Application of a joint and iterative MMSEbased estimation of SNR and frequencyselective channel for OFDM systems
 Vincent Savaux^{1, 2}Email author,
 Yves Louët^{2},
 Moïse DjokoKouam^{1} and
 Alexandre Skrzypczak^{3}
https://doi.org/10.1186/168761802013128
© Savaux et al.; licensee Springer. 2013
Received: 26 March 2013
Accepted: 17 June 2013
Published: 12 July 2013
Abstract
This article presents an iterative minimum mean square error (MMSE) based method for the joint estimation of signaltonoise ratio (SNR) and frequencyselective channel in an orthogonal frequency division multiplexing (OFDM) context. We estimate the SNR thanks to the MMSE criterion and the channel frequency response by means of the linear MMSE (LMMSE). As each estimation requires the other one to be performed, the proposed algorithm is iterative. In this article, a realistic case is considered; i.e., the channel covariance matrix used in LMMSE is supposed to be totally unknown at the receiver and must be estimated. We will theoretically prove that the algorithm converges for a relevantly chosen initialization value. Furthermore simulations show that the algorithm quickly converges to a solution that is close to the one in which the covariance matrix is perfectly known. Compared to existing SNR estimation methods, the algorithm improves the tradeoff between the number of required pilots and the SNR estimation quality.
Keywords
1 Introduction
The multipath channel and the additive noise are two important sources of distortion in wireless communication systems. Firstly, the channel impulse or frequency response provides information about the selectivity of the channel. Secondly, the noise is usually characterized by means of its comparison with the signal level by the signaltonoise ratio (SNR). The knowledge of these parameters (channel and noise) allows to design more accurately both transmitter and receiver. For instance, at the transmitter side, the constellation type and its size can be adapted according to the SNR level [1]. Yet, at the transmitter, the timereversal method [2] can be performed, thanks to the channel impulse response. At the receiver side, many algorithms such as the LMMSE channel estimation [3] or the turbodecoder [4] require the knowledge of the SNR, and an accurate channel state information (CSI) allows to perform a simple onetap equalization in orthogonal frequency division multiplexing (OFDM) systems.
Some SNR estimation methods are proposed in [5–7] for single carrier systems that are being used in additive white Gaussian noise (AWGN) channels. The second and fourthorder moment (M_{2}M_{4}) estimator, firstly mentioned in [8], does not require any channel estimation. In addition, M_{2}M_{4} has a low complexity. However, its efficiency is degraded in frequencyselective channels. The maximum likelihood (ML) estimator, whose developments are given in [9], offers a good efficiency, but has a prohibitive complexity in the case of frequencyselective channels. The minimum mean square error (MMSE) estimator, from which we derive our proposed method, requires the estimation of the transmission channel. In references [5, 10, 11] only a theoretical expression of the MMSE can be found, but the authors do not propose any practical solution to reach it. Reference [10] covers the usual M_{2}M_{4}, ML, and MMSE estimators in an OFDM case, and presents a method so as to estimate the SNR in frequencyselective channels. This latter method is based on the autocorrelation function given by the model of the channel (Rayleigh or Rice models). The authors of [11–13] also present SNR estimation methods for OFDM transmissions in frequencyselective channels. In order to avoid the need for the channel estimation, author in [11] proposes a method for a 2×2 multiinput and multioutput (MIMO) configuration which features a two pilotsymbols preamble and assumes that the channel coefficients are invariant over two consecutive carriers. Author in [12] also proposes a preamblebased method featuring two pilot symbols for the estimation of the noise variance. The SNR’s estimation is performed, thanks to the combination of this noise estimation with the second moment order (M _{2}) of the received signal. The methods of [11] and [12] require a two pilotsymbols preamble, which reduces the useful data rate of the transmission, especially if the preamble is regularly repeated. The authors of [13] present a SNR estimation based on the properties of the channel covariance matrix, estimated thanks to a one pilotsymbol preamble. This method is limited by the channel’s insufficient statistics, which degrades the estimation performance.
The literature is very extensive concerning channel estimation. A wide range of usual channel estimation methods is based on ML [9], least square (LS) [14, 15], or MMSE [16, 17]. Here we focus on recursive and iterative methods. The recursive least square algorithm (RLS), described in [18] or [19], uses the estimations of the previous channel frequency response to perform the estimation of the current one. Similarly to the RLS principle, authors in [20, 21] propose a recursive MMSE method that does not require an a priori need for channel statistics. In [22], the channel variations are tracked by employing the Kalman filter estimator. As presented in [23–26], the iterative channel estimation methods are combined with equalization, data detection, decoding, or even interference cancellation. In this case, a soft or hard feedback from the detection block to the estimator block performs an iterative channel and data estimation. The iterative expectation maximization (EM) algorithm [27, 28] has been developed so that the ML estimator is an appropriate tool in frequencyselective channels when the observed data are not complete, i.e., when the size of the observation is smaller than the vector to be estimated. An adaptation of this algorithm for both channel and noise estimation is presented in [29, 30], and joint iterative EM data detection and recursive channel tracking are proposed in [31]. However, when a preamble is used, the sizes of the observation and that of the vector of the channel frequency response to be estimated are the same, so the EM algorithm is not necessary. Furthermore, under this condition and considering a Gaussian channel, the ML estimator is equivalent to the usual LS estimator [32, 33]. In [34], we proposed an MMSEbased iterative algorithm for both SNR and channel estimations. However, it was a theoretical approach, in which the channel covariance matrix was supposed to be known at the receiver.
In the present article, we propose an approach in which the channel covariance matrix is estimated at the receiver. As a consequence, this paper is considered as an application of the theoretical approach developed in [34]. We estimate the SNR thanks to the MMSE criterion, which requires an estimation of the frequencyselective channel. Since we use the LMMSE method for the channel estimation, the noise variance is required. We clearly notice that one estimation feeds the other one. Then, it seems natural to propose an iterative algorithm. We show that it converges, thanks to a relevant choice of the initialization. Since we suppose no a priori CSI at the receiver, this algorithm is also valid for communications systems such as WiFi or LTE, broadcast systems featuring standards such as Digital Radio Mondiale DRM/DRM+ [35], or digital video broadcastingterrestrial DVBT [36]. Although we use the term iterative to describe our method, it differs from the usual iterative methods such as [25, 26, 31] since it does not require a feedback from the data detection block.
The rest of this paper is organized as follows: Section 2 presents the used OFDM system model, the noise variance, and the SNR and channel estimations. The proposed algorithm is developed in Section 3, and we prove in Section 4 that it converges for a relevant choice of initialization. Simulations presented in Section 5 verify the convergence of the method. As for the SNR estimation, we compare our method to two others presented in the literature [12, 13], the wellknown M_{2}M_{4}[10], and to the estimation performed in the perfect case [34]. The channel estimation performance is compared to the perfect one and to the usual LS. We draw our conclusions in Section 6.
2 Background and system model
2.1 Notations
In the following, the normal font x is used for scalar variables, the bold font x is used for vectors, and the underlined bold font $\underline{\mathbf{\text{x}}}$ for matrices. Furthermore, small letter x refers to the variables in the time domain and capital letter X to the variables in the frequency domain.
2.2 System model
where m denotes the subcarrier subscript, L the length of the impulse response, and h _{ l,n } the zeromean complex process of the l th path of the channel. Each τ _{ l } is the discrete expression of the delay. All L paths are considered to be independent. We also assume a quasistatic channel, i.e., the coefficients H _{ m,n } are supposed to be invariant over a frame including a one pilotsymbol preamble and OFDM data symbols.
2.3 Noise variance estimation
where ${\widehat{\mathbf{\text{H}}}}_{p}$ denotes the channel estimation performed on the pilot symbols. The index p refers to the pilot preamble of a given frame. In practice, the expectation can only be approximated by the mean over a sufficiently large number of subcarriers, leading to ${\widehat{\sigma}}^{2}=\frac{1}{M}\sum _{m=0}^{M1}{U}_{m,p}{C}_{m,p}{\u0124}_{m,p}{}^{2}$.
2.4 SNR estimation
where ${\widehat{\sigma}}^{2}$ is defined in (3) and ${\widehat{M}}_{2}$ by ${\widehat{M}}_{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\frac{1}{M}\sum _{m=0}^{M1}{U}_{m,n}{}^{2}$.
2.5 Channel estimation
where ${\underline{\mathbf{\text{R}}}}_{\phantom{\rule{0.3em}{0ex}}H}$ denotes the channel covariance matrix. The LMMSE channel estimation is more efficient than that of the LS but requires a matrix inversion. Without loss of generality, we assume in the rest of the paper that ∀m=0,…,M−1,C _{ m,p }=1 on a given preamble position p. Consequently, the pilot matrix ${\underline{\mathbf{\text{C}}}}_{\phantom{\rule{0.3em}{0ex}}p}$ is equal to the identity matrix noted $\underline{\mathbf{\text{I}}}$, which leads to ${\widehat{\mathbf{\text{H}}}}_{p}^{\text{LMMSE}}={\underline{\mathbf{\text{R}}}}_{\phantom{\rule{0.3em}{0ex}}H}{({\underline{\mathbf{\text{R}}}}_{\phantom{\rule{0.3em}{0ex}}H}+{\sigma}^{2}\underline{\mathbf{\text{I}}})}^{1}{\widehat{\mathbf{\text{H}}}}_{p}^{\text{LS}}$. As ${\underline{\mathbf{\text{R}}}}_{\phantom{\rule{0.3em}{0ex}}H}$ and σ ^{2} are usually unknown at the receiver, we propose an iterative algorithm for both noise variance and channel estimation.
3 Proposed algorithm
The characterization of the initialization ${\widehat{\sigma}}_{(i=0)}^{2}$ will be discussed in Section 4. However, it is already obvious that ${\widehat{\sigma}}_{(i=0)}^{2}$ must be strictly positive. Indeed, if its value is equal to 0, then ${\widehat{\mathbf{\text{H}}}}_{p(i=1)}^{\text{LMMSE}}$ is equal to ${\widehat{\mathbf{\text{H}}}}_{p}^{\text{LS}}$ in (8).
The algorithm given in the realistic case (considering an unknown channel covariance matrix) is summarized in Algorithm 1.
Algorithm 1 MMSEbased joint estimation of both channel and SNR
4 Convergence of the algorithm in realistic case
This section aims at proving that the algorithm converges in the realistic case (i.e., using an a priori unknown channel covariance matrix) if a relevant initialization ${\widehat{\sigma}}_{(i=0)}^{2}$ is chosen. Thanks to the expressions (12) and (14), it is obvious that the channel estimation and the SNR converge, since the convergence of $\left({\widehat{\sigma}}_{\left(i\right)}^{2}\right)$ is established.
4.1 Scalar expression of the sequence $\left({\widehat{\sigma}}_{\left(i\right)}^{2}\right)$
4.2 Necessary condition for the convergence of the sequence $\left({\widehat{\sigma}}_{\left(i\right)}^{2}\right)$
4.3 Sufficient condition for the convergence of the sequence $\left({\widehat{\sigma}}_{\left(i\right)}^{2}\right)$
However, the choice of a relevant value Λ is not obvious. Indeed, since the channel frequency response and the noise variance can take an infinite number of values, the design of an abacus of f _{2} is computationally prohibitive. Furthermore, we assume that the receiver has no a priori knowledge of the set of the parameters {λ _{ m },σ ^{2}}, which makes the choice of the optimal initialization impossible. In order to overcome the issue of the complexity of the choice of ${\widehat{\sigma}}_{(i=0)}^{2}$, we propose in Section 4.4 a simple characterization which does not require any abacus.
4.4 Optimal choice of the initialization ${\widehat{\sigma}}_{(i=0)}^{2}$
The conditions on ${\widehat{\sigma}}_{(i=0)}^{2}$ given in previous Section 4.3 are either not relevant enough (${\widehat{\sigma}}_{(i=0)}^{2}=\Lambda {M}_{2}$ with Λ>>1), or too complex (use of abacus of f _{2}). Here, we propose a simple characterization of ${\widehat{\sigma}}_{(i=0)}^{2}$ made thanks to the noise variance estimation ${\widehat{\sigma}}^{2}$, which was performed on the last frame. If we note $\mathcal{F}$ the index of the current frame, the proposed method is as follows:

For the first frame $\mathcal{F}=1$, perform the algorithm thanks to the arbitrary initialization ${\widehat{\sigma}}_{(i=0)}^{2}=\Lambda {M}_{2}$ chosen with the sufficient condition Λ>>1.

For $\mathcal{F}>1$, get the noise variance ${\widehat{\sigma}}^{2}$ and the eigenvalues of the channel covariance matrix ${\underline{\stackrel{~}{\mathbf{\text{R}}}}}_{\phantom{\rule{0.3em}{0ex}}H}^{\text{LMMSE}}$ (11), estimated at the previous frame $\mathcal{F}1$.

Considering the expression of f _{2} given in (23), look for ${\widehat{\sigma}}_{(i=0),\text{opt}}^{2}$ so that$\frac{{\widehat{\sigma}}^{4}}{M}\sum _{m=0}^{M1}\frac{{\lambda}_{m}+{\widehat{\sigma}}^{2}}{{\left(\frac{{\left({\lambda}_{m}+{\widehat{\sigma}}^{2}\right)}^{3}}{{\left({\lambda}_{m}+{\widehat{\sigma}}^{2}+{\widehat{\sigma}}_{(i=0),\mathit{\text{opt}}}^{2}\right)}^{2}}+{\widehat{\sigma}}^{2}\right)}^{2}}{\widehat{\sigma}}^{2}=0.$(32)
The direct solving of (32) is very complex, but in practice, the receiver can use a simple binary search algorithm to approach the optimal solution. This optimal solution ${\widehat{\sigma}}_{(i=0),\text{opt}}^{2}$ can then be found at the frame $\mathcal{F}$.
For a relevant choice of the initialization ${\widehat{\sigma}}_{(i=0)}^{2}$, we have given a sufficient condition so that the algorithm converges to a nonnull solution. Additionally, an optimal value of ${\widehat{\sigma}}_{(i=0)}^{2}$ can be found, which allows the convergence to take place at the expected noise and channel values. Section 5 depicts the performance of our algorithm and finally shows that the estimated couple (${\widehat{\sigma}}^{2},{\widehat{\mathbf{\text{H}}}}_{n}$) is close to the perfect estimation one (σ ^{2},H _{ n }).
5 Simulations results
This section aims at confirming, by means of simulations, the theoretical results developed in the previous sections. Furthermore, it characterizes the algorithm performance, such as the speed of convergence, the bias of the noise variance estimation, or the bit error rate (BER), thanks to the proposed channel estimation compared to the perfect one. The simulation parameters are based on the Digital Radio Mondiale (DRM/DRM+) standard [35], designed for the digital audio broadcasting over the current AM/FM frequency bands. We consider a 201subcarrier OFDM modulation with a sampling frequency equal to 12 kHz. Each OFDM frame is composed of 20 symbols, each symbol being filled with data symbols from a 16QAM constellation. The added CP featuring a T _{CP} duration of 2.66 ms is supposed to avoid the intersymbol interferences, i.e., it is longer than the maximum delay of the channel. Although the DRM standard recommends a distribution of the pilot tones in staggered rows in the OFDM frame, we considered a preamble distribution for the purpose for our method. As previously mentioned, each preamble is composed of one pilot symbol only. We consider the US Consortium channel model, also described in the DRM standard. It is a fourpath channel in which the maximum delay is τ _{max}=2.2 ms and the maximum Doppler spread is equal to 2 Hz. Here, the channel is supposed to be quasistatic, which means that it varies very slowly during a frame duration. In the following, the term ‘perfect case’ refers to the algorithm proposed in [34], in which the channel covariance matrix is supposed to be known at the receiver, whereas ‘practical case’ refers to the proposed algorithm.
5.1 Convergence of the noise variance estimation
For iterations i≥1, we remark that the sequence $\left({\widehat{\sigma}}_{\left(i\right)}^{2}\right)$ is monotonous and converges to a single nonnull value, which verifies the theoretical developments given in Section 4. We also observe a fast convergence to the single limit, which will be confirmed in Section 5.2. Figure 4 characterizes the noise variance estimation thanks to the normalized bias β calculated by $\beta =\left\right({\widehat{\sigma}}_{\left(i\right)}^{2}){\sigma}^{2}/{\sigma}^{2}$, β being averaged over 4,000 runs. Expressed in percentage, the bias of the proposed estimation is equal to 5.9% for ρ=0 dB and 1.2% for ρ=10 dB. These results are very close to the estimation performed in the perfect case.
5.2 Speed of convergence of the algorithm
For example, in order to reach a fixed value e _{ σ }=0.01, three iterations are required in the practical case and two in the perfect case. For e _{ σ }=0.0001, seven iterations are required in the practical case and three in the perfect case. These results confirm the highspeed convergence of the algorithm.
5.3 Comparison of SNR estimation with other methods
As mentioned in [10], we remark that the performance of the M_{2}M_{4} method is degraded in Rayleigh channels, which is the case here. Whatever the SNR is, the proposed method outperforms the one from Xu’s. In Figure 6, the performance of the algorithm is degraded compared to that obtained with the Ren’s method for low SNR values (<3 dB). It confirms that for low values of SNR,${\widehat{\sigma}}_{(i=0)}^{2}$ is not large enough compared to the value of the noise variance σ ^{2}. However, when the algorithm is used with an updated initialization (Figure 7), the method outperforms Ren’s one whatever the SNR is, and the SNR gap with the perfect case is less than 1 dB from S N R=0 dB. This proves the efficiency of the proposed algorithm and the validity of the improvement with regard to the choice of${\widehat{\sigma}}_{(i=0)}^{2}$, when performed with an update in each frame. Furthermore, our method requires only one pilotsymbol preamble while Ren’s requires two. We conclude that we can improve both the rate of the required pilots and the efficiency of the estimation.
5.4 Channel estimation
We observe on Figure 8 that the channel estimation converges to a value that is close to the perfect estimation. Indeed, zooming around SNR = 25 dB, we can see that the gap between the perfect channel estimation and our method is less than 0.2 dB after seven iterations. It confirms the high speed of convergence and illustrates the efficiency of the channel estimation algorithm.
6 Conclusions
In this article, we presented a practical algorithm for a joint and iterative MMSEbased estimation of the SNR and the frequencyselective channel in an OFDM context. The SNR is estimated, thanks to the MMSE noise variance estimation combined with the second moment order of the signal, and the channel, thanks to the LMMSE method. Since each estimation requires the other one, the algorithm is iterative, as proposed in [34] for a theoretical case in which the channel covariance matrix is supposed to be known at the receiver. However, we considered in this paper a practical case which assumes that the channel covariance matrix is a priori unknown at the receiver side. We theoretically proved that for a wellchosen initialization value, the algorithm converges. Furthermore, simulations showed that the proposed method has a very good quality of estimation for both the SNR and the channel frequency response. Compared to existing methods, the algorithm improves the ratio between the required number of pilots and the efficiency of the SNR estimation. Further works will concern the application of the proposed algorithm to the domain of the cognitive radio, in particular, for free bands detection.
Appendix
Declarations
Authors’ Affiliations
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