1

Sampling Theorem

© Ammar Abu-Hudrouss Islamic

University Gaza

Slide 2

Digital Signal Processing

Signal is any physical quantity that varies with time, space, or any other variable.

System is a physical device or software that performs an operation on a signal

Slide 3

Digital Signal Processing

Basic Elements of DSP System

Analogue electronic systems are continuous

Electronic System are increasingly digitalized

Signals are converted to numbers, processed, and converted back

Analogue System x(t) y(t)

Digital System A/D D/A y(t) x(t) y(n) x(n)

Slide 4

Digital Signal Processing

Advantages of Digital over Analogue

Advantages

Flexibility (simply changing program).

Accuracy.

Storage Capability.

Cheap

Ability to apply highly sophisticated algorithms.

Disadvantages

It has certain limitations (very fast sample rate is needed when the bandwidth of signal is very large)

It has a larger time delay compared to the analogue.

Slide 5

Digital Signal Processing

Classification of signals

Mono-channel versus Multi-channel

One Dimensional versus Multidimensional

Continues Time versus Discrete Time

Continuous Values and Discrete Valued

Deterministic versus Random

Slide 6

Digital Signal Processing

Periodic Continuous Signal

21

FT

tAtx cos)(

We will take sinusoidal signals for example. Continuous sinusoidal signal has the form

The signal can be characterised by three parameters A: Amplitude, frequency in radian and : phase

The period is defined as,

Slide 7

Digital Signal Processing

Periodic Continuous Signal

7

In analogue signal, increasing the frequency will always lead to increase the rate of the oscillation.

In analogue signals with distinct frequencies are themselves distinct from each other.

Slide 8

Digital Signal Processing

Periodic Discrete Signal

)22cos()2cos(

)()(

fNfnfn

Nnxnx

nAnx cos)(

N

kf

kkfN

,......2,1,022

8

Discrete sinusoidal signal has the form

1) Discrete time sinusoid is periodic only if its frequency in hertz ( f = / 2) is a rational number

From the definition of a periodic discrete signal

This is only true if

Slide 9

Digital Signal Processing

Periodic Discrete Signal

)()cos())2cos((1 nxnAnAnx

nAnx cos)(

,......2,1,02 kkk )cos()( nAnx kk

9

2) Discrete time sinusoid whose radian frequencies are separated by integer multiples of 2 are identical

To prove this, we start from the signal

As a result, all the following signals are identical

3) All signal in the range - < are unique.

So the range of the discrete frequency f is [-0.5 0.5]

Slide 10

Digital Signal Processing 10

Slide 11

Digital Signal Processing

Harmonically related Complex Exponential

The basic signals for continuous-time harmonically related exponentials are

tjk

k ets 0

Nfk /1,2,1,0 0

The basic signals for discrete-time harmonically related exponentials are

nkfj

k ens 02

00 2,2,1,0 Fk

Slide 12

Digital Signal Processing

Analogue to Digital Conversion

Sampler Quantizer Coder xa(t) x(n) xq(n)

Analog Signal

Discrete-time Signal

Quantized Signal

Digital Signal

101101…

1) Sampling: Conversion of analogue signal into a discrete signal by taking sample at every Ts s.

2) Quantization: Conversion of discrete signal into discrete signals with discrete values. (the value of each sample is represented by a value selected from a finite set of possible value)

3) Coding: is process of assigning each quantization level a unique binary code of b bits.

Slide 13

Digital Signal Processing

Sampling of Analog Signal

We will focus on uniform sampling where

x (n) = xa(nTs) -∞ < n < ∞

Fs = 1/Ts is the sampling rate given in samples per second

As we can see the discrete signal is achieved by replacing the continuous variable t by nTs.

Consider the analog signal xa (t ) = A cos(2Ft + ) The sampled signal is xa(nT) = A cos(2FnTs + ) x (n) = A cos(2fn + ) The digital frequency = analog freq. X sampling time

f = FTs

Slide 14

Digital Signal Processing

Sampling of Analog Signal

But from previous discussion , for the analogue frequency

-∞< F <∞ or -∞< <∞

And for the digital frequency

-0.5 ≤ f < 0.5 or - ≤ <

From the above argument the infinite analog frequency is mapped into finite digital frequency.

This mapping is one-to-on as long as the resultant digital frequency is between the limits of [-0.5 o.5]

Slide 15

Digital Signal Processing

Sampling of Analog Signal

Which leads that

-1/2 ≤ FTs < 1/2 or - ≤ Ts <

OR

-1/(2Ts) ≤ F < 1/(2Ts) or - /Ts ≤ < /Ts

Hence that highest possible analogue frequency is

Fmax = Fs /2 = 1/(2Ts) and < Fs = /Ts .

Fs /2 is called the folding frequency

Slide 16

Digital Signal Processing

Sampling of Analog Signal

Example Consider the two analog sinusoidal signals

X1(t ) = cos [2(10)t ] and X2(t ) = cos [2(50)t ]. Both are sampled with sampling rate Fs = 40 sample/s, find the

corresponding discrete sequences

X1 (n) = cos [2(10/40)t] = cos [n/2]

X2(t) = cos [2(50/40)t] = cos [5n/2] = cos [n/2]

a 1Hz and a 6Hz sinewave are sampled at a rate of 5Hz.

Slide 17

Digital Signal Processing

Sampling of Analog Signal

All sinusoids with frequency

Fk = F0 + k Fs, k = 1,2,3,………

Leads to unique signal if sampled at Fs sample/s.

proof

xa(t) = cos (2 Fk t + ) = cos (2 (F0 + k Fs )t +)

x(n) = xa(nTs) = cos (2 (F0 + k Fs )/Fs n + ) = cos (2 F0 /Fs n + 2 k n + ) = cos (2 F0 /Fs n + ) = cos (2 f0 n + )

Slide 18

Digital Signal Processing

Sampling Theorem

Sampling Theorem

A continuous-time signal x(t) with frequencies no higher than fmax (Hz) can be reconstructed EXACTLY from its samples x[n] = x(nTs), if the samples are taken at a rate fs = 1/Ts that is greater than 2 fmax.

Consider a band-limited signal x(t) with Fourier Transform X()

Slide 19

Digital Signal Processing

Sampling Theorem

Sampling x(t) is equivalent to multiply it by train of impulses

X

Slide 20

Digital Signal Processing

Sampling Theorem

In mathematical terms

Converting into Fourier transform

)()()( tstxtxs

n

ss nTttxtx )()()(

n

s

s

s nT

XX )(1

*)(

n

s

s

s nXT

X )(1

)(

Slide 21

Digital Signal Processing

Sampling Theorem

By graphical representation in the frequency domain

X

Slide 22

Digital Signal Processing

Sampling Theorem

Therefore, to reconstruct the original signal x(t), we can use an ideal lowpass filter on the sampled spectrum

This is only possible if the shaded parts do not overlap. This means that fs must be more than TWICE that of B.

Slide 23

Digital Signal Processing

Sampling Theorem

Example x(t ) and its Fourier representation is shown in the Figure.

If we sample x(t) at Fs = 20,10,5

1) Fs = 20 x (t ) can be easily

recovered by LPF

Slide 24

Digital Signal Processing

Sampling Theorem

2) Fs = 10 x(t ) can be

recovered by sharp LPF

3) Fs = 5 x(t) can not be

recovered

Compare fs with 2 B in each case

Slide 25

Digital Signal Processing

Anti-aliasing Filter

To avoid corruption of signal after sampling, one must ensure that the signal being sampled at fs is band-limited to a frequency B, where B < fs/2.

Consider this signal spectrum:

After sampling:

After reconstruction:

Slide 26

Digital Signal Processing

Anti-aliasing Filter

Apply a lowpass filter before sampling:

Now reconstruction can be done without distortion or corruption to lower frequencies:

Sampler Anti-aliasing

filter x(t)

y(n) x'(t)

Slide 27

Digital Signal Processing

Homework

Students are encouraged to solve the following questions from the main textbook

1.2, 1.3, 1.4, 1.5, 1.7, and 1.9