---
abstract: 'This paper presents a novel efficient receiver design for wireless communication systems that incorporate orthogonal frequency division multiplexing (OFDM) transmission. The proposed receiver does not require channel estimation or equalization to perform coherent data detection. Instead, channel estimation, equalization, and data detection are combined into a single operation, and hence, the detector is denoted as a direct data detector ($D^{3}$). The performance of the proposed system is thoroughly analyzed theoretically in terms of bit error rate (BER), and validated by Monte Carlo simulations. The obtained theoretical and simulation results demonstrate that the BER of the proposed $D^{3}$ is only $3$ dB away from coherent detectors with perfect knowledge of the channel state information (CSI) in flat fading channels, and similarly in frequency-selective channels for a wide range of signal-to-noise ratios (SNRs). If CSI is not known perfectly, then the $D^{3}$ outperforms the coherent detector substantially, particularly at high SNRs with linear interpolation. The computational complexity of the $D^{3}$ depends on the length of the sequence to be detected, nevertheless, a significant complexity reduction can be achieved using the Viterbi algorithm.'
author:
- 'A. Saci, , A. Al-Dweik, , A. Shami, [^1][^2][^3]'
title: Direct Data Detection of OFDM Signals Over Wireless Channels
---

OFDM, fading channels, data detection, Viterbi, sequence detection, channel estimation, equalization.

Introduction
============

Consequently, a low-complexity single-tap equalizer can be utilized to eliminate the impact of the multipath fading channel. Under such circumstances, the OFDM demodulation process can be performed once the fading parameters at each subcarrier, commonly denoted as channel state information (CSI), are estimated.

In general, channel estimation can be classified into blind [@One-Shot-CFO-2014]-[@blind-massive-mimo-acd], and pilot-aided techniques [@Robust-CE-OFDM-2015]-[@pilot-ce-pilot-freq-domain]. Blind channel estimation techniques are spectrally efficient because they do not require any overhead to estimate the CSI, nevertheless, such techniques have not yet been adopted in practical OFDM systems. Conversely, pilot-based CSI estimation is preferred for practical systems, because typically it is more robust and less complex. In pilot-based CSI estimation, the pilot symbols are embedded within the subcarriers of the transmitted OFDM signal in time and frequency domain; hence, the pilots form a two dimensional (2-D) grid [@LTE-A]. The channel response at the pilot symbols can be obtained using the least-squares (LS) frequency domain estimation, and the channel parameters at other subcarriers can be obtained using various interpolation techniques [@Rayleigh-Ricean-Interpolation-TCOM2008]. Optimal interpolation requires a 2-D Wiener filter that exploits the time and frequency correlation of the channel, however, it is substantially complex to implement [@Interpolation-TCOM-2010], [@Wiener]. The complexity can be reduced by decomposing the 2-D interpolation process into two cascaded 1-D processes, and then, using less computationally-involved interpolation schemes [@Adaptive-Equalization-IEEE-Broadcasting-2008], [@Comp-Pilot-VTC2007]. Low complexity interpolation, however, is usually accompanied by error rate performance degradation [@Comp-Pilot-VTC2007]. It is also worth noting that most practical OFDM-based systems utilize a fixed grid pattern structure [@LTE-A].

Once the channel parameters are obtained for all subcarriers, the received samples at the output of the fast Fourier transform (FFT) are equalized to compensate for the channel fading. Fortunately, the equalization for OFDM is performed in the frequency domain using single-tap equalizers. The equalizer output samples, which are denoted as the decision variables, will be applied to a maximum likelihood detector (MLD) to regenerate the information symbols.

In addition to the direct approach, several techniques have been proposed in the literature to estimate the CSI or detect the data symbols indirectly, by exploiting the correlation among the channel coefficients. For example, the per-survivor processing (PSP) approach has been widely used to approximate the maximum likelihood sequence estimator (MLSE) for coded and uncoded sequences [@PSP-Raheli], [@PSP-Zhu], [@Rev-1]. The PSP utilizes the Viterbi algorithm (VA) to recursively estimate the CSI without interpolation using the least mean squares (LMS) algorithm. Although the PSP provides superior performance when the channel is flat over the entire sequence, its performance degrades severely if this condition is not satisfied, even when the LMS step size is adaptive [@PSP-Zhu]. Multiple symbol differential detection (MSDD) can be also used for sequence estimation without explicit channel estimation. In such systems, the information is embedded in the phase difference between adjacent symbols, and hence, differential encoding is needed. Although differential detection is only $3$ dB worse than coherent detection in flat fading channels, its performance may deteriorate significantly in frequency-selective channels [@Divsalar], [@Diff-Xhang]. Consequently, Wu and Kam [@Wu; @2010] proposed a generalized likelihood ratio test (GLRT) receiver whose performance without CSI is comparable to the coherent detector. Although the GLRT receiver is more robust than differential detectors in frequency-selective channels, its performance is significantly worse than coherent detectors.

The signal at channel output is estimated with a minimum mean square error (MMSE) estimator from the knowledge of the received signal and the second order statistics of the channel and noay provide BER that is about $1$ dB from the ML coherent detector in flat fading channels but at the expense of a large number of pilots. Decision-directed techniques can also be used to avoid conventional channel estimation. For example, the authors in [@Saci-Tcom] proposed a hybrid frame structure that enables blind decision-directed channel estimation. Although the proposed system manages to offer reliable channel estimates and BER in various channel conditions, the system structure follows the typical coherent detector design where equalization and symbol detection are required.

Motivation and Key Contributions
--------------------------------

Unlike conventional OFDM detectors, this work presents a new detector to regenerate the information symbols directly from the received samples at the FFT output, which is denoted as the direct data detector ($D^{3})$. By using the $D^{3}$, there is no need to perform channel estimation, interpolation, equalization, or symbol decision operations. The $D^{3}$ exploits the fact that channel coefficients over adjacent subcarriers are highly correlated and approximately equal. Consequently, the $D^{3}$ is derived by minimizing the difference between channel coefficients of adjacent subcarriers. The main limitation of the $D^{3}$ is that it suffers from a phase ambiguity problem, which can be solved using pilot symbols, which are part of a transmission frame in most practical standards [@WiMax], [@LTE-A]. To the best of the authors’ knowledge, there is no work reported in the published literature that uses the proposed principle.

The $D^{3}$ performance is evaluated in terms of complexity, computational power, and bit error rate (BER), where analytic expressions are derived for several channel models and system configurations. The $D^{3}$ BER is compared to other widely used detectors such as the maximum likelihood (ML) coherent detector [@Proakis-Book-2001] with perfect and imperfect CSI, multiple symbol differential detector (MSDD) [@Divsalar], the ML sequence detector (MLSD) with no CSI [@Wu; @2010], and the per-survivor processing detector [@PSP-Raheli]. The obtained results show that the $D^{3}$ is more robust than all the other considered detectors in various cases of interest, particularly in frequency-selective channels at moderate and high SNRs. Moreover, the computational power comparison shows that the $D^{3}$ requires less than $35\%$ of the computational power required by the ML coherent detector.

Paper Organization and Notations
--------------------------------

The rest of this paper is organized as follows. The OFDM system and channel models are described in Section \[sec:Signal-and-Channel\]. The proposed $D^{3}$ is presented in Section \[sec:Proposed-System-Model\], and the efficient implementation of the $D^{3}$ is explored in Section \[sec:Efficient-Implementation-of-D3\]. The system error probability performance analysis is presented in Section \[sec:System-Performance-Analysis\]. Complexity analysis of the conventional pilot based OFDM and the $D^{3}$ are given in Section \[sec:Complexity-Analysis\]. Numerical results are discussed in Section \[sec:Numerical-Results\], and finally, the conclusion is drawn in Section \[sec:Conclusion\].

In what follows, unless otherwise specified, uppercase boldface and blackboard letters such as $\mathbf{H}$ and $\mathbb{H}$, will denote $N\times N$ matrices, whereas lowercase boldface letters such as $\mathbf{x}$ will denote row or column vectors with $N$ elements. Uppercase, lowercase, or bold letters with a tilde such as $\tilde{d}$ will denote trial values, and symbols with a hat, such as $\hat{\mathbf{x}}$, will denote the estimate of $\mathbf{x}$. Letters with apostrophe such as $\acute{v}$ are used to denote the next value, i.e., $\acute{v}\triangleq v+1$. Furthermore, $\mathrm{E}\left[\cdot\right]$ denotes the expectation operation.

Signal and Channel Models \[sec:Signal-and-Channel\]
====================================================

Consider an OFDM system with $N$ subcarriers modulated by a sequence of $N$ complex data symbols $\mathbf{d}=[d_{0}$, $d_{1}$, $....$, $d_{N-1}]^{T}$. The data symbols are selected uniformly from a general constellation such as $M$-ary phase shift keying (MPSK) or quadrature amplitude modulation (QAM). In conventional pilot-aided OFDM systems [@IEEE-AC], $N_{P}$ of the subcarriers are allocated for pilot symbols, which can be used for channel estimation and synchronization purposes. The modulation process in OFDM can be implemented efficiently using an $N$-point inverse FFT (IFFT) algorithm, where its output during the $\ell$th OFDM block can be written as $\mathbf{x(\ell)=F}^{H}\mathbf{d(\ell)}$ where $\mathbf{F}$ is the normalized $N\times N$ FFT matrix, and hence, $\mathbf{F}^{H}$ is the IFFT matrix. To simplify the notation, the block index $\ell$ is dropped for the remaining parts of the paper unless it is necessary to include it. Then, a CP of length $N_{\mathrm{CP}}$ samples, no less than the channel maximum delay spread ($\mathcal{D}_{\mathrm{h}}$), is appended to compose the OFDM symbol with a total length $N_{\mathrm{t}}=N+N_{\mathrm{CP}}$ samples and duration of $T_{\mathrm{t}}$s.

At the receiver front-end, the received signal is down-converted to baseband and sampled at a rate $T_{\mathrm{s}}=T_{\mathrm{t}}/N_{\mathrm{t}}$. In this work, the channel is assumed to be composed of $\mathcal{D}_{\mathrm{h}}+1$ independent multipath components each of which has a gain $h_{m}\sim\mathcal{CN}\left(0,2\sigma_{h_{m}}^{2}\right)$ and delay $m\times T_{\mathrm{s}}$, where $m\in\{0$, $1$,$...$, $\mathcal{D}_{\mathrm{h}}\}$. A quasi-static channel is assumed throughout this work, and thus, the channel taps are considered constant over one OFDM symbol, but they may change over two consecutive symbols. Therefore, the received sequence after dropping the CP samples and applying the FFT can be expressed as, $$\mathbf{r}=\mathbf{Hd+w}\label{eq:rx_sig_FD}$$ where $\left\{ \mathbf{r,w}\right\} \in\mathbb{C}^{N\times1}$, $w_{v}\sim\mathcal{CN}\left(0\text{, }2\sigma_{w}^{2}\right)$ is the additive white Gaussian noise (AWGN) vector and $\mathbf{\mathbf{H}}$ denotes the channel frequency response (CFR) $$\mathbf{\mathbf{H}}=\text{diag}\left\{ \left[H_{0}\text{, }H_{1}\text{,}\ldots\text{, }H_{N-1}\right]\right\} .$$ By noting that $\mathbf{r|}_{\mathbf{H,d}}\sim\mathcal{CN}\left(\mathbf{Hd}\text{, }2\sigma_{w}^{2}\mathbf{I}_{N}\right)$ where $\mathbf{I}_{N}$ is an $N\times N$ identity matrix, then it is straightforward to show that the MLD can be expressed as $$\mathbf{\hat{d}}=\arg\text{ }\min_{\tilde{\mathbf{d}}}\text{ }\left\Vert \mathbf{r-H}\tilde{\mathbf{d}}\right\Vert ^{2}\label{E-MLD-01}$$ where $\left\Vert \mathbf{\cdot}\right\Vert $ denotes the Euclidean norm, and $\tilde{\mathbf{d}}=\left[\tilde{d}_{0}\text{, }\tilde{d}_{1}\text{,}\ldots\text{, }\tilde{d}_{N1}\right]^{T}$ denotes the trial values of $\mathbf{d}$. As can be noted from (\[E-MLD-01\]), the MLD requires the knowledge of $\mathbf{\mathbf{H}}$. Moreover, because (\[E-MLD-01\]) describes the detection of more than one symbol, it is typically denoted as maximum likelihood sequence detector (MLSD). If the elements of $\mathbf{d}$ are independent, the MLSD can be replaced by a symbol-by-symbol MLD $$\hat{d}_{v}=\arg\text{ }\min_{\tilde{d}_{v}}\text{ }\left\vert r_{v}\mathbf{-}H_{v}\tilde{d}_{v}\right\vert ^{2}\text{.}\label{E-MLD-02}$$ Since perfect knowledge of $\mathbf{H}$ is infeasible, an estimated version of $\mathbf{H}$, denoted as $\hat{\mathbf{H}}$, can be used in (\[E-MLD-01\]) and (\[E-MLD-02\]) instead of $\mathbf{H}$**.** Another possible approach to implement the detector is to equalize $\mathbf{r}$, and then use a symbol-by-symbol MLD. Therefore, the equalized received sequence can be expressed as, $$\check{\mathbf{r}}=\left[\hat{\mathbf{H}}^{H}\hat{\mathbf{H}}\right]^{-1}\hat{\mathbf{H}}^{H}\mathbf{r}$$ and $$\hat{d}_{v}=\arg\min_{\tilde{d}_{v}}\left\vert \check{r}_{v}-\tilde{d}_{v}\right\vert ^{2}\text{, }\forall v\text{.}$$

It is interesting to note that solving (\[E-MLD-01\]) does not necessarily require the explicit knowledge of $\mathbf{H}$ under some special circumstances. For example, Wu and Kam [@Wu; @2010] noticed that in flat fading channels, i.e., $H_{v}=H$ $\forall v$, it is possible to detect the data symbols using the following MLSD, $$\mathbf{\hat{d}}=\arg\text{ }\max_{\tilde{\mathbf{d}}}\text{ }\frac{\left\vert \tilde{\mathbf{d}}^{H}\mathbf{r}\right\vert ^{2}}{\parallel\tilde{\mathbf{d}}\Vert}.\label{E-Wu}$$ Although the detector described in (\[E-Wu\]) is efficient in the sense that it does not require the knowledge of $\mathbf{H}$, its BER is very sensitive to the channel variations.

Proposed $D^{3}$ System Model\[sec:Proposed-System-Model\]
==========================================================

One of the distinctive features of OFDM is that its channel coefficients over adjacent subcarriers in the frequency domain are highly correlated and approximately equal. The correlation coefficient between two adjacent subcarriers can be defined as $$\begin{aligned}
\varrho_{f} & \triangleq & \mathrm{E}\left[H_{v}H_{\acute{v}}^{\ast}\right]\nonumber \\
 & = & \mathrm{E}\left[\sum_{n=0}^{\mathcal{D}_{\mathrm{h}}}h_{n}e^{-j2\pi\frac{nv}{N}}\sum_{m=0}^{\mathcal{D}_{\mathrm{h}}}h_{m}^{\ast}e^{j2\pi\frac{m\acute{v}}{N}}\right]=\sum_{m=0}^{\mathcal{D}_{\mathrm{h}}}\sigma_{h_{m}}^{2}e^{j2\pi\frac{m}{N}}\label{eq:rho-f}\end{aligned}$$ where $\sigma_{h_{m}}^{2}=\mathrm{E}\left[\left\vert h_{m}\right\vert ^{2}\right]$. The difference between two adjacent channel coefficients is $$\Delta_{f}=\mathrm{E}\left[H_{v}-H_{\acute{v}}\right]=\mathrm{E}\left[\sum_{m=0}^{\mathcal{D}_{\mathrm{h}}}h_{n}e^{-j2\pi\frac{mv}{N}}\left(1-e^{-j2\pi\frac{m}{N}}\right)\right]$$ For large values of $N$, it is straightforward to show that $\varrho_{f}\rightarrow1$ and $\Delta_{f}\rightarrow0$. Similar to the frequency domain, the time domain correlation defined according to the Jakes’ model can be computed as [@Jakes-Model], $$\varrho_{t}=\mathrm{E}\left[H_{v}^{\ell}\left(H_{v}^{\acute{\ell}}\right)^{\ast}\right]=J_{0}\left(2\pi f_{d}T_{\mathrm{s}}\right)\label{eq:rho-t}$$ where $J_{0}\left(\cdot\right)$ is the Bessel function of the first kind and $0$ order, $f_{d}$ is the maximum Doppler frequency. For large values of $N$, $2\pi f_{d}T_{\mathrm{s}}\ll1$, and hence $J_{0}\left(2\pi f_{d}T_{\mathrm{s}}\right)\approx1$, and thus $\varrho_{t}\approx1$. Using the same argument, the difference in the time domain $\Delta_{t}\triangleq\mathrm{E}\left[H_{v}^{\ell}-H_{v}^{\acute{\ell}}\right]\approx0$. Although the proposed system can be applied in the time domain, frequency domain, or both, the focus of this work is the frequency domain.

Based on the aforementioned properties of OFDM, a simple approach to extract the information symbols from the received sequence $\mathbf{r}$ can be designed by minimizing the difference of the channel coefficients between adjacent subcarriers, which can be expressed as $$\mathbf{\hat{d}}=\arg\min_{\tilde{\mathbf{d}}}\sum_{v=0}^{N-2}\left\vert \frac{r_{v}}{\tilde{d}_{v}}-\frac{r_{\acute{v}}}{\tilde{d}_{\acute{v}}}\right\vert ^{2}.\label{E-DDD-00}$$ As can be noted from (\[E-DDD-00\]), the estimated data sequence $\mathbf{\hat{d}}$ can be obtained without the knowledge of $\mathbf{H}$. Moreover, there is no requirement for the channel coefficients over the considered sequence to be equal, and hence, the $D^{3}$ should perform fairly well even in frequency-selective fading channels. Nevertheless, it can be noted that (\[E-DDD-00\]) does not have a unique solution because $\mathbf{d}$ and $-\mathbf{d}$ can minimize (\[E-DDD-00\]). To resolve the phase ambiguity problem, one or more pilot symbols can be used as a part of the sequence $\mathbf{d}$**.** In such scenarios, the performance of the $D^{3}$ will be affected indirectly by the frequency selectivity of the channel because the capability of the pilot to resolve the phase ambiguity depends on its fading coefficient. Another advantage of using pilot symbols is that it will not be necessary to detect the $N$ symbols simultaneously. Instead, it will be sufficient to detect $\mathcal{K}$ symbols at a time, which can be exploited to simplify the system design and analysis.

Using the same approach of the frequency domain, the $D^{3}$ can be designed to work in the time domain as well by minimizing the channel coefficients over two consecutive subcarriers, i.e., two subcarriers with the same index over two consecutive OFDM symbols, which is also applicable to single carrier systems. It can be also designed to work in both time and frequency domains, where the detector can be described as $$\mathbf{\hat{D}}_{\mathcal{L}\text{,}\mathcal{K}}\mathbf{=}\arg\min_{\mathbf{\tilde{\mathbf{D}}}_{\mathcal{L}\text{,}\mathcal{K}}}\text{ }J\left(\tilde{\mathbf{D}}_{\mathcal{L}\text{,}\mathcal{K}}\right)\label{eq:opt-D}$$ where $\mathbf{D}_{\mathcal{L}\text{,}\mathcal{K}}$ is an $\mathcal{L}\times\mathcal{K}$ data matrix, $\mathcal{L}$ and $\mathcal{K}$ are the time and frequency detection window size, and the objective function $J\left(\tilde{\mathbf{D}}\right)$ is given by $$J\left(\tilde{\mathbf{D}}_{\mathcal{L}\text{,}\mathcal{K}}\right)=\sum_{\ell=0}^{\mathcal{L}-1}\sum_{v=0}^{\mathcal{K}-2}\left\vert \frac{r_{v}^{\ell}}{\tilde{d}_{v}^{\ell}}-\frac{r_{\acute{v}}^{\ell}}{\tilde{d}_{\acute{v}}^{\ell}}\right\vert ^{2}+\left\vert \frac{r_{v}^{\ell}}{\tilde{d}_{v}^{\ell}}-\frac{r_{v}^{\acute{\ell}}}{\tilde{d}_{v}^{\acute{\ell}}}\right\vert ^{2}\text{.}\label{eq:objective-function}$$ For example, if the detection window size is chosen to be the LTE resource block, then, $\mathcal{L}=14$ and $\mathcal{K=}12$. Moreover, the system presented in can be extended to the multi-branch receiver scenarios, single-input multiple-output (SIMO) as, $$\begin{aligned}
\hat{\mathbf{D}} & =\arg\min_{\tilde{\mathbf{d}}}\sum_{n=1}^{\mathcal{N}}\sum_{\ell=0}^{\mathcal{L}-1}\sum_{v=0}^{\mathcal{K}-2}\left\vert \frac{r_{v}^{\ell,n}}{\tilde{d}_{v}}-\frac{r_{\acute{v}}^{\ell,n}}{\tilde{d}_{\acute{v}}^{\ell}}\right\vert ^{2}+\left\vert \frac{r_{v}^{\ell,n}}{\tilde{d}_{v}^{\ell}}-\frac{r_{v}^{\acute{\ell},n}}{\tilde{d}_{v}^{\acute{\ell}}}\right\vert ^{2}\end{aligned}$$ where $\mathcal{N}$ is the number of receiving antennas.

Efficient Implementation of $D^{3}$\[sec:Efficient-Implementation-of-D3\]
=========================================================================

It can be noted from and that solving for $\hat{\mathbf{D}}$, given that $N_{P}$ pilot symbols are used, requires an $M^{\mathcal{K}\mathcal{L-}N_{P}}$ trials if brute force search is adopted, which is prohibitively complex, and thus, reducing the computational complexity is crucial.

![Example of a 1-D segmentation over the frequency domain for an LTE-A resource block.\[fig:2D-to-1D\] ](graphics/fig_01_digram_1D_sigmentation_modified)

\[subsec:The-Viterbi-Algorithm\]The Viterbi Algorithm (VA)
----------------------------------------------------------

By noting that the expression in (\[E-DDD-00\]) corresponds to the sum of correlated terms, which can be modeled as a first-order Markov process, then MLSD techniques such as the VA can be used to implement the $D^{3}$ efficiently. For example, the trellis diagram of the VA with binary phase shift keying (BPSK) is shown in Fig. \[fig:Viterbi-D3\], and can be implemented as follows:

1.  Initialize the path metrics $\left\{ \Gamma_{0}^{U},\acute{\Gamma}_{0}^{U},\Gamma_{0}^{L},\acute{\Gamma}_{0}^{L}\right\} =0$, where $U$ and $L$ denote the upper and lower branches, respectively. Since BPSK is used, the number of states is $2$.

2.  Initialize the counter, $c=0$.

3.  Compute the branch metric $J_{m,n}^{c}=\left\vert \frac{rc}{m}-\frac{r_{\acute{c}}}{n}\right\vert ^{2}$, where $m$ is current symbol index, $m=0\rightarrow\tilde{d}=-1$, and $m=1\rightarrow\tilde{d}=1$, and $n$ is the next symbol index using the same mapping as $m$.

4.  Compute the path metrics using the following rules, $$\begin{array}{ccc}
    \Gamma_{\acute{c}}^{U}=\min\left[\Gamma_{c}^{U}\text{, }\acute{\Gamma}_{c}^{U}\right]+J_{00}^{c} &  & \Gamma_{\acute{c}}^{L}=\min\left[\Gamma_{c}^{L}\text{, }\acute{\Gamma}_{c}^{L}\right]+J_{01}^{c}\\
    \acute{\Gamma}_{\acute{c}}^{U}=\min\left[\Gamma_{c}^{U}\text{, }\acute{\Gamma}_{c}^{U}\right]+J_{10}^{c} &  & \acute{\Gamma}_{\acute{c}}^{L}=\min\left[\Gamma_{c}^{L}\text{, }\acute{\Gamma}_{c}^{L}\right]+J_{11}^{c}
    \end{array}$$

5.  Track the surviving paths, $2$ paths in the case of BPSK.

6.  Increase the counter, $c=c+1$.

7.  if $c=\mathcal{K}$, the algorithm ends. Otherwise, go to step 3.

![Trellis diagram of the $D^{3}$ detector for BPSK.\[fig:Viterbi-D3\]](graphics/fig_02_digram_trellis)

It is worth mentioning that placing a pilot symbol at the edge of a segment terminates the trellis. To simplify the discussion, assume that the pilot value is $-1$, and thus we compute only $J_{0,0}$ and $J_{1,0}$. Consequently, long data sequences can be divided into smaller segments bounded by pilots, which can reduce the delay by performing the detection over the sub-segments in parallel without sacrificing the error rate performance.

1.  2.  

System Design with an Error Control Coding
------------------------------------------

Forward error correction (FEC) coding can be integrated with the $D^{3}$ in two ways, based on the decoding process, i.e., hard or soft decision decoding. For the hard decision decoding, the integration of FEC coding is straightforward where the output of the $D^{3}$ is applied directly to the hard decision decoder (HDD).

For the soft decision decoding, we can exploit the coded data to enhance the performance of the $D^{3}$, and then use the $D^{3}$ output to estimate the channel coefficients in a decision-directed manner. The $D^{3}$ with coded data can be expressed as $$\mathbf{\hat{d}}=\arg\min_{\tilde{\mathbf{u}}\in\mathbb{U}}\sum_{v=0}^{N-2}\left\vert \frac{r_{v}}{\tilde{u}_{v}}-\frac{r_{\acute{v}}}{\tilde{u}_{\acute{v}}}\right\vert ^{2}\label{E-D3-Joint}$$ where $\mathbb{U}$ is the set of all codewords modulated using the same modulation used at the transmitter. Therefore, the trial sequences $\tilde{\mathbf{u}}$ are restricted to particular sequences. For the case of convolutional codes, the detection and decoding processes can be integrated smoothly since both of them are using the VA. Such an approach can be adopted with linear block codes as well because trellis-based decoding can be also applied to block codes [@Trellis-Block].

Error Rate Analysis of the $D^{3}$\[sec:System-Performance-Analysis\]
=====================================================================

The system BER analysis is presented for several cases according to the pilot and data arrangements. For simplicity, each case is discussed in separate subsections. To make the analysis tractable, we consider BPSK modulation in the analysis while the BER of higher-order modulations is obtained via Monte Carlo simulations.

![Double-sided pilot segment. \[fig:Double-sided-pilot\]](graphics/fig_03_single_sided_diagram){width="0.48\columnwidth"}

![Double-sided pilot segment. \[fig:Double-sided-pilot\]](graphics/fig_04_double_sided_diagram){width="0.48\columnwidth"}

Single-Sided Pilot \[subsec:Single-Sided-Pilot\]
------------------------------------------------

To detect a data segment that contains $\mathcal{K}$ symbols, at least one pilot symbol should be part of the segment in order to resolve the phase ambiguity problem. Consequently, the analysis in this subsection considers the case where there is only one pilot within the $\mathcal{K}$ symbols, as shown in Fig. \[fig:Single-sided-pilot\]. Given that the FFT output vector $\mathbf{r}=\left[r_{0}\text{, }r_{1}\text{,}\ldots,r_{N-1}\right]$ is divided into $L$ segments each of which consists of $\mathcal{K}$ symbols, including the pilot symbol, then the frequency domain $D^{3}$ detector can be written as, $$\hat{\mathbf{d}}_{l}=\arg\min_{\tilde{\mathbf{d}}}\sum_{v=l}^{\mathcal{K-}2+l}\left\vert \frac{r_{v}}{\tilde{d_{v}}}-\frac{r_{\acute{v}}}{\tilde{d}_{\acute{v}}}\right\vert ^{2}\,\,\,\,\mathcal{K}\in\left\{ 2,3,\dots,N-1\right\} \label{eq:d_hat}$$ where $l$ denotes the index of the first subcarrier in the segment, and without loss of generality, we consider that $l=0$. Therefore, by expanding we obtain, $$\begin{gathered}
\hat{\mathbf{d}}_{0}=\arg\min_{\tilde{\mathbf{d}}}\left(\frac{r_{0}}{\tilde{d_{0}}}-\frac{r_{1}}{\tilde{d}_{1}}\right)\left(\frac{r_{0}}{\tilde{d_{0}}}-\frac{r_{1}}{\tilde{d}_{1}}\right)^{\ast}+\cdots+\left(\frac{r_{\mathcal{K}-2}}{\tilde{d}_{\mathcal{K}-2}}-\frac{r_{\mathcal{K}-1}}{\tilde{d}_{\mathcal{K}-1}}\right)\left(\frac{r_{\mathcal{K}-2}}{\tilde{d}_{\mathcal{K}-2}}-\frac{r_{\mathcal{K}-1}}{\tilde{d}_{\mathcal{K}-1}}\right)^{\ast}\label{eq:analysis-expansion-01}\end{gathered}$$ which can be simplified to, $$\begin{gathered}
\hat{\mathbf{d}}_{0}=\arg\min_{\tilde{\mathbf{d}}}\left\vert \frac{r_{0}}{\tilde{d_{0}}}\right\vert ^{2}+\left\vert \frac{r_{1}}{\tilde{d_{1}}}\right\vert ^{2}+\dots+\left\vert \frac{r_{\mathcal{K}-1}}{\tilde{d}_{\mathcal{K}-1}}\right\vert ^{2}-\frac{r_{0}}{\tilde{d_{0}}}\frac{r_{1}}{\tilde{d}_{1}^{\ast}}-\frac{r_{0}}{\tilde{d_{0}^{\ast}}}\frac{r_{1}}{\tilde{d}_{1}}-\cdots\\
-\frac{r_{\mathcal{K}-2}}{\tilde{d}_{\mathcal{K}-2}}\frac{r_{\mathcal{K}-1}}{\tilde{d}_{\mathcal{K}-1}^{\ast}}-\frac{r_{\mathcal{K}-2}}{\tilde{d}_{\mathcal{K}-2}^{\ast}}\frac{r_{\mathcal{K}-1}}{\tilde{d}_{\mathcal{K}-1}}.\label{eq:analysis-expansion-02}\end{gathered}$$ For BPSK, $\left\vert r_{v}/\tilde{d_{v}}\right\vert ^{2}=\left\vert r_{v}\right\vert ^{2}$, which is a constant term with respect to the maximization process in , and thus, they can be dropped. Therefore, the detector is reduced to $$\hat{\mathbf{d}}_{0}=\arg\max_{\tilde{\mathbf{d}_{0}}}\sum_{v=0}^{\mathcal{K-}2}\Re\left\{ \frac{r_{v}r_{\acute{v}}}{\tilde{d_{v}}\tilde{d}_{\acute{v}}}\right\} .$$ Given that the pilot symbol is placed in the first subcarrier and noting that $d_{v}\in\left\{ -1,1\right\} $, then $\tilde{d_{0}}=1$ and $\hat{\mathbf{d}}_{0}$ can be written as $$\hat{\mathbf{d}}_{0}=\arg\max_{\tilde{d}_{0}\notin\tilde{\mathbf{d}}_{0}}\frac{1}{\tilde{d_{1}}}\Re\left\{ r_{0}r_{1}\right\} +\sum_{v=1}^{\mathcal{K-}2}\frac{1}{\tilde{d_{v}}\tilde{d}_{\acute{v}}}\Re\left\{ r_{v}r_{\acute{v}}\right\} .\label{eq:d_hat_single_sided}$$ The sequence error probability ($P_{S}$), conditioned on the channel frequency response over the $\mathcal{K}$ symbols ($\mathbf{H}_{0})$ and the transmitted data sequence $\mathbf{d}_{0}$ can be defined as, $$P_{S}|_{\mathbf{H}_{0},\mathbf{d}_{0}}\triangleq\left.\Pr\left(\hat{\mathbf{d}_{0}}\neq\mathbf{d}_{0}\right)\right\vert _{\mathbf{H}_{0},\mathbf{d}_{0}}\label{eq:SEP-definition}$$ which can be also written in terms of the conditional probability of correct detection $P_{C}$ as, $$P_{C}|_{\mathbf{H}_{0},\mathbf{d}_{0}}=1-\Pr\left(\hat{\mathbf{d}_{0}}=\mathbf{d}_{0}\right)\mid_{\mathbf{H}_{0},\mathbf{d}_{0}}.\label{eq:SEP-Analysis-01}$$ Without loss of generality, we assume that $\mathbf{d}_{0}\mathbf{=}[1$, $1$,…$,1]\triangleq\mathbf{1}$ . Therefore, $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=\Pr\left(\sum_{v=0}^{\mathcal{K-}2}\Re\left\{ r_{v}r_{\acute{v}}\right\} =\max_{\tilde{\mathbf{d}_{0}}}\left\{ \sum_{v=0}^{\mathcal{K-}2}\frac{\Re\left\{ r_{v}r_{\acute{v}}\right\} }{\tilde{d_{v}}\tilde{d}_{\acute{v}}}\right\} \right).\label{eq:probability-correct-sequence}$$ Since $\mathbf{d}_{0}$ has $\mathcal{K-}1$ data symbols, then there are $2^{\mathcal{K-}1}$ trial sequences, $\tilde{\mathbf{d}}_{0}^{(0)}$, $\tilde{\mathbf{d}}_{0}^{(1)}$,$\ldots$, $\tilde{\mathbf{d}}_{0}^{(\psi)}$, where $\psi=2^{\mathcal{K-}1}-1$, and $\tilde{\mathbf{d}}_{0}^{(\psi)}\mathbf{=}[1$, $1$,…$,1]$ . The first symbol in every sequence is set to $1$, which is the pilot symbol. By defining $\sum_{v=0}^{\mathcal{K-}2}\frac{\Re\left\{ r_{v}r_{\acute{v}}\right\} }{\tilde{d_{v}}\tilde{d}_{\acute{v}}}\triangleq A_{n}$, where $\tilde{d_{v}}\tilde{d}_{\acute{v}}\in\tilde{\mathbf{d}}_{0}^{(n)}$, then (\[eq:probability-correct-sequence\]) can be written as, $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=\Pr\left(A_{\psi}>A_{\psi-1},A_{\psi-2},\ldots,A_{0}\right)\label{E-PC-00}$$ which, as depicted in Appendix I, can be simplified to $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=\prod\limits _{v=0}^{\mathcal{K-}2}\Pr\left(\Re\left\{ r_{v}r_{\acute{v}}\right\} >0\right).\label{eq:pc-expansion-2}$$

To evaluate $P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}$ given in , it is necessary to compute $\Pr\left(\Re\left\{ r_{v}r_{\acute{v}}\right\} >0\right)$, which can be written as $$\Pr\left(\Re\left\{ r_{v}r_{\acute{v}}\right\} >0\right)=\Pr\left(\underbrace{r_{v}^{I}r_{\acute{v}}^{I}-r_{v}^{Q}r_{\acute{v}}^{Q}}_{r_{v,\acute{v}}^{\mathrm{SP}}}>0\right).\label{E-rSP}$$ Given that $\mathbf{d}_{0}\mathbf{=}[1$, $1$,…$,1]$ , then $r_{v}^{I}=\Re\left\{ r_{v}\right\} =H_{v}^{I}+w_{v}^{I}$ and $r_{v}^{Q}=\Im\left\{ r_{v}\right\} =H_{v}^{Q}+w_{v}^{Q}$. Therefore, $r_{v}^{I},$ $r_{v}^{Q}$, $r_{\acute{v}}^{I}$ and $r_{\acute{v}}^{Q}$ are independent conditionally Gaussian random variables with averages $H_{v}^{I}$, $H_{v}^{Q}$, $H_{\acute{v}}^{I}$ and $H_{\acute{v}}^{Q}$, respectively, and the variance for all elements is $\sigma_{w}^{2}$. To derive the PDF of $r_{v,\acute{v}}^{\mathrm{SP}}$, the PDFs of $r_{v}^{I}r_{\acute{v}}^{I}$ and $r_{v}^{Q}r_{\acute{v}}^{Q}$ should be evaluated, where each of which corresponds to the product of two Gaussian random variables. Although the product of two Gaussian variables is not usually Gaussian, the limit of the moment-generating function of the product has Gaussian distribution. Therefore, the product of two variables $X\sim\mathcal{N}(\mu_{x},\sigma_{x}^{2})$ and $Y\sim\mathcal{N}(\mu_{y},\sigma_{y}^{2})$ tends to be $\mathcal{N}(\mu_{x}\mu_{y},\mu_{x}^{2}\sigma_{y}^{2}+\mu_{y}^{2}\sigma_{x}^{2})$ as the ratios $\mu_{x}/\sigma_{x}$ and $\mu_{y}/\sigma_{y}$ increase [@Product; @of; @2RV]. By noting that in in (\[E-rSP\]) $\mathrm{E}\left[r_{y}^{x}\right]=H_{y}^{x}$, $x\in\left\{ I,Q\right\} $ and $y\in\left\{ v,\acute{v}\right\} $ and $\sigma_{r_{y}^{x}}=\sigma_{w}$, thus $\mathrm{E}\left[r_{y}^{x}\right]/\sigma_{r_{y}^{x}}\gg1$ $\forall\left\{ x,y\right\} $. Moreover, because the PDF of the sum or difference of two Gaussian random variables is also Gaussian, then, $r_{v,\acute{v}}^{\mathrm{SP}}\sim\mathcal{N}\left(\bar{\mu}_{\mathrm{SP}},\bar{\sigma}_{\mathrm{SP}}^{2}\right)$ where $\bar{\mu}_{\mathrm{SP}}=H_{v}^{I}H_{\acute{v}}^{I}+H_{v}^{Q}H_{\acute{v}}^{Q}$ and $\bar{\sigma}_{\mathrm{SP}}^{2}=\sigma_{w}^{2}\left(\left\vert H_{v}\right\vert ^{2}+\left\vert H_{\acute{v}}\right\vert ^{2}+\sigma_{w}^{2}\right)$. Consequently, $$P_{C}|_{\mathbf{H}_{0},\mathbf{1}}=\prod_{v=0}^{\mathcal{K}-2}\Pr\left(r_{v,\acute{v}}^{\mathrm{SP}}>0\right)=\prod_{v=0}^{\mathcal{K}-2}\left[1-Q\left(\sqrt{\frac{2\bar{\mu}_{\mathrm{SP}}}{\bar{\sigma}_{\mathrm{SP}}^{2}}}\right)\right]$$ and $$P_{S}|_{\mathbf{H}_{0},\mathbf{1}}=1-\prod_{v=0}^{\mathcal{K}-2}\left[1-Q\left(\sqrt{\frac{2\bar{\mu}_{\mathrm{SP}}}{\bar{\sigma}_{\mathrm{SP}}^{2}}}\right)\right]\label{eq:SEP}$$ where $Q\left(x\right)\triangleq\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\exp\left(-\frac{t^{2}}{2}\right)dt$. Since $H_{v}^{I}$ and $H_{v}^{Q}$ are independent, then, the condition on $\mathbf{H}_{0}$ in can be removed by averaging $P_{S}$ over the PDF of $\mathbf{H}_{0}^{I}$ and $\mathbf{H}_{0}^{Q}$ as, $$\begin{gathered}
\mathrm{SEP}\mid_{\mathbf{d}=1}=\underbrace{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}}_{2\mathcal{K}\text{ fold}}\mathrm{SEP}\mid_{\mathbf{H}_{0},\mathbf{d}=1}f_{\mathbf{H}_{0}^{I}}\left(H_{0}^{I},H_{1}^{I},\dots,H_{\mathcal{K}-1}^{I}\right)\times\\
f_{\mathbf{H}_{0}^{Q}}\left(H_{0}^{Q},H_{1}^{Q},\dots,H_{\mathcal{K}-1}^{Q}\right)dH_{0}^{I}dH_{1}^{I}\dots dH_{\mathcal{K}-1}^{I}dH_{0}^{Q}dH_{1}^{Q}\dots dH_{\mathcal{K}-1}^{Q}\text{.}\label{eq:unconditional-SER}\end{gathered}$$ Because the random variables $H_{i}^{I}$ and $H_{i}^{Q}$ $\forall i$ in are real and Gaussian, their PDFs are multivariate Gaussian distributions [@Proakis-Book-2001], $$f_{\mathbf{X}}\left(X_{0},X_{1},\dots,X_{\mathcal{K}-1}\right)=\frac{\exp\left(-\frac{1}{2}(\mathbf{X}-\boldsymbol{\mu})^{\mathrm{T}}\boldsymbol{\Sigma}^{-1}(\mathbf{X}-\boldsymbol{\mu})\right)}{\sqrt{(2\pi)^{\mathcal{K}}|\boldsymbol{\Sigma}|}}\label{eq:multi-variate-gaussian}$$ where $\boldsymbol{\mu}$ is the mean vector, which is defined as, $$\boldsymbol{\mu}=\mathrm{E}\left[\mathbf{X}\right]=\left[\mathrm{E}\left[X_{1}\right],\mathrm{E}\left[X_{2}\right],\dots,\mathrm{E}\left[X_{\mathcal{K}-1}\right]\right]^{T}$$ and $\boldsymbol{\Sigma}$ is the covariance matrix, $\boldsymbol{\Sigma}=\mathrm{E}\left[\left(\mathbf{X}-\mu\right)\left(\mathbf{X}-\mu\right)^{T}\right].$

Due to the difficulty of evaluating $2\mathcal{K}$ integrals, we consider the special case of flat fading, which implies that $H_{v}=H_{\acute{v}}\triangleq H$ and $\left(H^{I}\right)^{2}+\left(H^{Q}\right)^{2}\triangleq\alpha^{2}$, where $\alpha$ is the channel fading envelope, $\alpha=\left\vert H\right\vert $. Therefore, the SEP expression in becomes, $$P_{S}|_{\alpha,\mathbf{1}}=1-\left[1-Q\left(\sqrt{\frac{\alpha^{2}}{\sigma_{w}^{2}\left(\alpha^{2}+\sigma_{w}^{2}\right)}}\right)\right]^{\mathcal{K}-1}.\label{eq:SEP-conditional-general}$$ Recalling the Binomial Theorem, we get $$\left(a+b\right)^{n}=\sum_{v=0}^{n}\binom{n}{v}a^{n-v}b^{v}\text{, }\binom{n}{v}\triangleq\frac{n!}{\left(n-v\right)!v!}.\label{eq:binomial-theorem}$$ Then the SEP formula in using the Binomial Theorem in can be written as, $$P_{S}|_{\alpha,\mathbf{1}}=1-\sum_{v=0}^{\mathcal{K}-1}\binom{\mathcal{K}-1}{v}\left(-1\right)^{v}\left[Q\left(\sqrt{\frac{\alpha^{2}}{\sigma_{w}^{2}\left(\alpha^{2}+\sigma_{w}^{2}\right)}}\right)\right]^{v}.\label{eq:sep_cond_higher_k}$$ The conditioning on $\alpha$ can be removed by averaging over the PDF of $\alpha$, which is Rayleigh. Therefore, $$f\left(\alpha\right)=\frac{\alpha}{\sigma_{H}^{2}}e^{-\frac{\alpha^{2}}{2\sigma_{H}^{2}}}.\label{eq:rayleigh-pdf}$$ And hence, $$P_{S}|_{\mathbf{1}}=\int_{0}^{\infty}P_{S}|_{\alpha,\mathbf{1}}\text{ }f\left(\alpha\right)d\alpha.\label{E-Averaging}$$ Because the expression in contains high order of $Q$-function $Q^{n}\left(x\right)$, evaluating the integral analytically becomes intractable for $\mathcal{K}>2$. For the special case of $\mathcal{K}=2$, $P_{S}$ can be evaluated by substituting (\[eq:sep\_cond\_higher\_k\]) and (\[eq:rayleigh-pdf\]) into (\[E-Averaging\]) and evaluating the integral yields the following simple expression, $$P_{S}|_{\mathbf{1}}=\frac{1}{2\left(\bar{\gamma}_{s}+1\right)}\text{, \ \ }\bar{\gamma}_{s}\triangleq\frac{\mathrm{E}\left[\left\vert d_{v}\right\vert ^{2}\right]\mathrm{E}\left[\left\vert H\right\vert ^{2}\right]}{2\sigma_{w}^{2}}\label{E-Pe_K2}$$ where $\bar{\gamma}_{s}$ is the average signal-to-noise ratio (SNR). Moreover, because all data sequences have an equal probability of error, then $P_{S}|_{\mathbf{1}}=P_{S}$, which also equivalent to the bit error rate (BER). It is interesting to note that (\[E-Pe\_K2\]) is similar to the BER of the differential binary phase shift keying (DBPSK) [@Proakis-Book-2001]. However, the two techniques are essentially different as $D^{3}$ does not require differential encoding, has no constraints on the shape of the signal constellation, and performs well even in frequency-selective fading channels.

To evaluate $P_{S}$ for $\mathcal{K}>2$, we use an approximation for $Q\left(x\right)$ in [@Q-Func-Approx-02], which is given by $$Q\left(x\right)\approx\frac{1}{\sqrt{2\pi\left(x^{2}+1\right)}}e^{-\frac{1}{2}x^{2}},\text{ }x\in\lbrack0,\infty).\label{eq:Q-func-Approx}$$ Therefore, by substituting into the conditional SEP and averaging over the Rayleigh PDF , the evaluation of the SEP becomes straightforward. For example, evaluating the integral for $\mathcal{K}=3$ gives, $$P_{S}|_{\mathbf{1}}=\frac{\zeta_{1}}{\pi}\mathit{\mathrm{Ei}}\left(1,\zeta_{1}+1\right)e^{\zeta_{1}+1}\text{, \ \ }\zeta_{1}\triangleq\frac{1}{2\bar{\gamma}_{s}}\left(\frac{1}{\bar{\gamma}_{s}}+1\right)$$ where $\mathrm{Ei}\left(x\right)$ is the exponential integral (EI), $\mathrm{Ei}\left(x\right)\triangleq-\int_{-x}^{\infty}\frac{e^{-t}}{t}dt$. Similarly, $P_{S}$ for $\mathcal{K}=7$ can be evaluated to, $$P_{S}|_{\mathbf{1}}=\frac{\zeta_{2}}{64\pi^{3}}\left[e^{\zeta+3}\left(2\zeta_{2}+6\right)^{2}\text{ }\mathit{\mathrm{Ei}}\left(1,\zeta_{2}+3\right)-4\left(\zeta_{2}+1\right)\right]\text{, \ }\zeta_{2}\triangleq\frac{1}{2\bar{\gamma}_{s}}\left(\frac{1}{4\bar{\gamma}_{s}}+1\right).$$

Although the SEP is a very useful indicator for the system error probability performance, the BER is actually more informative. For a sequence that contains $\mathcal{K}_{D}$ information bits, the BER can be expressed as $P_{B}=\frac{1}{\Lambda}P_{S}$, where $\Lambda$ denotes the average number of bit errors given a sequence error, which can be defined as $$\Lambda=\sum_{m=1}^{\mathcal{K}_{D}}m\Pr\left(m\right).$$ Because the SEP is independent of the transmitted data sequence, then, without loss of generality, we assume that the transmitted data sequence is $\mathbf{d}_{0}^{(0)}$. Therefore, $$\Lambda=\sum_{m=1}^{\mathcal{K}_{D}}m\Pr\left(\left\Vert \mathbf{\hat{d}}_{0}\right\Vert ^{2}=m\right)$$ where $\left\Vert \mathbf{\hat{d}}_{0}\right\Vert ^{2}$, in this case, corresponds to the Hamming weight of the detected sequence $\mathbf{\hat{d}}_{0}$, which can be expressed as $$\Pr\left(\left\Vert \mathbf{\hat{d}}_{0}\right\Vert ^{2}=m\right)=\Pr\left(\mathbf{d}_{0}^{(0)}\rightarrow\bigcup\limits _{i}\mathbf{d}_{0}^{(i)}\right)\text{, }\left\Vert \mathbf{d}_{0}^{(i)}\right\Vert ^{2}=m$$ where $\mathbf{d}_{0}^{(0)}\rightarrow\mathbf{d}_{0}^{(i)}$ denotes the pairwise error probability (PEP). By noting that $\Pr\left(\mathbf{d}_{0}^{(0)}\rightarrow\mathbf{d}_{0}^{(i)}\right)\neq\Pr\left(\mathbf{d}_{0}^{(0)}\rightarrow\mathbf{d}_{0}^{(j)}\right)$ $\forall i\neq j$, then deriving the PEP for all cases of interest is intractable. As an alternative, a simple approximation is derived.

For a sequence that consists of $\mathcal{K}_{D}$ information bits, the BER is bounded by $$\frac{1}{\mathcal{K}_{D}}P_{S}\leq P_{B}\leq P_{S}\text{.}\label{E-Bounds}$$ In practical systems, the number of bits in the detected sequence is generally not large, which implies that the upper and lower bounds in (\[E-Bounds\]) are relatively tight, and hence, the BER can be approximated as the middle point between the two bounds as, $$P_{B}\approx\frac{P_{S}}{0.5\left(1+\mathcal{K}_{D}\right)}.\label{E_PB}$$

The analysis of the general $1\times\mathcal{N}$ SIMO system is a straightforward extension of the single-input single-output (SISO) case. To simplify the analysis, we consider the flat channel case where the conditional SEP can be written as, $$P_{S}|_{\mathbf{\alpha}}=1-\left[1-Q\left(\sqrt{\frac{\sum_{i=1}^{\mathcal{N}}\alpha_{i}^{2}}{\sigma_{w}^{2}\left(\mathcal{N}\sigma_{w}^{2}+\sum_{i=1}^{\mathcal{N}}\alpha_{i}^{2}\right)}}\right)\right]^{\mathcal{K}-1}.$$ Given that all the receiving branches are independent, the fading envelopes will have Rayleigh distribution $\alpha_{i}\sim\mathcal{R}\left(2\sigma_{H}^{2}\right)$ $\forall i$, and thus, $\sum_{i=1}^{\mathcal{N}}\alpha_{i}^{2}\triangleq a$ will have Gamma distribution, $a\sim\mathcal{G}\left(\mathcal{N},2\sigma_{H}^{2}\right)$, $$f\left(a\right)=\left(2\sigma_{H}^{2}\right)^{\mathcal{N}}e^{-2\sigma_{H}^{2}a}\frac{_{a^{\mathcal{N}-1}}}{\Gamma\left(\mathcal{N}\right)}.$$ Therefore, the unconditional SEP can be evaluated as, $$P_{S}=\int_{0}^{\infty}P_{S}|_{\mathbf{\alpha}}\text{ }f_{A}\left(a\right)da.$$ For the special case of $\mathcal{N=}2$, $\mathcal{K}=2$, $P_{S}$ can be evaluated as, $$P_{S}=\frac{1}{2}+Q\left(\frac{\varkappa}{\sqrt{\bar{\gamma}_{s}}}\right)\left[2\bar{\gamma}_{s}\left(\frac{\bar{\gamma}_{s}}{\sqrt{2}}+2\right)-e^{\varkappa^{2}}\right]-\bar{\gamma}_{s}\frac{\varkappa}{\sqrt{2\pi}}$$ where $\varkappa\triangleq\sqrt{2+\bar{\gamma}_{s}}.$ Computing the closed-form formulas for other values of $\mathcal{N}$  and $\mathcal{K}$ can be evaluated following the same approach used in the SISO case.

Double-Sided Pilot \[subsec:Double-Sided-Pilot\]
------------------------------------------------

Embedding more pilots in the detection segment can improve the detector’s performance. Consequently, it worth investigating the effect of embedding more pilots in the SEP analysis. More specifically, we consider double-sided segment, $\tilde{d}_{0}=1$, $\tilde{d}_{\mathcal{K}-1}=1$, as illustrated in Fig. \[fig:Double-sided-pilot\]. In this case, the detector can be expressed as, $$\hat{\mathbf{d}_{0}}=\arg\max_{\tilde{\mathbf{d}}_{0}}\frac{\Re\left\{ r_{0}r_{1}\right\} }{\tilde{d}_{1}}+\frac{\Re\left\{ r_{\mathcal{K}-2}r_{\mathcal{K}-1}\right\} }{\tilde{d}_{\mathcal{K}-2}}+\sum_{v=1}^{\mathcal{K-}3}\frac{\Re\left\{ r_{v}r_{\acute{v}}\right\} }{\tilde{d_{v}}\tilde{d}_{\acute{v}}},\text{\thinspace\thinspace\thinspace\thinspace\ensuremath{\mathcal{K}\in\left\{ 3,4,\dots,N-1\right\} .}}\label{eq:d_hat_double_sided}$$ From the definition in , the probability of receiving the correct sequence can be derived based on the reduced number of trials as compared to . Therefore, $$\begin{gathered}
P_{C}|_{\mathbf{H}_{0},\mathbf{1}}=\Pr\Big(\left(\Re\left\{ r_{0}r_{1}\right\} +\Re\left\{ r_{\mathcal{K}-2}r_{\mathcal{K}-1}\right\} \right)\cap\\
\Re\left\{ r_{1}r_{2}\right\} \cap\Re\left\{ r_{2}r_{3}\right\} \cap\dots\cap\Re\left\{ r_{\mathcal{K}-4}r_{\mathcal{K}-3}\right\} >0\Big)\label{eq:pc-double-sided}\end{gathered}$$ which, similar to the single-sided case, can be written as, $$P_{C}|_{\mathbf{H}_{0},\mathbf{1}}=\Pr\left(\left[\prod_{v=0}^{\mathcal{K}-3}\Pr\left(\Re\left\{ r_{v}r_{\acute{v}}\right\} \right)+\prod_{v=1}^{\mathcal{K}-2}\Pr\left(\Re\left\{ r_{v}r_{\acute{v}}\right\} \right)\right]>0\right).$$ Therefore, $$P_{S}|_{\mathbf{H}_{0},\mathbf{1}}=1-\left[1-Q\left(\sqrt{\frac{2\sqrt{2}\bar{\mu}_{\mathrm{SP}}}{\bar{\sigma}_{\mathrm{SP}}^{2}}}\right)\right]\times\prod_{v=1}^{\mathcal{K}-3}\left[1-Q\left(\sqrt{\frac{2\bar{\mu}_{\mathrm{SP}}}{\bar{\sigma}_{\mathrm{SP}}^{2}}}\right)\right].\label{eq:SEP-1}$$ For flat fading channels, the SEP expression in can be simplified by following the same procedure in Subsection \[subsec:Single-Sided-Pilot\], for the special case of $\mathcal{K}=3$, the SEP becomes, $$P_{S}=\left(\frac{\Upsilon}{2}-\sqrt{2}\right)\frac{1}{\Upsilon}\text{, \ }\Upsilon\triangleq\sqrt{8\bar{\gamma}_{s}+\sqrt{2}\left(4+\frac{1}{\bar{\gamma}_{s}}\right)}.$$ For $\mathcal{K}>3$, the approximation of $Q^{n}\left(x\right)$, as illustrated in Subsection \[subsec:Single-Sided-Pilot\], can be used in to average over the PDF in . For example, the case $\mathcal{K}=4$ can be evaluated as, $$P_{S}=\frac{1}{8\pi\bar{\gamma}_{s}}\left(\Omega_{1}-1\right)e^{\Omega_{1}}\mathit{\mathrm{Ei}}\left(1,\Omega_{1}\right)\text{, \ }\Omega_{1}\triangleq1+\frac{\sqrt{2}}{4\bar{\gamma}_{s}}\left(1+\frac{1}{4\bar{\gamma}_{s}}\right).$$ For $\mathcal{K}=6$, $$P_{S}=\frac{\Omega_{1}-1}{4\pi^{2}}\left[1-\left[\left(\Omega_{1}-1\right)e^{\Omega_{2}}+2\right]\mathit{\mathrm{Ei}}\left(1,\Omega_{2}\right)\right]\text{, \ }\Omega_{2}\triangleq2+\frac{\sqrt{2}}{\bar{\gamma}_{s}}\left(8+\frac{1}{32\bar{\gamma}_{s}}\right)$$ For the double-sided pilot, $P_{B}=P_{S}$ for the case of $\mathcal{K}=3$, while it can be computed using (\[E\_PB\]) for $\mathcal{K}>3$.

Complexity Analysis\[sec:Complexity-Analysis\]
==============================================

The computational complexity is evaluated as the total number of primitive operations needed to perform the detection. The operations that will be used are the number of real additions ($R_{A}$), real multiplications ($R_{M}$), and real divisions ($R_{D}$) required to produce the set of detected symbols $\hat{\mathbf{d}}$ for each technique. It worth noting that one complex multiplication ($C_{M}$) is equivalent to four $R_{M}$ and three $R_{A}$ operations, while one complex addition ($C_{A}$) requires two $R_{A}$. To simplify the analysis, we first assume that constant modulus (CM) constellations such as MPSK is used, then, we evaluate the complexity for higher-order modulation such as quadrature amplitude modulation (QAM) modulation.

Complexity of Conventional OFDM Detectors\[subsec:Complexity-of-Conventional\]
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The complexity of the conventional OFDM receiver that consists of the following main steps with the corresponding computational complexities:

1.  Channel estimation of the pilot symbols, which computes $\hat{H}_{k}$ at all pilot subcarriers. Assuming that the pilot symbol $d_{k}$ is selected from a CM constellation, then $\hat{H}_{k}=r_{k}d_{k}^{*}$ and hence, $N_{P}$ complex multiplications are required. Therefore, $R_{A}^{\left(1\right)}=4N_{P}$ and $R_{M}^{\left(1\right)}=4N_{P}$.

2.  Interpolation, which is used to estimate the channel at the non-pilot subcarriers. The complexity of the interpolation process depends on the interpolation algorithm used. For comparison purposes, we assume that linear interpolation is used, which is the least complex interpolation algorithm. The linear interpolation requires one complex multiplication and two complex additions per interpolated sample. Therefore, the number of complex multiplications required is $N-N_{P}$ and the number of complex additions is $2\left(N-N_{P}\right)$. And hence, $R_{A}^{\left(2\right)}=7\left(N-N_{P}\right)$ and $R_{M}^{\left(2\right)}=4\left(N-N_{P}\right)$.

3.  Equalization, a single-tap equalizer requires $N-N_{P}$ complex division to compute the decision variables $\check{r}_{k}=\frac{r_{k}}{\hat{H}_{k}}=r_{k}\frac{\hat{H}_{k}^{*}}{\left|\hat{H}_{k}^{*}\right|^{2}}$. Therefore, one complex division requires two complex multiplications and one real division. Therefore, $R_{A}^{\left(3\right)}=6\left(N-N_{P}\right)$, $R_{M}^{\left(3\right)}=8\left(N-N_{P}\right)$ and $R_{D}^{\left(3\right)}=\left(N-N_{P}\right)$.

4.  Detection, assuming symbol-by-symbol minimum distance detection, the detector can be expressed as $\hat{d}_{k}=\arg\min_{\tilde{d}_{i}}J\left(\tilde{d}_{i}\right),\,\,\forall i\in\left\{ 0,1,\dots,M-1\right\} $ where $J\left(\tilde{d}_{i}\right)=\left|\check{r}_{k}-\tilde{d}_{i}\right|^{2}$ . Assuming CM modulation is used, expanding the cost function and dropping the constant terms we can write $J\left(\tilde{d}_{k}\right)=-\check{r}_{k}\tilde{d}_{k}^{*}-\check{r}_{k}^{*}\tilde{d}_{k}$. We can also drop the minus sign from the cost function, and thus, the objective becomes maximizing the cost function $\hat{d}_{k}=\arg\min_{\tilde{d}_{i}}J\left(\tilde{d}_{i}\right)$. Since the two terms are complex conjugate pair, then $-\check{r}_{k}\tilde{d}_{k}^{*}-\check{r}_{k}^{*}\tilde{d}_{k}=2\Re\left\{ \check{r}_{k}\tilde{d}_{k}^{*}\right\} $, and thus we can write the detected symbols as, $$\hat{d}_{k}=\arg\max_{\tilde{d}_{k}}\left(\Re\left\{ \check{r}_{k}\right\} \Re\left\{ \tilde{d}_{k}^{*}\right\} -\Im\left\{ \check{r}_{k}\right\} \Im\left\{ \tilde{d}_{k}^{*}\right\} \right)$$ Therefore, the number of real multiplications required for each information symbol is $2M$, and the number of additions is $M$. Therefore, $R_{A}^{\left(4\right)}=\left(N-N_{P}\right)M$ and $R_{M}^{\left(4\right)}=2\left(N-N_{P}\right)M$.

Finally, the total computational complexity per OFDM symbol can be obtained by adding the complexities of the individual steps $1\rightarrow4$, as:

$$\begin{aligned}
R_{A}^{CM} & ={\displaystyle \sum_{i=1}^{4}R_{A}^{\left(i\right)}=\left(13+M\right)N-\left(10+M\right)N_{P}}\\
R_{M}^{CM} & =\sum_{i=1}^{4}R_{M}^{\left(i\right)}=2N\left(6+M\right)-2N_{P}\left(4+M\right)\\
R_{D}^{CM} & =\sum_{i=1}^{4}R_{D}^{\left(i\right)}=N-N_{P}.\end{aligned}$$

Complexity of the $D^{3}$
-------------------------

The complexity of the $D^{3}$ based on the VA is mostly determined by the branch and path metrics calculation. The branch metrics can be computed as $$J_{m,n}^{c}=\frac{\left\vert r_{c}\right\vert ^{2}}{\left\vert \tilde{d}_{m}\right\vert ^{2}}-\frac{r_{c}r_{\acute{c}}^{\ast}}{\tilde{d}_{m}\tilde{d}_{n}^{\ast}}-\frac{r_{c}^{\ast}r_{\acute{c}}}{\tilde{d}_{m}^{\ast}\tilde{d}_{n}}+\frac{\left\vert r_{c}\right\vert ^{2}}{\left\vert \tilde{d}_{n}\right\vert ^{2}}.$$ For CM constellation, the first and last terms are constants, and hence, can be dropped. Therefore, $$J_{m,n}^{c}=-\frac{r_{c}r_{\acute{c}}^{\ast}}{\tilde{d}_{m}\tilde{d}_{n}^{\ast}}+\frac{r_{c}^{\ast}r_{\acute{c}}}{\tilde{d}_{m}^{\ast}\tilde{d}_{n}}.\label{eq:branch-metric-viterbi}$$ By noting that the two terms in are the complex conjugate pair, then $$J_{m,n}^{c}=-2\Re\left\{ \frac{r_{c}r_{\acute{c}}^{\ast}}{\tilde{d}_{m}\tilde{d}_{n}^{\ast}}\right\} .\label{eq:branch-metric-viterbi-02}$$ From the expression in , the constant “$-2$ can be dropped from the cost function, however, the problem with be flipped to a maximization problem. Therefore, by expanding , we get, $$J_{m,n}^{c}=\Re\left\{ \frac{\Re\left\{ r_{c}\right\} \Re\left\{ r_{\acute{c}}^{\ast}\right\} -\Im\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} +j\left[-\Re\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} +\Im\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} \right]}{\Re\left\{ \tilde{d}_{m}\tilde{d}_{n}^{\ast}\right\} +j\Im\left\{ \tilde{d}_{m}\tilde{d}_{n}^{\ast}\right\} }\right\} .\label{eq:branch-metric-03}$$ By defining $\tilde{d}_{m}\tilde{d}_{n}^{\ast}\triangleq\tilde{u}_{m,n},$ and using complex numbers identities, we get , $$J_{m,n}^{c}=\frac{\left[\Re\left\{ r_{c}\right\} \Re\left\{ r_{\acute{c}}^{\ast}\right\} +\Im\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} \right]\Re\left\{ \tilde{u}_{m,n}\right\} -\left[-\Re\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} +\Im\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} \right]\Im\left\{ \tilde{u}_{m,n}\right\} }{\Re\left\{ \tilde{u}_{m,n}\right\} ^{2}+\Im\left\{ \tilde{u}_{m,n}\right\} ^{2}}.\label{eq:branch-metric-04}$$ For CM, $\Re\left\{ \tilde{u}_{m,n}\right\} ^{2}+\Im\left\{ \tilde{u}_{m,n}\right\} ^{2}$ is constant, and hence, it can be dropped from the cost function, which implies that no division operations are required.

To compute $J_{m,n}^{c}$, it is worth noting that the two terms in brackets are independent of $\left\{ m,n\right\} $, and hence, they are computed only once for each value of $c$. Therefore, the complexity at each step in the trellis can be computed as $R_{A}=3\times2^{M}$, $R_{M}=4+2\times2^{M}$ and $R_{D}=0$, where $2^{M}$ is the number of branches at each step in the trellis. However, if the trellis starts or ends by a pilot, then only $M$ computations are required. By noting that the number of full steps is $N-2N_{P}-1$, and the number of steps that require $M$ computations is $2\left(N_{P}-1\right)$, then the total computations of the branch metrics (BM) are: $$\begin{aligned}
R_{A}^{BM} & =\left(3\times2^{M}\right)\left(N-2N_{P}-1\right)+2\left(3\times M\right)\left(N_{P}-1\right)\\
R_{M}^{BM} & =\left(4+2^{M+1}\right)\left(N-2N_{P}-1\right)+2\left(N_{P}-1\right)\left(4+2M\right)\\
R_{D}^{BM} & =0\end{aligned}$$ The path metrics (PM) require $R_{A}^{PM}=\left(N-2N_{P}-1\right)+M\left(N_{P}-1\right)$ real addition. Therefore, the total complexity is: $$\begin{aligned}
R_{A}^{CM} & =\left(N-2N_{P}-1\right)\left(5\times2^{M}\right)+7M\left(N_{P}-1\right)\\
R_{M}^{CM} & =\left(N-2N_{P}-1\right)\left(4+2^{M+1}\right)+2\left(N_{P}-1\right)\left(4+2M\right)\\
R_{D}^{CM} & =0\end{aligned}$$

  [$N$]{}              [$128$]{}    [$256$]{}    [$512$]{}    [$1024$]{}   [$2048$]{}
  -------------------- ------------ ------------ ------------ ------------ ------------
  [$\eta_{R_{A}}$]{}   [$0.58$]{}   [$1.07$]{}   [$1.21$]{}   [$1.27$]{}   [$1.31$]{}
  [$\eta_{R_{M}}$]{}   [$0.77$]{}   [$0.72$]{}   [$0.68$]{}   [$0.64$]{}   [$0.61$]{}
  [$R_{D}$]{}          [$96$]{}     [$192$]{}    [$384$]{}    [$768$]{}    [$1536$]{}
  [$\eta_{P}$]{}       [$0.20$]{}   [$0.21$]{}   [$0.22$]{}   [$0.26$]{}   [$0.31$]{}

  : Computational complexity comparison using different values of $N$, $N_{P}=N/4$, for BPSK.\[tab:Computational-power-analysis\]

To compare the complexity of the $D^{3}$, we use the conventional detector using LS channel estimation, linear interpolation, zero-forcing (ZF) equalization, and MLD, denoted as coherent-L, as a benchmark due to its low complexity. The relative complexity is denoted by $\eta$, which corresponds to the ratio of the $D^{3}$ complexity to the conventional detector, i.e., $\eta_{R_{A}}$ denotes the ratio of real additions and $\eta_{R_{M}}$ corresponds to the ratio of real multiplications. As depicted in Table \[tab:Computational-power-analysis\], $R_{A}$ for $D^{3}$ less than coherent-L only using BPSK for $N=128$, and then it becomes larger for all the other considered values of $N$. For $R_{M}$, $D^{3}$ is always less than the coherent-L, particularly for high values of $N$, where it becomes 0.61 for $N=2048$. It is worth noting that $R_{D}$ in the table corresponds to the number of divisions in the conventional OFDM since the $D^{3}$ does not require any division operations. For a more informative comparison between the two systems, we use the computational power analysis presented in [@computational_power], where the total power for each detector is estimated based on the total number of operations. Table \[tab:Computational-power-analysis\] shows the relative computational power $\eta_{P}$, which shows that the $D^{3}$ detector requires only $0.2$ of the power required by the coherent-L detector for $N=128$ and $0.31\%$ for $N=2048$.

  -- -- -- -- --
              
              
              
              
              
  -- -- -- -- --

Besides, it is worth noting that linear interpolation has lower complexity as compared to more accurate interpolation schemes such as the spline interpolation [@spline-interpolation], [@Spline], which comes at the expense of the error rate performance. Therefore, the results presented in Table \[tab:Computational-power-analysis\] can be generally considered as upper bounds on the relative complexity of the $D^{3}$, when more accurate interpolation schemes are used, the relative complexity will drop even further as compared to the results in Table \[tab:Computational-power-analysis\].

Complexity with Error Correction Coding
---------------------------------------

To evaluate the impact of the complexity reduction of the $D^{3}$ in the presence of FEC coding, convolutional codes are considered with soft and hard decision decoding using the VA. BPSK is the modulation considered for the complexity evaluation and the code rate is assumed to be $1/2$. For decoding of convolutional codes, the soft VA requires $n\times2^{K}$ addition or subtractions and multiplications per decoded bit, where $1/n$ is the code rate and $K$ is the constraint length [@P-Wu]. Therefore, for $1/2$ code rate, $R_{A}=R_{M}=2^{K+1}$. Given that each OFDM symbol has $N$ coded bits and $N/2$ information bits, the complexity per OFDM symbol becomes $R_{A}=R_{M}=N\times2^{K}$. For the hard VA, $N\times2^{K}$ XOR operations are required for the branch metric computation, while $N\times2^{K-1}$ additions are required for the path metric computations. Because the XOR operation is a bit operation, it’s complexity is much less than the addition. Assuming that addition is using an 8-bit representation, then the complexity of an addition operation is about eight times the XOR. Therefore, $R_{A}$, in this case, can be approximated as $N\left(2^{K}+2^{K-2}\right)$.

   [$K$ ]{}    [$3$ ]{}      [$4$ ]{}       [$5$]{}       [$6$]{}       [$7$]{}
  ---------- ------------- ------------- ------------- ------------- -------------
              [$0.96$ ]{}   [$0.97$ ]{}   [$0.97$ ]{}   [$0.98$ ]{}   [$0.99$ ]{}
              [$0.24$ ]{}   [$0.26$ ]{}   [$0.28$ ]{}   [$0.33$ ]{}   [$0.41$ ]{}

  : Computational complexity comparison using hard and soft VA for different values of $K$, $N=2048$. \[T-coded\]

As can be noted from Table \[T-coded\], the complexity reduction when soft VA is used less significant as compared to the hard VA. Such a result is obtained because the soft VA requires the CSI to compute the reliability factors, which requires $N-N_{P}$ division operations when the $D^{3}$ is used. For hard decoding, the advantage of the $D^{3}$ is significant even for high constraint length values.

Numerical Results\[sec:Numerical-Results\]
==========================================

This section presents the performance of the $D^{3}$ detector in terms of BER for several operating scenarios. The system model follows the LTE-A physical layer (PHY) specifications [@LTE-A], where the adopted OFDM symbol has $N=512$, $N_{\mathrm{CP}}=64$, the sampling frequency $f_{s}=7.68$ MHz, the subcarrier spacing $\Delta f=15$ kHz, and the pilot grid follows that of Fig. \[fig:2D-to-1D\]. The total OFDM symbol period is $75$ $\mu\sec$, and the CP period is $4.69$ $\mu\sec$. The channel models used are the flat Rayleigh fading channel, the typical urban (TUx) multipath fading model [@Typical; @Urban] that consists of $6$ taps with normalized delays of $\left[0,2,3,9,13,29\right]$ and average taps gains are $\left[0.2,0.398,0.2,0.1,0.063,0.039\right]$, which corresponds to a severe frequency-selective channel. The TUx model is also used to model a moderate frequency-selective channel where the number of taps in the channel is $9$ with normalized delays of $[0$, $1$, $\ldots$, $8]$ samples, and the average taps gains are $[0.269$, $0.174$, $0.289$, $0.117$, $0.023$, $0.058$, $0.036$, $0.026$, $0.008]$. The channel taps gains are assumed to be independent and Rayleigh distributed. The Monte Carlo simulation results included in this work are obtained by generating $10^{6}$ OFDM symbols per simulation run. Throughout this section, the ML coherent detector with perfect CSI will be denoted as coherent, while the coherent with linear and spline interpolation will be denoted as coherent-L and coherent-S, respectively. Moreover, the results are presented for the SISO system, $\mathit{\mathcal{N}\mathrm{=1}}$, unless it is mentioned otherwise.

Fig. \[fig:BER-Single-Double-Sided-Flat\] shows the BER of the single-sided (SS) and double-sided (DS) $D^{3}$ over flat fading channels for $\mathcal{K}=2,6$ and $3,7$, respectively, and using BPSK. The number of data symbols $\mathcal{K}_{D}=\mathcal{K}-1$ for the SS and $\mathcal{K}_{D}=\mathcal{K}-2$ for the DS because there are two pilot symbols at both ends of the data segment for the DS case. The results in the figure for the SS show that $\mathcal{K}$ has a noticeable impact on the BER where the difference between the $\mathcal{K}=2$ and $6$ cases is about $1.6$ dB at BER of $10^{-3}$. For the DS segment, the BER has the same trends of the SS except that it becomes closer to the coherent case because having more pilots reduces the probability of sequence inversion due to the phase ambiguity problem. The figure shows that the approximated and simulation results match very well for all cases, which confirms the accuracy of the derived approximations.

The effect of the frequency selectivity is illustrated in Fig. \[fig:BER-SISO-D3-SS-6-taps\] for the SS and DS configurations using$\mathcal{K}_{D}=1$. As can be noted from the figure, frequency-selective channels introduce error floors at high SNRs, which is due to the difference between adjacent channel values caused by the channel frequency selectivity. Furthermore, the figure shows a close match between the simulation and the derived approximations. The approximation results are presented only for $\mathcal{K}=2$ because evaluating the BER for $\mathcal{K}>2$ becomes computationally prohibitive. For example, evaluating the integral for the $\mathcal{K}=3$ requires solving a $6$-fold integral. The results for the frequency-selective channels are quite different from the flat fading cases. In particular, the BER performance drastically changes when the DS pilot segment is used. Moreover, the impact of the frequency selectivity is significant, particularly for the SS pilot case.

![BER in frequency-selective channels using BPSK, $\mathcal{K}_{D}=1$ and $\mathcal{N}=1$. \[fig:BER-SISO-D3-SS-6-taps\] ](graphics/fig_05_ber_ss_ds_flat){width="0.4\paperwidth"}

![BER in frequency-selective channels using BPSK, $\mathcal{K}_{D}=1$ and $\mathcal{N}=1$. \[fig:BER-SISO-D3-SS-6-taps\] ](graphics/fig_07_ber_ss_ds_6_taps_9_taps_awgn){width="0.4\paperwidth"}

Fig. \[fig:BER-SIMO-D3-flat\] shows the BER of the $1\times2$ SIMO $D^{3}$ over flat fading channels for SS and DS pilot segments. It can be noted from the figure that the maximum ratio combiner (MRC) BER with perfect CSI outperforms the DS and SS systems by about $2$ and $3$ dB, respectively. Moreover, the figure shows that the MLSD [@Wu; @2010] and the $D^{3}$ have equivalent BER for the SISO and SIMO scenarios.

![BER of the SISO $D^{3}$ and MLSD [@Wu; @2010] over the 6-taps frequency-selective channel using QPSK, $\mathcal{K}_{D}=1$, $\mathcal{N}=1$, $2.$\[fig:BER-SISO-D3-QPSK\] ](graphics/fig_08_ber_ss_ds_flat_simo){width="0.4\paperwidth"}

![BER of the SISO $D^{3}$ and MLSD [@Wu; @2010] over the 6-taps frequency-selective channel using QPSK, $\mathcal{K}_{D}=1$, $\mathcal{N}=1$, $2.$\[fig:BER-SISO-D3-QPSK\] ](graphics/fig_10_ber_ds_6_taps_qpsk_siso_simo){width="0.4\paperwidth"}

Figs. \[fig:BER-SISO-D3-QPSK\] shows the BER of the SISO and $1\times2$ SIMO MLSD, coherent, coherent-S and coherent-L systems over frequency-selective channels. For both SISO and SIMO, the BER of all the considered techniques converges at low SNRs because the AWGN dominates the BER in the low SNR range. For moderate and high SNRs, the $D^{3}$ outperforms all the other considered techniques except for the coherent, where the difference is about $3.5$ and $2.75$ dB at BER of $10^{-3}$ for the SISO and SIMO systems, respectively.

![ ](graphics/fig_12_ber_comparasion){width="0.4\paperwidth"}

![ ](graphics/fig_11_ber_ds_6_taps_16qam_siso){width="0.4\paperwidth"}

![](graphics/fig_14_ber_d3_full_rb){width="0.4\paperwidth"}

![](graphics/fig_13_ber_d3_coded){width="0.4\paperwidth"}

Conclusion and Future Work\[sec:Conclusion\]
============================================

This work proposed a new receiver design for OFDM-based broadband communication systems. The new receiver performs the detection process directly from the FFT output symbols without the need of experiencing the conventional steps of channel estimation, interpolation, and equalization, which led to a considerable complexity reduction. Moreover, the $D^{3}$ system can be deployed efficiently using the VA. The proposed system was analyzed theoretically where simple closed-form expressions were derived for the BER in several cases of interest. The analytical and simulation results show that the $D^{3}$ BER outperforms the coherent pilot-based receiver in various channel conditions, particularly in frequency-selective channels where the $D^{3}$ demonstrated high robustness.

Although the $D^{3}$ may perform well even in severe fading conditions, it is crucial to evaluate its sensitivity to various practical imperfections. Thus, we will consider in our future work the performance of the $D^{3}$ in the presence of various system imperfections such as phase noise, synchronization errors and IQ imbalance. Moreover, we will evaluate the $D^{3}$ performance in mobile fading channels, where the channel variation may introduce intercarrier interference.

Appendix I {#appendix-i .unnumbered}
==========

By defining the events $A_{\psi}>A_{n}\triangleq E_{\psi,n}$, $n\in\left\{ 0\text{, }1\text{, }\ldots,\psi-1\right\} $, then, $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=P\left(\bigcap\limits _{n=0}^{\psi-1}E_{\psi,n}\right).\label{E-PC-01}$$ Using the chain rule, $P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}$ can be written as, $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=\Pr\left(\left.E_{\psi,\psi-1}\right\vert \bigcap\limits _{n=0}^{\psi-2}E_{\psi,n}\right)\Pr\left(\bigcap\limits _{n=0}^{\psi-2}E_{\psi,n}\right).$$ For $\mathcal{K}=2$, $\psi=1$, $\tilde{\mathbf{d}}_{0}^{(0)}=[1$, $-1]$, $\tilde{\mathbf{d}}_{0}^{(1)}=[1$,$1]$, and thus, $$\begin{aligned}
P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}} & = & \Pr\left(E_{1,0}\right)\nonumber \\
 & = & \Pr\left(\Re\left\{ r_{1}r_{2}\right\} >\Re\left\{ -r_{1}r_{2}\right\} \right)=\Pr\left(\Re\left\{ r_{0}r_{1}\right\} >0\right).\end{aligned}$$ For $\mathcal{K}=3$, $\psi=4$, $\tilde{\mathbf{d}}_{0}^{(0)}=[1$, $1$, $-1]$, $\tilde{\mathbf{d}}_{0}^{(1)}=[1$, $-1$, $-1]$, $\tilde{\mathbf{d}}_{0}^{(2)}=[1$, $-1$, $1]$ and $\tilde{\mathbf{d}}_{0}^{(3)}=[1$, $1$,…$,1]$ . Using the chain rule $$\begin{aligned}
P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}} & = & \Pr\left(E_{3,2}|E_{3,1}\text{, }E_{3,0}\right)\Pr\left(E_{3,1},E_{3,0}\right)\nonumber \\
 & = & \Pr\left(E_{3,2}|E_{3,1}\text{, }E_{3,0}\right)\Pr\left(E_{3,1}|E_{3,0}\right)\Pr\left(E_{3,0}\right)\label{E-PrA3A0}\end{aligned}$$ However, $\Pr\left(E_{3,0}\right)=\Pr\left(A_{3}>A_{0}\right)$, and thus $$\begin{aligned}
\Pr\left(E_{3,0}\right) & = & \Pr\left(\Re\left\{ r_{0}r_{1}+r_{1}r_{2}\right\} >\Re\left\{ r_{0}r_{1}-r_{1}r_{2}\right\} \right)\nonumber \\
 & = & \Pr\left(\Re\left\{ r_{1}r_{2}\right\} >\Re\left\{ -r_{1}r_{2}\right\} \right)=\Pr\left(\Re\left\{ r_{1}r_{2}\right\} >0\right).\end{aligned}$$ The second term in (\[E-PrA3A0\]) can be evaluated by noting that the events $E_{3,1}$ and $E_{3,0}$ are independent. Therefore $\Pr\left(E_{3,1}|E_{3,0}\right)=\Pr\left(E_{3,1}\right)$, which can be computed as $$\begin{aligned}
\Pr\left(E_{3,1}\right) & = & \Pr\left(\Re\left\{ r_{0}r_{1}+r_{1}r_{2}\right\} >\Re\left\{ -r_{0}r_{1}+r_{1}r_{2}\right\} \right)\nonumber \\
 & = & \Pr\left(\Re\left\{ r_{0}r_{1}\right\} >\Re\left\{ -r_{0}r_{1}\right\} \right)=\Pr\left(\Re\left\{ r_{0}r_{1}\right\} >0\right).\end{aligned}$$ The first term in (\[E-PrA3A0\]) $\Pr\left(E_{3,2}|E_{3,1}\text{, }E_{3,0}\right)=1$ because if $A_{3}>\left\{ A_{1},A_{0}\right\} $, then $A_{3}>A_{2}$ as well. Consequently, $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=\Pr\left(\Re\left\{ r_{0}r_{1}\right\} >0\right)\Pr\left(\Re\left\{ r_{1}r_{2}\right\} >0\right).$$ By induction, it is straightforward to show that $P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}$ can be written as, $$P_{C}\mid_{\mathbf{H},\mathbf{\mathbf{d}=\mathbf{1}}}=\prod\limits _{n=0}^{\mathcal{K-}2}\Pr\left(\Re\left\{ r_{n}r_{\acute{n}}\right\} >0\right).$$

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[^1]: A. Saci, A. Al-Dweik and A. Shami are with the Department of Electrical and Computer Engineering, Western University, London, ON, Canada, (e-mail: {asaci, aaldweik, abdallah.shami}@uwo.ca).

[^2]: A. Al-Dweik is also with the Department of Electrical and Computer Engineering, Khalifa University, Abu Dhabi, UAE, (e-mail: dweik@kustar.ac.ae).

[^3]: Part of this work is protected by the US patent: A. Al-Dweik “Signal detection in a communication system.” U.S. Patent No. 9,596,119. 14 Mar. 2017.
