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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.

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 [

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) [

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 [

The convolution of two vectors is transformed by DCT1 into a pointwise product of their transforms (under some symmetry conditions on the vectors) [

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

Let us consider a multicarrier modulation communications system that performs an inverse transform

Block diagram of a multicarrier modulation communications system, including the channel estimation in the receiver.

In multicarrier systems, in order to eliminate interblock interference, we often add to the original symbol

Now, the question is if we know the training symbol

The existence of such transform

As OFDM systems present poor behaviour under carrier frequency offset, other multicarrier modulation (MCM) techniques have been investigated, which are related to other transformations different from DFT. Among them, the eight types of Discrete Cosine Transforms (DCTs) have been studied in the literature, and for each one of them the corresponding extension technique has been proposed [

The DCT1 even of an

The first contribution of this work is the demonstration of the following theorem regarding the invertibility of some submatrices of the DCT1 matrix. This is a key property which guarantees that the channel filter can be obtained by means of a small amount of received data; this will be applied in the following section, when using compressed sensing techniques. Let us now state and prove this important property.

Any

The submatrix formed by the first

To show that

To this aim, let us now introduce the auxiliary self-reciprocal polynomial

Our strategy is to prove that

By denoting the complex numbers

it is easy to see that

where we have used (

As already mentioned, for any

So there are

The union of the set of roots of (

In case

Let us assume that the channel filter presents whole-point (WS) symmetry:

Symmetries in

It is proved in [

Thus, we have been able to find an easy solution to the channel estimation problem in DCT1 MCM communication systems. Following the general statement of the problem given in Section

As the components of the training signal

Now, the question is what can we do if a component of

Let us explain this idea in detail. If there are only

Besides, we can exploit the structure of

Compressed sensing techniques show that it is possible to achieve the sparsest vector

As

Alternatively, as

Choose a training signal

Apply a whole-point symmetry of length

Transmit

Take the

Apply the DCT1 block:

Compute

In case

In case some components of

that can be solved via CS techniques or simply defining

In any case, from

In this section, we analyse the behaviour of the proposed compressed channel sensing (CCS) scheme by testing it on three channels: a fixed simple channel of length

The inverse DCT1 is then performed and the time-domain transmitted vector

As a first example, we select the following

Figure

Channel reconstruction SNR (

Figure

(a), (c), and (e) Estimated channel’s impulse response for SNR = 0 dB, SNR = 10 dB, and SNR = 20 dB, respectively. (b), (d), and (f) Estimated channel’s frequency response for SNR = 0 dB, SNR = 10 dB, and SNR = 20 dB, respectively. In all cases the length of the channel is

We have also tested the effect of the number of subcarriers,

Channel reconstruction SNR (

As a second example, we consider a length

(a), (c), and (e) Estimated channel’s impulse response for SNR = 0 dB, SNR = 10 dB, and SNR = 20 dB, respectively. (b), (d), and (f) Estimated channel’s frequency response for SNR = 0 dB, SNR = 10 dB, and SNR = 20 dB, respectively. In all cases the length of the channel is

Finally, we test our approach on a perturbed symmetric version of the ITU-T M.1225 pedestrian channel A. The pedestrian channel A was generated using Matlab’s

Channel reconstruction SNR (

In this work, we have presented a general procedure for the estimation of any symmetric channel filter for multicarrier communication systems based on the Discrete Cosine Transform Type-I (DCT1) even. For any training signal transmitted through the channel, at the receiver, we show how to take into account the information of the training symbol so as to estimate the channel filter. The main contribution of this work is that it is possible to estimate the channel filter with a small amount of training signals, just knowing a small amount of the received samples, and regardless of the location of these samples. This is an important consequence of the good properties of the DCT1 matrix that have been also proved here for the first time. Thus, our proposed procedure with the DCT1 formulation meets the conditions that guarantee perfect estimation of the channel filter in absence of noise, whereas in noisy scenarios a very good estimation can also be achieved. We have also designed specific sparse training signals for our DCT1 procedure and showed that it can also be applied to channels whose impulse response is only approximately symmetric with good results. Future research lines include extending these procedure to nonsymmetric channels.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work has been partially supported by the Spanish Ministry of Economy and Competitiveness through Project TEC2012-38058-C03-01. David Luengo has also been supported by the BBVA Foundation through Project MG-FIAR (“I Convocatoria de Ayudas Fundación BBVA a Investigadores, Innovadores y Creadores Culturales”). All the authors are members of the UPM TACA Research Group and also thank the Universidad Politécnica de Madrid for its support.