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Above the Nyquist Rate, Modulo Folding Does Not Hurt

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Above the Nyquist Rate, Modulo Folding Does Not Hurt 1 Above the Nyquist Rate, Modulo Folding Does Not Hurt Elad Romanov and Or Ordentlich Abstract Q Q m −(R−1) —We consider the problem of recovering a continuous- choice will be xn . After all, xn xn < 2 2 , time bandlimited signal from the discrete-time signal obtained {res } | −m | −R · whereas xn xn can be as large as 2 (1 2 ). However, from sampling it every Ts seconds and reducing the result modulo somewhat| surprisingly,− | this letter shows that− the opposite is ∆, for some ∆ > 0. For ∆ = ∞ the celebrated Shannon- Nyquist sampling theorem guarantees that perfect recovery is true, provided that Ts < 1/2W . In particular, we improve possible provided that the sampling rate 1/Ts exceeds the so- upon the recent results of Bhandari et al. [1] and show called Nyquist rate. Recent work by Bhandari et al. has shown that whenever the sampling frequency exceeds the Shannon- ∆ 0 that for any > perfect reconstruction is still possible if the Nyquist frequency, for any R N, the signal xn can be sampling rate exceeds the Nyquist rate by a factor of πe. In perfectly reconstructed from x∈res . { } this letter we improve upon this result and show that for finite { n } energy signals, perfect recovery is possible for any ∆ > 0 and To be more concrete, the setup we consider is that of any sampling rate above the Nyquist rate. Thus, modulo folding sampling a modulo-reduced signal. For a given ∆ > 0, and does not degrade the signal, provided that the sampling rate a real number x R, we define x∗ = [x] mod ∆ as the exceeds the Nyquist rate. This claim is proved by establishing a ∈ ∆ ∆ ∗ Z unique number in [ 2 , 2 ) such that x x ∆ . For connection between the recovery problem of a discrete-time signal a complex number x− = a + ib C, the− modulo∈ operation from its modulo reduced version and the problem of predicting ∈ the next sample of a discrete-time signal from its past, and corresponds to reducing both the real and the imaginary parts ∗ ∗ ∗ ∗ leveraging the fact that for a bandlimited signal the prediction modulo ∆, i.e., x = a + ib . The signal x (t) is obtained error can be made arbitrarily small. by modulo reducing x(t) at all times, and the discrete-time ∗ ∗ ∗ signal xn = x (nTs) is obtained by sampling x (t) every { } ∗ ∗ Z Ts seconds. Note that xn = [xn] for all n , where I. INTRODUCTION x = x(nT ) . The question we address in this∈ letter is { n s } The Shannon-Nyquist sampling theorem guarantees that any under what conditions on Ts and ∆ is it possible to exactly signal x(t) whose Fourier transform is supported on [ W, W ] recover x(t) from x∗ ? { n} can be perfectly reconstructed from the discrete-time− signal Our main result is that for all finite energy signals whose x = x(nT ) as long as T < 1/2W . However, in order to Fourier transform is supported on [ W, W ], and whose am- { n s } s − be digitally stored/processed, the signal xn must be further plitude vanishes for t large enough, perfect reconstruction is { } | | 1 digitized, which is achieved by quantizing it to a set of finite possible for any ∆ > 0 and any Ts < 2W . See Section II-D cardinality. for the formal statement. In practice, quantization of the process xn is invariably done via a scalar uniform quantizer. For a signal{ } with bounded A. Related Work m m dynamic range, say xn ( 2 , 2 ) for all n Z, the In [1], it was shown that under essentially the same assump- operation of an R-bit scalar∈ uniform− quantizer can be∈ roughly tions as above on the class of signals, perfect reconstruction 1 1 described as that of writing the binary expansion of each is possible for any ∆ > 0 and any Ts < πe 2W . Thus, our arXiv:1903.12289v1 [cs.IT] 28 Mar 2019 m ∞ m−i sample as xn = an,02 + i=1 an,i2 , and then dis- results improve the sampling rate obtained in [1] by a factor of − 1 carding all but the R most significant bits (MSBs), i.e., xn is πe samples per seconds. Clearly, the sampling rate Ts < P 2W represented by an,0,an,1,...,an,R−1. Clearly, after this form found here is the best possible, since even without modulo of quantization, as well as any other finite bit-rate form of reduction (alternatively, ∆= ) this condition is required in quantization, x(t) can no longer be perfectly reconstructed. order to guarantee perfect recovery∞ within the considered class Q Let xn be the signal obtained by quantizing xn using a of signals. { } res Q { } R-bit uniform quantizer, and xn = xn xn be the residual The results of Bhandari et al. [1] had already sparked a lot { res − } signal. Note that the signal xn corresponds to discarding of follow up work, see e.g. [2]–[8]. A closely related recent the R MSBs of each sample{ and} keeping all the remaining line of work is that of modulo ADCs [9]–[12], which are finite least significant bits (LSBs). If one has to choose between bit-rate ADCs that reduce their input signal modulo ∆ prior Q res reconstructing xn from either xn or xn , the intuitive to quantization. The main difference between the so-called { } { } { } unlimited sampling framework of [1] and the modulo ADC framework is that the former does not consider the effect Elad Romanov and Or Ordentlich are with the School of Computer Science and Engineering, Hebrew University of Jerusalem, Israel (emails: of quantization noise added to the modulo reduced signal, {elad.romanov,or.ordentlich}@mail.huji.ac.il). whereas the latter explicitly studies the tradeoff between This work was supported, in part, by ISF under Grant 1791/17, and by the quantization rate and distortion after modulo reduction. GENESIS consortium via the Israel Ministry of Economy and Industry. ER acknowledges support from ISF grant 1523/16. The setup considered in this letter falls within the framework of unlimited sampling. Namely, we are assuming x∗ is { n} 2 Sampler Decoder available at perfect precision, and we are only interested in recovered xn mod∆ Σ mod∆ Σ x characterizing the optimal tradeoff between Ts and ∆ for n − which xn (and consequently also x(t)) can be perfectly reconstructed{ } despite the modulo folding. Our results indicate Filter that modulo reduction per se, incurs no loss on our ability to reconstruct x(t), provided that the Shannon-Nyquist condition is satisfied. Therefore, the main question of interest is what can be gained from modulo folding in terms of quantization Fig. 1. Schematic architecture for the proposed recovery algorithm. rate. The results in [9] show that for stationary Gaussian processes, oversampled modulo ADCs achieve rate-distortion 1 tradeoff which is close to the information theoretic limits. [ γ,γ], where γ = W Ts < . To see this, note that − · 2 Similar results were also obtained in [9] for spatially correlated ∞ processes. e = x h x = x ⋆c , n n − i n−i n n Despite the differences between the unlimited sampling and i=1 X modulo ADCs frameworks, the insights gained from develop- where cn = δn hn is a monic filter, i.e., a causal filter with ing recovery algorithms for oversampled modulo ADCs [9] − first tap equal to 1. Since the coefficients h1,h2,... are not turn out to also be suitable for recovery under the unlimited constrained, we are free to take cn as any monic filter. In sampling framework. In particular, the recovery algorithm { } particular, for any ǫ> 0, we may choose cn to satisfy outlined below is inspired by that of [9, Section 3]. { } ∞ 2 2 −i2πfk ǫ f γ C(f) = cke = 2γ | |≤ . | | ( 1 1−2γ f (γ, 1 ] B. Recovery Through Prediction k=1 ǫ | | ∈ 2 X The existence of a monic filter with this frequency response Our main result may look somewhat absurd at first, as it is guaranteed by Paley-Wiener’s spectral factorization theo- seems that as we drive ∆ to 0, we should end up with no rem [13], and the fact that 1/2 log C(f) 2df = 0. Now, information left. However, the result becomes very intuitive −1/2 | | since x has no spectral energy in f >γ, we have that once a connection is established between the problem of { n} R | | modulo unwrapping of a discrete-time signal and the problem e 2 = x ⋆c 2 = ǫ x 2. (1) k{ n}k2 k{ n n}k2 k{ n}k of predicting the next sample of a discrete-time signal from 2 ∆ ∆ Thus, if we take ǫ < 2 we get that en 2 < , its past. 4k{xn}k 2 ∆ k{ }k ∗ and in particular en < for all n Z. The argument is In particular, our algorithm for recovering xn from xn | | 2 ∈ is of a sequential nature. Since we assume x{(t) }is small{ for} made precise in Section II, where the infinite length filter used all large enough t , there exists some negative| integer| N Z above is replaced with a finite length one, whose coefficients such that x < |∆| for all n<N, which implies that x∗ =∈x are designed based on Chebyshev’s polynomials. | n| 2 n n for all n<N. Next, we would like to recover xn, for n = N, We note that the recovery algorithm proposed in [1] relied ∗ on th-order discrete differentiation of ∗ reduced modulo which is no longer guaranteed to satisfy xn = xn. To this end, L xn ∞ ∆, which is equivalent to Lth-order discrete{ } differentiation we first compute a linear predictor xˆn = i=1 hixn−i for xn from the past samples, that are available to us unfolded.
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