Communication Systems, 5e

Chapter 6: Sampling and pulse modulation

A. Bruce CarlsonPaul B. Crilly

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Pop Quiz1) To accurately reconstruct a baseband signal from its

samples, we must sample at a frequency at least _____ times higher than the highest frequency in the signal.

a) two b) three c) four

2) It is possible to reconstruct a narrowband signal by sampling at less than the Nyquist rate.

a) TRUE b) FALSE

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Pop Quiz1) To accurately reconstruct a baseband signal from its

samples, we must sample at a frequency at least _____ times higher than the highest frequency in the signal.

a) two b) three c) four

2) It is possible to reconstruct a narrowband by sampling at less than the Nyquist rate.

a) TRUE b) FALSE

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Chapter 6: Sampling and pulse modulation

Sampling theory and practice

Pulse-amplitude modulation

Pulse-time modulation

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Sampling theory and practice

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Practical sampling and reconstruction

Real samplers have finite duration pulses (e.g.

chopper sampling)

Practical reconstruction filters are not ideal

Sampled signals are time limited ⇒ not bandlimited

⇒ cannot avoid some aliasing

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Non-ideal sampling

As seen earlier, no loss of info with non-impulse

sampling

Non-ideal LPF reconstruction can be overcome by

prefiltering the original message ⇒ equalization

We can minimize aliasing by band limiting the input

signal ⇒ anti-alias LPF

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Practical reconstruction filter

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Reconstruction using a zero order hold (ZOH) interpolation

y(t)=∑k

x (k T S)∏ [ t−k T S−T S /2

T S ]⇒ reconstruction filter

⇒∣H ZOH( f )∣ = ∣T S sinc( f T S)∣

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Reconstruction using a first order hold (FOH) interpolation

y(t)=∑k

x (k T S)Λ[ t−k T S−T S /2

T S ]⇒ reconstruction filter

⇒∣H FOH( f )∣=∣T S √1+(2π f T S)2sinc2

( f T S)∣

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Signal reconstruction from sampled signal (a) ZOH (b) FOH

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Signal Reconstruction

H FOH f and H ZOH f are low pass filter functions,but they are not ideal LPFs.The difference in frequency response between an idealLPF and the ZOH or FOH introduces aperture errors inthe reconstruction process.These aperture errors can be compensated by prefilteringthe data to emphasize the higher frequencies.Prefiltering⇒equalization

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RC anti-alias filter

1. Time limited signals have large bandwidth⇒ to reduce errors due to aliased componentsspilling over into the message bandpass weincorporate an anti-aliasing filter.

2. Anti-aliasing filter often implemented via

an RC LPF with B=1

2 RC3. Because RC LPFs do not have sharp cutoff ⇒ BW

⇒oversample the message signal .

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Message spectrum (a) output of RC filter, (b) after sampling

(a) Original message spectrum, (b) spectrum after sampling

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Assuming a first order LPF is usedthe maximum aliasing error is

Error%= 1/0.707

1 f a /B2

with B=1

2 RC, f a= f S−B

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Oversampling

If a signal has been over-sampled with an acceptable amount of aliasing, we then

Feed it to our DSP for digital filtering to remove the components above W.

Down-sample the signal to some desired rate.

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Upsampling

Some cases we can only sample a signal at the Nyquist rate;

but need more samples⇒ upsample the data.

Upsampling ⇒ insert correct samples between the original

set ⇒ decrease effective sample interval ⇒ T'S = T

S / M

Time domain: interpolation

Frequency domain: zero padding

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Upsampling via linear interpolation

, t k

sT

( )x t

( )x k

k 1k + 2k + 3k +, 't k

'sT

( )x t

'( ')x k

'k ' 2k + ' 4k + ' 6k +

upsampling

⇒

(a) (b)

(a) Original signal sampled, (b) signal upsampled by factor of 2

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Why upsampling?

Upsampling ⇒ greater time resolution

Adaptive filtering methods may require more samples

than obtained by the Nyquist rate

Other DSP algorithms that depend on time resolution may require more than the minimum number of samples

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Upsampling does not add information

Upsampling

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In Class Exercise

Problem 6.1-11