chapter 1 Sampling Theorm.ppt - الصفحات الشخصية | الجامعة...

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Sampling Theorem

Spring 2009

© Ammar Abu-Hudrouss Islamic University Gaza

Slide 2Digital Signal Processing

Continuous Versus DigitalAnalogue 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)

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

Take a sample every sampling period Ts – uniform sampling


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 when the bandwidth of signal is very large)

It has a larger time delay compared to the analogue.

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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


Periodic Continuous Signal

21f

T

tAtx cos)(

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

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

The period is defined as

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Periodic Continuous Signal

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


Periodic Discrete Signal

)22cos()2cos()()(

fNfnfnNnxnx

nAnx cos)(

Nkf

kkfN

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

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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

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Periodic Discrete Signal

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

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

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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]


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Analogue to Digital Conversion

Sampler Quantizer Coderxa(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.


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 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

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But from previous discussion , for the analoge 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]



Which leads that -1/2< FTs <1/2 or - < Ts < OR -1/(2Ts) < F < 1/(2Ts) or - /Ts < < /Ts

Hence that highest possible analoge frequency is Fmax = Fs/2 = 1/(2Ts) and < Fs = /Ts

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ExampleConsider the two analog sinusoidal signalsX1(t) = cos 2(10)t and X2(t) = cos 2(50)tBoth are sampled with sampling rate Fs = 40, 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.



All sinusoids with frequency Fk = F0 + k Fs, k= 1,2,3,……… Leads to unique signal if sampled at Fs Hz.

proofxa(t) = cos (2 Fk 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 (2 F0/Fs n +)

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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 2fmax.

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


Sampling Theorem

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

X

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Sampling Theorem

In mathematical terms

Converting into Fourier transform

)()()( tstxnx

n

snTttxnx )()()(

n

ss

nT

XX )(1*)(

n

ss

nXT

X )(1)(


Sampling Theorem

By graphical representation in the frequency domain

X

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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.


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 = 20x(t) can be easily

recovered by LPF

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Sampling Theorem

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

by sharp LPF

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

recovered

Compare fs with 2B in each case


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:

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Anti-aliasing Filter

Apply a lowpass filter before sampling:

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

SamplerAnti-aliasing filterx(t)

y(n)x'(t)


Homework

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

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

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