Sampling

Chang-Su Kim

Some figures have been excerpted from

1. The lecture notes of Dr. Benoit Boulet in McGill University

(http://www.cim.mcgill.ca/~boulet/304-304A/304-304A.htm)

2. Signals and Systems by M. J. Roberts

3. http://www.ics.uci.edu/~majumder/CG/classes/sampling_1nov.pdf

Sampling

Sampling is a procedure

to extract a DT signal

from a CT signals

(b), (c), (d) are obtained

by sampling (a)

Is (b) enough to

represent (a)?

What is the adequate

sampling rate to

represent a given CT

signal without information

loss?

In general, DT signal cannot represent CT signal

perfectly

Are these sample enough to reconstruct the original blue curve?

Sampling Theorem

Under certain conditions, a CT signal can be completely

represented by and recoverable from samples

A lowpass signal can be reconstructed from samples, if the

sampling rate is high enough. Because it is a lowpass signal,

the change between two close samples is constrained (or

expected).

Let x(t) be a band-limited signal with X(jw) = 0 for

|w| > wM. Then, x(t) is uniquely determined by its

samples x(nT), if

ws > 2 wM

where the sampling rate ws is defined as

ws = 2p/T

Information Equivalence

[ ] ( )x n x nT

( ) ( ) ( )p

n

x t x nT t nT

Given [ ], we can generate ( ).

Given ( ), we can generate [ ].

Therefore, [ ] and ( ) have

the same information.

p

p

p

x n x t

x t x n

x n x t

Let x(t) be a band-limited signal with X(jw) = 0 for

|w| > wM. Then, x(t) is uniquely determined by the

modulated impulse train

if

ws > 2 wM

where the sampling rate ws is defined as

ws = 2p/T

( ) ( ) ( )p

n

x t x nT t nT

Restated Sampling Theorem

2 wM: Nyquist rate

Frequency Domain Interpretation

Show that

( ) ( ) ( )

1( ) ( ( ))

p

n

p s

k

x t x nT t nT

X jw X j w kwT

p(t)

Reconstruction: ideal lowpass filter

( ) 2

0 otherwise

sin sin2( ) sinc( )

s

s

wT w

H jw

wt t

tTh t Tt T

tT

p

pp

2

sw

2

sw

T

swsw

Reconstruction of Signal from Its Samples

1( ) ( ( ))p s

k

X jw X j w kwT

( )X jw

w w

( ) ( )* ( )

sinc( / )* ( ) ( )

( )sinc

r p

n

n

x t h t x t

t T x nT t nT

t nTx nT

T

Reconstruction of Signal from Its Samples

This is the way to reconstruct

or interpolate xr(t) from

samples x(nT)’s

Note that xr(t) = x(t), if the

sampling rate is higher than

the Nyquist rate

Practical Interpolation Filters

Ideal interpolation filters –sinc function

Infinite duration

Not implementable

Practical interpolation filtersZero order holding (repetition)

First order holding (linear)

Practical Interpolation Filters

Undersampling Causes Aliasing

Undersampling: sampling rate is less than Nyquist

rate

Undersampling Causes Aliasing

Undersampling Causes Aliasing

Rotating disk

1 rotation/second

To avoid aliasing, it should be

motion-pictured with at least

2 frames/s.

0 1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12

Undersampling Causes Aliasing

12 frames/s

0 1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12

Undersampling Causes Aliasing

6 frames/s

0 1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12

Undersampling Causes Aliasing

3 frames/s

0 1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12

Undersampling Causes Aliasing

2 frames/s

0 1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12

Undersampling Causes Aliasing

12/11 = 1.09 frames/s

0 1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12

DT Processing of CT Signals

We assume that the sampling rate is higher than Nyquist rate

So yc(t) = xc(t)

Relation between modulated impulse train and DT signal

[ ] [ ]d dx n y n

( ) ( / ) or ( ) ( ),

( ) ( / ) or ( ) ( )

j jwT

d p p d

j jwT

d p p d

X e X j T X jw X e

Y e Y j T Y jw Y e

DT Processing of CT Signals

CT system Hc(jw)

( ) ( ) ( )

or ( ) ( )

jwT

c c d

jwT

c d

Y jw X jw H e

H jw H e

This is true if xc(t) is band-limited and T satisfies the Nyquist condition

DT Processing of CT Signals

Refer to Figure 7.25