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Oct 02, Alfred Hero, Univ. Michigan

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Digitization and Reconstruction of Continuous Time Signals

• Digitization/Reconstruction block diagram• Reconstruction methods• Reconstruction via interpolation• Examples

Prof Alfred HeroEECS206 F02

Lect 19

Oct 02, Alfred Hero, Univ. Michigan

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Digital Signal Processing System

t

X(t)

Signal Digitizer SignalReconstructionDigital processor

y(t)

X(t) y(t)X[n]

t

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Sources of ambiguity and error• Signal Digitization:

– Sampling rate too slow– Quantization too coarse

• Digital processing:– Roundoff errors/register overflow– Transmission/memory-storage errors– Buggy code!

• Signal Reconstruction:– Interpolation errors

Oct 02, Alfred Hero, Univ. Michigan

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Sampling Considerations • Consider time sampling/reconstruction without quantization:

• sampling period (secs/sample)

• sampling rate or frequency (samples/sec)

Ideal Sampler Reconstruction

x(t)y(t)x[n]

x(t)

x[n]

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Reconstruction of periodic signals from their time samples

• Time domain methods: estimate x(t) by simple ZOH or by applying interpolation to x[n]

• Frequency domain methods: estimate line spectrum of x(t) from line spectrum of x[n] and use estimate to re-synthesize x(t).

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Interpolation Methods of Reconstruction

• Square interpolation

• Triangular interpolation

Amplitude = 1

Amplitude = 1

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Interpolation Methods of Reconstruction

• Truncated sinc interpolation Amplitude = 1

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Example

• Consider cts time periodic signal

• Fundamental frequency: ?

• Highest freq component: ?

Oct 02, Alfred Hero, Univ. Michigan

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Example

• Consider cts time periodic signal

• Fundamental frequency:

• Highest freq component:

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Example (ctd)X(t)

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Sample x(t) at 1850 samples/sec

Note: max frequency sinusoid gets sampled at 1850/900=2.05 samples/cycle Oct 02, Alfred Hero,

Univ. Michigan12

Square wave interpolation(ZOH)

(Note sampling is time shifted wrt last slide)

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Square wave interpolation(ZOH)

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ZOH Reconstruction: 2 samples/cycle

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Triangular wave (linear) interpolation

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Triangular wave (linear) interpolation

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Linear interpolator reconstruction:2 samples/cycle

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Truncated Sinc Interpolation

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Oct 02, Alfred Hero, Univ. Michigan

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Truncated sinc reconstruction2 samples/cycle

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Can reduce error by increasing sampling rate to 7250 samples/sec

Note: max frequency sinusoid gets sampled at 7250/900=8.0 samples/cycle

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ZOH Reconstruction8 samples/cycle

Oct 02, Alfred Hero, Univ. Michigan

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Linear interpolator reconstruction8 samples/cycle

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Truncated Sinc Reconstruction8 samples/cycle

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Poor choice of sample frequency and/or interpolation function cause

errors• Sampling a signal below a certain rate (the Nyquist

rate) depending on the signal dynamics leads to fundamental ambiguity due to a phenomenon called “aliasing.”

• Over-sampling a signal above Nyquist rate never leads to aliasing but may be computationally expensive.

• Even if adequately sampled, use of poor interpolators can lead to distortion in reconstructed signal.

• There is a distortion tradeoff between low sampling rate and low complexity of interpolator