In this overview article, we review the literature on design and analysis of recursive algorithms for reconstructing a time sequence of sparse signals from compressive measurements. The signals are assumed to be sparse in some transform domain or in some dictionary. Their sparsity patterns can change with time, although, in many practical applications, the changes are gradual. An important class of applications where this problem occurs is dynamic projection imaging, e.g., dynamic magnetic resonance imaging (MRI) for real-time medical applications such as interventional radiology, or dynamic computed tomography.

Recursive Recovery of Sparse Signal Sequences From Compressive Measurements: A Review

Timing and carrier synchronization is a fundamental requirement for any wireless communication system to work properly. Timing synchronization is the process by which a receiver node determines the correct instants of time at which to sample the incoming signal. Carrier synchronization is the process by which a receiver adapts the frequency and phase of its local carrier oscillator with those of the received signal. In this paper, we survey the literature over the last 5 years (2010–2014) and present a comprehensive literature review and classification of the recent research progress in achieving timing and carrier synchronization in single-input single-output (SISO), multiple-input multiple-output (MIMO), cooperative relaying, and multiuser/multicell interference networks. Considering both single-carrier and multi-carrier communication systems, we survey and categorize the timing and carrier synchronization techniques proposed for the different communication systems focusing on the system model assumptions for synchronization, the synchronization challenges, and the state-of-the-art synchronization solutions and their limitations. Finally, we envision some future research directions.

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Timing and carrier synchronization in wireless communication systems: a survey and classification of research in the last 5 years

In this paper, a very efficient semi-blind approach for the detection of most significant taps (MSTs) in sparse orthogonal frequency-division multiplexing (OFDM) channel estimation is developed. The least square (LS) estimation problem of sparse OFDM channels is first formulated, showing that the key to sparse channel estimation lies in the detection of the MSTs. An in-depth study of the second-order statistics of the signal received through a noise-free sparse OFDM channel reveals the sparsity and other properties of the correlation functions of the received signal. These properties lead to a direct relationship between the positions of the MSTs of the sparse channel and the most significant lags of the correlation functions, which is then used in conjunction with a pilot-assisted LS estimation to detect the MSTs in a semi-blind fashion. It os also shown that the new MST detection algorithm can be extended for the estimation of multiple-input–multiple-output (MIMO)–OFDM channels. A number of computer-simulation-based experiments for various sparse channels are carried out to confirm the effectiveness of the proposed semi-blind approach.

Semi-Blind Most Significant Tap Detection for Sparse Channel Estimation of OFDM Systems

In this paper, OFDM data-aided channel estimation based on the decimation of the Channel Impulse Response (CIR) through the selection of the Most Significant Samples (MSS) is addressed. Our aim is to approach the Minimum Mean Square Error (MMSE) channel estimation performance, while avoiding the need for a-priori knowledge of channel statistics (KCS). The optimal set of samples is defined in the instantaneous and average senses. We derive lower bounds on the estimation mean-square error (MSE) performance for any MSS selection strategy. We show how MSS-based channel estimation can approach these MSE lower bounds. We introduce novel MSS strategies oriented towards instantaneous decimation (Instantaneous Energy Selection - IES), and windowed decimation (Average Energy Selection - AES). We also consider decimation via Threshold Crossing Selection (TCS), which we characterize analytically, to derive the optimum threshold in the minimum MSE sense. We also propose a sub-optimal method for threshold setting that does not require KCS. Finally, we provide numerical results in terms of both MSE estimation performance and Bit Error Rate (BER) of a coded OFDM system using the proposed channel estimators, to show that they indeed approach MMSE performance.

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OFDM Channel Estimation Based on Impulse Response Decimation: Analysis and Novel Algorithms

In this paper, Alamouti space–frequency block coding, applied over the carriers of an orthogonal frequency-division multiplexing (OFDM) system, is considered for obtaining transmit diversity in an underwater acoustic channel. This technique relies on the assumptions that there is sufficient spatial diversity between the channels of the two transmitters, and that each channel changes slowly over the carriers, thus satisfying the basic Alamouti coherence requirement and allowing simple data detection. We propose an adaptive channel estimation method based on Doppler prediction and time smoothing, whose decision-directed operation allows for reduction in the pilot overhead. System performance is demonstrated using real data transmitted in the 10–15-kHz acoustic band from a vehicle moving at 0.5–2 m/s and received over a shallow-water channel, using quadrature phase-shift keying (QPSK) and a varying number of carriers ranging from 64 to 1024. Results demonstrate an average mean squared error gain of about 2 dB as compared to the single-transmitter case and an order of magnitude decrease in the bit error rate when the number of carriers is chosen optimally.

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Space–Frequency Block Coding for Underwater Acoustic Communications

A threshold-based procedure to estimate sparse channels in an orthogonal frequency division multiplexing (OFDM) system is proposed. An optimal threshold is derived by maximising the probability of correct detection between significant and zero-valued taps estimated by the least squares (LS) estimator. Improved LS estimates are obtained by pruning the LS estimates with the statistically derived threshold.

Sparse channel estimation in OFDM systems by threshold-based pruning

A Krylov subspace based low-rank channel estimation in OFDM systems

The authors consider the design of pilot sequences for channel estimation in the presence of carrier frequency offset (CFO) in systems that employ orthogonal frequency division multiplexing (OFDM). The CFO introduces intercarrier interference (ICI) which degrades the accuracy of the channel estimation. In order to minimise this effect, the authors design pilot sequence that minimises the mean square error (MSE) of the modified least squares (mLS) channel estimator. Since the identical pilot sequence, which minimises this MSE, has high peak-to-average power ratio of the OFDM signal, an alternative approach is proposed for channel estimation. The authors first introduce a new estimator as an alternative to the mLS estimator and design a low PAPR pilot sequences tailored to this new estimator. They show that the proposed procedure completely eliminates the effect of the ICI on the channel estimate. They then extend their design of pilot sequences for realistic sparse channels. Both analytical and computer simulation results presented in this study demonstrate the superiority of the proposed approach over conventional methods for channel estimation in the presence of ICI.

Improved least squares channel estimation for orthogonal frequency division multiplexing

We consider the design of pilot sequences for channel estimation in the presence of carrier frequency offset (CFO) in systems that employ orthogonal frequency division multiplexing (OFDM). The CFO introduces intercarrier interference (ICI) which degrades the accuracy of the channel estimation. To minimize this effect, we solve for the pilot sequence that minimizes the mean square error of the modified least squares (mLS) channel estimator. The transmission of this sequence results in high peak-to-average power ratio (PAPR) of the OFDM signal. To solve this problem, we first introduce a new estimator as an alternative to the mLS estimator. We then design a low PAPR pilot sequence tailored to this new estimator. We show that the proposed procedure completely eliminates the effect of the ICI on the channel estimate. Both analytical and computer simulation results demonstrate the superiority of the proposed scheme over the mLS estimator.

Pilot sequence design for improving OFDM channel estimation in the presence of CFO

We consider the problem of pilot-aided accurate least squares (LS) channel estimation in the presence of carrier frequency offset (CFO). We first formulate the mean square error (MSE) of the LS channel estimator in the presence of CFO and then show how to select the pilot sequence (with element values constrained to plus or minus one) that minimizes the MSE. Our main conclusion is that, in the presence of CFO accurate channel estimates can be obtained by transmitting identical pilot symbol on all subcarriers. We demonstrate our results via computer simulations.

J. Oliver

Papers

Sparse channel estimation in OFDM systems by threshold-based pruning

A Krylov subspace based low-rank channel estimation in OFDM systems

Improved least squares channel estimation for orthogonal frequency division multiplexing

Pilot sequence design for improving OFDM channel estimation in the presence of CFO

Improved channel estimation in OFDM systems in the presence of CFO