Signals and Systems

6552111 Signals and Systems

Sopapun Suwansawang

1

Lecture #7

Sampling System

Checkpoint

Sampling theorem,

Nyquist theorem,

Aliasing,

Signal Reconstruction

2Sopapun Suwansawang

6552111 Signals and Systems

Digital Signal Processing System

3

Analog

to Digital

Digital SignalProcessing

Digital

to

Analog

)(tx )(nx )(ny )(ty

Sopapun Suwansawang

6552111 Signals and Systems

Impulse Sampling

4

Multiplying x(t) by the sampling function

Sopapun Suwansawang

6552111 Signals and Systems

5

Sampling

5Sopapun Suwansawang

6552111 Signals and Systems

Analog

to Digital

Digital SignalProcessing

Digital

to

Analog

)(tx )(nx )(ny )(ty

Hold Quantizer

A/D Converter

Sample and Hold

nTt

6Sopapun Suwansawang

6552111 Signals and Systems

Sampling (cont.)

Hold Quantizer

A/D Converter

Sample and Hold

nTt

7

6552111 Signals and Systems

Sopapun Suwansawang

Sampling (cont.)

The quantization process introduces errors that

cannot be removed

Figure shows the digital samples using a 2-bit quantizer.

8Sopapun Suwansawang

6552111 Signals and Systems

Sampling (cont.)

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

Time (mS)

Input and Q

uantized s

ignals

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

Time (mS)

Input and Q

uantized s

ignals

8 Bit3 Bit

Sampling (cont.)

Sopapun Suwansawang 9

6552111 Signals and Systems

( )x t ( )x n

Sampling frequency:sf

...1

s

Tf

t

( )x t

1( ) ( ) ,

t nTs

x n x t Tf

10

Sampling Theorem

6552111 Signals and Systems

Sopapun Suwansawang

From , is sampling frequency in hertz

(or sampling rate in cycle per second)

In order to represent an analog signal x(t) by a discrete-

time signal x(nT) accurately, two conditions must be met:

1. x(t) must be bandlimited : is maximum frequency.

2. The sampling frequency must be at least :

Shannon ‘s theorem

sfT

1 sf

Mf

Ms ff 2

11Sopapun Suwansawang

6552111 Signals and Systems

Sampling Theorem

The minimum frequency : the Nyquist rate,

is the Nyquist frequency (or folding frequency)

is the Nyquist interval

Ms ff 2

20

sff

2

sf

2,

2

ss ff

sfsf2

sf

12Sopapun Suwansawang

6552111 Signals and Systems

Example 1

Given a sampling rate for a specific application,

a) In narrowband telecommunication system, the sampling is

given as fs = 8 kHz, thus T = 1/8000 = 125 s.

b) In wideband telecommunication system, the sampling is

given as fs = 16 kHz, thus T = ?.

c) In audio CDs, the sampling is given as fs = 44.1 kHz,

thus T = ?.

d) In professional audio system, the sampling rate fs = 48 kHz,

thus T = ?.

13Sopapun Suwansawang

6552111 Signals and Systems

Aliasing

02sf f

0f sf

alias