SIGNALS AND SYSTEMS

EEE F243 / INSTR F243

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Lecture 16

Sampling

Todays Session

Fourier Analysis of Sampled Signals

Ideal Low Pass Filter

Signal Reconstruction

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Signal Reconstruction

Practical Difficulties in Sampling

Fourier Analysis of Sampled Signals

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Sampling Realization

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Spectrum of Sampled Signal

( ) ( ) ( ) ( ) ( )

===

nTS nTtnTfttftf

( ) ( )

= SS nFF 1

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( ) ( )=

=n

SS nFT

F 1

Sampling Theorem: Nyquist Rate ,

Nyquist Interval,

( ) BFS 2=

BT

2

1=

Example

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Interpolation

The Process of signal reconstruction or signal recovery of a continuous-

time signal from its samples is known as interpolation.

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Signal reconstruction is achieved through ideal low pass filter (LPF).

Ideal Low Pass Filter

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Physical Realization of

Interpolation

Simple Interpolation using a zero- order hold circuit:

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Improvement over Interpolation

First order hold circuit:

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( )

=B

TrectH

4

( ) ( ) ( ) =k

kBtctftf 2sin

Generalized Interpolation

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Signal values between samples as a weighted sum at the sample values.

Practical Difficulties

Causality of the ideal low pass filter.

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Filter gain must be zero beyond the first cycle.

Practical Difficulties

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Treachery of Aliasing

Solution:Anti Aliasing Filter: Band widthT

FS

2

1

2=

Practical Difficulties

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Features of Sampling

Sampling a signal in time domain introduces periodicity in the

frequency domain.

Sampling at a rate less than the Nyquist rate yield Aliasing.

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Sampling at a rate less than the Nyquist rate yield Aliasing.

Signal should be band - limited before it is send to sampler.

Band limiting is achieved by passing the signal through a low pass

filter.