Chapter 1 Sampling Theorm2

8/10/2019 Chapter 1 Sampling Theorm2

1/13

1

1

Sampling Theorem

Spring 2009

Ammar Abu-Hudrouss Islamic

University Gaza

Slide 2

Digital Signal Processing

Continuous Versus Digital

Analogue electronic systems are continuous

Electronic System are increasingly digitalized

Signals are converted to numbers, processed, and converted back

Analogue Systemx(t) y(t)

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


2/13

2

Slide 3


Sampling Theorem

Use A-to-D converters to turn x(t) into numbers x[n]

Take a sample every sampling period Ts uniform sampling

Slide 4


Advantages of Digital over Analogue

Advantages

Flexibility (simply changing program)

Accuracy

Storage

Ability to apply highly sophisticated algorithms.

Disadvantages

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

It has a larger time delay compared to the analogue.


3/13

3

Slide 5


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


Periodic Continuous Signal

21

f

T

tAtx cos)(

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

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

The period is defined as


4/13

4

Slide 7


Periodic Continuous Signal

7

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

Slide 8


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


5/13

5

Slide 9


Periodic Discrete Signal

)()cos())2cos(( nxnAnA nAnx cos)(

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

9

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

To prove this, we start from the signal

As a result, all the following signals are identical

3) All signal in the range -


6/13

6

Slide 11


Analogue to Digital Conversion

Sampler Quantizer Coderxa(t)x(n) xq(n)

AnalogSignal

Discrete- timeSignal

QuantizedSignal

DigitalSignal

101101

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

2) Quantization: Conversion of discrete signal into discretesignals with discrete values. (the value of each sample is

represented by a value selected from a finite set ofpossible value)

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

Slide 12


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 sample per second

As we can see the discrete signal is achieved by replacing thecontinuous 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


7/13

7

Slide 13



But from previous discussion , for the analoge frequency

-< F


8/13

8

Slide 15



ExampleConsider 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, find the correspondingdiscrete 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 16



All sinusoids with frequency

Fk = F0 + k Fs, k= 1,2,3,

Leads to unique signal if sampled at Fs Hz.

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

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

= cos (2 F0/Fs n + 2 k n +)

= cos (2F0/Fs n +)


9/13

9

Slide 17


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 isgreater than 2fmax.

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

Slide 18


Sampling Theorem

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

X


10/13

10

Slide 19


Sampling Theorem

In mathematical terms

Converting into Fourier transform

)()()( tstxnx

n

snTttxnx )()()(

n ss

nT

XX )(1

*)(

n

s

s

nXT

X )(1

)(

Slide 20


Sampling Theorem

By graphical representation in the frequency domain

X


11/13

11

Slide 21


Sampling Theorem

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

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

Slide 22


Sampling Theorem

Examplex(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 easilyrecovered by LPF


12/13

12

Slide 23


Sampling Theorem

2) fs = 10x(t) can be recovered

by sharp LPF

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

recovered

Compare fs with 2B in each case

Slide 24


Anti-aliasing Filter

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

Consider this signal spectrum:

After sampling:

After reconstruction:


13/13

13

Slide 25


Anti-aliasing Filter

Apply a lowpass filter before sampling:

Now reconstruction can be done without distortion or corruptionto lower frequencies:

SamplerAnti-aliasing

filterx(t)

y(n)x'(t)

Slide 26


Homework

Students are encouraged to solve the following questions fromthe main textbook

1.2, 1.3, 1.7, 1.8, 1.9, 1.11 and 1.15

Documents

Chapter 1 Sampling Theorm2