A new approach to near-theoretical sampling rate for modulated wideband converter
22 Jul 2014-pp 1-5
TL;DR: A new greedy algorithm is proposed, which exploits the clustered sparse structure of the multiband signals to sample at near-theoretical rates and the simulation results supporting the better performance of the algorithm are presented.
Abstract: For a multiband signal, the minimum sampling rate required for an arbitrary sampling method, which allows perfect reconstruction, is NB, where N is the number of bands and B is the maximum bandwidth. It has been proposed in the literature that, if the carrier frequency information of a multiband signal is not known apriori, then we require a minimum sampling rate of 2NB for perfect reconstruction. Modulated wideband converter (MWC) is a recently introduced blind sampling method. Unlike the traditional sampling methods, where the continuous-time signal can be expressed in terms of samples using simple Whittaker-Shannon interpolation, there is no closed-form expression relating the samples generated by MWC and the continuous-time signal. In order to reconstruct the signal, we require compressive sensing (CS) algorithm. The CS algorithm, simultaneous orthogonal matching pursuit (SOMP) used in the reconstruction stage requires a minimum rate of 4N B log(M/2N), which is nearly twice the theoretical rate. In this paper, we propose a new greedy algorithm, which exploits the clustered sparse structure of the multiband signals to sample at near-theoretical rates. The simulation results supporting the better performance of our algorithm are also presented.
Citations
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TL;DR: The problem of wideband spectrum sensing/sampling in the sub-Nyquist domain is solved in this paper using sparse (low-density) binary-valued measurement matrices to achieve an efficient compression ratio and improve the signal reconstruction performance.
Abstract: The problem of wideband spectrum sensing/sampling in the sub-Nyquist domain is solved in this paper using sparse (low-density) binary-valued measurement matrices. Key objectives are (i) to achieve an efficient compression ratio, and (ii) improve the signal reconstruction performance. We propose a novel RF front-end with parallel branches that we have called Low-Density Wideband Converter (LDWC). We show that the LDWC implements a binary Low-Density Parity Check (LDPC) matrix as the compressive sensing (CS) measurement matrix. We evaluate, using an Information-Theoretic approach, the asymptotic bound on the required number of LDWC parallel branches for sparsity detection. We develop two new belief propagation (BP) algorithms that operate on the Tanner graph of the CS measurements. We have derived the first algorithm by assuming independence among the variable nodes (VNs) of the Tanner graph. For the second method, we have accounted for the joint probability distribution of the VNs. Analytical and simulated performance results prove the concepts of the LDWC and the proposed BP algorithms and quantify the attainment of objectives (i) and (ii) stated above.
11 citations
Cites background from "A new approach to near-theoretical ..."
...Several extensions of the RD-MWC, and in general the Analog to Information Converters (AIC) [20], aim to (i) improve the estimation of locations and number of active subbands, and reduce the false alarm rate [21]–[23], (ii) improve the energy requirement and the sensitivity [20], [24], (iii) reduce L or the bandwidth of the PR mixing signals [2], [25]–[27], or (iv) provide joint SNSS and DoA (Direction of Arrival) estimation [28]....
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TL;DR: In this paper, an aliased modulated wideband converter (AMWC) is proposed to induce intentional signal aliasing at the analog-to-digital converter (ADC).
Abstract: A modulated wideband converter (MWC) has been introduced as a sub-Nyquist sampler that exploits a set of fast alternating pseudo random (PR) signals. Through parallel analog channels, an MWC compresses a multiband spectrum by mixing it with PR signals in the time domain, and acquires its sub-Nyquist samples. Previously, the ratio of compression was fully dependent on the specifications of PR signals. That is, to further reduce the sampling rate without information loss, faster and longer-period PR signals were needed. However, the implementation of such PR signal generators results in high power consumption and large fabrication area. In this paper, we propose a novel aliased modulated wideband converter (AMWC), which can further reduce the sampling rate of MWC with fixed PR signals. The main idea is to induce intentional signal aliasing at the analog-to-digital converter (ADC). In addition to the first spectral compression by the signal mixer, the intentional aliasing compresses the mixed spectrum once again. We demonstrate that AMWC reduces the number of analog channels and the rate of ADC for lossless sub-Nyquist sampling without needing to upgrade the speed or the period of PR signals. Conversely, for a given fixed number of analog channels and sampling rate, AMWC improves the performance of signal reconstruction.
7 citations
References
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TL;DR: This work considers in depth the extension of two classes of algorithms-Matching Pursuit and FOCal Underdetermined System Solver-to the multiple measurement case so that they may be used in applications such as neuromagnetic imaging, where multiple measurement vectors are available, and solutions with a common sparsity structure must be computed.
Abstract: We address the problem of finding sparse solutions to an underdetermined system of equations when there are multiple measurement vectors having the same, but unknown, sparsity structure. The single measurement sparse solution problem has been extensively studied in the past. Although known to be NP-hard, many single-measurement suboptimal algorithms have been formulated that have found utility in many different applications. Here, we consider in depth the extension of two classes of algorithms-Matching Pursuit (MP) and FOCal Underdetermined System Solver (FOCUSS)-to the multiple measurement case so that they may be used in applications such as neuromagnetic imaging, where multiple measurement vectors are available, and solutions with a common sparsity structure must be computed. Cost functions appropriate to the multiple measurement problem are developed, and algorithms are derived based on their minimization. A simulation study is conducted on a test-case dictionary to show how the utilization of more than one measurement vector improves the performance of the MP and FOCUSS classes of algorithm, and their performances are compared.
1,454 citations
"A new approach to near-theoretical ..." refers background in this paper
...Therefore, the discretetime Fourier transform (DTFT) of the samples yi[n] can be expressed in terms of X(f) as Yi(e j2πfTs) = � n yi[n]e −j2πfnTs = L0� l=−L0 cilX(f − lfp), f ∈ Fs (3) where, L0 = � fNYQ + fs 2fp � − 1, fNYQ is the Nyquist fre- quency of the signal and L = (2L0 + 1)....
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TL;DR: The significance of the results presented in this paper lies in the fact that making explicit use of block-sparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem.
Abstract: We consider efficient methods for the recovery of block-sparse signals-ie, sparse signals that have nonzero entries occurring in clusters-from an underdetermined system of linear equations An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce We then show that a block-version of the orthogonal matching pursuit algorithm recovers block -sparse signals in no more than steps if the block-coherence is sufficiently small The same condition on block-coherence is shown to guarantee successful recovery through a mixed -optimization approach This complements previous recovery results for the block-sparse case which relied on small block-restricted isometry constants The significance of the results presented in this paper lies in the fact that making explicit use of block-sparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem
1,289 citations
"A new approach to near-theoretical ..." refers background in this paper
...It was proved in [3] that a matrix U satisfying the relation V = AU, ||U||0 ≤ 2N (9) shares the same support as z....
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TL;DR: This paper considers the challenging problem of blind sub-Nyquist sampling of multiband signals, whose unknown frequency support occupies only a small portion of a wide spectrum, and proposes a system, named the modulated wideband converter, which first multiplies the analog signal by a bank of periodic waveforms.
Abstract: Conventional sub-Nyquist sampling methods for analog signals exploit prior information about the spectral support. In this paper, we consider the challenging problem of blind sub-Nyquist sampling of multiband signals, whose unknown frequency support occupies only a small portion of a wide spectrum. Our primary design goals are efficient hardware implementation and low computational load on the supporting digital processing. We propose a system, named the modulated wideband converter, which first multiplies the analog signal by a bank of periodic waveforms. The product is then low-pass filtered and sampled uniformly at a low rate, which is orders of magnitude smaller than Nyquist. Perfect recovery from the proposed samples is achieved under certain necessary and sufficient conditions. We also develop a digital architecture, which allows either reconstruction of the analog input, or processing of any band of interest at a low rate, that is, without interpolating to the high Nyquist rate. Numerical simulations demonstrate many engineering aspects: robustness to noise and mismodeling, potential hardware simplifications, real-time performance for signals with time-varying support and stability to quantization effects. We compare our system with two previous approaches: periodic nonuniform sampling, which is bandwidth limited by existing hardware devices, and the random demodulator, which is restricted to discrete multitone signals and has a high computational load. In the broader context of Nyquist sampling, our scheme has the potential to break through the bandwidth barrier of state-of-the-art analog conversion technologies such as interleaved converters.
1,186 citations
"A new approach to near-theoretical ..." refers background or methods in this paper
...But the simulations results show that SOMP requires approximately 4N log(M/2N) channels for successful reconstruction, which is twice the minimum number of channels....
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...Note that z(f) is an L-dimensional vector and each element zi depends on f ....
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...However, if the Nyquist rate is high, these methods either require excessive analog preprocessing hardware or employ analog-to-digital converters (ADCs) with impractical analog bandwidth requirements [4]....
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...Numerical results are presented in Section IV and Section V concludes the paper....
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928 citations
"A new approach to near-theoretical ..." refers methods in this paper
...In order to sample the signals which fall under this model, we require a minimum sampling rate of 2NB [1]....
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...(3) can be arranged in the form of a vector y(f) in order to express it as a linear combination of X(f − lfp), f ∈ Fs as y(f) = Az(f), f ∈ Fs. (4) Here, A ∈ CmxL and its coefficients are given by: Ai,l = ci,−l+L0+1....
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TL;DR: Simulations show that the predictions made by the proved theorems tend to be very conservative; this is consistent with some recent advances in probabilistic analysis based on random matrix theory.
Abstract: The sparse representation of a multiple-measurement vector (MMV) is a relatively new problem in sparse representation. Efficient methods have been proposed. Although many theoretical results that are available in a simple case-single-measurement vector (SMV)-the theoretical analysis regarding MMV is lacking. In this paper, some known results of SMV are generalized to MMV. Some of these new results take advantages of additional information in the formulation of MMV. We consider the uniqueness under both an lscr0-norm-like criterion and an lscr1-norm-like criterion. The consequent equivalence between the lscr0-norm approach and the lscr1-norm approach indicates a computationally efficient way of finding the sparsest representation in a redundant dictionary. For greedy algorithms, it is proven that under certain conditions, orthogonal matching pursuit (OMP) can find the sparsest representation of an MMV with computational efficiency, just like in SMV. Simulations show that the predictions made by the proved theorems tend to be very conservative; this is consistent with some recent advances in probabilistic analysis based on random matrix theory. The connections will be discussed
821 citations
"A new approach to near-theoretical ..." refers background in this paper
...Therefore, the discretetime Fourier transform (DTFT) of the samples yi[n] can be expressed in terms of X(f) as Yi(e j2πfTs) = � n yi[n]e −j2πfnTs = L0� l=−L0 cilX(f − lfp), f ∈ Fs (3) where, L0 = � fNYQ + fs 2fp � − 1, fNYQ is the Nyquist fre- quency of the signal and L = (2L0 + 1)....
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