Sampling Theorem 主講者：虞台文

Content Periodic Sampling Sampling of Band-Limited Signals Aliasing --- Nyquist rate CFT vs. DTFT Reconstruction of Band-limited Signals Discrete-Time Processing of Continuous-Time Signals Continuous-Time Processing of Discrete-Time Signals Changing Sampling Rate Realistic Model for Digital Processing

Sampling Theorem Periodic Sampling

Continuous to Discrete-Time Signal Converter xc(t) x(n)= xc(nT) C/D T Sampling rate

Conversion from impulse train to discrete-time sequence C/D System Conversion from impulse train to discrete-time sequence xc(t) x(n)= xc(nT)  s(t) xs(t)

Sampling with Periodic Impulse train xc(t) T 2T 3T 4T T 2T 3T t xc(t) 2T 4T 8T 10T 2T 4T 8T n x(n) 1 2 3 4 1 2 3 n x(n) 2 4 6 8 2 4 6

Sampling with Periodic Impulse train We want to restore xc(t) from x(n). Sampling with Periodic Impulse train What condition has to be placed on the sampling rate? t xc(t) T 2T 3T 4T T 2T 3T n x(n) 1 2 3 4 1 2 3 8T 10T 4T 8T 6 8 4 6

Conversion from impulse train to discrete-time sequence C/D System Conversion from impulse train to discrete-time sequence xc(t) x(n)= xc(nT)  s(t) xs(t)

Conversion from impulse train to discrete-time sequence C/D System Conversion from impulse train to discrete-time sequence xc(t) x(n)= xc(nT)  s(t) xs(t)

C/D System s: Sampling Frequency

C/D System

Sampling of Band-Limited Signals Sampling Theorem Sampling of Band-Limited Signals

Band-Limited Signals Xc(j) Band-Limited  Band-Unlimited Yc(j)  N 1 Band-Limited  Yc(j) Band-Unlimited

Sampling of Band-Limited Signals  Xc(j) N N 1 Band-Limited Sampling with Higher Frequency  s s 2s 3s 2s 3s S(j) 2/T Sampling with Lower Frequency  4s 4s 2s 6s 2s 6s S(j) 2/T

Aliasing --- Nyquist Rate Sampling Theorem Aliasing --- Nyquist Rate

Recoverability s > 2N s < 2N Xc(j) Band-Limited  S(j) N 1 s s 2s 3s 2s 3s S(j) 2/T 4s 4s 6s 6s Sampling with Higher Frequency Lower Frequency Recoverability s > 2N s < 2N

Case 1: s > 2N Xc(j)  S(j)  Xs(j)  N N 1 s s 2s 3s 2/T  s s 2s 3s 2s 3s Xs(j) 1/T

Case 1: s > 2N Xs(j) is a periodic function with period s. Xc(j) N N 1 s s 2s 3s 2s 3s S(j) 2/T 1/T Xs(j) Case 1: s > 2N Passing Xs(j) through a low-pass filter with cutoff frequency N < c< s N , the original signal can be recovered. Xs(j) is a periodic function with period s.

Case 2: s < 2N Xc(j)  S(j)  Xs(j)  N N 1 2s 2s 4s 2/T  2s 2s 4s 6s 4s 6s Xs(j) 1/T

Aliasing Case 2: s < 2N No way to recover the original signal. Xc(j) N N 1 1/T 2s 2s 4s 6s 4s 6s S(j) 2/T Xs(j) Case 2: s < 2N Xs(j) is a periodic function with period s. No way to recover the original signal. Aliasing

Nequist Rate Xc(j) Band-Limited  Nequist frequency (N) 1 Band-Limited Nequist frequency (N) The highest frequency of a band-limited signal Nequist rate = 2N

Nequist Sampling Theorem  Xc(j) N N 1 Band-Limited s > 2N Recoverable s < 2N Aliasing

Sampling Theorem CFT vs. DTFT

Conversion from impulse train to discrete-time sequence C/D System Conversion from impulse train to discrete-time sequence xc(t) x(n)= xc(nT)  s(t) xs(t)

Continuous-Time Fourier Transform Conversion from impulse train to discrete-time sequence xc(t) x(n)= xc(nT)  s(t) xs(t)

Conversion from impulse train to discrete-time sequence CFT vs. DTFT Conversion from impulse train to discrete-time sequence xc(t) x(n)= xc(nT)  s(t) xs(t) x(n)

Conversion from impulse train to discrete-time sequence xc(t) x(n)= xc(nT)  s(t) xs(t) x(n) CFT vs. DTFT

CFT vs. DTFT

CFT vs. DTFT  Xc(j) 1  Xs(j) s s 1/T  X(ej) 2 1/T 2 4 4

CFT vs. DTFT s2 Xc(j)  Xs(j)  X(ej)  Amplitude scaling 1 Amplitude scaling & Repeating  Xs(j) s s 1/T Frequency scaling s2  X(ej) 2 1/T 2 4 4

Reconstruction of Band-limited Signals Sampling Theorem Reconstruction of Band-limited Signals

Key Concepts t  Xc(j) CFT ICFT Sampling C/D n DTFT  X(ej)   xc(t) T 2T 3T 4T T 2T 3T  Xc(j) /T /T CFT ICFT Sampling C/D Retrieve One period n x(n) 1 2 3 4 1 2 3 DTFT  X(ej)   IDTFT

Interpolation

Interpolation n(t) x(n)

Ideal D/C Reconstruction System x(n) xs(t) xr(t) Covert from sequence to impulse train T Ideal Reconstruction Filter Hr(j)

Ideal D/C Reconstruction System Obtained from sampling xc(t) using an ideal C/D system. Ideal D/C Reconstruction System x(n) xs(t) xr(t) Covert from sequence to impulse train T Ideal Reconstruction Filter Hr(j)  /T /T Hr(j) T

Ideal D/C Reconstruction System x(n) xs(t) xr(t) Covert from sequence to impulse train T Ideal Reconstruction Filter Hr(j)

Ideal D/C Reconstruction System x(n) xs(t) xr(t) Covert from sequence to impulse train T Ideal Reconstruction Filter Hr(j) D/C

Ideal D/C Reconstruction System xc(t) C/D T x(n) xr(t) D/C T In what condition xr(t) = xc(t)?

Discrete-Time Processing of Continuous-Time Signals Sampling Theorem Discrete-Time Processing of Continuous-Time Signals

The Model xc(t) yr(t) xc(t) x(n) y(n) yr(t) T C/D D/C T Discrete-Time System Continuous-Time System xc(t) yr(t)

The Model Heff(j) H (ej) y(n) yr(t) xc(t) x(n) D/C T C/D Discrete-Time System xc(t) C/D x(n) Continuous-Time H (ej) Heff(j)

LTI Discrete-Time Systems xc(t) C/D x(n) y(n) yr(t) H (ej) Discrete-Time System D/C T

LTI Discrete-Time Systems y(n) yr(t) D/C T Discrete-Time System xc(t) C/D x(n) H (ej)

LTI Discrete-Time Systems Continuous-Time System xc(t) yr(t) Heff(j)

Example:Ideal Lowpass Filter y(n) yr(t) D/C T Discrete-Time System xc(t) C/D x(n) 1  c c H(ej)

Example:Ideal Lowpass Filter 1  c c Heff(j) Continuous-Time System xc(t) yr(t)

Example: Ideal Bandlimited Differentiator Continuous-Time System xc(t)

Example: Ideal Bandlimited Differentiator  |Heff(j)| Continuous-Time System xc(t)

Example: Ideal Bandlimited Differentiator Continuous-Time System xc(t)  |Heff(j)|

Impulse Invariance xc(t) yc(t) Continuous-Time LTI system hc(t), Hc(j) xc(t) yc(t) y(n) yc(t) D/C T Discrete-Time LTI System h(n) H(ej) xc(t) C/D x(n) What is the relation between hc(t) and h(n)?

Impulse Invariance

Impulse Invariance

Impulse Invariance xc(t) yc(t) Continuous-Time LTI system hc(t), Hc(j) xc(t) yc(t) y(n) D/C T Discrete-Time LTI System h(n) H(ej) C/D x(n) What is the relation between hc(t) and h(n)?

Continuous-Time Processing of Discrete-Time Signals Sampling Theorem Continuous-Time Processing of Discrete-Time Signals

The Model x(n) y(n) yc(t) y(n) x(n) xc(t) C/D T D/C Discrete-Time Continous-Time System x(n) D/C xc(t) Discrete-Time System x(n) y(n)

The Model H (ej) Hc(j) yc(t) y(n) x(n) xc(t) C/D T D/C Discrete-Time Continous-Time System x(n) D/C xc(t) Discrete-Time Hc(j) H (ej)

The Model yc(t) y(n) C/D T Continous-Time System x(n) D/C xc(t) Hc(j)

The Model

The Model Discrete-Time System x(n) y(n) H (ej)

The Model H (ej) Hc(j) x(n) y(n) yc(t) y(n) x(n) xc(t) C/D T D/C Continous-Time System x(n) D/C xc(t) Hc(j) Discrete-Time System x(n) y(n) H (ej)

Changing Sampling Rate Using Discrete-Time Processing Sampling Theorem Changing Sampling Rate Using Discrete-Time Processing

The Goal Down/Up Sampling

Sampling Rate Reduction By an Integer Factor Down/Up Sampling Down Sampling

Sampling Rate Reduction By an Integer Factor

Sampling Rate Reduction By an Integer Factor Let r = kM + i

Sampling Rate Reduction By an Integer Factor Xc(j)  N N Xs(j), X (ejT)  2/T 2/T 1/T N=NT N X (ej)  2 2 1/T

Sampling Rate Reduction By an Integer Factor Xc(j)  Xs(j), X (ejT) 2/T 2/T 1/T N=NT N X (ej)  2 2 N <  : no aliasing Sampling Rate Reduction By an Integer Factor M=2 Xd (ej)  2 2 1/MT Xd (ejT)  1/T’ 2/T’ 2/T’ 4/T’ 4/T’

Antialiasing Aliasing M=3 N N X (ej)  2 2 1/T  Xd (ej) 1/MT

Antialiasing However, xd(n) x(nT’) M=3 N N X (ej)  2 2 1/T /3 Hd (ej)  2 2 1 /3 M=3  2 2 /3 /3  2 2

Decimator Lowpass filter Gain = 1 Cutoff = /M M

Increasing Sampling Rate By an Integer Factor Up Sampling T

Increasing Sampling Rate By an Integer Factor Up Sampling X (ej)    1/T X’ (ej)    L/T

Interpolator Lowpass filter Gain = L Cutoff = /L L

Interpolator

Interpolator L=3 X (ej)    1/T Xe(ej)    1/T   Hi(ej)  /3 /3 Xi(ej)    L/T

Changing the Sampling Rate By a Noninteger Factor Resampling

Changing the Sampling Rate By a Noninteger Factor Sampling Periods: Lowpass filter Gain = 1 Cutoff = /M M Gain = L Cutoff = /L L M Lowpass filter Gain = L Cutoff = min(/L, /M) L

Realistic Model for Digital Processing Sampling Theorem Realistic Model for Digital Processing

Ideal Discrete-Time Signal Processing Model y(n) yc(t) D/C T Discrete-Time LTI System xc(t) C/D x(n) Real world signal usually is not bandlimited Ideal continuous-to-discrete converter is not realizable Ideal discrete-to-continuous converter is not realizable

Compensated reconstruction filter More Realistic Model y(n) yc(t) D/C T Discrete-Time LTI System xc(t) C/D x(n) Anti-aliasing filter Sample and Hold A/D converter Discrete-time system D/A converter Compensated reconstruction filter T

Analog-to-Digital Conversion Sample and Hold A/D converter T T

Sample and Hold t T ho(t) t T

Sample and Hold t xo(t) t T ho(t) T

Sample and Hold t T ho(t) xo(t)

Sample and Hold Goal: To hold constant sample value for A/D converter. Zero-Order Hold ho(t)

A/D Converter C/D T Quantizer Coder

Typical Quantizer 2’s complement code Offset binary 011 010 001 000 111 110 101 100 2Xm (B+1)-bit Binary code

Analysis of Quantization Errors C/D T Quantizer Coder Quantizer Q[ ]

Analysis of Quantization Errors The error sequence e(n) is a stationary random process. e(n) and x(n) are uncorrelated. The random variables of the error process are uncorrelated, i.e., the error is a white-noise process. e(n) is uniform distributed.

SNR (Signal-to-Noise Ratio)

SNR (Signal-to-Noise Ratio) 每增加一個bit，SNR增加約6dB

SNR (Signal-to-Noise Ratio) Let x=Xm / 4  SNR  6B1.25 dB SNR (Signal-to-Noise Ratio) x大較有利，但不得過大(為何？) x過小不利 x每降低一倍SNR少6dB X~N(0, x2)  P(|X|<4x )0.00064