The Scientific World Journal

Volume 2015, Article ID 151370, 10 pages

http://dx.doi.org/10.1155/2015/151370

## Estimation of Symmetric Channels for Discrete Cosine Transform Type-I Multicarrier Systems: A Compressed Sensing Approach

^{1}ETSI Industriales, Universidad Politécnica de Madrid, C/José Gutiérrez Abascal 2, 28006 Madrid, Spain^{2}ETSIS de Telecomunicación, Universidad Politécnica de Madrid, Carretera de Valencia Km 7, 28031 Madrid, Spain

Received 15 May 2015; Accepted 20 September 2015

Academic Editor: Sandra Costanzo

Copyright © 2015 María Elena Domínguez-Jiménez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The problem of channel estimation for multicarrier communications is addressed. We focus on systems employing the Discrete Cosine Transform Type-I (DCT1) even at both the transmitter and the receiver, presenting an algorithm which achieves an accurate estimation of symmetric channel filters using only a small number of training symbols. The solution is obtained by using either matrix inversion or compressed sensing algorithms. We provide the theoretical results which guarantee the validity of the proposed technique for the DCT1. Numerical simulations illustrate the good behaviour of the proposed algorithm.

#### 1. Introduction

In wireless communications, the channel filter is usually time-varying; for this reason, it is necessary to estimate the channel filter from time to time. To this aim, some training symbols (i.e., symbols known both by the transmitter and by the receiver) are typically used. In this way, when the training symbols are transmitted by the channel, the received signal is used to extract the information about the channel filter. Some well-known techniques for channel estimation are based on the Discrete Fourier Transform (DFT); in this case, the training symbols are OFDM waveforms.

Additionally, if the channel filter is sparse (i.e., containing only a small amount of nonzero coefficients), then compressed sensing techniques can be applied. Compressed sensing (CS) algorithms approximate the sparsest solution to a linear system [1]. This is very useful when the solution depends on a small number of degrees of freedom and only a few measurements of the vector are observed. For this reason, in the last few years CS algorithms have been applied to a wide variety of scenarios in communications: cognitive radio, radar, antenna arrays, multicarrier communications, and so forth. When CS is applied to channel estimation problems, it is usually denoted as compressed channel sensing (CCS). Several CCS algorithms have been proposed in the literature for different types of channels arising in communication problems, such as ultrawideband channels, underwater acoustic communications, or multipath channels [2–5]. Most of these techniques are based on DFTs or spread spectrum signals.

In this work, we consider a multicarrier communications system that is based on the Discrete Cosine Transform Type-I (DCT1) even instead of the standard DFT. Some Discrete Cosine Transforms have been widely used in the context of multicarrier modulation (MCM), as an alternative to the DFT, due to their good properties (e.g., good performance under carrier frequency offset) [6–13]. In particular, in a very recent work [14] the DCT1 is applied for MCM communications. The main advantages of the DCT1 are as follows:(i)The inverse of the DCT1 is the same transform DCT1, up to a scaling factor; so we can use the same transform at both the transmitter and the receiver [15].(ii)The convolution of two vectors is transformed by DCT1 into a pointwise product of their transforms (under some symmetry conditions on the vectors) [15, 16]. This is analogous to the circular convolution property of the DFT. This is a key property for signal reconstruction in MCM communications [14].

For these reasons, we investigate the use of DCT1 for channel estimation; in particular, we address the problem of estimation of whole-point symmetric (WS) channels by means of CS techniques in the DCT1 transform domain. The strategy consists of using only a few training symbols, which are transmitted through the channel, and reconstructing the impulse response of the filter in the receiver by using the same small number of measurements. Thus, the economy of the data can be exploited by CS algorithms, which are able to provide sparse filters.

In this work we will provide not only a new estimation procedure but also the training signals valid for our algorithm, and we will show that this technique is both simple and theoretically correct. These are the main contributions of this paper. Numerical simulations also illustrate the effectiveness of our results.

The paper is organized as follows. Firstly, in Section 2 we recall the general channel estimation problem. Secondly, in Section 3 the DCT1 is introduced and we obtain new important properties of this transform. Then, the proposed procedure is presented in Section 4, where its theoretical justification is also provided. Section 6 contains some numerical examples that illustrate the behaviour of our algorithm. Finally, we highlight the main contributions of this work in Section 7.

#### 2. The Channel Estimation Problem

Let us consider a multicarrier modulation communications system that performs an inverse transform in the transmitter and a direct transform in the receiver , as shown in Figure 1. Let us consider also a channel with the following impulse response: The transmission of an information symbol through this channel results in a received symbol , such that where is a term related to the additive noise.