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1. The Concept and Representation of Periodic

Sampling of a CT Signal

2. Analysis of Sampling in the Frequency Domain

3. The Sampling Theorem the Nyquist Rate

4. In the Time Domain: Interpolation

5. Undersampling and Aliasing

Signals and SystemsFall 2003

Lecture #13

21 October 2003

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We live in a continuous-time world: most of the signals we

encounter are CT signals, e.g.x(t). How do we convert them into DTsignalsx[n]?

SAMPLING

How do we perform sampling?

Sampling, taking snap shots ofx(t) every Tseconds.

T sampling period

x[n] x(nT), n = ..., -1, 0, 1, 2, ... regularly spaced samples

Applications and Examples

Digital Processing of Signals

Strobe

Images in Newspapers Sampling Oscilloscope

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Why/When Would a Set of Samples Be Adequate?

Observation:Lots of signals have the same samples

By sampling we throw out lots of information

all values ofx(t) between sampling points are lost.

Key Question for Sampling:

Under what conditions can we reconstruct the original CT signalx(t) from its samples?

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Impulse Sampling Multiplyingx(t) by the sampling function

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Illustration of sampling in the frequency-domain for a

band-limited (X(j)=0 for | |> M

) signal

No overlap between shifted spectra

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Reconstruction ofx(t) from sampled signals

If there is no overlap betweenshifted spectra, a LPF can

reproducex(t) fromxp(t)

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The Sampling Theorem

Supposex(t) is bandlimited, so that

Thenx(t) is uniquely determined by its samples {x(nT)} if

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Observations on Sampling

(1) In practice, we obviously

dont sample with impulses

or implement ideal lowpass

filters. One practical example:

The Zero-Order Hold

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Observations (Continued)

(2) Sampling is fundamentally a time-varyingoperation, since wemultiplyx(t) with a time-varying functionp(t). However,

is the identity system (which is TI) for bandlimitedx(t) satisfying

the sampling theorem (s > 2M).

(3) What ifs 2M? Something different: more later.

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Time-Domain Interpretation of Reconstruction of

Sampled Signals Band-Limited Interpolation

The lowpass filter interpolates the samples assuming x(t) containsno energy at frequencies c

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Interpolation Methods

Bandlimited Interpolation Zero-Order Hold

First-Order Hold Linear interpolation

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Undersampling and Aliasing

When s 2 M Undersampling

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Undersampling and Aliasing (continued)

Higher frequencies ofx(t) are folded back and take on the

aliases of lower frequencies

Note that at the sample times,xr(nT) =x(nT)

Xr(j) X(j)

Distortion because

ofaliasing

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Demo: Sampling and reconstruction of cosot

A Simple Example

Picture would be

Modified