Lecture 16:Sampling of Continuous-Time Signals

Instructor: Dr. Ghazi Al SukkarDept. of Electrical EngineeringThe University of JordanEmail: ghazi.alsukkar@ju.edu.jo

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Outline

Continuous-time processing of Discrete-time signals: Sampling rate reduction by an integer factor (downsampling) Increasing the sampling rate by an integer factor (upsampling) Changing the sampling rate by Non-integer factor

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Continuous-time processing of Discrete-time signals: It is not practically used, but provides a useful interpretation of

certain of certain discrete-time systems that have no simple interpretation in the discrete domain.

and

Where and

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C/D

𝑇

𝑦 𝑐(𝑡) 𝑦 [𝑛 ]D/C

𝑇𝑥𝑐(𝑡)𝑥 [𝑛 ] 𝐻𝑐 ( 𝑗Ω )

continuous-time system

𝐻 (𝑒 𝑗 𝜔 )

Cont..

And

Also

Then:

Or equivalently, the overall impulse response will equal to a given if:

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Application: Non-integer delay One of the most application of the continuous-time processing

of discrete-time signals is the non-integer delay. Consider a discrete-time system with frequency response:

When is integer then When is non-integer then has no formal meaning, because we

cannot shift the sequence by non-integer amount. However, using the continuous-time processing of discrete-time

signals, then

Hence,

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𝑦 [𝑛 ]𝑥 [𝑛 ] 𝐻 (𝑒 𝑗 𝜔 )

Cont..

Where

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x[n]

-3 -2 2 3 4-1 10n

y[n],

-3 -2 2 3 4-1 10n

Sampling rate reduction by an integer factor (downsampling) Reducing the sampling period by a factor of means that

i.e., is identical to the sequence that would be obtained from by sampling period .

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Sampling periodSampling period↓𝑀

compressor

0 5 10-5

𝑥 [𝑛 ]

-10

0 5-5

𝑥 [3𝑛 ]

-10

Down sampling

Cont.. If is band-limitied i.e., then is an exact representation of if

Nyquist theorem is satisfied

Which means, that the sampling rate can be reduced to without causing any aliasing if the original sampling rate is at least times the Nyquist rate or the bandwidth of the sequence is first reduced by a factor of . Frequency Domain representation:Recall that the DTFT of is:

The DTFT of is:

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Cont..Let Where and

But since

Then:

Aliasing can be avoided by insuring that is band-limited:

And

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Cont.. If this condition does not hold, aliasing occurs. In some cases, downsampling can be done without aliasing if

we are willing to reduce the bandwidth of the signal before sampling

i.e., if is filtered by an ideal low-pass filter with cutoff frequency then the output can be downsampled without aliasing (decimation).

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Sampling periodSampling period↓𝑀Low-pass filter

Gaincutoff freq.

Sampling period

Decimator

Cont..

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No aliasing

aliasing

Increasing the sampling rate by an integer factor (upsampling) Given increase its sampling rate by a factor If Then Where

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Sampling periodSampling period↑𝐿 Low-pass filter

Gaincutoff freq.

Sampling period

Expander

interpolator

0 5 10-5

𝑥 [𝑛 ]

-10

0 5 10-5

𝑥 [𝑛 /2]

-10

Up Sampling

Cont..

Then we can write as:

The frequency domain representation:

Hence The magnitude of should be instead of for , hence the Gain of

the low-pass filter should be .

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The frequency-scaled images of except at integer multiples of should be removed, hence the low-pass filter cutoff frequency should be .

The whole system (expander + low-pass filter) is called interpolator since it fills in the missing samples.

The process is called interpolation.

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Cont..

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Ω𝑁=𝜋 /𝑇

Changing the sampling rate by Non-integer factor: By combining the decimation and interpolation, it is possible to

change the sampling rate by a noninteger factor.

The interpolator decreases the sampling period from to , followed by a decimator that increases the sampling period by , producing an output that has an effective sampling period of .

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↑𝐿 Low-pass filter Gain

cutoff freq.

Expander↓𝑀Low-pass filter

Gaincutoff freq.

Decimatorinterpolator

↑𝐿 Low-pass filter Gain

cutoff freq.

Expander↓𝑀

Compressor

Cont.. The signal is sampled at the Nyquist rate. L=2, M=3, The net sampling period is Notice the aliasing that wouldOccur if the cut off frequency taken As

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