M7. Introduction to Sampling of Continuous-Time Signals

3/22/2011 I. Discrete-Time Signals and Systems 1

M7. Introduction to Sampling of

Continuous-Time Signals

Reading material: p.140-160


Content

What’s the sampling mechanism?

What’s the aliasing problem?

Nyquist sampling theorem

Frequency analysis of sampling


Introduction

Discrete-time signals can be commonly obtained by sampling

the continuous-time signals

Digital computers can only deal with digital signals, which is

a special class of discrete-time signals

Digital signal processing, digital control systems …

Plant

Computer

input

Input noise Output disturbance

output

A/DA/DD/A

Sampling and holdingSignal processingconstruction


Periodic Sampling A sequence of samples x[n], a continuous-time signal xc(t)

x[n]=xc(nT), -∞∞∞∞<n < ∞∞∞∞

Sampling period T, sampling frequency f=1/T (Herz); or sampling frequency ΩΩΩΩs=2ππππ/T (rad/sec)

Practically, the operation of sampling is implemented by analog-to-digital (A/D) converters

Quantization of the sampled outputs

Linearity of quantization steps

Sample-and-hold circuits

Limitations on the sampling rate

Principle of sampling

Generating a impulse train from the continuous-time signal

Converting the impulse train to a discrete-time sequence


Principle of Sampling


Frequency Analysis of Sampling

Generation of a impulse train xs(t) through modulation of the

uniformed periodic impulse train

The Fourier transform of a periodic impulse train is a

periodic impulse train in frequency domain

∑∞

−∞=

−=

n

nTtts )()( σ

)()()()()()()( nTtnTxnTttxtstxtxn

c

n

ccs −=−== ∑∑∞

−∞=

∞

−∞=

σσ

Tn

TjS s

n

s

πσ

π 2,)(

2)( =ΩΩ−Ω=Ω ∑

∞

−∞=


Frequency Analysis of Sampling (Cont’d)

The frequency-domain relation between xs(t) and xc(t)

Xs(jΩΩΩΩ) consists of periodically repeated copies of Xc(jΩΩΩΩ)!!!

xc(t) could be recovered possibly from xs(t) with a ideal

lowpass filter if …

)()()()()()()( nTtnTxnTttxtstxtxn

c

n

ccs −=−== ∑∑∞

−∞=

∞

−∞=

σσ

Tn

TjS s

n

s

πσ

π 2,)(

2)( =ΩΩ−Ω=Ω ∑

∞

−∞=

∑∞

−∞=

Ω−Ω=ΩΩ=Ω

n

sccs njXT

jSjXjX ))((1

)(*)(2

1)(

π


Effect of Sampling in Frequency Domain

Consider a bandlimited signal with highest nonzero frequency ΩΩΩΩN

If ΩΩΩΩs >2 ΩΩΩΩN, there is no overlap with replicas of Xc(jΩΩΩΩ), so,

xc(t) could be recovered

possibly from xs(t)

If ΩΩΩΩs<2 ΩΩΩΩN, there is overlap with replicas of Xc(jΩΩΩΩ), then,

Aliasing distortion (aliasing)

Exists in the recovered

signal through …


Exact Recovery from Sampling

Consider bandlimited signals

Under condition ΩΩΩΩs>2 ΩΩΩΩN

Use an ideal lowpass filter

Hr(jΩΩΩΩ) with gain T and cutoff

frequency ΩΩΩΩc such that

ΩΩΩΩN <ΩΩΩΩc < (ΩΩΩΩs-ΩΩΩΩN)

The continuous signal can be

exactly recovered as

Xr(jΩΩΩΩ)= Xc(jΩΩΩΩ)


Alias and Nyquist Sampling Theorem

If ΩΩΩΩs<2 ΩΩΩΩN, aliasing phenomenon occurs…

Consider a cosine signal xc(t)=cos ΩΩΩΩ0t …

Nyquist Sampling

Theorem

Let xc(t) be a bandlimited

signal with Xr(jΩΩΩΩ)= 0 for

|ΩΩΩΩ|> ΩΩΩΩN, then xc(t) is

uniquely determined by its

samples x[n]=xc(nT),

-∞∞∞∞<n < ∞∞∞∞, if ΩΩΩΩs= 2ππππ/T ≥≥≥≥2 ΩΩΩΩN


Relation between X(ejωωωω) and Xc(jΩΩΩΩ) Xc(jΩΩΩΩ) is the (continuous) Fourier transform of xc(t)

X(ejωωωω) is the (discrete) Fourier transform of x[n]= xc(nT)

The Fourier transform of impulse train function xs(t)

Then there is

i.e., a frequency scaling or normalization …ωωωω=ΩΩΩΩT

dtetxjXtj

∫∞

∞−

Ω−=Ω )()(

jwk

k

jekxeX

−

∞

−∞=

∑= ][)( ω

nTj

n

cs

n

ccs

enTxjX

nTtnTxtstxtx

Ω−

∞

−∞=

∞

−∞=

∑

∑

=Ω

−==

)()(

)()()()()( σ

)(|)()( Tj

T

j

s eXeXjXΩ

Ω===Ω

ω

ω ∑∞

−∞=

Ω−Ω=Ω

n

scs njXT

jX ))((1

)(

∑∞

−∞=

−=

n

c

j

Tn

TjX

TeX ))

2((

1)(

πωω


Examples: Sampling and Aliasing

Example 4.1 (p.147) sampling and reconstruction …

Example 4.2, 4.3 (p.148-9) Aliasing in reconstruction …

! Should be ππππ/T


Reconstruction of Bandlimited Sampled Signals

Principle of reconstruction

Generating a impulse train from

discrete-time sequence

Converting the impulse train to a

continuous-time signal

Reconstruction (lowpass) filter

∑∞

−∞=

−=

n

s nTtnxtx )(][)( σ

∑∞

−∞=

−=

n

rr nTthnxtx )(][)(

Tt

Ttth r

/

)/sin()(

π

π=

∑∞

−∞= −

−=

n

rTnTt

TnTtnxtx

/)(

)/)(sin(][)(

π

π


Discrete-Time Processing of Continuous-Time Signals

C/D converter produces a discrete-time sequence

D/C converter creates a continuous-time signal

LTI discrete-time systems

The overall system is equivalent to a LTI continuous-time

system

∑∞

−∞=

−==

n

c

j

cT

nT

jXT

eXnTxnx ))2

((1

)(),(][πωω

<Ω

=Ω=Ω−

−=

Ω

Ω

∞

−∞=

∑otherwsie

TeTYeYjHjY

TnTt

TnTtnyty

Tj

Tj

rr

n

r,0

/||),()()()(,

/)(

)/)(sin(][)(

π

π

π

)()()(],[*][][ ωωω jjjeXeHeYnxnhny ==

≥Ω

<Ω=ΩΩΩ=Ω

Ω

T

TeHjHjXjHjY

Tj

effceffr/||,0

/||),()(),()()(

π

π


Example of Discrete Processing

Example 4.4 p.155…

LTI discrete-time system: a ideal lowpass filter

Bandlimited input signal

Sampling rate satisfies Nyquist Theorem

The effective system is a LTI continuous-time system with …

The effective cutoff frequency

ΩΩΩΩc=ωωωωc/T


Practical Issues in ImplementationIn practice, …

Continuous-time signals are not precisely bandlimited

Ideal lowpas filters can not be realized

Ideal C/D and D/C converters can only be approximated by devices called A/D and D/A converters

Therefore, …

Prefiltering to avoid aliasing

A/D conversion: sample and hold circuits, quantization…

D/A conversion ….


Exercise Seven

Problem 4.4 on page214 of the textbook;

Problem 4.5 on page214 of the textbook;


Course Summary M1. Discrete-Time Signal and System

M2. The Z-Transform and its Properties

M3. The Inverse Z-Transform and Frequency-

Domain Representation

M4. Fourier Transform and Its Properties

M5. LTI Systems Described by Linear Constant

Coefficient Different Equations

M6. Frequency Response of LTI Systems

M7. Introduction to Sampling of Continuous-Time

Signals

Documents

M7. Introduction to Sampling of Continuous-Time Signals