1

Dr. Tahir Zaidi

Advanced Digital Signal Processing

Lecture 7

Sampling of Continuous-Time Signals

1. Analog-to-Digital Conversion and Sampling

C

D DSP

D

C

)(txc

)(

][

nTx

nx

c )(

][

nTy

ny

r

)(tyr

A-to-D Conversion D-to-A Conversion

Sampling

2

Periodic sampling

Periodic sampling: Time domain

3

Periodic sampling: Frequency domain

Periodic sampling: Frequency domain

4

1 2

ms5.02000

1

3

2

3

4f0

signal

f0'

signal

ms2500

1

0

ttxc 4000cos)( ttxc 1000cos)(

Original signal Aliased signal

Nyquist Theorem

5

Oversampling

Undersampling (Aliasing Distortion)

6

Reconstruction (Frequency Domain)

Nyquist Theorem

7

Band-limited interpolation

Reconstruction Time Domain

8

Band-limited interpolation

Ideal D/C and C/D

9

DT processing of continuous time signals

Downsampling in time-domain

decim

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Downsampling

Downsampling

WN -WN

1 Xc(jW)

W

/2 -/2

1/T X (w)

wWT 2 -2

X d(w)

wWT’

Downsampling by M

generates baseband

plus M-1 copies of

baseband per period of

frequency domain

Sample the analog

bandlimited signal

every T time units

Aliasing occurs: avoid aliasing by pre-

filtering with lowpass filter with gain of

1 and cutoff of /M to extract baseband

3/2 -3/2 2

M=3

MT

1

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Downsampling in Frequency-domain

Aliasing and pre-filtering

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Upsampling

Upsampling

13

Aliasing and pre-filtering

One-Dimensional Upsampling

WN -WN

1 Xc(jW)

W

-

1/T X (w)

wWT 2 -2

/L

1/T

X u(w) = X(L w)

wWT’ 3/L -/L -3/L -5/L

Upsampling by L gives

L images of baseband

per 2 period of w

Sample the analog

bandlimited signal

every T time units

Apply lowpass

interpolation filter

with gain of L and

cutoff of /L to

extract baseband

/L

1/T = L/T X i(w)

2 -/L -2 wWT’ Fig. 3.22 Oppenheim &

Schafer, 1989.

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Zero insertion

Zero insertion

15

Zero insertion

Zero insertion

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Zero insertion