasing increases as the sampling rate decreases. X(j) X p (j)= s P (j) s (X(j ) P (j ))() s s 2 2

53 Anti-Aliasing Filter o avoid aliasing, remove frequency components that alias before sampling. x(t) Anti-aliasing Filter s 2 s 2 x p (t) Reconstruction Filter s 2 s 2 x r (t) p(t)

54 Aliasing Aliasing increases as the sampling rate decreases. X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

55 Aliasing Aliasing increases as the sampling rate decreases. Anti-aliased X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

56 Aliasing Aliasing increases as the sampling rate decreases. Anti-aliased X(j) s P (j) s X p (j)= (X(j ) P (j ))() s s 2 2

57 Aliasing Aliasing increases as the sampling rate decreases. Anti-aliased X(j) X p (j)= s P (j) s (X(j ) P (j ))() s s 2 2

58 Anti-Aliasing Demonstration Sampling Music s = =f s f s = khz without anti-aliasing f s = khz with anti-aliasing f s =5.5 khz without anti-aliasing f s =5.5 khz with anti-aliasing f s =2.8 khz without anti-aliasing f s =2.8 khz with anti-aliasing J.S. Bach, Sonata No. in G minor Mvmt. IV. Presto Nathan Milstein, violin

59 Sampling: Summary Effects of sampling are easy to visualize with Fourier representations. Signals that are bandlimited in frequency (e.g., W <<W) can be sampled without loss of information. he minimum sampling frequency for sampling without loss of information is called the Nyquist rate. he Nyquist rate is twice the highest frequency contained in a bandlimited signal. Sampling at frequencies below the Nyquist rate causes aliasing. Aliasing can be eliminated by pre-filtering to remove frequency components that would otherwise alias.

60 MI OpenCourseWare Signals and Systems Spring 200 For information about citing these materials or our erms of Use, visit: