Richard Baraniuk Rice University dsp.rice.edu/cs Lecture 2: Compressive Sampling for Analog 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, … wideband signal samples large Gabor (TF) coefficients time frequency time

Local Fourier Sparsity (Spectrogram) time frequency

Signal Sparsity wideband signal samples large Gabor (TF) coefficients Fourier matrix

Compressive Sampling Compressive sampling “random measurements” measurements sparse signal information rate

Compressive Sampling Universality

Streaming Measurements measurements Nyquist rate information rate 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 analog signal digital measurements information statistics A2IDSP 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-sparse vector analog signal digital measurements information statistics A2IDSP Operator takes discrete vector and generates analog signal from a (wideband) subspace

A Simple Analog CS Model K-sparse vector analog signal digital measurements information statistics A2IDSP Operator takes analog signal and generates discrete vector

Analog CS K-sparse vector analog signal digital measurements information statistics A2IDSP 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-Nyquist average 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)y(t)

Comparison to Full Gaussian Fourier-sparse signals N = 128, K = 10 y(t)y(t) B = length of filter h in terms of Nyquist rate samples = horizontal width of band of A2I conv

Architectures for A2I: 3. 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 dsp.rice.edu/cs