Hardware-Efficient Random Sampling of Fourier-Sparse Signals Patrick Maechler, Norbert Felber, and Hubert Kaeslin

Andreas Burg

Integrated Systems Laboratory, ETH Zurich Gloriastr. 35, 8092 Zurich, Switzerland

Telecommunications Circuits Laboratory, EPFL Station 11, 1015 Lausanne, Switzerland

Abstract—Spectrum sensing, i.e. the identification of occupied frequencies within a large bandwidth, requires complex sampling hardware. Measurements suggest that only a small fraction of the available spectrum is actually used at any time and place, which allows a sparse characterization of the frequency domain signal. Compressed sensing (CS) can exploit this sparsity and simplify measurements. We investigate the performance of a very simple hardware architecture based on the slope analogto-digital converter (ADC), which allows to sample signals at unevenly spaced points in time. CS algorithms are used to identify the occupied frequencies, which can be continuously distributed across a large bandwidth.

I. I NTRODUCTION Scanning a large bandwidth to identify occupied bands is required in different applications. One of them is cognitive radio (CR) [1]. The idea behind CR is to allow secondary users to transmit using the same spectrum as licensed primary users, whenever unoccupied bands in the spectrum of the primary users can be detected. Thus, CR can significantly increase spectrum utilization. Unfortunately, identifying unused frequency bands by scanning large bandwidths is a challenging task requiring complex and power-hungry hardware. An opportunity to simplify spectrum sensing opens up if the spectrum is only sparsely occupied. Indeed, measurements show that often licensed spectrum is heavily underutilized. Spectrum occupancy of around 10% for TV bands or even less than 0.5% for the lower L bands is reported in [2]. A promising approach to exploit the sparsity assumption for reducing the complexity of the sampling process comes from the area of compressed sensing (CS) [3]. In essence, the corresponding recent results explain why and how signals with a sparse representation in some domain can be sampled below their Nyquist rate and how the original signal can be reconstructed. Related work: One CS-based approach to efficiently sample sparse signals is to project the signal onto a incoherent basis in the analog domain. For example, the random demodulator [4] first multiplies the signal with a high-frequency pseudo-noise sequence, accumulates the result in an integrator, and then samples at a reduced frequency. Similarly, the modulated wideband converter (MWC) [5] projects on periodic high-frequency waveforms and uses low-pass filters instead of integrators. These projection-based approaches were developed for Fourier sparse and frequency-band sparse signals, respectively.

For signals sparse in the Fourier-basis, a much simpler sampling scheme becomes an interesting alternative. The basic idea is to sample at randomly distributed points in time. Traditional ADC architectures are not suitable for random sampling without introducing artificial delays and therefore waisting hardware resources. Two possible implementation strategies are described in [6]: one uses a bank of lowrate ADCs with shifted starting points, the other one utilizes capacitors to store analog values until they are sampled by a low-rate ADC. In a previous paper, we introduced the random sampling slope ADC (RSS-ADC) [7], which samples at unevenly distributed points in time and employs greedy CS algorithms for recovery. Our ADC is based on a small modification of the well-known slope ADC and achieves full hardware utilization since no artificial delays enforcing a regular sampling grid are necessary. A random sampling ADC very similar to ours was developed independently and concurrently [8], which is also based on a slope ADC. In contrast to our implementation, wait periods of random length are introduced before a new slope is started. This, however, reduces the hardware efficiency since all components stay idle during the waiting periods. Contributions: In this paper, we further elaborate on our previously introduced RSS-ADC. We extend the reconstruction such that it can cope with continuously distributed frequencies instead of only frequencies on a grid. We also introduce hardware constraints such as propagation delays in our design and evaluate their effects. Outline: The rest of the paper is organized as follows: We first briefly review the basics of sparse signal recovery and random sampling with an adapted reconstruction algorithm in Sec. II. In Sec. III, the RSS-ADC is explained. The detection of off-grid frequencies is studied in Sec. IV and hardware aspects are treated in Sec. V. II. R ANDOM S AMPLING CS provides a framework with reconstruction algorithms and convergence criteria that allows to reconstruct sparse or compressible signals from fewer measurements than the dimension of the unknown signal suggests [3]. A signal is called K-sparse if it can be represented by K coefficients in a given basis. If x ∈ CM is the sparse representation of this signal, only K out of its M coefficients have non-zero values. For a measurement vector y ∈ CN with potentially N  M ,

Algorithm 1 CoSaMP with model restrictions 1: r ← y, Γ ← ∅ 2: while stopping criterion not met do ¯Hr 3: c←Φ 4: Γ ← Γ ∪ findLargestIndices(applyModel(|c|2 ), 2K) ¯ Γx 5: x ¯temp ← argminx ||Φ ¯Γ − y||22 6: Γ ← findLargestIndices(applyModel(|¯ xtemp |2 ), K) 7: r ← r − ΦΓ x ¯temp,Γ 8: end while 9: x ¯ ← 0, x ¯Γ ← x ¯temp

voltage

A

reference slope

-A

(2B − 1)T0

A

voltage

yn

time

yn+1

-A

cn T0

time

Figure 1. Slopes generated for given input signal (red) in a traditional slope ADC (top) and a RSS-ADC (bottom)

and a measurement matrix (or dictionary) Φ ∈ CN ×M , one can reconstruct x from y = Φx under certain conditions on Φ. Columns of Φ are called atoms or dictionary elements. With noise-free measurements, the signal x can be reconstructed by finding the sparsest possible solution which fulfills y = Φx according to x ˆ = argminx ||x||0 , subject to y = Φx.

input signal

(1)

In the spectrum sensing application considered in this paper, the continuous time signal y(t) is assumed to be band-limited to a frequency f with a sufficiently sparse frequency domain representation x. The measurements y are taken at random points in the time domain with a time-resolution T0 < 1/fN with fN = 2f being the Nyquist frequency. By reducing the number of measurements, i.e., N < M , the signal y(t) becomes undersampled with an effective sampling rate of feff = N/(M T0 ).

A formal description of the CoSaMP algorithm using (3) and (4) is provided in Alg. 1. On line 4, the 2K best fitting atoms are chosen by the function findLargestIndices(a, 2K) which returns the indices of the 2K largest values of a. Since real and imaginary part were separated in (3), one has to make sure that the two parts corresponding to the same frequency are always selected together. This model assumption is enforced in the function applyModel(x). On line 5, a least squares optimization in the sub-space spanned by the 2K atoms is performed. Only the K best atoms are then kept and will be used as a starting point for the next iteration, where new atoms can be added and obsolete ones can be discarded. III. R ANDOM S AMPLING S LOPE ADC (RSS-ADC)

The RSS-ADC [7] performs a pseudo-random sampling of an input signal. It is based on the simplest possible slope yn = y(t)|t=kn T0 , kn ∈ {1, M }, kn > kn−1 (2) ADC, which generates a linear reference slope and detects its intersection point with the input signal. The reference n = 1, . . . , N slope of a traditional slope ADC is periodically reset and The measurement matrix Φ corresponds to an inverse discrete restarted (Fig. 1, top). Thus, the exact acquisition time is Fourier transform (DFT) matrix where only the rows with signal-dependent but the periodic reset of the reference slope allows no more than one sample per period. indices in the set {kn }N n=1 are kept and all other rows In contrast to this ADC, the RSS-ADC resets the reference discarded. slope right after the reference slope reaches the input signal. A. Reconstruction This modification allows to obtain a more uneven spacing, We use the Compressive Sampling Matching Pursuit since the periodic reset is removed, and it permits collecting (CoSaMP) algorithm [9] to find a solution for (1). CoSaMP is a larger number of samples (Fig. 1, bottom) without complicomputationally less complex than l0 - or l1 -minimization but cating the circuit. Together with the Fourier domain sparsity only approximates the optimal solution. assumption, CS reconstruction algorithms allow better idenSince we consider a real-valued time domain signal, the tification of occupied frequencies with less (or less accurate) corresponding spectrum x ∈ CM is conjugate symmetric. The samples than uniformly sampling ADCs. measurement matrix can be adapted in order to ensure that the The frequency domain signal is reconstructed by the modleast squares optimization step of CoSaMP returns only real- ified CoSaMP algorithm described in Alg. 1. With ideal valued estimates. The conjugate symmetry of the spectrum x hardware, the lowest possible sampling interval is given by allows to compute only half of its components. A new real- the time-to-digital resolution T0 . The sampling instances kn ¯ x can thus be defined. valued system y = Φ¯ are determined by the time cn at which the cross-overs of the reference slopes with the input signal were detected. The ¯ = [Re{x2 } Re{x3 } . . . Re{xM/2 } x series of sampling times in multiples of T0 is then given Im{x2 } Im{x3 } . . . Im{xM/2 }]T . (3) by k = Pn0 c 0 . New measurements are acquired until n n =1 n    the end of the sampling window is reached at t = M T0 . M  √2 cos 2π (kn −1)m if 1 < m < 2 M N ¯   The measurement values yn are determined according to Φn,m = 3 M  √2 sin 2π (kn −1)( 2 M −m) if ≤ m ≤ M − 2 y = c m + m . n n 0 The increment m within time T0 of a M 2 N (4) reference slope raising from −A to A with a resolution of

Perfect support recovery rate

0.5

RSS−ADC 0

Eff. sampling rate feff [in multiples of fN]

Slope ADC

1

2 3 4 5 f0 = 1/T0 [in multiples of the Nyquist frequency]

6

2

Perfect support recovery rate

1

1

Discrete grid

0.8

Cont. distr.: not overcomplete Cont. distr.: 2x overcomplete

0.6

Cont. distr.: 4x overcomplete Cont. distr.: 6x overcomplete

0.4 0.2 0

0.05

0.1

1

0

Figure 3. 1

2 3 4 5 f0 = 1/T0 [in multiples of the Nyquist frequency]

0.15 Sparsity

0.2

0.25

Overcomplete dictionaries for continuous frequency distribution

6

Figure 2. Perfect support recovery rate of a signal with sparsity K = 25: Comparison of a traditional slope ADC to the RSS-ADC

B bits is given by m = 2A/(2B − 2) and the initial offset is m0 = −A − m/2. Simulations: In Fig. 2, the performance of the RSS-ADC is compared to the traditional slope ADC. We set T0 = 1/(3fN ), M = 1024 and use a resolution of B = 3 bit, which turns out to be optimal for the RSS-ADC [7]. The signal to be acquired is a discrete multi-tone signal where all possible frequencies lie on a grid and each corresponds to a single entry in x. The K-sparse support set is randomly chosen. The perfect support recovery rate is used as a quality metric. It is defined by the ratio of successful scans to the total number of scans in a Monte Carlo simulation. A scan is called successful when all the occupied frequency bins were identified correctly. In Fig. 2, one thus sees that by using the new RSS-ADC instead of the traditional slope ADC, a lower time resolution T0 is required and thus the hardware implementation is less complex. IV. O FF -G RID F REQUENCIES For all the initial investigations so far, all active tones were placed on a grid, where each entry in the vector x corresponds to one frequency bin. In real-world applications however, the sparse frequency components are continuously distributed on arbitrary frequencies. When continuously distributed frequencies need to be reconstructed using a dictionary with components on a grid of size M , the result is not sparse anymore due to the heavy sidelobes introduced by the Dirichlet sampling kernel. Possible remedies are described in [10]. One approach is to increase the resolution in the frequency domain by adding dictionary elements to Φ for intermediate frequencies. This corresponds to computing a DFT of a zero-padded signal and results in a highly redundant dictionary. The more redundant dictionary entries are added, the higher is the sparsity of a signal, which increases the recovery performance. However, with an increasing dictionary size, the dictionary elements become more and more coherent, which reduces the recovery performance. This trade-off can be mitigated by introducing the additional model constraint that frequencies must not be too close to each other: whenever a frequency is selected, its immediate (and highly correlated) neighbors can not be

selected anymore. Our reconstruction algorithm needs only a slight adaptation, which is to include the additional model constraint in the applyModel(x) function in Alg. 1. Numerical simulations (Fig. 3) show that going from a spectrum, which is sparse on a discrete grid with resolution 1/T0 , to a continuous distribution results in a large performance loss. However, by using overcomplete dictionaries, some of the lost performance can be recovered. The simulations were performed for 2x, 4x, and 6x overcomplete dictionaries. It turns out that an overcompletion factor of 4 is optimal and higher factors do not further increase the performance. V. H ARDWARE A SPECTS A major advantage of the RSS-ADC is its simple hardware architecture, which is sketched in Fig. 4. The RSS-ADC contains a ramp generator, a comparator, and a time-to-digital converter. Until now all these components were considered ideal with no significant delays. In order to evaluate how well the RSS-ADC is working with less ideal hardware components, hardware propagation delays need to be considered. The hardware requirements of the RSS-ADC can be described by mainly one parameter at each stage, Td and T0 . The first stage is the analog-to-time conversion. This includes the ramp generator and the comparator. The most important limitation is the propagation delay Tpd of the comparator, which determines the time needed for the comparator output to switch after the differential of the two input signals changed its sign. Also, the generated ramp might not be ideal when non-ideal current sources or reset transistors are considered. Especially the discharging of the capacitor is not immediate but will cause an additional delay. The sum of all delays from the input signal crossing until a new ramp can be started is denoted Td . This includes the propagation delay of the comparator Tpd , after which the ramp reset starts, and the time needed to switch the comparator output back. InputSignal Ramp generator

time-to-digital Comparator

Clock

Reset

Figure 4.

Schematic of a slope ADC

Td in between two samples but is not signal-dependent. Both random samplers use exactly the same number of samples as the RSS-ADC, which results in the same effective sampling rate for all three samplers. As the simulations show, delays of up to Td = 7T0 do not decrease the perfect support recovery rate for a sparsity of K = 16 but significantly reduce speed requirements and reduce the effective sampling rate to around 40% of the Nyquist frequency. The comparison with a perfect random sampler shows what could be achieved with no constraints on the sampling instances. In order to identify which part of the performance difference is due to the delay constraint and which part is due to the signal-dependence of the RSS-ADC, the intermediate curve of a random sampler with delay constraint needs to be examined. It turns out that the delay constraint has a smaller influence than the signal dependence of the sampling points.

Tpd Td

Figure 5.

Simulated reference slope including propagation delays

Perfect support recovery rate

1

RSS−ADC RS with delay RS

0.5

Eff. sampling rate feff [in multiples of fN]

0

0

5

10

15 Delay T /T d

20

25

30

0

1 0.5 0

0

5

10

15 20 Delay Td/T0

25

30

Figure 6. Perfect support recovery rate of the RSS-ADC and random samplers (RS) with and without delay constraint (top) and effective sampling rate (bottom) with varying comparator delay

The second stage is the time-to-digital conversion, which measures the time at which the comparator switches. It can be characterized by the time resolution T0 at which the comparator output is sampled. Given the resolution in time and the slope steepness m, the quantization of the amplitude is fixed. An illustration of the RSS-ADC reference signal with those hardware delays included is depicted in Fig. 5. A first order low-pass filter is applied to the ideal ramp in order to include an exponential discharging during reset. The most interesting parameter for our purpose is the time Td since it determines the speed requirements of the comparator and thus the hardware complexity. Td indicates the smallest possible spacing in between two samples but it does not limit the time resolution T0 , which might still be much higher and allows for a supported bandwidth higher than 1/Td . A practical example of a very fast 250 MS/s slope ADC [11] shows a Tpd /T0 ratio of 4. In order to quantify the effect of this delay on the system performance, different values for Td were simulated with 600 random sparse patterns (Fig. 6). The series of sampling points now include a slope reset delay of r = Td /T0 . kn = nr +

n X

cn0

(5)

n0 =1

We fix T0 = 1/(3fN ), the resolution to B = 3 bit and the sparsity to K = 16. A higher Td simplifies the hardware but reduces the number of measurements. In Fig. 6, the RSS-ADC is compared to a perfect random sampler and a random sampler which also requires a minimum spacing of

VI. C ONCLUSIONS The random sampling slope ADC provides a simple realization of a device acquiring unevenly spaced samples. When the spectrum is only sparsely occupied, this approach allows to reduce the hardware complexity by applying compressed sensing reconstruction algorithms. High hardware efficiency is achieved by a fast reference slope restart. Also, no extra analog hardware is necessary compared to the traditional slope ADC. We have shown that propagation delays in the analog frontend can be tolerated, which further reduces hardware requirements. Using overcomplete dictionaries and model constraints in the CS reconstruction algorithm allows to detect continuously distributed frequency components, which makes this device more suitable for real-world applications. R EFERENCES [1] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 2, pp. 201–220, Feb. 2005. [2] Z. Wang and S. Salous, “Spectrum occupancy statistics and time series models for cognitive radio,” Journal of Signal Processing Systems, vol. 62, pp. 145–155, 2011. [3] D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289–1306, April 2006. [4] J. Tropp, J. Laska, M. Duarte, J. Romberg, and R. Baraniuk, “Beyond Nyquist: Efficient sampling of sparse bandlimited signals,” IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 520–544, 2010. [5] M. Mishali and Y. Eldar, “From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals,” IEEE Journal of Selected Topics in Signal Processing, vol. 4, no. 2, pp. 375–391, 2010. [6] J. Laska, S. Kirolos, Y. Massoud, R. Baraniuk, A. Gilbert, M. Iwen, and M. Strauss, “Random sampling for analog-to-information conversion of wideband signals,” in IEEE Dallas/CAS Workshop on Design, Applications, Integration and Software, 2006, pp. 119 –122. [7] P. Maechler, N. Felber, and A. Burg, “Random sampling ADC for sparse spectrum sensing,” in Proc. Eusipco, Sept. 2011. [8] P. Yenduri, A. Gilbert, M. Flynn, and S. Naraghi, “Rand PPM: A low power compressive sampling analog to digital converter,” in ICASSP 2011, 2011, pp. 5980–5983. [9] D. Needell and J. Tropp, “CoSaMP: Iterative signal recovery from incomplete and inaccurate samples,” Applied and Computational Harmonic Analysis, vol. 26, no. 3, pp. 301–321, May 2009. [10] M. F. Duarte and R. G. Baraniuk, “Spectral compressive sensing,” 2010, preprint, http://www.math.princeton.edu/~mduarte/images/SCS-TSP.pdf. [11] P. Harpe, C. Zhou, K. Philips, and H. de Groot, “A 0.8-mW 5-bit 250MS/s time-interleaved asynchronous digital slope ADC,” IEEE Journal of Solid-State Circuits, vol. 46, no. 11, pp. 2450–2457, Nov. 2011.

Hardware-Efficient Random Sampling of Fourier ...

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