Richard Baraniuk

Rice Universitydsp.rice.edu/cs

Lecture 2:CompressiveSampling forAnalog Time Signals

Analog-to-Digital Conversion

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

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

“Analog-to-Information” Conversion

[Dennis Healy, DARPA]

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

equ

ency

time

Local Fourier Sparsity (Spectrogram)

time

frequ

ency

Signal Sparsity

widebandsignalsamples

largeGabor (TF)coefficients

Fourier matrix

Compressive Sampling

• Compressive sampling“random measurements”

measurements sparsesignal

informationrate

Compressive Sampling

• Universality

Streaming Measurements

measurementsNyquist

rate

informationrate

streaming requires special

• Streaming applications: cannot fit entire signal into a processing buffer at one time

A Simple Model for Analog Compressive Sampling

Analog CS

analogsignal

digitalmeasurements

informationstatistics

A2I 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

A Simple Analog CS Model

K-sparsevector

analogsignal

digitalmeasurements

informationstatistics

A2I DSP

• Operator takes discrete vector and generates analog signal from a(wideband) subspace

A Simple Analog CS Model

K-sparsevector

analogsignal

digitalmeasurements

informationstatistics

A2I DSP

• Operator takes analog signal and generates discrete vector

Analog CS

K-sparsevector

analogsignal

digitalmeasurements

informationstatistics

A2I DSP

is a CS matrix

Architectures for A2I:

1. Random Sampling

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

Sparsogram

• Spectrogram computed using random samples

Example: Frequency Hopper

• Random sampling A2I at 13x sub-Nyquistaverage sampling rate

spectrogram sparsogram

Architectures for A2I:

2. Random Filtering

A2I via Random Filtering

• Analog LTI filter with “random impulse response”

• Quasi-Toeplitz measurement system

y(t)

Comparison to Full Gaussian

Fourier-sparse signalsN = 128, K = 10

y(t)

B = length of filter hin terms of

Nyquist rate samples = horizontal width of

band of A2I conv

Architectures for A2I:

3. Random Demodulation

A2I via Random Demodulation

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

Empirical Results

Example: Frequency Hopper

• Random demodulator AIC at 8x sub-Nyquist

spectrogram sparsogram

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

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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