sources of ambiguity and error sampling considerationsrate) depending on the signal dynamics leads...
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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
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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
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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: ?
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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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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
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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