Compressive Sensing for Spread Spectrum Receivers Fyhn, Karsten; Jensen, Tobias Lindstrøm; Larsen, Torben; Jensen, Søren Holdt - PDF

Aalborg Universitet Compressive Sensing for Spread Spectrum Receivers Fyhn, Karsten; Jensen, Tobias Lindstrøm; Larsen, Torben; Jensen, Søren Holdt Published in: I E E E Transactions on Wireless Communications

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Aalborg Universitet Compressive Sensing for Spread Spectrum Receivers Fyhn, Karsten; Jensen, Tobias Lindstrøm; Larsen, Torben; Jensen, Søren Holdt Published in: I E E E Transactions on Wireless Communications DOI (link to publication from Publisher): /TWC Publication date: 2013 Document Version Submitted manuscript Link to publication from Aalborg University Citation for published version (APA): Fyhn, K., Jensen, T. L., Larsen, T., & Jensen, S. H. (2013). Compressive Sensing for Spread Spectrum Receivers. I E E E Transactions on Wireless Communications, 12(5), DOI: /TWC General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.? You may not further distribute the material or use it for any profit-making activity or commercial gain? You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us at providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from on: March 14, 2017 1 Compressive Sensing for Spread Spectrum Receivers Karsten Fyhn, Member, IEEE, Tobias L. Jensen, Member, IEEE, Torben Larsen, Senior Member, IEEE, and Søren Holdt Jensen, Senior Member, IEEE Abstract With the advent of ubiquitous computing there are two design parameters of wireless communication devices that become very important: power efficiency and production cost. Compressive sensing enables the receiver in such devices to sample below the Shannon-Nyquist sampling rate, which may lead to a decrease in the two design parameters. This paper investigates the use of Compressive Sensing (CS) in a general Code Division Multiple Access (CDMA) receiver. We show that when using spread spectrum codes in the signal domain, the CS measurement matrix may be simplified. This measurement scheme, named Compressive Spread Spectrum (CSS), allows for a simple, effective receiver design. Furthermore, we numerically evaluate the proposed receiver in terms of bit error rate under different signal to noise ratio conditions and compare it with other receiver structures. These numerical experiments show that though the bit error rate performance is degraded by the subsampling in the CS-enabled receivers, this may be remedied by including quantization in the receiver model. We also study the computational complexity of the proposed receiver design under different sparsity and measurement ratios. Our work shows that it is possible to subsample a CDMA signal using CSS and that in one example the CSS receiver outperforms the classical receiver. Index Terms Compressive sensing, sparse sampling, spread spectrum receivers, multiuser decoding I. INTRODUCTION As wireless communication devices are becoming more and more widespread and ubiquitous, the need for power efficiency and low production cost becomes paramount. A power costly operation in wireless communication is the conversion from analog to digital signals - the Analog to Digital Converter (ADC). The classic ADC uses the Shannon-Nyquist sampling theorem to represent an analog signal in digital form. The Shannon-Nyquist sampling theorem states that to perfectly represent an analog signal, it must be sampled at a frequency higher than twice the signal s bandwidth. When this theorem is obeyed, the original analog signal may be reconstructed perfectly from its sampled representation. The Shannon-Nyquist sampling theorem has been the foundation of digital signal processing for more than half a century and is considered a fundamental building block of digital signal processing systems. Recently, a new concept termed Compressive Sensing (CS) [1], [2] has been attracting more and more attention in the signal processing community as it provides an exception to the lower bound on the sampling rate by exploiting sparsity The authors are with Aalborg University, Faculty of Engineering and Science, Department of Electronic Systems, DK-9220 Aalborg, Denmark. The authors s This work is supported by The Danish Council for Strategic Research under grant number in the signal. If a signal is sparse in some arbitrary basis, it may be sampled at a rate lower than the Nyquist frequency. Sparsity in CS is when a signal is comprised of only a few atoms from a given basis. The sampled signal must be acquired through some linear measurement scheme. Examples of these are random Gaussian, Bernoulli and Rademacher measurement schemes, as well as the Random Demodulator (RD) [3], [4] and the Modulated Wideband Converter [5]. Compressive sensing has primarily been studied in the general signal processing area, and relatively few researchers have looked into its application in communication systems. In [6], [7] the authors examine the use of CS in Ultra-Wideband (UWB) communication systems for channel estimation where the sparsity of the signal lies in the time domain. Others have used compressive sensing for source coding in communication networks, together with network coding [8]. In the spread spectrum area, some researchers have studied the general use of CS for spread spectrum communication systems [9]. However, their work is mainly focused on using CS for fast multi-user detection, rather than subsampling. Another example is in [10], where the authors use CS to decrease the sampling rate of a GPS receiver by exploiting sparsity in the number of possible signal components at the receiver. Their receiver structure is based on possibly complicated hardware filters, which may make their implementation very difficult considering the impact of hardware filters to CS performance [11]. In [12] the authors treat a similar topic where they design spread spectrum codes to enable a base station to perform multi-user detection on a large number of users, of which only a few are active at a time. Their work focuses on simple on-off signalling, i.e. the existence of nodes, rather than communication with them, and solves the multiuser detection problem using an adapted convex optimization algorithm. Their motivation is on increasing the number of active users in a network, rather than decreasing the sampling rate of the ADC. A more applied approach is taken in [13] where compressive signal processing [14] is applied to enable subsampling of an IEEE Direct Sequence Spread Spectrum (DSSS) signal. In [15] the authors also solve a multiuser detection problem using compressive sensing, but in their work the focus is on the design of possibly complex analog filters. For this paper we focus on keeping the analog part as simple as possible and process the signals in the digital domain instead. In our work we apply CS to a general CDMA system. We show that a RD implementation may be used to subsample the CDMA signal, but we also develop a simplified version of 2 the RD which performs equally well for CDMA signals but is simpler and cheaper to implement. Our motivation is that by taking fewer samples we may be able to conserve power in the receiver, as can be seen in e.g. Eqn. 13 in [16]. We show the performance of the proposed receiver structure for the simple discrete case, when compared to a classic receiver structure and an RD receiver structure. Then we extend our results to a full RF numerical simulation and demonstrate that the performance is identical in this setting. Due to noise folding the CS approach suffers a penalty for downsampling, but we show that if quantization is taken into account CS outperforms the classic receiver in some cases. We finally investigate the complexity of the developed algorithms and compare the computational cost of the numerical experiments with the theoretically calculated computation cost. Following the paradigm of Reproducible Research [17], all our results and code are made available at To define our notation, let all vectors and matrices be denoted using lower and upper case letters in bold, x and X, respectively. The Penrose-Moore pseudo-inverse is denoted as X, the transpose of a real matrix as X T and the conjugate transpose of a matrix as X. The expectation operator is denoted by E[ ]. In the following, we first develop a simple signal model in Section II, based on a dictionary of Gold sequences. We then elaborate on what CS is and which reconstruction algorithm we use in the numerical experiments in Section III. Furthermore, we define a novel measurement matrix design for spread spectrum receivers and demonstrate numerically how this measurement matrix performs with a Gold dictionary and the Subspace Pursuit reconstruction algorithm. This performance is compared to that of a Rademacher measurement matrix and a RD measurement matrix. This is followed by Section IV, which includes a simple numerical experiment of the different receiver structures. In Section V we extend the experiment to a full RF simulation with and without quantization. We then analyze the computational complexity of the proposed method in Section VI, after which we conclude the paper in Section VII. II. SIGNAL MODEL First, we consider a purely discrete model of a spread spectrum communication system. Uncoded information bits are sent in a slotted fashion, with each slot containing a single CDMA signal. The system is assumed to be synchronized, which may be obtained by e.g. having a central node or base station transmit beacons, which signify the beginning and end of slots. This is how mobile phone networks and some wireless sensor networks operate. The receiver is considered non-coherent, as information is encoded in the phase, but we do assume that there is no carrier frequency offset between the transmitter and receiver oscillators. This is of course not a realistic assumption but it is done to keep the system simple. Future work should investigate the impact of oscillator drift on performance. Each slot contains an independent CDMA signal and the slots are decoded sequentially and independently of each other. For one slot, define a discrete QPSK baseband signal, x C N 1 as: x = Ψα, (1) where Ψ S Ψ {±1} N N is an orthogonal or nearorthogonal dictionary, containing spreading waveforms for transmission, S Ψ is the subset of {±1} N N that contains orthogonal or near-orthogonal dictionaries and α {±1 ± j,0} N 1 is a sparse vector, that selects which spreading waveform(s) and what QPSK constellation point(s) to send. α is a vector here because we only process one slot at a time and we assume that within a slot, the signal amplitude for each user is constant. That α is assumed to be sparse is justified in some scenarios, which is demonstrated shortly. An example of a system using the above signal model could be a wireless sensor network in which one node must gather information from any possible neighbors. Each node has a unique CDMA sequence assigned, which it uses to transfer information and each node does not know which neighbors it has, but it knows all possible CDMA sequences. Note that in this signal model α is defined so that all users have identical amplitude. This is not realistic as the distance between nodes might vary a lot, resulting in differences between amplitude in the received signal components. We choose this simplification here but the reconstruction algorithm is not limited by this and also works for sparse vectors with different amplitude components. In cases where the number of active nodes or users in a network is smaller than the total number of possible users, the vector α may be assumed sparse, which is the enabling factor for CS. Cases such as these arise in e.g. mobile phone networks and wireless sensor networks, where the number of surrounding nodes may be large, but is often small. At the receiver the following signal is observed: y = Θ(x+w) = ΘΨα+Θw, (2) where Θ is a measurement matrix, which we shall treat later, and w C N 1 is Additive White Gaussian Noise (AWGN). Notice here that we take into account noise folding as the noise is folded down into the compressed domain together with the signal. This makes the noise colored and has an impact on the demodulation performance, because each time the sampling rate is reduced by one half, the Signal to Noise Ratio (SNR) is decreased by 3 db [18], [19]. A. Spread Spectrum Dictionary of Gold Sequences In spread spectrum signals, a possible dictionary Ψ is a set of Gold sequences, as used in e.g. GPS technology [20]. A set of Gold sequences is a special dictionary of binary sequences with very low auto and cross-correlation properties [21]. To generate a Gold dictionary, two maximum length sequences must be generated by two linear feedback shift registers (LFSR). A maximum length sequence is often denoted an m- sequence (it has m elements), and is a special kind of pseudorandom noise sequence generated by a LFSR, such that it is periodic and produces a sequence of length 2 m 1. It is called a maximum length sequence as its period is at maximum 3 length. The reason for the length being 2 m 1 rather than 2 m is that the state where all cells are zero must be avoided. To obtain an m-sequence, the LFSR must be carefully chosen as there is no algorithm for ensuring maximum length. However, there are many known LFSR setups for varying choices of m. Furthermore, the two m sequences must be chosen so that their periodic cross-correlation is three-valued and takes on only the values { 1, t,t 2}, where: { 2 (m+1)/2 +1 for odd m and t = 2 (m+2)/2 (3) +1 for even m. The set of Gold sequences are then generated using two m-sequences: g 1 and g 2, both of length N = 2 m 1. Each Gold sequence in the set is generated as g 1 g i (exclusive or), where g i is g 2 cyclically shifted by the parameter i. As i can take on values between 1 i 2 m 1, each shift constitutes a candidate for the set, resulting in a dictionary as follows: Define a N N dictionary of Gold sequences as Ψ, with each column signifying a possible code sequence. When using such a CDMA dictionary, the received signal must be sampled at a rate corresponding to the chip rate, where a chip is one entry in the received Gold sequences. If α is sparse the information rate of the signal is much lower and it may be possible to decrease the sampling rate by using CS. In this paper, we use three Gold dictionary sizes: m = 5,m = 7 and m = 10. The m-sequence feedback sets used to generate these are: m = 5: X 5 +X 2 +1 and X 5 +X 4 +X 3 +X 2 +1 m = 7: X 7 +X 6 +1 and X 7 +X 4 +1 m = 10: X 10 +X 3 +1 and X 10 +X 9 +X 8 +X 6 +X 3 +X 2 +1 The chosen polynomials may be validated by calculating the auto and cross-correlation of the generated dictionaries and verifying that they follow the structure listed in the above. III. COMPRESSIVE SENSING CS is a novel sampling scheme, developed to lower the number of samples required to obtain some desired signal. At the heart of CS is the linear sampling scheme, called the measurement matrix. In classic receivers the measurement matrix Θ may be modelled as the identity matrix, such that x is sampled at the chip rate of each channel (I and Q). Here, we shall denote a classic receiver using Θ 1 = I, where the subscript 1 denotes no subsampling and I is the identity matrix of size N N. In CS another measurement matrix is used. Denote by Θ κ R M N a CS measurement matrix, where κ N 1 is the subsampling ratio when compared to the Nyquist rate and M = N/κ (If κ does not divide N, M is rounded to the nearest integer). This measurement matrix is then responsible for mapping the N-dimensional signal x to a M- dimensional signal y. Normally this would make it impossible to recover the original signal, but under the assumption that x is sparse in some basis, it is possible to reconstruct the original signal from the sampled, M-dimensional signal y [1], [2]. Notice that we are not interested in the reconstructed signal, x, but in the sparse vector α, which allows us to demodulate the data in the signal. We may obtain an estimate of α by reconstructing the signal from y. Such a reconstruction may be obtained using e.g. a convex optimization problem solver or a greedy algorithm. For this work, we choose the greedy algorithm Subspace Pursuit [22]. This algorithm is chosen due to its good performance in terms of both reconstruction accuracy and running time, as shown in Section III-B. Before explaining the reconstruction algorithm, we return to the measurement matrix and introduce a new measurement scheme which is enabled by the use of CDMA. This new measurement scheme is easier to implement than the RD, but performs almost identically for spread spectrum systems. We call this a Compressive Spread Spectrum (CSS) measurement matrix and explain it further in the following. A. Compressive Spread Spectrum Measurement Matrix In most CS literature a choice of measurement matrix or structure must be made. The Bernoulli or Rademacher distributed measurement matrix is often seen in the theoretical literature, but it is not well suited for practical implementation in a wireless receiver. The Random Demodulator (RD) sampling structure [3], [4] is one of the most well-known measurement matrix structures developed, which is well suited for practical implementation. In the RD a Pseudo-Random Noise (PRN) sequence is mixed with the received signal followed by low-pass filtering. Because a spread spectrum transmitter has already spread the signal before transmission, we show that the RD structure can be improved so that the mixing with a PRN sequence at the receiver may be skipped. This is similar to what is done in [13] with IEEE signals, which uses Direct-Sequence-Spread-Spectrum (DSSS) signals. These can be viewed as a special class of CDMA signals, which are used to counter interference from blockers in the same frequency band, rather than to distinguish between users or signals. The proposed measurement matrix may therefore be defined similarly to the definition of the RD matrix in [4]. In their work, the measurement matrix is based on two matrices, D and H. First, let ǫ 0,ǫ 1,...,ǫ N {±1} be the chipping sequence used in the RD for a signal of length N. The mapping x Dx signifies the demodulation mapping with the chipping sequence, where D is the diagonal matrix: ǫ 0 ǫ 1 D =. (4)... ǫn Second, the H matrix denotes the accumulate-and-dump action performed after mixing. Let M denote the number of samples taken and assume here that M divides N. Then each sample is the sum of N/M consecutive entries of the demodulated signal. The matrix performing this sampling action may therefore be defined as an M N matrix, with N/M consecutive unit entries in therth row starting in column rn/m + 1 for each r = 0,1,...,M 1. An example with M = 3 and N = 6 is: 1 1 H = 1 1. (5) 1 1 4 The RD is therefore designed to sample an analog signal, so that in a discrete representation this is the equivalent to: y = HDx, (6) where x is the Nyquist sampled input signal and y is the compressively sampled output signal. The reason for applying a chipping sequence is to spread the signal across the frequency spectrum, so that information is aliased down into the lower frequency area, which is left untouched by the low-pass filtering. In the proposed receiver this mixing is unnecessary because the signal has already been spread at the transmitter. The proposed receiver may therefore be simplified to: y = Hx. (7) This is significantly simpler to implement in hardware than the RD. Comparing to the notation introduced for the measurement matrix in Section II we therefore have: Θ κ = H. To justify the use of no PRN sequence in the measurement matrix, consider the following. The use of a CDMA dictionary int
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