Estimation of Compressible Channel Impulse Response for OFDM Modulated Transmissions

Abstract

Channel estimation scheme for OFDM modulated transmissions usually combines an initial block-pilot-assisted stage with a tracking one based on comb or scattered pilots distributed among user data in the signal frame. The channel reconstruction accuracy in the former stage has a significant impact on tracking efficiency of the channel variations and the overall transmission quality. The paper presents a new block-pilot-assisted channel reconstruction procedure based on the DFT-based approach and the Least Square impulse response estimation. The proposed method takes into account a compressibility feature of the channel impulse response and restores its coefficients in groups of automatically controlled size. The proposition is analytically explained and tested in a OFDM simulation environment. The popular DFT-based methods including compressed sensing oriented one were used as references for comparison purposes. The obtained results show a quality improvement in terms of Bit Error Rate and Mean Square Error measures in low and mid ranges of signal-to-noise ratio without significant computational complexity growth in comparison to the classical DFT-based solutions. Moreover, additional multiplication operations can be eliminated, compared to the competitive, in terms of estimation quality, compressed sensing reconstruction method based on greedy approach.

1. Introduction

OFDM (Orthogonal Frequency Division Multiplexing) has become the primary modulation for physical layer design in many modern digital transmission systems since the beginning of the century. The main OFDM feature behind its popularity is the division of the entire frequency transmission spectrum into narrowband flat subchannels using DFT (Discrete Fourier Transform). When the modulation parameters such as the subchannel spacing, guard interval, and symbol lengths are well matched to the occupied frequency band, channel characteristics, and system mobility, the subchannels are orthogonal, and the transmission within them is mutually independent in consecutive OFDM symbols [1].

The subchannel orthogonality as a result of FFT makes data decoding in the individual subchannels straightforward with ZF (Zero Forcing) approach, provided that the channel frequency characteristic is known at the time when the OFDM symbols are received. In practice, the CSI (Channel State Information) is acquired prior to data reception by means of pilots, i.e., training complex symbols (e.g., belonging to QPSK constellation) known to the receiver and defined in the frequency domain. If all active subchannels of a single OFDM symbol convey pilots, they form a block pilots. On the other hand, the pilots transmitted only in some dedicated subchannels are named comb or/and scattered ones [2,3]. It is worth mentioning that estimation precision of CSI is crucial not only for high data reception quality but also for effective users management in multiuser MIMO-OFDM systems [4].

The transmitted data are usually arranged in frames. Each frame starts with a synchronization preamble followed by one or more OFDM training symbols with block pilots for channel estimation. Comb pilots transmitted in the remaining OFDM symbols are necessary to keep synchronization and track channel variations using interpolation methods. An implementation example of the abovementioned frame structure can be found in 802.11 standards’ family [5,6]. A similar frame configuration seems to be appropriate in any simple OFDM transmission scheme including various proprietary solutions designed for IoT, industrial, or military-oriented applications. The good accuracy of the channel reconstruction with block pilots is one of the key factors that determines the overall transmission quality. Moreover, it can be even the sole channel reconstruction stage if the coherence time is larger than the frame duration (i.e., the channel is considered fixed during frame transmission). Considering the above system configuration, the block-pilot oriented methods implemented in the channel estimation stage are the main concern of this study. Because this initial processing is executed in receiver regardless of channel mobility, all tests were performed for fixed or slow-fading channel environments to simplify the simulations. This provides an in-depth properties understanding of the considered methods without excluding their application in mobile systems.

The simplest estimation procedure of the frequency channel characteristic is based on the LS (Least Squares) method [7]. Each subchannel transfer function is estimated independently using the pilots transmitted and received only in that subchannel, but its variance is high due to the presence of noise during signal transmission. To reduce the noise impact, the LMMSE (Linear Minimum Mean Square Error) approach can be applied, but it requires statistical knowledge about the reconstructed channel and higher computational resources [8]. Both methods originally reconstruct the channel in the frequency domain; therefore, their estimates can be directly applied to restore the transmitted symbols using the ZF approach.

In [9], a simple extension of the LS method is proposed to smooth its channel characteristic estimate in the frequency domain. It uses the fact that in a properly designed OFDM transmission, the maximum delay spread of a CIR (Channel Impulse Response) should not be longer then the guard period between symbols. Hence, the impulse response obtained as IFFT (Inverse FFT) of the LS result is divided into two time-continuous parts. The first one of the length of the guard period is the channel part and consists of the highest energy impulse response coefficients. The second, a noise part, is formed with the remaining coefficients that can be further used for noise variance estimation. Finally, only the channel part is used to recalculate the channel frequency characteristic. In FFT processing, the coefficients considered as noise are replaced with zero values thus contributing to the smoothing/averaging effect in the frequency domain. All channel refinement is performed in the time domain; therefore, the double use of the Fourier transform is the inherited feature of the methods known as DFT-based.

A detailed inspection of the channel part reveals that the number of significant impulse response coefficients, whose reconstruction is necessary, may be even smaller. Numerous studies indicate that it can be a typical situation especially in a wireless environment where only a few signal propagation paths transport most of the energy between the transmitter and receiver [10,11]. These types of channels have a sparse or, even more frequently, a compressible impulse response (see a comparison in Figure 1). The channel compressibility means that the impulse response coefficients form clusters around the dominant ones with rapidly decreasing amplitudes within the cluster. The compressible CIR can be successfully reconstructed supplementing the DFT-based estimation idea with iterative greedy algorithms belonging to the broad class of compressed sensing (CS) methods [12]. For example, a simple modification of the classical OMP method that forms groups of the reconstructed coefficients was proposed in [13]. Unfortunately, its quite good reconstruction efficiency comes at the cost of high computational complexity due to the inherent iterative procedure of the greedy algorithms.

This paper presents a solution that reduces the computational complexity and retains the group-based reconstruction principle. The proposition is a modification of the estimation procedures described in [14,15], where the restored CIR coefficients have been individually selected by comparing their energy to the power of noise. However, the channel compressibility nature has not been exploited there. The authors’ method is mathematically explained using a relation between estimation and approximation errors—the idea with a detailed description presented in [16]. A simulation comparative analysis with other popular DFT-based methods shows quality improvement of the uncoded transmission in terms of Bit Error Rate (BER) and Mean Square Error (MSE) measures in low and middle ranges of signal-to-noise ratio (SNR). On the other hand, a slightly worse performance for high SNR is not a serious obstacle to use this proposition because error correction codes commonly used in practical systems can compensate this loss. Especially implementation efficient LDPC (Low Density Parity Check) codes including nonbinary ones [17,18] and polar codes [19] are designated for use in modern communications systems.

The original contributions of the paper are as follows:

• to complement the DFT-based scheme with an additional moving average filter in order to perform the final CIR coefficients selection in groups (Section 3.3);
• to determine the moving average filter order using the relationship between approximation and estimation parts of the MSE error (Section 3.3);
• to perform a comparison analysis between the proposed method and the related ones using simulation tests in OFDM system model (Section 4)

In order to outline a relation between the proposed and the most closely related solutions (discussed here for comparison purposes),

Table 1. Two main distinctive features of the compared methods (+ present/− absent).

CIR Estimation Method	Single/In groups CIR Coefficients Reconstruction Approach	Noise/Residual Energy Based Error Metric
Classical DFT-based [9]	−/+	−/−
Modified DFT-based [14,15]	+/−	+/−
Classical CS [12]	+/−	−/+
Block CS [13]	−/+	−/+
Proposed	–/+	+/–

highlights their main distinctive features. All the methods indicated in the table belong to the DFT-based solutions with the initial Least Squares channel frequency response estimation but differ in the principle of the final CIR coefficients selection (single or in groups) and in the design of the error metrics (noise energy or residual error energy) used for the control of the selection procedure.

The remaining parts of the paper are organized as follows: The next section outlines an OFDM transmission model and data reception rules applied during the studies. Section 3 presents general principles of DFT-based methods and briefly describes their popular implementations that are based on appropriate selection of the LS impulse response coefficients. The mathematical explanation of the proposed modification is elaborated in the last part of this section. The simulation-based performance and comparison analyses are discussed in Section 4. Section 5 provides computational complexity considerations. The last section contains final remarks on the obtained results and possible development steps for future research.

Notation: Matrices/vectors are denoted by bold fonts (e.g., $\mathbf{C}$ or $\mathbf{c}$). Their elements are indexed by italic lowercase letters in parentheses (e.g., $\mathbf{c}\left(n\right)$). Superscript in parentheses denotes the current iteration number. The symbols ${(·)}^H$, $\left|\right|·{\left|\right|}_2^2$, and $\widehat{(·)}$ denote the conjugate transpose, the square of L2-norm, and an estimated value, respectively. Italic uppercase and lowercase letters represent signal variables in the frequency and time domain, respectively. Their particular frequency or time moments are described in parentheses (e.g., $H\left(n\right)$, $h\left(n\right)$). In addition, this type of font can describe system parameters (e.g., L, N, M, etc.). To avoid ambiguity, explanations in the text clearly indicate the meaning of the variables (whether this is a signal or a system parameter).

2. System and Channel Models

Consider an OFDM symbol in the baseband. This consists of N frequency samples of complex values usually taken from a QPSK or QAM constellation sets, which describe binary data encoding rules. The symbol is transformed by N-point IFFT into the time domain. Before it is transmitted through channel with the sampling period of $T_S$, the last M samples of the IFFT result are copied and appended to the beginning of the symbol to form a cyclic prefix. This facilitates symbol reception in the presence of linear channel distortion and time synchronization offset by preserving the mutual independence of the adjacent OFDM symbols in the time.

Let F consecutive OFDM symbols form a single information frame, starting with a preamble s of P-element pseudorandom synchronization sequence followed by a training OFDM symbol with block pilots in all subchannels (see the frame structure in Figure 2). The training symbol is used for the reconstruction of a linear multipath channel described by the impulse response h of length $L<M$ samples. Because, in these studies, the channel characteristic is considered constant during the single frame transmission, the obtained channel estimate is applied to decode the information in all OFDM data symbols of the given frame.

The transmission environment is characterized by an impulse response of the wireless channel within B = 10 MHz channel band conforming with the paths models for indoor and outdoor-to-indoor transmissions presented in [20]. It is worth mentioning that the time separation between the subsequent paths is sometimes under the resolution of the sampling period $T_S=1/B$. This enhances the compressibility feature of the channel and makes it impossible to resolve the individual paths in the estimated CIR. Other causes of compressibility are as follows: the delays of the individual paths differ slightly from multiples of the sampling period $T_S$, or they are all shifted in time to simulate a time synchronization offset.

The system model assumes operation in the baseband with perfect frequency synchronization (no interchannel interferences; therefore, subchannel orthogonality is preserved), whereas the frame timing $t_s$ is acquired using correlation between preamble in received signal y and the original synchronization sequence s:

$$t_s=\underset{n}{\arg \max }\left(\left|\sum _{i=0}^{P-1}{s}^{*}\left(i\right)y(n+i)\right|\right)-\Delta .$$

(1)

The parameter $\Delta$ slightly advances the moment when the frame starts to be analyzed. It reduces a risk of ISI (intersymbol interferences) because individual OFDM symbols extracted from the frame structure have their origin in the region of cyclic prefix that is free from preceding symbol interference.

Due to the subchannel orthogonality, the information transmitted in the respective subchannels can be processed independently and simultaneously. The signal received in kth subchannel is described as follows:

$$Y\left(k\right)=X\left(k\right)H\left(k\right)+W\left(k\right),$$

(2)

where X is the original information symbol, H is the frequency characteristic of a real channel, and W is a complex, zero-mean, white Gaussian noise signal. To restore the original information, ZF (Zero Forcing) or MMSE (Minimum Mean Square Error) equalization techniques can be applied. For the former, widely used due to its simplicity, the equalized data $\widehat{X}$ can be calculated in the receiver using the following formula:

$$\widehat{X}\left(k\right)=\frac{Y\left(k\right)}{\widehat{H}\left(k\right)}.$$

(3)

where $\widehat{H}$ represents a channel estimate determined with the block pilots. The above system description is outlined in Figure 3. In this paper, DFT-based channel estimation methods are under investigation. The next section presents their overview and the proposition of a new one.

3. DFT-Based LS Channel Estimation Methods

3.1. General Principle

The classical LS estimate ${\widehat{\mathbf{h}}}_{LS}$ of the CIR can be an entry point to development of various DFT-based methods. It is represented in matrix notation by the following $N\times 1$ vector:

$${\widehat{\mathbf{h}}}_{LS}=\frac{1}{N}{\mathbf{F}}^H{\mathbf{X}}_p^{-1}{\mathbf{Y}}_p,$$

(4)

where $\mathbf{F}$ is $N\times N$ DFT matrix with entries $\mathbf{F}(n,k)=e^{-j2\pi nk/N}$, ${\mathbf{X}}_p$ is the diagonal matrix of block pilot tones, and ${\mathbf{Y}}_p$ is vector of the received training symbol. According to the common rule of thumb, the pilot tones are unit amplitude.

The DFT-based methods are developed under the assumption that in a properly designed OFDM system, the length of the channel impulse response (L) should not be longer then length of the cyclic prefix (M). Thus, the number $(N-M)$ of remaining, nonzero amplitude coefficients in the reconstructed impulse response ${\widehat{\mathbf{h}}}_{LS}$ can be interpreted as a noise-induced distortion. Therefore, to improve the channel estimation, only M successive elements of ${\widehat{\mathbf{h}}}_{LS}$ that have the largest total energy should be considered as the ones used to determine the pure channel part of impulse response. In this article, special attention is directed toward methods where the estimation quality improvement is obtained by simple selection of the appropriate LS coefficients. The basic assignment complies with the following formula [9]:

$${\widehat{\mathbf{h}}}_C={\mathbf{I}}_{N\left\{\mathbb{M}\right\}}{\widehat{\mathbf{h}}}_{LS},$$

(5)

where ${\mathbf{I}}_{N\left\{\mathbb{M}\right\}}$ is a $N\times N$ diagonal matrix with the entries defined as follows:

$${\mathbf{I}}_{N\left\{\mathbb{M}\right\}}(n,n)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\hspace{1em}n\in \mathbb{M}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right..$$

(6)

Due to possible time offset in frame synchronization, taking into account the circular shift property of FFT, the set:

$$\mathbb{M}=\{m,m+1,\dots ,(m+M-1)\hspace{1em}\mathrm{mod}N\},$$

(7)

contains M successive integers modulo N and

$$m=\underset{n}{\arg \max }\left(\sum _{i=n}^{(n+M-1)\hspace{1em}modN}{\left|{\widehat{\mathbf{h}}}_{LS}\left(i\right)\right|}^2\right)$$

(8)

assures maximum energy of ${\mathbf{h}}_C$ out of all other shifts. The last processing step of the channel reconstruction is Fourier transform:

$${\widehat{\mathbf{H}}}_C=\mathbf{F}{\widehat{\mathbf{h}}}_C$$

(9)

to obtain the channel estimate ${\widehat{\mathbf{H}}}_C\equiv \widehat{\mathbf{H}}$ in the frequency domain (each entry of the vector $\widehat{\mathbf{H}}$ corresponds to the particular subchannel characteristic).

3.2. Improvements in the Related Methods

The main reason of the further modifications of the DFT-based methods is the observation that among M nonzero coefficients of ${\widehat{\mathbf{h}}}_C$, there are often still the ones with relatively low amplitudes. These can be probably other noise-like components inside the channel part, as can be inferred from the channel models with possible sparse or compressible impulse responses. Methods that replace their low amplitudes with zero values are the simplest form of the channel characteristic refinement. However, the issue is how to select the coefficients from ${\widehat{\mathbf{h}}}_C$ to form a new impulse response? Two popular approaches are briefly outlined below.

3.2.1. Methods Controlled with Noise Energy

These type of methods compare a noise-based threshold level $\alpha$ to the estimated energy of the particular impulse response coefficients ${\mathbf{h}}_C$ in order to select the ones with energy amplitudes above the threshold. Consequently, the new impulse response can be described as follows:

$${\widehat{\mathbf{h}}}_N={\mathbf{I}}_{N\left\{\mathbb{D}\right\}}{\widehat{\mathbf{h}}}_C,$$

(10)

where

$${\mathbf{I}}_{N\left\{\mathbb{D}\right\}}(n,n)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\hspace{1em}n\in \mathbb{D}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.,$$

(11)

and

$$\mathbb{D}=\left\{n|n\in \mathbb{M}\wedge \left(|{\widehat{\mathbf{h}}}_C{\left(n\right)|}^2\ge \alpha \right)\right\}.$$

(12)

In [14], the research on the optimal selection rule resulted in the following threshold value:

$$\alpha =2{\sigma }_W^2,$$

(13)

where

$${\sigma }_W^2=\frac{1}{N-M}\sum _{n\in \{0,1,\dots ,N-1\}\backslash \mathbb{M}}{\left|{\widehat{\mathbf{h}}}_{LS}\left(n\right)\right|}^2$$

(14)

is the variance of the estimated noise component in the time domain.

3.2.2. Methods Controlled with Residual Error Energy

Another approach to select the final set of the impulse response coefficients is based on the iterative testing of the residual error energy. This error is estimated as follows:

$${\mathbf{R}}^{\left(i\right)}=\mathbf{Y}-\mathbf{X}\mathbf{F}{\widehat{\mathbf{h}}}_R^{\left(i\right)}$$

(15)

and provides an instant but indirect information about reconstruction quality of the channel impulse response ${\mathbf{h}}_R$ in the ith iteration step.

Generally, the LS subvector ${\widehat{\mathbf{h}}}_R$ has nonzero values for the selected indices from the set $\mathbb{M}$. The specific number of the index in the ith iteration is determined according to the formula:

$$r^{\left(i\right)}=\underset{r\in \mathbb{M}}{\arg \max }\left(|{\mathbf{f}}_r^H{\mathbf{X}}^H{\mathbf{R}}^{(i-1)}|\right)$$

(16)

and then appended to the set of coefficients’ indices obtained in previous iterations. ${\mathbf{f}}_r$ denotes the rth column of the Fourier Transform matrix $\mathbf{F}$. The procedure of CIR estimation stops when the residual error energy drops below a predefined threshold. This working principle is typical for the OMP algorithm that belongs to the family of greedy methods in the compressed sensing domain [21]. If all N subchannels are active, the procedure can be significantly simplified. The nonzero part of ${\widehat{\mathbf{h}}}_R$ is formed by the smallest number of ${\widehat{\mathbf{h}}}_C$ coefficients with the largest amplitudes for which the residual error energy meets the following condition:

$$\left|\right|{\mathbf{R}}^{\left(i\right)}{\left|\right|}_2^2\le N\left(N{\sigma }_W^2\right),$$

(17)

where $N{\sigma }_W^2$ is the noise variance in the frequency domain.

The analysis conducted in [13] shows that the above simple selection rule may be inefficient in the case of compressible channel impulse response. The improvement introduced there relies on the idea that for such channels, the reconstructed impulse response coefficients should gather in groups around the dominant ones. The proposed modification has two distinct stages. In the first one, the largest elements of ${\widehat{\mathbf{h}}}_C\left(n\right)$ are selected until $\left|\right|{\mathbf{R}}^{\left(i\right)}{\left|\right|}_2^2\le 2N\left(N{\sigma }_W^2\right)$ is met [13]. Then, additional coefficients, adjacent to the dominant ones (selected in the current and previous iterations), are added. The final stop occurs when condition (17) or another one based on difference between the successive residual errors

$$\left|\right|{\mathbf{R}}^{\left(i\right)}-{\mathbf{R}}^{(i-1)}{\left|\right|}_2^2\le 10\left(N{\sigma }_W^2\right).$$

(18)

are met. The use of supplementary condition (18) can improve the accuracy of impulse response estimation but at the cost of increasing computational complexity of the already complex core of the greedy procedure.

3.3. Proposed Modification

Implementation simplicity is a big advantage of noise energy controlled method presented in Section 3.2.1. However, in the case of compressible channel impulse response, the individual decisions about assigning a given coefficient of ${\widehat{\mathbf{h}}}_C$ to the vector ${\widehat{\mathbf{h}}}_N$ (see Equation (10)) can be a shortcoming of the solution. Two unfavorable situations can be distinguished. In the first one, a coefficient is rejected only because its amplitude unexpectedly took a value below the reference threshold due to noise interference. In the second, the opposite one, when a large value of a wrongly assigned coefficient is caused by the influence of noise.

To minimize the risk of the above incorrect interpretations, a modification with the use of an additional moving average filter of length U is proposed in this article. The energy values of U-element groups of the subsequent coefficients of ${\widehat{\mathbf{h}}}_C$, obtained at the filter output using circular convolution, is tested against a new threshold level $\beta$. The coefficients of ${\mathbf{h}}_C$ that belong to the groups with total energy larger than the threshold $\beta$ are assigned to a new impulse response vector ${\mathbf{h}}_G$. Thus, not a single but a group of coefficients is selected to ${\mathbf{h}}_G$ after every positive verification against the threshold. Referring to the previous notations, the impulse response can be defined as follows:

$${\widehat{\mathbf{h}}}_G={\mathbf{I}}_{N\left\{\mathbb{G}\right\}}{\widehat{\mathbf{h}}}_C,$$

(19)

where

$${\mathbf{I}}_{N\left\{\mathbb{G}\right\}}(n,n)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\hspace{1em}n\in \mathbb{G}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.,$$

(20)

and

$$\mathbb{G}=\left\{n|n\in \mathbb{M}\wedge \sum _{i=n}^{(n+U-1)\hspace{1em}\mathrm{mod}N}{\left|{\widehat{\mathbf{h}}}_C\left(i\right)\right|}^2\ge \beta \right\}.$$

(21)

Next important issue is the determination of the threshold value $\beta$ and the length U. The former can be defined in relation to the value in (14). Because U impulse response coefficients are analyzed by the filter at the same time, a direct and straightforward proposition of the threshold level is:

$$\beta =2U{\sigma }_W^2.$$

(22)

This takes into account the cumulative effect of potential noise samples in the window of analysis. If the energy of U adjacent impulse response coefficients is less then (22), they all are considered as noise. Otherwise, it is possible that among them can be real channel coefficients so they are all appended to the reconstructed impulse response.

The choice of the length U requires a little bit more investigation. The research in [16] reveals that the best estimate of compressible signal in terms of MSE is its sparse approximation. This means that only large-amplitude components of the signal should be recovered, and their number is inversely related to noise power. Finally, the low amplitude ones are set to zero because if they were reconstructed, it would reduce the overall quality of the estimation. The above explanation implies that the window length U should be smaller for low SNR than in the opposite situation. A trade-off solution for U can be found using the relation between two error components—estimation error ${ϵ}_E$ and approximation error ${ϵ}_A$—that are responsible for the total MSE value of the channel impulse response ${\mathbf{h}}_U$ estimate, i.e.:

$$\mathrm{MSE}={ϵ}_E+{ϵ}_A.$$

(23)

Consider a hypothetical compressible impulse response ${\mathbf{h}}_C$ of length limited to M (please note that the remaining N-M coefficients are considered 0; therefore, they have no effect on further processing). Only U of its largest coefficients are recovered in a sparse counterpart ${\mathbf{h}}_U$. The estimation error ${ϵ}_E$ is defined as the variance of the LS estimation of ${\widehat{\mathbf{h}}}_U$. Owing to the subchannel orthogonality, its value can be determined as follows:

$${ϵ}_E=U{\sigma }_W^2,$$

(24)

where ${\sigma }_W^2$ is the noise power in the time domain. The approximation error defined as:

$${ϵ}_A=\sum _{n\in {\mathbb{U}}^{\prime }}{\left|{\mathbf{h}}_C\left(n\right)\right|}^2,$$

(25)

expresses how much the actual impulse response ${\mathbf{h}}_C$ differs from its sparse representation ${\mathbf{h}}_U$. The set ${\mathbb{U}}^{\prime }$ consists of $M-U$ indices of the CIR that are not recovered. Both components of MSE (23) are monotonic but in the opposite direction. If a longer and longer sparse approximation of ${\mathbf{h}}_C$ is selected, the approximation error decreases but the estimation one increases.

The length U that balances the contribution of both errors (${ϵ}_E={ϵ}_A$) is the target condition adopted in the proposed modification. Unfortunately, the actual values of the ${\mathbf{h}}_C$ are unknown. Hence, the estimated energy of every coefficient in ${\widehat{\mathbf{h}}}_C$ minus the noise variance is used instead. If the pilot symbols have unit amplitude, the LS estimation variance of the individual coefficient equals the noise variance [22] represented in this study by ${\sigma }_W^2$ (14). Using this, the approximation error can be estimated as follows:

$${ϵ}_A\approx \sum _{n\in {\mathbb{U}}^{\prime }}{\left|{\widehat{\mathbf{h}}}_C\left(n\right)\right|}^2-(M-U){\sigma }_W^2$$

(26)

Consequently, the following condition is proposed to find the suitable length U of the moving average filter:

$$U=\underset{u\in \mathcal{N}}{\min }\left(u\right|M{\sigma }_W^2\ge \sum _{i=1}^{M-u}|{\widehat{\mathbf{h}}}_C\left(m\left(i\right)\right){|}^2\wedge m\in {\mathbb{U}}^{\prime })$$

(27)

where $\mathcal{N}$ stands for the set of natural numbers and ${\mathbb{U}}^{\prime }$ is the set of $(M-u)$ indices pointing to the smallest amplitude of nonzero coefficients in the vector ${\widehat{\mathbf{h}}}_C$. Expression $m\left(i\right)$ represents the indices of ${\widehat{\mathbf{h}}}_C$, arranged in the order corresponding to the ascending amplitudes of the CIR coefficients, i.e., for $i=1$, the estimated channel coefficient ${\widehat{\mathbf{h}}}_C\left(m\left(1\right)\right)$ has the smallest amplitude, and vice versa, for $i=M-u$, the amplitude is the largest. An explanation for the above condition is as follows: if the inequality in (27) is not satisfied, it means that the sum on its right-hand side may include some coefficients which are significant enough in the impulse response reconstruction. The proof of inequality (27) is presented in Appendix A.

4. Numerical Experiments and Performance Analysis

4.1. Simulation Parameters

The proposed modification and the representative DFT-based reference methods were tested in a purpose-made OFDM modulated transmission system. The analysis was performed in baseband assuming perfect frequency synchronization. The frame synchronization was recovered using the standard preamble-based crosscorrelation technique (see Equation (1)), and the LS CIR estimate was the base for further processing both in the reference methods and in the proposed one. The frame synchronization fluctuations resulting in a cyclic shift of the CIR estimate were compensated by the detection of its the most significant part according to Formula (8) using a moving average filter of length M.

The block-pilot symbols were randomly generated from the QPSK unit-energy symbol constellation and transmitted in all N subchannels. The synchronization preamble was a double repetition of the sequence obtained as the result of IFFT transformation of an N-element random QPSK vector. The system noise was modeled as the AWGN.

The major assumptions of the system model are summarized in

Table 2. System model parameters.

Parameter	Value
Carrier frequency	$f_C$ = 2 GHz
Sampling frequency	$f_S$ = 10 MHz
Active subchannels	N = 128
FFT size	N = 128
Cyclic prefix length	M = 32 (samples)
Synchronization sequence	P = 256 (samples)
Synchronization factor	$\Delta =M/2$
Block-pilot OFDM symbols per frame	1
Pilot constellation	QPSK
Data OFDM symbols per frame	F = 50
Frame duration	$T_F=P+(M+N)F{T}_S=825.6$ μs
Data constellation	16-QAM, 64-QAM
Data frames per transmission	50
Maximum speed	v = 10 km/h
Doppler spread	$f_d=\frac{v}{c}{f}_C\approx 18.5$ Hz
Coherence time [23]	$T_C\approx \frac{0.423}{f_d}=22.9$ ms
SNR range	$〈0,25〉$ dB with 5 dB step

.

According to the assumed mobility parameter (maximum speed is v = 10 km/h), the channel characteristic can be considered fixed during frame transmission because the coherence time $T_C$ is significantly larger then the frame duration $T_F$. This communication scenario can be typical, for example, for sensor networks and related systems designed to handle at most pedestrian traffic (including recreational running too).

The benchmark channel models used in the simulation analysis are enumerated in

Table 3. Chanel models.

Channel	Average Relative Paths Delays [ns]	Average Paths Gains [dB]
Indoor A [20]	[0, 50, 110, 170, 290, 310]	[0, −3, −10, −18, −26, −32]
Indoor B [20]	[0, 100, 200, 300, 500, 700]	[0, −3.6, −7.2, −10.8, −18, −25.2]
Indoor B Shift	[50, 150, 250, 350, 450, 750]	[0, −3.6, −7.2, −10.8, −18, −25.2]
Outdoor-To-Indoor A [20]	[0, 110, 190, 410]	[0, −9.7, −19.2, −22.8]

. They are a part of the recommendation for the evaluation of radio transmission systems operating in 2 GHz band [20]. The selected channels represent propagation environments typical for indoor transmission (Indoor A and B) and outdoor-to-indoor transmission when the base station is located outdoors. They cover cases of the most frequently occurring propagation conditions including also indoor-to-outdoor transmission scenarios on the basis of channel reciprocity property [24]. Every channel model is characterized by a tapped-delay line model in the time domain. In the mentioned recommendation, the particular taps are the coefficients of CIR directly. In the simulations presented here, a more realistic scenario was assumed. The taps were considered propagation paths, and the target CIR was a result of low pass filtering in baseband. The paths delay varied randomly about 3% around average value according to the model definitions. Additionallyn the shifted version of Indoor B model was proposed to enhance channel compressibility—the sampling moments (i.e., integer multiples od $T_S$) were not consistent with the paths delay.

The quality metrics, MSE and BER, were evaluated in Monte Carlo experiments using 300 independent uncoded baseband transmissions over CIRs randomly generated by MATLAB built-in function. There was assumed a single-channel realization per transmission.

During the simulation, the following channel estimation methods were analyzed (with the labels used in figures’ legends and the rest of the paper for the sake of clarity):

CSI 

—full CSI available;

LS 

  —classic LS estimation method [7];

C 

    —the basic DFT-based method. The LS impulse response limited to M samples [9];

N 

    —noise energy-controlled DFT-based method with individual assignment of impulse response coefficients [14];

R 

    —residual error-controlled DFT-based method with impulse response reconstruction in groups (the authors’ modification proposed in [13]);

G 

    —the modification proposed in this article.

4.2. Results and Discussion

The simulation results show that coefficient clustering mechanism during impulse response reconstruction (in methods R and G) has a generally positive influence on the transmission quality. The reference classic LS estimate has the worst performance in terms of MSE and BER among tested methods, as might be expected, but this result was an entry point of those modifications. The DFT-based approach, by simply removing the noise part from the LS impulse response, clearly improves estimation and transmission quality. The gain is about 1.5–2.0 dB for BER = 0.01 in relation to LS method. However, this simplest solution (C) can be improved. Both authors’ propositions (R and G) have the competitive quality measures in low and mid range of SNR (up to about 20 dB). Their performance results are graphically depicted in Figure 4, Figure 5, Figure 6 and Figure 7. The performance of the proposed method G shows quality improvement comparable to the one existing between C and N. This is more apparent for the MSE measure. Looking at BER, the additional gain in regard to the N is about 0.2–0.3 dB, which translates into approximately 5–7 percent saving in transmission energy. More specific values for BER = 0.1 and 0.01 are listed in

Table 4. SNR gap between N and G methods ($SN{R}_N-SN{R}_G$) for the particular BERs and channel models with a resolution 0.1 dB.

BER	Indoor A	Indoor B	Indoor B Shift	Outdoor-to-Indoor A
0.1	0.3 dB	0.2 dB	0.2 dB	0.3 dB
0.01	0.1 dB	0.2 dB	0 dB	0.2 dB

.

The main part of the proposed method is a moving average filter with dynamic adjusted length U.

Table 5. Median values of the length U of the additional averaging filter for the particular SNRs and channel models.

SNR	Indoor A	Indoor B	Indoor B Shift	Outdoor-to-Indoor A
0 dB	5	5	7	4
5 dB	6	5	9	4
10 dB	8	6	11	5
15 dB	11	7	14	6
20 dB	14	7	15	7
25 dB	15	8	17	8

depicts median values of U obtained during the simulations for the particular SNR and channel models.

As transmission conditions improve, the filter length U increases. The maximum possible value is $U_{max}=M$. In good transmission conditions (high SNR values), the estimation quality for G and N solutions are roughly similar. Because the N method does not have to implement the additional averaging filter (detailed complexity issues are elaborated in next section), a noise-controlled automatic switching between these methods could be considered, but such a procedure is out of the scope of this study.

Another interesting conclusion can be drawn comparing the simulation results between the proposed and R methods. The estimation quality is almost the same in the entire tested SNR range. Moreover, the performance of R can be slightly better for a specific channel, as presented in

Table 6. SNR gap between G and R methods ($SN{R}_G-SN{R}_R$) for the particular BERs and channel models with a resolution 0.01 dB.

BER	Indoor A	Indoor B	Indoor B Shift	Outdoor-to-Indoor A
0.1	0 dB	0.05 dB	0.01 dB	0.1 dB
0.01	0.04 dB	0.07 dB	0.11 dB	0.05 dB

. Nevertheless, a decisive advantage of the proposed method is the lower computational complexity manifested in particular by the absence of multiplication operations.

5. Computational Complexity Issues

In order to evaluate the numerical complexity of the proposed method, a relative analysis to references methods is presented. A justification of this methodology is that all the methods (beside LS) have the same preliminary processing as C method. These common processing steps are as follows:

C1

LS estimate of the frequency characteristics. It requires N complex multiplications;

C2

CIR calculation using N-point IFFT. The complexity is proportional to $\mathcal{O}\left(N\log N\right)$;

C3

Initial selection of M consecutive CIR coefficients according to energy maximization rule. $2N$ real number multiplications and N additions for the coefficients energy calculation, and $2(N-1)$ additions and $N+M$ comparisons for the final coefficients selection are required;

C4

Noise power estimation according to (14) ($N-M-1$ real number additions);

The proposed CIR selection method adds the following processing steps which all are performed for real numbers:

S1

Coefficients sorting. The coefficients of ${\widehat{h}}_C$ are sorted in terms of their energy. The maximum time complexity depends on implemented sorting methods and varies from $\mathcal{O}\left(M\log M\right)$ to $\mathcal{O}\left(M^2\right)$ [25];

S2

Length U of the moving average filter. This step is performed iteratively until inequality (27) is met. Every iteration consumes a single addition and comparison. The number of iterations is proportional to SNR and corresponds to the length U.

S3

Fine coefficients selection out of M-element set. This processing step implements (21). $2(M-1)$ additions, and the $M+U$ comparisons are required.

It is worth noting that in the proposed G solution, no multiplications are required (besides simple thresholds determination in Equations (22) and (27). This is a clear complexity reduction compared to the quality-related R method, where two residual errors along with their energies (Equations (17) and (18)) are iteratively calculated. In addition, the processing step S1 has to be implemented there.

The efficiency of the sorting operation is crucial in both discussed approaches which rely on greedy CS methods and the MSE components analysis. This step needs some sort of additional computational resources (e.g., time and memory space) that depend on an implementation of a target sorting algorithm in the system. However, further implementations studies are not considered here, because this problem is strongly related to a hardware and software platform design of the receiver (the detailed design issues are beyond the scope of this paper).

6. Conclusions

The new CIR estimation procedure for OFDM transmission was proposed and evaluated in the paper. The clustered coefficients reconstruction was the main idea of the method motivated by the compressibility feature of CIR. The relation between the estimation and approximation components of MSE measure was the entry point to develop the reconstruction procedure. A moving average filter with automatically adjusted length was used as the main processing component. The simulation tests with the benchmark channel models show that for the proposed solution it is possible to achieve the transmission quality results comparable or slightly worse to those obtained for more numerically complex method based on the greedy approach (R) and better than in the case of individual coefficients selection (N). The estimation gain in the latter case allows the transmitter to save energy in range of 5–7%.

Future works can cover detailed implementation analysis focusing especially on sorting procedures to fit their time and space complexity to a particular system design platform. In addition, a switching mechanism can be considered that changes the CIR estimation algorithm from the clustered coefficients selection in low and medium SNRs to individual selection in high SNRs. Other interesting research directions are to test the adaptability of the DFT-based channel estimation approach to operate in systems with various CIR characterization and arbitrary subchannel allocation. The latter problem is relevant in OFDM-based cognitive radio systems, where spectrum sensing is performed prior to data transmission to detect unused subchannels [26].