2.2 The design
2.2.3 Related literature
2.2.3.1 Other RMPI systems
RMPI error improves as more samples are collected. With the Nyquist-rate receiver, there is no estimation occurring, and more samples do not improve the estimate.
The RMPI penalty compared to an oracle pays only a penalty of 3.11ρ−1 ' 15.9 dB. This is quite reasonable, since the oracle result requires knowing the frequency of the signals and having an ADC that samples atfswith very high ENOB. This comparison is only valid whenkis sufficiently small.
farther from DC, although this may have been using a non-calibrated model, in which case this is due to model mismatch and not any intrinsic effect of the pole. Their dominant non-ideality was the non-linearity in the mixer, which was more significant than clock jitter in the mixer. Our RMPI has been designed carefully to keep the mixer very linear, and we find that the LNA non-linearity is just as significant. The work of [KLW+06,KRL+06,LKD+07] is due to a Rice University team. Rice University has a contract with the same DARPA BAA as the Caltech team, and they are actively working to implement a type of compressed sensing system in hardware, though their device has very different bandwidth and recovery goals than the RMPI [RLN+08].
The first published work on a multi-channel design appears in a report by a team at Texas A & M University [YHS08]; this application is tailored for spectrum sensing in cognitive radio.
These authors also have a contract with the DARPA BAA, so it is not surprising that their design, the Rice University design, and the Caltech design are all related. Their methodology is called the
“Parallel Segmented Compressed Sensing” (PSCS) structure, and at a high level, it is identical to the RMPI. They cite worries over the “10th order” integrating filter in [LKD+07] (though this may be due to a misinterpretation), and suggest that the single-channel design’s samples are redundant, which can be fixed with the multi-channel design. Regarding the number of channels, they suggest that an upper limit is necessary, due to practical implementation issues. Notably, they introduce a stagger to the channels, suspecting that samples would have higher fidelity in the interior of a sampling period and thus the edges of the channels should not align. Using orthogonal matching pursuit (OMP), they demonstrate basic feasibility with Matlab simulations using on-grid tones and a signed Bernoulli matrix.
The same team recently published [CYH+11], detailing a 90 nm CMOS chip they plan to fabri- cate. The layout was fully simulated in Cadence Spectre. The chip consumes 120 mW per channel, which includes specs for ADCs, although they assume that a LNA is off-chip and that clock distri- bution, which typically accounts for 30% or so of power, is also off-chip. Their signal model assumes on-grid tones, of which 4% are occupied. They predict a 44 dB SNR. The effective bandwidth is 1.5 GHz, using 8 channels, each sampling at 110 MHz. This is very similar to the Caltech design, though less ambitious in terms of the undersampling ratio (they undersample by×.587, our design undersamples by×.08). Similar to what we will find in this chapter, they conclude that mixing jitter is a limiting non-ideality, and assume their jitter will have 0.5 ps rms (same as our design). Non-DC poles in the integrator are not discussed, but they mention using a dual-integrator design which samples on one branch while resetting the other branch, then swapping, so that railing/clipping of the integrator is not an issue. They briefly discuss calibration. Like in the earlier paper, signal recovery is done straightforwardly via OMP and a Fourier dictionary, since off-grid tones are not considered. A thorough analysis of noise at different blocks is presented. Due to the similarities with the Caltech RMPI, it will be interesting to see the results once their chip is fabricated, since
both teams face similar problems.
We note recent work by [TV11] that considers parallel branches of mixers and integrators, which is similar to already proposed designs except for an extra step at the sampling which introduces extra samples; the extra samples are correlated to earlier samples, but they claim it gives processing gains. Another new A2I device has been mentioned in the dissertation of Michael Trakimas at Tufts, but we are unaware of any papers on the subject and so do not know the details. The recent work of [MPAL11] also proposes a RMPI design, and they have constructed a prototype in 500 nm CMOS with 200 MHz of bandwidth, and appear to be planning construction of a 130 nm device. They demonstrate on-grid recovery with Matlab simulated data.
Beyond Nyquist. The paper [TLD+10] called “Beyond Nyquist” is probably the most influential A2I paper, so we give it an extended discussion. Though authors of the paper are associated with both the Rice and Caltech hardware projects, the paper itself is mainly theoretical. Their setup is identical to the model considered in this chapter, except they only consider a single channel, and with different assumptions on non-idealities and practical considerations. Using a particular signal model, they prove the first rigorous theorems, which state, in effect, that the RMPI mixing operations are not much worse than using a standard compressed sensing matrix like a random signed Bernoulli matrix. This signal model is not overly realistic, but it is important because the result suggests that the operations of the system adhere to the compressed sensing principle, and follow a phase-transition similar to the Donoho-Tanner phase transition for Gaussian matrices.
Specifically, assume x is a periodic signal with K on-grid tones. For example, amplitude- modulation of on-grid signals falls into this class, while frequency-modulation does not. Their model uses a single channel which undersamples the Nyquist rate by the amountM/N. Empirically, they suggest that recovery happens when
M = 1.7Klog(N/K+ 1).
This is complemented by two theoretical results, which we state here. Fix a timeT and let N = T /∆T. WriteX for the unitary DFT of the discrete signalx.
Theorem 2.2.4 (Noiseless recovery of a random on-grid random signals, [TLD+10]). Suppose x hasK on-grid frequencies chosen uniformly at random, soX has up to2K non-zeros, including the negative frequencies. Let the non-zero entries of X have arbitrary amplitudes, and random complex phases. Pick a PRBS sequence at random. Then a single-channel ideal RMPI withM measurements (and assumeM divides N)
M ≥C Klog(N) + log3(N)
will exactly recover the signal from the noiseless measurements
b= Φx, Φ∈RM×N
by solving basis pursuit (1.1.5), except with probability O(N−1). The constant C is independent of N andK.
This theorem might be interpreted as saying that most PRBS sequences will work to reconstruct most inputs, as long asM is large enough. However, since the PRBS sequence is typically fixed and not changed, we might wonder if there is a big risk of choosing a “bad” PRBS. It turns out as long as the PRBS is initially selected at random, then most likely, recovery will still happen. Furthermore, this is robust to additive noise, and adapts to compressible signals. To be specific:
Theorem 2.2.5 (Noisy recovery of arbitrary signals, [TLD+10]). Fix any PRBS sequence. Let M ≥CKlog6(N)
and assumeM dividesN. Take measurements
b= Φx+η
and estimatexˆbe the solution to the basis pursuit denoising problem(1.1.6)with parameterε=kηk2. Then except with probabilityO(N−1),
kxˆ−xk2≤Cmax
ε, 1
√KkX−XKk1
whereXK is any K-sparse vector inCN, i.e., the best approximation of X.
Another improvement of this theorem is that it doesn’t require x to be exactly K sparse in frequency; good reconstruction occurs as long as its DFT is compressible. However, a major issue still happens for off-grid tones. These tones are not periodic, so instead of the DFT, it is appropriate to analyze their continuous Fourier transform. Because only a finite time sampleT is observed, this has the effect of a boxcar filter in the time domain, which means the Fourier transform of the time- limited function will show a sinc decay away from the active tone. Since the sinc function decays like 1/f in frequency, it is not well-approximated by a sparse DFT.
This is a common problem in signal processing, and the usual solution is windowing, as will be discussed in detail in §2.7.2.6. The “Beyond Nyquist” paper proposes implementing a window in hardware via a time-varying amplifier; furthermore, because two shifted windows are needed, the number of channels must double. Whether such a scheme is practical remains to be seen.
Though calibration is not discussed in detail, the paper suggests that non-ideal integrators should be acceptable as long as they can be characterized well, which agrees with the findings presented in this chapter. Some Matlab simulations are presented using IRLS.