3GPP 5G New Radio (NR) wireless access technology will support communication at bands ranging from sub GHz to millimeter wave. For communication at millimeter wave, e.g., at 60 GHz, the transmitted signal experiences severe attenuation, due to outdoor-to-indoor penetration, oxygen absorption, etc., significantly degrading the link budget performance of the NR system. In order to achieve good coverage for NR systems, operating at all bands, it is important to ensure the support of power efficient waveforms for physical channels carrying data and reference signals. Demodulation reference signals (DM-RS) are used in 5G NR systems to assist receivers in estimation of the wireless channel for subsequent coherent processing of the corresponding received data on the physical channel. In this paper, we propose a novel design of DM-RS that achieves channel estimation performance similar to NR Release 15 DM-RS, while providing higher power efficiency. The proposed DM-RS design in conjunction with power efficient waveforms for physical channel can thus be used to enhance coverage of NR systems operating at higher carrier frequency ranges of millimeter wave bands, e.g. 60 GHz. We corroborate our claims via computer simulations.

We study the support recovery problem for compressed sensing, where the goal is to reconstruct the sparsity pattern of a high-dimensional K-sparse signal x is an element of R-N, as well as the corresponding sparse coefficients, from low-dimensional linear measurements with and without noise. Our key contribution is a new compressed sensing framework through a new family of carefully designed sparse measurement matrices associated with minimal measurement costs and a low-complexity recovery algorithm. Specifically, the measurement matrix in our framework is designed based on the well-crafted sparsification through capacity-approaching sparse-graph codes, where the sparse coefficients can be recovered efficiently in a few iterations by performing simple error decoding over the observations. We formally connect this general recovery problem with sparse-graph decoding in packet communication systems and analyze our framework in terms of the measurement cost, computational complexity, and recovery performance. Specifically, we show that in the noiseless setting, our framework can recover any arbitrary K-sparse signal in O(K) time using 2K measurements asymptotically with a vanishing error probability. In the noisy setting, when the sparse coefficients take values in a finite and quantized alphabet, our framework can achieve the same goal in time O(K log(N/K)) using O(K log(N/K)) measurements obtained from measurement matrix with elements {-1, 0, 1}. When the sparsity K is sub-linear in the signal dimension K = O(N-delta) for some 0 < delta < 1, our results are order-optimal in terms of measurement costs and run-time, both of which are sub-linear in the signal dimension N. The sub-linear measurement cost and run-time can also be achieved with continuous-valued sparse coefficients, with a slight increment in the logarithmic factors. More specifically, in the continuous alphabet setting, when K = O(N-delta) and the magnitudes of all the sparse coefficients are bounded below by a positive constant, our algorithm can recover an arbitrarily large (1 - p)-fraction of the support of the sparse signal using O(K log(N/K) log log(N/K)) measurements, and O(K log(1+r) (N/K)) run-time, where r is an arbitrarily small constant. For each recovered sparse coefficient, we can achieve O(epsilon) error for an arbitrarily small constant epsilon. In addition, if the magnitudes of all the sparse coefficients are upper bounded by O(K-c) for some constant c < 1, then we are able to provide a strong l(1) recovery guarantee for the estimated signal <(x)over cap>: parallel to(x) over cap - x parallel to(1) <= kappa parallel to x parallel to(1), where the constant kappa can be arbitrarily small. This offers the desired scalability of our framework that can potentially enable real-time or near-realtime processing for massive datasets featuring sparsity, which are relevant to a multitude of practical applications.

The Fast Fourier Transform (FFT) is the most efficiently known way to compute the Discrete Fourier Transform (DFT) of an arbitrary n-length signal, and has a computational complexity of O(n log n). If the DFT X of the signal x has only k non-zero coefficients (where k < n), can we do better? In [1], we addressed this question and presented a novel FFAST (Fast Fourier Aliasing-based Sparse Transform) algorithm that cleverly induces sparse graph alias codes in the DFT domain, via a Chinese-Remainder-Theorem (CRT)-guided sub-sampling operation of the time-domain samples. The resulting sparse graph alias codes are then exploited to devise a fast and iterative onion-peeling style decoder that computes an n length DFT of a signal using only O(k) time-domain samples and O(klog k) computations. The FFAST algorithm is applicable whenever k is sub-linear in n (i.e. k = o(n)), but is obviously most attractive when k is much smaller than n. In this paper, we adapt the FFAST framework of [1] to the case where the time-domain samples are corrupted by a white Gaussian noise. In particular, we show that the extended noise robust algorithm R-FFAST computes an n-length k-sparse DFT X using O(klog ^3 n) noise-corrupted time-domain samples, in O(klog^4n) computations, i.e., sub-linear time complexity. While our theoretical results are for signals with a uniformly random support of the non-zero DFT coefficients and additive white Gaussian noise, we provide simulation results which demonstrates that the R-FFAST algorithm performs well even for signals like MR images, that have an approximately sparse Fourier spectrum with a non-uniform support for the dominant DFT coefficients.

It is a well-known fact that the Discrete Fourier Transform (DFT) (X) over right arrow of an arbitrary n-length input signal (x) over right arrow, can be computed from all the n time-domain samples in O(n log n) operations via a Fast Fourier Transform (FFT) algorithm. If the spectrum (X) over right arrow is k-sparse (where k << n), can we do better? We show that asymptotically in k and n, when k is sub-linear in n (precisely, k = O(n(delta)), where 0 <= delta < 1), and the support of the non-zero DFT coefficients is uniformly random, the fast fourier aliasing-based sparse transform (FFAST) algorithm, proposed in this paper, computes, with asymptotically high probability, the k non-zero DFT coefficients of <(X)over right arrow> from O(k) samples of (x) over right arrow in O(k log k) arithmetic operations. Further, the constants in the big Oh notation for both sample and computational cost are small, e.g., when delta < 0.99, which essentially covers a wide range of sublinear sparsity cases, the sample cost is less than 4k. Although, in this paper we assume that the samples of the signal <(x)over right arrow> observed by the FFAST are noise-free, a noise-robust extension of the FFAST is provided in a companion (Part II) paper [ 1]. Our approach is based on filter-less sub-sampling of the input signal using a set of carefully chosen uniform sub-sampling patterns guided by the Chinese Remainder Theorem (CRT). The idea is to cleverly exploit, rather than avoid, the resulting aliasing artifacts induced by sub-sampling. Specifically, the sub-sampling operation on the time-domain signal (x) over right arrow is designed to create aliasing patterns of the non-zero coefficients of the spectrum (X) over right arrow to "look like" parity-check constraints of a good erasure-correcting sparse-graph code. Next, we show that computing the sparse DFT (X) over right arrow is equivalent to decoding of an appropriate sparse-graph code. The sparse-graph codes further permit a fast peeling-style decoding. Consequently, the resulting DFT computation is low in both the sample and the decoding complexity. We analytically connect our proposed CRT-based aliasing framework to random sparse-graph codes, and analyze the performance of our algorithm using density evolution techniques from coding theory. We provide extensive simulation results, that are in tight agreement with our theoretical findings.

In this paper, we study the data exchange problem, where a set of users is interested in gaining access to a common file, but where each has only partial knowledge about it as side-information. Assuming that the file is broken into packets, the side-information considered is in the form of linear combinations of the file packets. Given that the collective information of all the users is sufficient to allow recovery of the entire file, the goal is for each user to gain access to the file, while minimizing some communication cost. We assume that the users can communicate over a noiseless broadcast channel, and that the communication cost is a sum of each user's cost function over the number of bits it transmits. For instance, the communication cost could simply be the total number of bits that needs to be transmitted. In the most general case studied in this paper, each user can have any arbitrary convex cost function. We provide deterministic, polynomial-time algorithms (in the number of users and packets), which find an optimal communication scheme that minimizes the communication cost. To further lower the complexity, we also propose a simple randomized algorithm inspired by our deterministic algorithm, which is based on a random linear network-coding scheme.

We present a novel algorithm, named the 2D-FFAST (Two-dimensional Fast Fourier Aliasing-based Sparse Transform), to compute a sparse 2D-Discrete Fourier Transform (2D-DFT) featuring both low sample and computational complexity. The proposed algorithm is based on diverse concepts from signal processing (sub-sampling and aliasing), coding theory (sparse-graph codes) and number theory (Chinese-remainder-theorem) and generalizes the 1D-FFAST algorithm recently proposed by Pawar and Ramchandran to the 2D setting. Concretely, our proposed 2D-FFAST algorithm computes a k-sparse 2D-DFT, with a uniformly random support, of size N = N x × N y using O(k) noiseless spatial-domain measurements in O(k log k) computational time. Our results are attractive when the sparsity is sub-linear with respect to the signal dimension, that is, when k → ∞ and k/N → 0. For the case when the spatial-domain measurements are corrupted by additive noise, our 2D-FFAST framework extends to a noise-robust version of computing a 2D-DFT using O(k log 3 N) measurements in sub-linear time of O(k log 4 N). Empirically, we show that the 2D-FFAST can compute a k = 3509 sparse 2D-DFT of a 508 × 508-size phantom image using only 4.75k measurements. We also empirically evaluate the 2D-FFAST algorithm on a real-world magnetic resonance brain image using a total of 60.18% of Fourier measurements to provide an almost instant reconstruction with SNR=4.5 dB. This provides empirical evidence that the 2D-FFAST architecture is applicable to a wider class of input signals than analyzed theoretically in the paper.

We consider the problem of computing the Walsh-Hadamard Transform (WHT) of some $N$-length input vector in the presence of noise, where the $N$-point Walsh spectrum is $K$-sparse with $K = {O}(N^{\delta})$ scaling sub-linearly in the input dimension $N$ for some $0<\delta<1$. Over the past decade, there has been a resurgence in research related to the computation of Discrete Fourier Transform (DFT) for some length-$N$ input signal that has a $K$-sparse Fourier spectrum. In particular, through a sparse-graph code design, our earlier work on the Fast Fourier Aliasing-based Sparse Transform (FFAST) algorithm computes the $K$-sparse DFT in time ${O}(K\log K)$ by taking ${O}(K)$ noiseless samples. Inspired by the coding-theoretic design framework, Scheibler et al. proposed the Sparse Fast Hadamard Transform (SparseFHT) algorithm that elegantly computes the $K$-sparse WHT in the absence of noise using ${O}(K\log N)$ samples in time ${O}(K\log^2 N)$. However, the SparseFHT algorithm explicitly exploits the noiseless nature of the problem, and is not equipped to deal with scenarios where the observations are corrupted by noise. Therefore, a question of critical interest is whether this coding-theoretic framework can be made robust to noise. Further, if the answer is yes, what is the extra price that needs to be paid for being robust to noise? In this paper, we show, quite interestingly, that there is {\it no extra price} that needs to be paid for being robust to noise other than a constant factor. In other words, we can maintain the same sample complexity ${O}(K\log N)$ and the computational complexity ${O}(K\log^2 N)$ as those of the noiseless case, using our SParse Robust Iterative Graph-based Hadamard Transform (SPRIGHT) algorithm.

We present a novel algorithm, named the 2D-FFAST, to compute a sparse 2D-Discrete Fourier Transform (2D-DFT) featuring both low sample complexity and low computational complexity. The proposed algorithm is based on mixed concepts from signal processing (sub-sampling and aliasing), coding theory (sparse-graph codes) and number theory (Chinese-remainder-theorem) and generalizes the 1D-FFAST 2 algorithm recently proposed by Pawar and Ramchandran [1] to the 2D setting. Concretely, our proposed 2D-FFAST algorithm computes a k-sparse 2D-DFT, with a uniformly random support, of size N = Nx x Ny using O(k) noiseless spatial-domain measurements in O(k log k) computational time. Our results are attractive when the sparsity is sub-linear with respect to the signal dimension, that is, when k -> infinity and k/N -> 0. For the case when the spatial-domain measurements are corrupted by additive noise, our 2D-FFAST framework extends to a noise-robust version in sub-linear time of O(k log4 N ) using O(k log3 N ) measurements. Simulation results, on synthetic images as well as real-world magnetic resonance images, are provided in Section VII and demonstrate the empirical performance of the proposed 2D-FFAST algorithm.

We consider the problem of interpolating a high-degree sparse polynomial, where the sparsity is in the number of monomial terms with non-zero coefficients. We propose a probabilistic algorithm that requires only O(k) evaluations of a polynomial with complex coefficients, on the unit circle at specified points and has a complexity O(k log k), where k is the sparsity of the polynomial. Thus the evaluation complexity as well as the computational complexity are independent of the maximum degree n in contrast to existing algorithms in the literature. We extend our algorithm to polynomials defined over the finite field using fast algorithms in the literature to compute discrete logs for certain field sizes.

Fifth generation wireless systems will heavily rely on available bandwidth in millimeter wave frequencies to achieve the very ambitious data rate targets that have been set forth. Besides overcoming challenging propagation conditions with reasonable implementation complexity and power consumption, careful system design is needed. In this paper we address a fundamental component, the waveform. We first characterize the requirements, and then explore possible multi-carrier and single carrier waveforms. Our analysis shows that design tradeoffs for millimeter wave are different from lower frequency bands, and that DFT-spread waveforms may represent a balanced solution between spectral efficiency and robustness to implementation impairments. Furthermore, we propose a newly introduced waveform, GI-DFT-s-OFDM, as a suitable candidate.

We consider the problem of recovering the support of an arbitrary K-sparse N-length vector in the presence of noise, where the sparsity K = O(N δ ) is sub-linear in N for some 0 < δ < 1. A new family of sparse measurement matrices is introduced with a low-complexity recovery algorithm, which achieves a sub-linear measurement cost O(K log 1.3̇ N) and sub-linear computational complexity O(K log 1.3̇ N). Our measurement system is designed to capture observations of the signal through the parity constraints of sparse-graph codes, and to recover the signal by using a simple peeling decoder. We formally connect general sparse recovery problems with sparse-graph decoding, and showcase our design in terms of the measurement cost, computational complexity and recovery performance.

Cloud Radio Access Network (C-RAN) is a novel mobile network architecture that allows for efficient and economic building, upgrading and maintenance of RAN networks to cope with increasing mobile traffic data. The main idea behind C-RAN is to maintain distributed remote radio head (RRH) units for transmission and reception of radio signals, while pool the Baseband Units (BBUs) from multiple base stations into a centralized location for statistical multiplexing gain. The BBUs are connected to RRH via front-haul links.Although the C-RAN architecture enjoys significant advantages in terms of both the capital expenditure (CAPEX) and the operational expenditure (OPEX) over traditional networks, it also exerts considerable burden on the capacity-constrained front-haul links of transporting the high bandwidth complex baseband data between RRH and BBUs. Fortunately, the LTE signal is inherently redundant and compressible. In this paper, we propose a low-complexity, frequency domain, front-haul compression algorithm, that leverages the user scheduling side-information available at the BBU pool, in addition to the LTE signal redundancy, to achieve a compression ratio of 10X for a full-buffer traffic model. We also provide extensive link-level and system-level simulations to validate the performance of our proposed front-haul compression algorithm on simulated LTE signal.

The Fast Fourier Transform (FFT) is the most efficiently known way to compute the Discrete Fourier Transform (DFT) of an arbitrary n-length signal, and has a computational complexity of O(n log n). If the DFT X⃗ of the signal x⃗ has only k non-zero coefficients (where k <; n), can we do better? In [1], we presented a novel FFAST (Fast Fourier Aliasing-based Sparse Transform) algorithm that cleverly induces sparse graph codes in the DFT domain, via a Chinese-Remainder-Theorem (CRT)-guided sub-sampling operation of the time-domain samples. The resulting sparse graph code is then exploited to devise a simple and fast iterative onion-peeling style decoder that computes an n length DFT of a signal using only O(k) time-domain samples and O(k log k) computations, in the absence of any noise. In this paper, we extend the FFAST framework of [1] to the case where the time-domain samples are corrupted by white Gaussian noise. In particular, we show that the extended noise robust algorithm R-FFAST computes an n-length k-sparse DFT X⃗ using O(k log n)1 noise-corrupted time-domain samples, in O(n log n) computations2. While our theoretical results are for signals with a uniformly random support of the non-zero DFT coefficients and additive white Gaussian noise, we provide simulation results which demonstrates that the R-FFAST algorithm performs well even for signals like MR images, that have an approximately sparse Fourier spectrum with a non-uniform support for the dominant DFT coefficients.

In this paper, we consider the problem of computing a K-sparse N-point Hadamard Transforms (HT) from noisy time domain samples, where K = O(Nα) scales sub-linearly in N for some α ∈ (0; 1). The SParse Robust Iterative Graph-based Hadamard Transform (SPRIGHT) algorithm is proposed to recover the sparse HT coefficients in a stable manner that is robust to additive Gaussian noise. In particular, it is shown that the K-sparse HT of the signal can be reconstructed from noisy time domain samples with a vanishing error probability using the same sample complexity O(K logN) as in the noiseless case of [1] and computational complexity1 O(N logN). Last but not least, given the complexity orders of the SPRIGHT algorithm, our numerical experiments further validate that the big-Oh constants in the complexity are small.

Given an n-length input signal x, it is well known that its Discrete Fourier Transform (DFT), X, can be computed in O(nlogn) complexity using a Fast Fourier Transform. If the spectrum X is exactly k-sparse (where k <;<; n), can we do better? We show that asymptotically in k and n, when k is sub-linear in n (i.e., k ∝ nδ where 0 <; δ <; 1), and the support of the non-zero DFT coefficients is uniformly random, we can exploit this sparsity in two fundamental ways (i) sample complexity: we need only M = rk deterministically chosen samples of the input signal x (where r <; 4 when 0 <; δ <; 0.99); and (ii) computational complexity: we can reliably compute the DFT X using O(k log k) operations, where the constants in the big Oh are small. Our algorithm succeeds with high probability, with the probability of failure vanishing to zero asymptotically in the number of samples acquired, M. Our approach is based on filterless subsampling of the input signal x using a small set of carefully chosen uniform subsampling patterns guided by the Chinese Remainder Theorem (CRT). Specifically, our subsampling operation on x is designed to create aliasing patterns on the spectrum X that "look like" parity-check constraints of good erasure-correcting sparse-graph codes. We show how computing the sparse DFT X is equivalent to decoding of these sparse-graph codes and is low in both sample complexity and decoding complexity. We accordingly dub our algorithm the FFAST (Fast Fourier Aliasing-based Sparse Transform) algorithm. In our analysis, we rigorously connect our CRT based graph constructions to random sparse-graph codes based on a balls-and-bins model and analyze the convergence behavior of the latter using well-studied density evolution techniques from coding theory. We provide simulation results in Section IV that corroborate our theoretical findings, and validate the empirical performance of the FFAST algorithm.

The sampling of spectrally sparse high-bandwidth signals below the Nyquist rate is of importance in many applications. In order to keep the front-end analog-to-digital converter sampling rate low, there has been a lot of recent interest in the literature to develop efficient and practical architectures for low-rate sampling and recovery of spectrally sparse signals. Our main contribution is a simple, implementation-friendly front-end sampling-architecture and a computationally efficient back-end algorithm for blind sub-Nyquist sampling of a class of band-limited sparse signals. In particular, we show that signals band-limited to W Hz consisting of k tones can be reliably recovered from sets of uniform samples obtained at a rate 4k Hz for asymptotic values of k and W, when k is sub-linear in W for noiseless observations of the signal. Further, the proposed reconstruction algorithm is computationally efficient, requiring only O(k log k) computations to recover the sinusoidal components. Our algorithm succeeds with high probability, with the probability of failure vanishing to zero asymptotically in the number of samples acquired. We provide simulation results that corroborate our theoretical findings, and also analyze the noise-robustness of the system.

Author(s): Pawar, Sameer | Advisor(s): Ramchandran, Kannan | Abstract: An emerging challenge in recent years for engineers, researchers, and data scientists across the globe is to acquire, store, and analyze ever-increasing amounts of data. In the past decade, a new paradigm in data acquisition called ``compressed-sensing" has emerged to cope with this data explosion. Compressed sensing exploits the fact that in many applications, although the signal of interest has a large ambient dimension, the relevant information resides in a significantly lower dimensional space. For example, the Magnetic-Resonance-Imaging (MRI) data is sparse in the wavelet-domain. In this thesis, we consider the problem of computing a sparse Discrete-Fourier-Transform of a high-dimensional signal from its time-domain samples, as a representative example of compressed-sensing problems. We use this problem to investigate the tradeoff between the number of measurements, noise robustness, and the computational complexity of the recovery algorithm in compressed sensing problems. We propose a new family of deterministic sparse sensing matrices, obtained by blending together diverse ideas from sparse graph codes, digital signal processing, and number-theoretic concepts like the Chinese-remainder-theorem (CRT). The specific sparse structure of the proposed family of measurement matrices further enables a Peeling-based Ultra-Low complexity algorithms for Sparse signal Estimation, that are accordingly dubbed PULSE algorithms. Further, using the CRT, we establish an intimate connection between the problem of computing a sparse DFT of a signal and decoding over an appropriately designed sparse graph code. This connection is then exploited 1) to design a sample efficient measurement matrix and a low-complexity peeling-style iterative recovery algorithm, and 2) to perform a rigorous analysis of the recovery algorithm by wielding powerful and well-established analytical tools like density-evolution, martingales, and expander graphs from the coding theory literature. In particular, we show that under some mild conditions a k-sparse n-length DFT of a signal can be computed using (nearly optimal) 4k measurements and O(k log k) computations. As a concrete example, when k=300, and n = 3.8 x 10^6, our algorithm achieves computational savings by a factor of more than 6000, and savings in the number of input samples by a factor of more than 3900 over the standard Fast-Fourier-Transform (FFT) algorithms. This can be a significant advantage in many existing applications and can enable new classes of applications that were not thought to be practical so far. Next, we extend these results to the case of noise-corrupted samples, computing sparse 2D-DFTs as well as to interpolation of multi-variate sparse polynomials over the complex field and finite fields. We also demonstrate an application of the proposed PULSE algorithm to acquire the Magnetic-Resonance-Imaging of the Brain. This provides some empirical evidence that PULSE algorithms are applicable for acquiring morerealistic signals. The proposed sensing framework and the recovery algorithm are sample efficient and robust against observation noise. We believe that this framework provides a possible direction towards designing efficient and low-power engineering solutions for sparse signal acquisition.

In this paper we study the data exchange problem where a set of users is interested in gaining access to a common file, but where each has only partial knowledge about it as side-information. Given that the collective information of all the users is sufficient to allow recovery of the entire file, the goal is to minimize the sum of convex functions over the number of bits each user has to transmit in order to achieve the goal. A special case of this objective is the uniform cost, which essentially leads to a “fair” communication load distribution among the users. We consider the following side-information settings: (i) side-information in the form of uncoded packets of the file, where the users' side-information consists of subsets of the file packets; and (ii) side-information in the form of linearly correlated packets, where the users have access to linear combinations of the file packets. We provide a polynomial-time deterministic algorithm that finds an optimal communication scheme w.r.t. the transmission cost. To further lower the complexity, we also give a simple randomized algorithm inspired by our deterministic algorithm.

In this paper we construct a deterministic polynomial time algorithm for the problem where a set of users is interested in gaining access to a common file, but where each has only partial knowledge of the file. We further assume the existence of another set of terminals in the system, called helpers, who are not interested in the common file, but who are willing to help the users. Given that the collective information of all the terminals is sufficient to allow recovery of the entire file, the goal is to minimize the (weighted) sum of bits that these terminals need to exchange over a noiseless public channel in order achieve this goal. Based on established connections to the multi-terminal secrecy problem, our algorithm also implies a polynomial-time method for constructing the largest shared secret key in the presence of an eavesdropper. We consider the following side-information settings: (i) side-information in the form of uncoded packets of the file, where the terminals' side-information consists of subsets of the file packets; (ii) side-information in the form of linearly correlated packets, where the terminals have access to linear combinations of the file packets; and (iii) the general setting where the the terminals' side-information has an arbitrary (i.i.d.) correlation structure. We provide a polynomial-time algorithm (in the number of terminals) that finds the optimal rate allocations for these terminals, and then determines an explicit optimal transmission scheme for cases (i) and (ii).

We use a hybrid mix of the Discrete Fourier Transform (DFT), an old workhorse in digital signal processing, and Low Density Parity Check (LDPC) codes, a recent workhorse in coding theory, to generate a linear measurement lens through which to perform compressive sensing (CS) of sparse high-dimensional signals. This novel hybrid DFT-LDPC framework represents a new family of sparse measurement matrices, and induces a fast algorithm (dubbed the Short-and-Wide Iterative Fast Transform based or SWIFT algorithm) for robustly recovering a high-dimensional k-sparse signal x, in Cn, from a near-optimal (upto a small constant multiple of k) number of linear observations, with a decoding complexity of k steps, under high signal-to-noise-ratio (SNR). A key attribute of our sensing framework and the SWIFT recovery algorithm is that both the sensing efficiency (number of measurements) and the recovery efficiency (decoding complexity) are independent of the ambient signal dimension n for the noiseless and high-SNR noisy regimes, which is particularly attractive when k <;<; n. This contrasts existing solutions1 in the CS literature based on the class of Linear Programming (LP)-based optimization techniques and the class of expander graph based greedy-pursuit (sketching) algorithms; for both these classes, both the number of measurements and the decoding complexity depend explicitly on the ambient signal dimension n, even for the noiseless and high-SNR regimes. We provide both an explicit constructions of the DFT-LDPC based matrix framework and an analytical characterization of the SWIFT recovery algorithm. Although our analytical characterization is for asymptotic values of k, n and for high SNR regime, our simulation results validate the efficiency of the SWIFT algorithm even for the low-to-moderate SNR regimes and modest problem dimensions (e.g., k = 250 and n = 20000), beyond what we can currently prove analytically.