Beyond Nyquist ❦ Joel A. Tropp Applied and Computational Mathematics California Institute of Technology jtropp@acm.caltech.edu With M. Duarte, J. Laska, R. Baraniuk (Rice DSP), D. Needell (UC-Davis), and J. Romberg (Georgia Tech) Research supported in part by NSF, DARPA, and ONR 1
The Sampling Theorem Theorem 1. Suppose f is a continuous-time signal whose highest frequency is at most W/ 2 Hz. Then � n � � f ( t ) = n ∈ Z f sinc( Wt − n ) . W where sinc( x ) = sin( πx ) /πx . ❧ The Nyquist rate W is twice the highest frequency ❧ The cardinal series represents a bandlimited signal by uniform samples taken at the Nyquist rate Reference: [Oppenheim et al. 2000] Beyond Nyquist (UC-Bolder, Sept. 2008) 2
Analog-to-Digital Converters (ADCs) ❧ An ADC consists of a low-pass filter , a sampler and a quantizer ❧ For sampling rate R , low-pass filter has cutoff R/ 2 to prevent aliasing ❧ Ideal sampler produces a sequence of amplitude values: f �− → { f ( nT ) : n ∈ Z } where the sampling interval T = R − 1 ❧ The quantizer maps the real sample values to a discrete set of levels ❧ Commonly, analog signals are acquired by sampling at the Nyquist rate and processing information with digital technology Beyond Nyquist (UC-Bolder, Sept. 2008) 3
ADCs: State of the Art ❧ The best current technology (2005) gives ❧ 18 effective bits at 2.5 MS/s (MegaSamples/sec) ❧ 13 effective bits at 100 MS/s ❧ Performance degradation about 1 effective bit per frequency octave ❧ The standard performance metric is P = 2 # effective bits · sampling frequency ❧ At all sampling rates, one effective bit improvement every 6 years References: [Walden 1999, 2006] Beyond Nyquist (UC-Bolder, Sept. 2008) 4
24 P=9.13x10 11 Data from 1978 to 1999 22 Data from 2000 to 2005 SNRbits (effective number of bits) 20 P=4.1x10 11 Analog Devices: 18 24 bit 2.5MS/s 16 16 bit 100 MS/s 14 12 10 8 6 4 2005 2 1999 0 0 2 4 6 8 10 10 10 10 10 10 10 F sample (HZ) Beyond Nyquist (UC-Bolder, Sept. 2008) 5
Train Wreck ❧ Modern applications already exceed ADC capabilities ❧ The Moore’s Law for ADCs is too shallow to help Conclusion: We need fundamentally new approaches Beyond Nyquist (UC-Bolder, Sept. 2008) 6
Idea: Exploit Structure ❧ Absent additional structure, Nyquist-rate sampling is optimal ❧ Need to identify and exploit other properties of signals ❧ Signals of interest do not contain much information relative to their bandwidth ❧ In communications applications, signals often contain few significant frequencies Beyond Nyquist (UC-Bolder, Sept. 2008) 7
Example: An FM Signal 0.01 Frequency (MHz) 0.02 0.04 0.05 0.06 0.07 40.08 80.16 120.23 160.31 200.39 Time ( µ s) Data provided by L3 Communications Beyond Nyquist (UC-Bolder, Sept. 2008) 8
Sparse, Bandlimited Signals A normalized model for signals sparse in time–frequency: ❧ Let W exceed the signal bandwidth (in Hz) ❧ Let Ω ⊂ {− W/ 2 + 1 , . . . , − 1 , 0 , 1 , . . . , W/ 2 } be integer frequencies ❧ For each one-second time interval, signal has the form � a ( ω ) e 2 π i ωt f ( t ) = for t ∈ [0 , 1) ω ∈ Ω ❧ The set Ω of frequencies can change every second ❧ In each time interval, number of frequencies | Ω | = K ≪ W Beyond Nyquist (UC-Bolder, Sept. 2008) 9
Information and Signal Acquisition ❧ Signals in our model contain little information ❧ In each time interval, have K frequencies and K coefficients ❧ Total: About K log W bits of information ❧ Idea: We should be able to acquire signals with about K log W nonadaptive measurements ❧ Challenge: Achieve goal with current ADC hardware ❧ Approach: Use randomness! Beyond Nyquist (UC-Bolder, Sept. 2008) 10
Random Demodulator: Intuition ❧ With clustered frequencies, demodulate to baseband and low-pass filter demodulation + low-pass filtering 0 0 ❧ Don’t know locations, so demodulate randomly and low-pass filter Beyond Nyquist (UC-Bolder, Sept. 2008) 11
input signal X( ! ) input signal x(t) pseudorandom modulating sequence p c (t) pseudorandom modulating sequence P c ( ! ) modulated signal X( ! ) and integrator (lowpass filter) modulated input signal x(t) Beyond Nyquist (UC-Bolder, Sept. 2008) 12
Exploded View of Passband Beyond Nyquist (UC-Bolder, Sept. 2008) 13
Random Demodulator: System Model Pseudorandom Seed Number Generator ❧ p c ( t ) alternates randomly between levels ± 1 at Nyquist rate W ❧ Sampler runs at rate R ≪ W Beyond Nyquist (UC-Bolder, Sept. 2008) 14
Matrix Formulation I ❧ The continuous signal has the form � ω ∈ Ω a ( ω ) e 2 π i ωt f ( t ) = for t ∈ [0 , 1) ❧ Time-averaging for 1 /W seconds at t n = n/W yields � e 2 π i ω/W − 1 � � t n +1 /W � e 2 π i ωt n f ( t ) d t = ω ∈ Ω a ( ω ) 2 π i ω t n � ω ∈ Ω s ( ω ) e 2 π i ωt n = ❧ Can express time-averaged signal as a vector x = F s ∈ C W ❧ s is sparse and supported on Ω ❧ F is essentially a DFT matrix ❧ x contains the same (discrete) frequencies as f Beyond Nyquist (UC-Bolder, Sept. 2008) 15
Matrix Formulation II ❧ The (ideal) action of the multiplier is given by ± 1 ± 1 ± 1 D = ... ± 1 ❧ The (ideal) action of the accumulate-and-dump sampler is given by 1 1 . . . 1 1 1 . . . 1 H = . ... ... 1 1 . . . 1 R × W Beyond Nyquist (UC-Bolder, Sept. 2008) 16
Reconstruction from Samples ❧ The matrix Φ summarizes the action of the random demodulator Φ = HDF : C W − → C R ❧ Maps a (sparse) amplitude vector s to a vector of samples y ❧ Given samples y = Φ s , signal reconstruction can be formulated as s = arg min � c � 0 subject to Φ c = y � ❧ The ℓ 0 function counts nonzero entries of a vector Beyond Nyquist (UC-Bolder, Sept. 2008) 17
Signal Reconstruction Algorithms Approach 1: Convex Relaxation ❧ Can often find sparsest amplitude vector by solving s = arg min � c � 1 subject to Φ c = y (P1) � Approach 2: Greedy Pursuit ❧ Identify a small set of significant frequencies and iteratively refine ❧ Examples: OMP and CoSaMP References: [Cand` es et al. 2006, Donoho 2006, Tropp–Gilbert 2007, Tropp–Needell 2008] Beyond Nyquist (UC-Bolder, Sept. 2008) 18
Shifting the Burden ❧ These algorithms are much more computationally intensive than linear reconstruction via cardinal series ❧ Move the work from the analog front end to the digital back end Moore’s Law for ICs saves us from Moore’s Law for ADCs! Beyond Nyquist (UC-Bolder, Sept. 2008) 19
Theoretical Analysis Theorem 2. [T 2007] Suppose the sampling rate satisfies R ≥ C · K · log 6 W Then the matrix Φ has the restricted isometry property (1 − c) � x � 2 2 ≤ � Φ x � 2 2 ≤ (1 + c) � x � 2 � x � 0 ≤ 2 K when 2 except with probability W − 1 . ❧ Abstract property supports efficient sampling and reconstruction ❧ Intuition: Sampling operator preserves geometry of sparse vectors Beyond Nyquist (UC-Bolder, Sept. 2008) 20
Recovery via Convex Optimization Theorem 3. [Cand` es–Romberg–Tao 2006] Suppose that ❧ the sampling matrix Φ has the RIP, ❧ the sample vector y = Φ s + e , and ❧ the error � e � 2 ≤ η . Then the solution � s to the program min � c � 1 � y − Φ c � 2 ≤ η subject to satisfies � 1 � √ � s − � s � 2 ≤ C � s − s K � 1 + η . K Beyond Nyquist (UC-Bolder, Sept. 2008) 21
Recovery via Greedy Pursuit Theorem 4. [Needell–T 2008] Suppose that ❧ the sampling matrix Φ has the RIP, ❧ the sample vector y = Φ s + e , ❧ η is a precision parameter, ❧ L bounds the cost of a matrix–vector multiply with Φ or Φ ∗ . Then CoSaMP produces a 2 K -sparse approximation � s such that � � 1 √ � s − � s � 2 ≤ C max η, � s − s K � 1 + � e � 2 K with execution time O( L · log( � s � 2 /η )) . Beyond Nyquist (UC-Bolder, Sept. 2008) 22
Simulations Goal: Estimate sampling rate R to achieve success probability 99% For each of 500 trials, ❧ Draw a random demodulator with dimensions R × W ❧ Choose a random set of K frequencies ❧ Set their amplitudes equal to one ❧ Take measurements of the signal ❧ Recover with ℓ 1 minimization (via IRLS) Define success at rate R when 99% of trials result in � s − � s � < ε mach Beyond Nyquist (UC-Bolder, Sept. 2008) 23
60 55 Sampling Rate Hz (R) 50 45 40 35 30 25 2 3 10 10 Signal Bandwidth Hz (W) K = 5 , regression line R = 1 . 69 K log( W/K + 1) + 4 . 51 Beyond Nyquist (UC-Bolder, Sept. 2008) 24
400 350 Sampling Rate Hz (R) 300 250 200 150 100 50 0 20 40 60 80 100 120 140 Number of Nonzero Components (K) W = 512 , regression line R = 1 . 71 K log( W/K + 1) + 1 . 00 Beyond Nyquist (UC-Bolder, Sept. 2008) 25
1 1 Sampling Efficiency (K/R) 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.2 0.4 0.6 0.8 1 Compression Factor (R/W) Beyond Nyquist (UC-Bolder, Sept. 2008) 26
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