lecture 2: compressive sampling for analog time signals · 2007. 6. 12. · sensing by sampling •...
TRANSCRIPT
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Richard Baraniuk
Rice Universitydsp.rice.edu/cs
Lecture 2:CompressiveSampling forAnalog Time Signals
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Analog-to-Digital Conversion
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Sensing by Sampling
• Foundation of Analog-to-Digital conversion:Shannon/Nyquist sampling theorem– periodically sample at 2x signal bandwidth
• Increasingly, signal processing systems rely on A/D converter at front-end– radio frequency (RF) applications have hit a performance
brick wall
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Sensing by Sampling
• Foundation of Analog-to-Digital conversion:Shannon/Nyquist sampling theorem– periodically sample at 2x signal bandwidth
• Increasingly, signal processing systems rely on A/D converter at front-end– RF applications have hit a performance brick wall– “Moore’s Law” for A/D’s: doubling in performance
only every 6 years”
• Major issues:– limited bandwidth (# Hz)– limited dynamic range (# bits)– deluge of bits to process downstream
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“Analog-to-Information”Conversion
[Dennis Healy, DARPA]
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Signal Sparsity
• Shannon was a pessimist
– sample rate N times/sec is worst-case bound
• Sparsity: “information rate” K per second, K<<N
• Applications: Communications, radar, sonar, …
widebandsignalsamples
largeGabor (TF)coefficients
timefr
equen
cytime
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Local Fourier Sparsity (Spectrogram)
time
freq
uen
cy
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Signal Sparsity
widebandsignalsamples
largeGabor (TF)coefficients
Fourier matrix
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Compressive Sampling
• Compressive sampling“random measurements”
measurements sparsesignal
informationrate
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Compressive Sampling
• Universality
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Streaming Measurements
measurementsNyquist
rate
informationrate
streaming requires special
• Streaming applications: cannot fit entire signal into a processing buffer at one time
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A Simple Model for Analog Compressive Sampling
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Analog CS
analogsignal
digitalmeasurements
informationstatisticsA2I DSP
• Analog-to-information (A2I) converter takes analog input signal and creates discrete (digital) measurements
• Much of CS literature involves exclusively discrete signals
• First, define an appropriate signal acquisition model
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A Simple Analog CS Model
K-sparsevector
analogsignal
digitalmeasurements
informationstatisticsA2I DSP
• Operator takes discrete vector and generates analog signal from a(wideband) subspace
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A Simple Analog CS Model
K-sparsevector
analogsignal
digitalmeasurements
informationstatisticsA2I DSP
• Operator takes analog signal and generates discrete vector
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Analog CS
K-sparsevector
analogsignal
digitalmeasurements
informationstatisticsA2I DSP
is a CS matrix
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Architectures for A2I:
1. Random Sampling
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A2I via Random Sampling[Gilbert, Strauss, et al]
• Can apply “random” sampling concepts from Anna Gilbert’s lectures directly to A2I
• Average sampling rate < Nyquist rate
• Appropriate for narrowband signals (sinusoids),wideband signals (wavelets), histograms, …
• Highly efficient, one-pass decoding algorithms
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Sparsogram
• Spectrogram computed using random samples
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Example: Frequency Hopper
• Random sampling A2I at 13x sub-Nyquistaverage sampling rate
spectrogram sparsogram
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Architectures for A2I:
2. Random Filtering
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A2I via Random Filtering
• Analog LTI filter with “random impulse response”
• Quasi-Toeplitz measurement system
y(t)
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Comparison to Full Gaussian
Fourier-sparse signalsN = 128, K = 10
y(t)
B = length of filter hin terms of Nyquist rate samples
= horizontal width ofband of A2I conv
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Architectures for A2I:
3. Random Demodulation
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A2I via Random Demodulation
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A2I via Random Demodulation
• Theorem [Tropp et al 2007]
If the sampling rate satisfies
then locally Fourier K-sparse signals can be recovered exactly with probability
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Empirical Results
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Example: Frequency Hopper
• Random demodulator AIC at 8x sub-Nyquist
spectrogram sparsogram
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Summary
• Analog-to-information conversion: Analog CS
• Key concepts of discrete-time CS carry over
• Streaming signals require specially structured measurement systems
• Tension between what can be built in hardware versus what systems create a good CS matrix
• Three examples:– random sampling, random filtering, random demodulation
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Open Issues
• New hardware designs
• New transforms that sparsity natural and man-made signals
• Analysis and optimization under real-world non-idealities such as jitter, measurement noise, interference, etc.
• Reconstruction/processing algorithms for dealing with large N
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