# Channel Estimation and Multipath Diversity Reception for RIS-Empowered
Broadband Wireless Systems Based on Cyclic-Prefixed Single-Carrier
Transmission

Qiang Li, , Miaowen Wen, , Ertugrul Basar, , George C. Alexandropoulos, ,
Kyeong Jin Kim, , and H. Vincent Poor, Qiang Li is with the College of
Information Science and Technology, Jinan University, Guangzhou 510632, China
(e-mail: qiangli@jnu.edu.cn).Miaowen Wen is with the School of Electronic and
Information Engineering, South China University of Technology, Guangzhou
510640, China (e-mail: eemwwen@scut.edu.cn).Ertugrul Basar is with the
Department of Electrical and Electronics Engineering, Koç University, Istanbul
34450, Turkey (e-mail: ebasar@ku.edu.tr).George C. Alexandropoulos is with the
National and Kapodistrian University of Athens, Panepistimiopolis Ilissia,
Greece (e-mail: alexandg@di.uoa.gr).Kyeong Jin Kim is with the Mitsubishi
Electric Research Laboratories, Cambridge, MA 02139, USA (e-mail:
kkim@merl.com).H. Vincent Poor is with the Department of Electrical and
Computer Engineering, Princeton University, Princeton, USA (e-mail:
poor@princeton.edu).

###### Abstract

In this paper, a cyclic-prefixed single-carrier (CPSC) transmission scheme
with phase shift keying (PSK) signaling is presented for broadband wireless
communications systems empowered by a reconfigurable intelligent surface
(RIS). In the proposed CPSC-RIS, the RIS is configured according to the
transmitted PSK symbols such that different cyclically delayed versions of the
incident signal are created by the RIS to achieve multipath diversity. A
practical and efficient channel estimator is developed for CPSC-RIS and the
mean square error of the channel estimation is expressed in closed-form. We
analyze the bit error rate (BER) performance of CPSC-RIS over frequency-
selective Nakagami-$m$ fading channels. An upper bound on the BER is derived
by assuming the maximum-likelihood detection. Furthermore, by resorting to the
concept of index modulation (IM), we propose an extension of CPSC-RIS, termed
CPSC-RIS-IM, which enhances the spectral efficiency. In addition to
conventional constellation information of PSK symbols, CPSC-RIS-IM uses the
full permutations of cyclic delays caused by the RIS to carry information. A
sub-optimal receiver is designed for CPSC-RIS-IM to aim at low computational
complexity. Our simulation results in terms of BER corroborate the performance
analysis and the superiority of CPSC-RIS(-IM) over the conventional CPSC
without an RIS and orthogonal frequency division multiplexing with an RIS.

###### Index Terms:

Channel estimation, cyclic delay diversity, cyclic-prefixed single-carrier,
index modulation, reconfigurable intelligence surface.

##  I Introduction

As an alternative to orthogonal frequency division multiplexing (OFDM),
cyclic-prefixed single-carrier (CPSC) shares some OFDM advantages, such as
low-complexity implementation, while avoiding several OFDM drawbacks, such as
a high peak-to-average power ratio and high sensitivity to carrier frequency
offsets [1, 2]. Moreover, uncoded CPSC with a minimum mean square error (MMSE)
receiver is able to extract multipath diversity in frequency-selective fading
channels for reasonably large values of block sizes and practical values of
bit error rate (BER) [3], and multipath diversity in identical [4] and non-
identical frequency selective fading channels [5] can be achieved by a
maximum-likelihood (ML) receiver irrespective of the uncoded block size. By
contrast, without channel coding or precoding techniques, OFDM cannot harvest
any multipath diversity even with the optimal ML receiver [3]. CPSC is a
promising solution to broadband wireless communications, and has been included
in the 3GPP Long Term Evolution standard for uplink transmission [1].

Rich multipath components, however, are required in the propagation
environment for CPSC to achieve a significant gain in terms of BER.
Unfortunately, this requirement may not be satisfied in general. Cyclic delay
diversity (CDD) [6] is a simple yet effective technique to solve this problem.
By transmitting the same signal from multiple antennas equipped in a single
transmitter with different cyclic delays, a multiple-input channel is
equivalently transformed to a single-input one with increasing multipath
diversity and without altering the receiver structure. For CDD-CPSC, the zero-
forcing (ZF) receiver fails to obtain any diversity gain, while the ML and
MMSE receivers are able to pick up diversity gains under some conditions, as
pure CPSC without CDD. To maximize the performance gain of applying CDD to
CPSC systems, non-linear equalizers, such as the frequency-domain Turbo
equalizer [7] and the block iterative generalized decision feedback equalizer
[8], were developed. In cooperative CPSC systems, distributed CDD can be
implemented via multiple single-antenna transmitters [9]. The idea of
distributed CDD was further applied to physical-layer security enhancement
[10] and spectrum sharing systems [11]. It is worth noting that all of the
above-mentioned CDD transmission schemes are based on multiple local or
distributed transmit antennas, each equipped with a radio frequency (RF)
chain. This configuration obviously complicates the system implementation,
expands the hardware cost, and increases the power consumption, which may be
unfavorable for energy-constrained and size-limited devices.

Recently, reconfigurable intelligent surface (RIS)-assisted transmission has
emerged as an easy-to-implement, low-cost, and green communication technique
[12, 13, 14]. On an RIS, there are a large number of passive reflecting
elements, each of which is able to reflect and exert adjustable amplitude-
phase changes on incident signals. In this sense, the amplitude-phase
responses of RISs and associated channels can be reconfigured intentionally to
achieve different purposes [15]. By exploiting this property, RISs have been
applied to enhance the wireless communications in terms of energy efficiency
[16, 17], weighted sum-rate [18], secrecy rate [19, 20, 21], network coverage
[22], error performance [23], localization accuracy [24], etc. In particular,
an RIS can be considered as an RF chain-free multi-antenna device. Inspired by
this observation, the authors in [25] and [26] designed RIS-based multiple-
input multiple-output transmission schemes for spatial multiplexing and
Alamouti space–time block coding, respectively. On the other hand, the idea of
index modulation (IM) [27], which uses the indices of some resources/building
blocks (e.g., antennas, subcarriers, time slots, and signal constellations) to
convey information, has been introduced into RIS-aided communications.
Obviously, RISs can be explicitly deployed to enhance the existing IM schemes
[28, 29, 30]. Moreover, new RIS-based IM schemes that use the indices of
reflection patterns to carry information were developed in [31, 32, 33, 34,
35, 36, 37, 38, 39]. In contrast to the above-mentioned RIS-related works that
focus on narrowband wireless communications over flat fading channels, the
authors in [40, 41, 42] studied RIS-aided OFDM broadband wireless
communications, in which the channel state information (CSI) is estimated in
the frequency domain without multipath diversity gains.

To the best of our knowledge, CPSC transmission techniques have not been
investigated for RIS-aided broadband wireless communications. Moreover, there
have not been any reports of time-domain channel estimation methods for RIS-
aided broadband frequency-selective channels in the literature. The fact that
an RIS acts as a multi-element reflector has also not been exploited to
implement CDD as well. Since OFDM and CPSC have different pros and cons, CPSC-
RIS can be considered as an alternative to OFDM-RIS. They can coexist and
complement each other for broadband communications. Against this background,
we design a CPSC transmission scheme for RIS-empowered broadband wireless
systems in this paper. However, the CP occupies a non-negligible bandwidth and
leads to a reduction of the spectral efficiency (SE), which is a notable
problem especially for short-packet communications. To solve this problem, we
further develop an extension of CPSC-RIS by resorting to the concept of IM.
Specifically, the main contributions of this paper are summarized as follows:

  * •

A CPSC transmission scheme with CDD and phase shift keying (PSK) signaling is
proposed for RIS-empowered broadband wireless systems to harvest multipath
diversity gains. An efficient pilot-aided channel estimator, which is able to
estimate the equivalent channel in the time domain via one transmission block,
is developed for CPSC-RIS. In this sense, CPSC-RIS provides a new framework of
channel estimation for RIS-aided broadband communications systems.

  * •

The proposed CPSC-RIS scheme provide a flexible design. By adjusting the block
size, CPSC-RIS can adapt to different degrees of variability in the CSI. In
particular, CPSC-RIS with a small block size can be dedicated to short-packet
communications at the cost of some diversity gains. On the other hand,
compared with conventional antenna-based CDD-CPSC, CPSC-RIS uses the RIS to
achieve CDD, avoiding multiple complicated and power-hungry RF chains.

  * •

Both ML and ZF/MMSE detectors are designed for CPSC-RIS. We analyze the BER
performance of CPSC-RIS over frequency-selective Nakagami-$m$ fading channels.
An upper bound on the BER of CPSC-RIS is derived in closed-form by assuming
the ML detection without channel estimation errors. From the BER analysis, the
diversity order and the performance improvement of CPSC-RIS over CPSC is
characterized.

  * •

An extension of CPSC-RIS, termed CPSC-RIS-IM, is also proposed to improve the
SE by resorting to the concept of IM. To be more specific, CPSC-RIS-IM encodes
the cyclic delays caused by the RIS for conveying additional information. A
low-complexity detector is then designed for CPSC-RIS-IM.

The remainder of this paper is organized as follows. Section II describes the
system model of CPSC-RIS, including the designs of channel estimator and
signal detectors. The performance of CPSC-RIS with the ML detection is
analyzed in Section III, followed by one IM-empowered extension of CPSC-RIS in
Section IV. Section V presents the computer simulation results, and finally,
Section VI concludes the paper.

Notation: Column vectors and matrices are denoted by lowercase and uppercase
boldface letters, respectively. Superscripts $(\cdot)^{*}$, $(\cdot)^{T}$, and
$(\cdot)^{H}$ stand for conjugate, transpose, and Hermitian transpose,
respectively. The $N\times N$ identity matrix is symbolized by
$\mathbf{I}_{N\times N}$. $\mathbf{1}_{1\times N}$ and $\mathbf{0}_{1\times
N}$ denote an all-one matrix and an all-zero matrix of size $1\times N$,
respectively. $\mathrm{diag}\\{\cdot\\}$ transforms a vector into a diagonal
matrix. $\Re\\{\cdot\\}$ and $\Im\\{\cdot\\}$ return the real and imaginary
parts of a complex number, respectively. $j=\sqrt{-1}$ is the imaginary unit.
$(\mathcal{C})\mathcal{N}(\mu,\sigma^{2})$ represents the (complex) Gaussian
distribution with mean $\mu$ and variance $\sigma^{2}$. The probability of an
event and the probability density function (PDF) of a random variable are
denoted by $\Pr(\cdot)$ and $p(\cdot)$, respectively. $E\\{\cdot\\}$ and
$Var\\{\cdot\\}$ denote expectation and variance, respectively. $\|{\cdot}\|$
stands for the Frobenius norm. $Q(\cdot)$, $\Gamma(\cdot)$, and
$\lfloor\cdot\rfloor$ represent the Gaussian $Q$-function, Gamma function, and
floor function, respectively. $\mathrm{rank}(\cdot)$ and $\mathrm{Tr}(\cdot)$
denote the rank and trace of a matrix, respectively. $\odot$ and $\star$
represent the Hadamard product and linear convolution, respectively.
$\angle(\cdot)$ denotes the phase of a complex number and
$\mathrm{cir}(\cdot)$ denotes right circulant operation.

##  II System Model

Figure 1: The block diagram of the proposed CPSC-RIS system comprising a
single-antenna transmitter, a multi-element RIS, and a single-antenna
receiver.

The block diagram of the proposed CPSC-RIS system is illustrated in Fig. 1,
where a CPSC transmitter communicates with a receiver over frequency-selective
block fading channels with the aid of an RIS. Both the transmitter and the
receiver are equipped with a single antenna, while the RIS that is connected
to a controller consists of $N_{R}$ reflecting elements. The RIS controller
follows the instructions from the transmitter to adjust the reflection
coefficients of the RIS [15]. However, $N_{R}$ is typically a large number,
which results in high overhead/complexity for channel estimation and
coefficient adjustment. Hence, we adopt the grouping method [40, 43], i.e.,
the total of $N_{R}$ reflecting elements are partitioned into $R$ reflecting
groups, each of which consists of $N_{G}=N_{R}/R$ adjacent elements sharing a
common reflection coefficient.111In practice, the grouping should be
implemented according to the statistical characteristics of channels. The
value of $N_{G}$ can be obtained by trial and error for achieving a good
trade-off between complexity, SE, and error performance. Also, the number of
reflecting elements in each group is not necessary to be equal. Although the
grouping that merges $N_{G}$ independent channels into a combined channel
results in a loss of some diversity gains, the numbers of reflection
coefficients and channel links that are required to be adjusted and estimated
respectively are reduced from $N_{R}$ to $R$, which significantly lowers the
complexity of RIS configuration and channel estimation. Accordingly, the
reflection coefficient for the $r$-th group of the RIS is expressed as
$\phi_{r}=a_{r}\exp(j\theta_{r})$, where $a_{r}$ is the amplitude coefficient
and $\theta_{r}$ is the phase shift with $r=1,\ldots,R$. For all $r$, we set
$a_{r}$ to unity and take $\theta_{r}\in[0,2\pi)$ from a discrete set
$\Theta$.

The direct channel link from the transmitter to the receiver is described by
the channel impulse response
$\mathbf{g}_{0}=[g_{0}(1),\ldots,g_{0}(L_{0})]^{T}$, where $L_{0}$ is the
number of channel taps for $\mathbf{g}_{0}$. The combined channels from the
transmitter to the $r$-th reflecting group and from the $r$-th reflecting
group to the receiver are denoted by
$\mathbf{h}_{r}=[h_{r}(1),\ldots,h_{r}(L_{r0})]^{T}$ with $L_{r0}$ channel
taps and $\mathbf{f}_{r}=[f_{r}(1),\ldots,f_{r}(L_{r1})]^{T}$ with $L_{r1}$
channel taps, respectively. The equivalent channel of the cascaded
transmitter-RIS-receiver link associated with the $r$-th reflecting group is
given by
$\mathbf{g}_{r}=[g_{r}(1),\ldots,g_{r}(L_{r})]^{T}=\mathbf{h}_{r}\star\mathbf{f}_{r}$
with $L_{r}=L_{r0}+L_{r1}-1$. In this paper, the amplitude of
$g_{\bar{r}}(l_{\bar{r}})$ is modeled as a Nakagami-$m$ distribution with
fading parameter $m_{\bar{r}}(l_{\bar{r}})\in[1,2,\ldots,\infty]$ and
spreading parameter $\Omega_{\bar{r}}(l_{\bar{r}})>0$, i.e.,

$\displaystyle
p_{|g_{\bar{r}}(l_{\bar{r}})|}(x)=\frac{2m_{\bar{r}}(l_{\bar{r}})^{m_{\bar{r}}(l_{\bar{r}})}x^{2m_{\bar{r}}(l_{\bar{r}})-1}}{\Omega_{\bar{r}}(l_{\bar{r}})^{m_{\bar{r}}(l_{\bar{r}})}\Gamma(m_{\bar{r}}(l_{\bar{r}}))}\exp\left(-\frac{m_{\bar{r}}(l_{\bar{r}})x^{2}}{\Omega_{\bar{r}}(l_{\bar{r}})}\right),\quad
x>0$ (1)

where $\bar{r}=0,1,\ldots,R$ and $l_{\bar{r}}=1,\ldots,L_{\bar{r}}$. Note that
the phase of $g_{\bar{r}}(l_{\bar{r}})$ is not uniformly distributed [44, 45];
its PDF is given in [45] and assumed to be the same for all $\bar{r}$ and
$l_{\bar{r}}$. Here, the channels associated with different reflecting groups
are approximately considered as independent and identically distributed
fading. We note that the practical case of correlated Nakagami-$m$ fading [46]
will be considered in future work.

###  II-A Transmitter Design

For each transmission, the CPSC transmitter modulates $b=N\log_{2}(M)$
information bits into a symbol vector $\mathbf{x}=[x(1),\ldots,x(N)]^{T}$,
where $x(n)$ is a normalized $M$-ary PSK symbol drawn from the constellation
$\mathcal{X}$ for $n=1,\ldots,N$. After adding an $L$-length CP to the
beginning of $\mathbf{x}$, the transmitted data vector can be given by

$\displaystyle\mathbf{x}_{CP}=[x(N-L+1),\ldots,x(N),x(1),\ldots,x(N)]^{T},$
(2)

where $L\geq\max\\{L_{0},\ldots,L_{R}\\}$.222The values of
$L_{0},\ldots,L_{R}$ are assumed to be known by using channel sounding schemes
[9]. At this point, the transmitter derives the values of $\theta_{r}$ for all
$r$ based on $\mathbf{x}_{CP}$ and convey them to the RIS controller that
further adjusts the $R$ reflecting groups. It is worth noting that since
$\mathbf{x}_{CP}$ is a PSK-modulated vector, cyclically delayed versions of
$\mathbf{x}_{CP}$ can be constructed by performing phase shifts on
$\mathbf{x}_{CP}$. To achieve CDD, $\theta_{r}$ is required to vary along with
the transmission of $\mathbf{x}_{CP}$ and thus denoted by the vector
$\bm{\theta}_{r}$ in the following. Moreover, by applying $\bm{\theta}_{r}$,
the signal reflected from the $r$-th group of the RIS is expected to be a
cyclically delayed version of $\mathbf{x}$, appended with an additional
$L$-length CP, i.e.,

$\displaystyle\mathbf{x}_{CP}(\Delta_{r})$
$\displaystyle=\Big{[}\underbrace{x(N-\Delta_{r}-L+1),\ldots,x(N-\Delta_{r})}_{\text{CP}},$
$\displaystyle\hskip
19.91684pt\underbrace{x(N-\Delta_{r}+1),\ldots,x(N),x(1),\ldots,x(N-\Delta_{r})}_{\mathbf{x}(\Delta_{r})}\Big{]}^{T},$
(3)

where $\mathbf{x}(\Delta_{r})$ is the cyclically delayed version of
$\mathbf{x}$ with the cyclic delay $\Delta_{r}$ for the $r$-th reflecting
group. In CPSC-RIS, we let $\Delta_{r}=r\Delta$, where
$L\leq\Delta\leq\lfloor{N/(R+1)}\rfloor$. Here, the constraint $\Delta\geq L$
is required to avoid inter-symbol interference (ISI) caused by multipath
channels, while the constraint $\Delta\leq\lfloor{N/(R+1)}\rfloor$ is required
to avoid ISI caused by simultaneous CPSC transmissions from different
reflecting groups considering the block length. From
$\mathbf{x}_{CP}\odot\exp(j\bm{\theta}_{r})=\mathbf{x}_{CP}(\Delta_{r})$, we
have

$\displaystyle\bm{\theta}_{r}=\Big{[}\angle x(N-\Delta_{r}-L+1)-\angle
x(N-L+1),\ldots,\angle x(N-\Delta_{r})-\angle x(N),$ $\displaystyle\angle
x(N-\Delta_{r}+1)-\angle x(1),\ldots,\angle x(N-\Delta_{r})-\angle
x(N)\Big{]}^{T},\quad r=1,\ldots,R$ (4)

and

$\displaystyle\Theta=\left\\{0,\frac{2\pi}{M},\cdots,\frac{2\pi(M-1)}{M}\right\\}.$
(5)

Figure 2: An example of $\mathbf{x}_{CP}$ and
$\mathbf{x}_{CP}(\Delta_{r}),r=1,\ldots,R$ for CPSC-RIS, where $N=8,R=3$, and
$\Delta=2$. To save space, $x(n)$ is denoted by $x_{n}$ in the figure and
later in Fig. 4.

In Fig. 2, we present an illustrative example of $\mathbf{x}_{CP}$ and
$\mathbf{x}_{CP}(\Delta_{r}),r=1,\ldots,R$, where $N=8,R=3,\Delta=2$, and
$\mathbf{x}_{CP}(\Delta_{r}),r=1,2,3$ are derived from $\mathbf{x}_{CP}$ via
phase shifts.

At the receiver, after discarding the CP, the channel output is given by

$\displaystyle\mathbf{y}=\mathbf{G}_{0}\mathbf{x}+\sum_{r=1}^{R}\mathbf{G}_{r}\mathbf{x}(\Delta_{r})+\mathbf{w}=\sum_{\bar{r}=0}^{R}\mathbf{G}_{\bar{r}}\mathbf{x}(\Delta_{\bar{r}})+\mathbf{w},$
(6)

where $\Delta_{0}=0$, $\mathbf{w}\in\mathbb{C}^{N\times 1}$ is the noise
vector with the distribution $\mathcal{CN}(\mathbf{0},N_{0}\mathbf{I}_{N\times
N})$, and $\mathbf{G}_{\bar{r}}$ is an $N\times N$ circulant channel matrix
given by $\mathbf{G}_{\bar{r}}=\mathrm{cir}(\mathbf{g}_{\bar{r}}^{0})$ with
$\mathbf{g}_{\bar{r}}^{0}=[\mathbf{g}_{\bar{r}}^{T},\mathbf{0}_{1\times(N-L_{\bar{r}})}]^{T}\in\mathbb{C}^{N\times
1}$. The signal-to-noise ratio (SNR) in this paper is defined as $E_{b}/N_{0}$
with $E_{b}=(N+L)/b$ being the average transmitted energy per bit. We can
rewrite (6) as

$\displaystyle\mathbf{y}=\sum_{{\bar{r}}=0}^{R}\mathbf{X}\mathbf{g}_{\bar{r}}^{0}(\Delta_{\bar{r}})+\mathbf{w}=\mathbf{X}\mathbf{g}_{eq}+\mathbf{w},$
(7)

where $\mathbf{g}_{\bar{r}}^{0}(\Delta_{\bar{r}})$ is the cyclicly delayed
version of $\mathbf{g}_{\bar{r}}^{0}$ with the cyclic delay
$\Delta_{\bar{r}}$, $\mathbf{X}$ is an $N\times N$ circulant matrix
represented by $\mathbf{X}=\mathrm{cir}(\mathbf{x})$, and $\mathbf{g}_{eq}$ is
given by

$\displaystyle\mathbf{g}_{eq}=\sum_{{\bar{r}}=0}^{R}\mathbf{g}_{\bar{r}}^{0}(\Delta_{\bar{r}})=[\mathbf{g}_{0}^{T},\mathbf{0}_{1\times(\Delta-
L_{0})},\mathbf{g}_{1}^{T},\mathbf{0}_{1\times(\Delta-
L_{1})},\ldots,\mathbf{g}_{R}^{T},\mathbf{0}_{1\times(N-R\Delta-L_{R})}]^{T}.$
(8)

From (8), provided that the cyclic delay constraints are satisfied, increasing
the value of $R$ enhances the channel impulse response (CIR). We also define
$\mathbf{g}_{eq}^{\prime}=[g_{eq}^{\prime}(1),\ldots,g_{eq}^{\prime}(L_{s})]^{T}=[\mathbf{g}_{0}^{T},\mathbf{g}_{1}^{T},\ldots,\mathbf{g}_{R}^{T}]^{T}$,
which consists of $L_{s}=\sum_{{\bar{r}}=0}^{R}L_{\bar{r}}$ non-zero entries
of $\mathbf{g}_{eq}$. As seen from (6) and (7), the cyclic delays on
$\mathbf{x}$ are converted into those on channels. By denoting
$\mathbf{G}_{eq}=\mathrm{cir}(\mathbf{g}_{eq})$, (7) can be further expressed
as

$\displaystyle\mathbf{y}=\mathbf{G}_{eq}\mathbf{x}+\mathbf{w}.$ (9)

It can be observed from (9) that CPSC-RIS can be regarded as a conventional
CPSC scheme with the enhanced CIR in (8). Employing CDD does not change the
characteristics of the transmitted signal. Obviously, the SE of CPSC-RIS is
given by

$\displaystyle\textsf{F}_{\text{CPSC-
RIS}}=\frac{b}{N+L}=\frac{N\log_{2}(M)}{N+L}\quad\text{bps/Hz}.$ (10)

###  II-B Receiver Design

In this subsection, we successively design a pilot-aided channel estimator and
various detectors for CPSC-RIS.

####  II-B1 Channel Estimator

For channel estimation, a pilot vector $\mathbf{x}_{p}\in\mathbb{C}^{N\times
1}$, instead of a data vector $\mathbf{x}$, is transmitted from the CPSC
transmitter, and the RIS is configured similarly to (II-A). Based on (7), the
least-square estimation of $\mathbf{g}_{eq}$ is given by

$\displaystyle\hat{\mathbf{g}}_{eq}=\mathbf{X}_{p}^{-1}\mathbf{y}=\mathbf{g}_{eq}+\mathbf{g}_{e},$
(11)

where $\mathbf{X}_{p}=\mathrm{cir}(\mathbf{x}_{p})$ and
$\mathbf{g}_{e}=\mathbf{X}_{p}^{-1}\mathbf{w}$ represents the vector of
channel estimation errors. $\mathbf{g}_{e}$ is independent of
$\mathbf{g}_{eq}$ and follows the Gaussian distribution
$\mathcal{CN}(\mathbf{0},\sigma_{e}^{2}\mathbf{I}_{N\times N})$. The MSE of
channel estimation can be derived as

$\displaystyle\epsilon$
$\displaystyle=E\\{\left\|{\mathbf{g}_{e}}\right\|^{2}\\}$
$\displaystyle=\mathrm{Tr}\left\\{\mathbf{X}_{p}^{-1}E\\{\mathbf{w}\mathbf{w}^{H}\\}\left(\mathbf{X}_{p}^{-1}\right)^{H}\right\\}$
$\displaystyle=N_{0}\mathrm{Tr}\left\\{\left(\mathbf{X}_{p}^{H}\mathbf{X}_{p}\right)^{-1}\right\\}.$
(12)

###### Proposition 1

To minimize the MSE, $\mathbf{X}_{p}$ should satisfy
$\mathbf{X}_{p}^{H}\mathbf{X}_{p}=N\mathbf{I}_{N\times N}$.

###### Proof:

See Appendix A. ∎

In particular, using Zadoff-Chu sequences [47] as $\mathbf{x}_{p}$ meets the
requirement. For $N$ being an even number, we take
$\mathbf{x}_{p}=[1,\exp({j\varpi\pi/N}),\exp({j4\varpi\pi/N}),\ldots,\exp({j\varpi\pi({N-1})}^{2}/N)]^{T}$,
where $\varpi$ is an arbitrary integer relatively prime to $N$. Obviously, in
this case, the achieved MSE is given by $\epsilon=N_{0}$ and we have
$\sigma_{e}^{2}=N_{0}/N$. In Fig. 3, we verify the theoretical analysis of MSE
by comparing it with simulation results, where
$N=16,R=4,m_{\bar{r}}(l_{\bar{r}})=3$, and $L_{\bar{r}}=2$ for all $\bar{r}$
and $l_{\bar{r}}$ with an exponential-decay power-delay profile (PDP). It can
be observed that the theoretical curve perfectly agrees with the simulation
results for our proposed channel estimator.

Figure 3: MSE versus $1/N_{0}$ for CPSC-RIS, where
$N=16,R=4,m_{\bar{r}}(l_{\bar{r}})=3$, and $L_{\bar{r}}=2$ for all $\bar{r}$
and $l_{\bar{r}}$ with an exponentially decaying PDP.

As described above, the channel estimation of CPSC-RIS can be completed via a
single block of pilots regardless of the value of $R$. This is much faster
than existing frequency-domain channel estimation methods for RIS-aided
broadband communications systems, such as the one in [41] that requires $R+1$
transmission blocks to extract the CSI. In this sense, CPSC-RIS can be
considered as a novel efficient framework of channel estimation in RIS-aided
broadband communications. With CSI, various detectors can be developed to
recover $\mathbf{x}$, which will be described in the following.

####  II-B2 ML Detector

From (7), the ML detector with the estimated CSI can be formulated as

$\displaystyle\hat{\mathbf{x}}=\arg\mathop{\min}\limits_{\bf{x}}{\left\|{{\bf{y}}-{\mathrm{cir}(\bf{x})\hat{\mathbf{g}}_{eq}}}\right\|^{2}},$
(13)

where $\hat{\mathbf{x}}=[\hat{x}(1),\ldots,\hat{x}(N)]^{T}$ is the estimate of
$\mathbf{x}$. Since the ML detector makes a joint decision on the $N$ $M$-ary
PSK symbols, the computational complexity in terms of complex multiplications
is of order $\sim\mathcal{O}(M^{N})$, which poses intolerable computational
burden to the receiver. Thus, a low-complexity detector is highly recommended.

####  II-B3 ZF/MMSE Detectors

Since $\mathbf{G}_{eq}$ is a circulant matrix, it has eigen decomposition
$\mathbf{G}_{eq}=\mathbf{F}^{H}\mathbf{\Lambda}\mathbf{F}$, where $\mathbf{F}$
is the unitary discrete Fourier transform (DFT) matrix with
$\mathbf{F}^{H}\mathbf{F}={\bf{I}}_{N\times N}$, and $\mathbf{\Lambda}$ is the
diagonal matrix whose diagonal elements are the $N$-point (non-unitary) DFT of
$\mathbf{g}_{eq}$, i.e.,

$\displaystyle\lambda(k)=\sum\limits_{n=1}^{N}{{g_{eq}}\left(n\right)\exp\left({-j\frac{{2\pi\left({n-1}\right)\left({k-1}\right)}}{N}}\right)},\quad
k=1,\ldots,N.$ (14)

Therefore, based on (9), the frequency-domain received signal can be written
as

$\displaystyle\mathbf{y}_{F}=\left[y_{F}(1),\ldots,y_{F}(N)\right]^{T}=\mathbf{F}\mathbf{y}=\mathbf{\Lambda}\mathbf{x}_{F}+\mathbf{w}_{F},$
(15)

where $\mathbf{x}_{F}$ and $\mathbf{w}_{F}$ are the frequency-domain
counterparts of $\mathbf{x}$ and $\mathbf{w}$, respectively. For frequency-
domain ZF/MMSE equalization, $\mathbf{y}_{F}$ is fed into $N$ single-tap
equalizers in parallel, each of which is simply realized by one complex-valued
multiplication, i.e.,

$\displaystyle\hat{x}_{F}(n)$ $\displaystyle=\phi(n)y_{F}(n),\quad
n=1,\ldots,N$ (16)

where $\hat{x}_{F}(n)$ is the output of the $n$-th equalizer, and
$\phi(n)=\lambda(n)^{*}/(|\lambda(n)|^{2}+cN_{0})$ with $c=1$ for MMSE
equalizer and $c=0$ for ZF equalizer. Finally, we arrive at the estimated
time-domain symbol vector via

$\displaystyle\hat{\mathbf{x}}=\mathbf{F}^{H}\hat{\mathbf{x}}_{F},$ (17)

where $\hat{\mathbf{x}}_{F}=[\hat{x}_{F}(1),\ldots,\hat{x}_{F}(N)]^{T}$.

Obviously, by using single-tap equalization and existing efficient algorithms
with the implementation of the DFT, the ZF/MMSE detectors achieve much lower
computational complexity than the ML detector of (13). As a compromise, the
ZF/MMSE detectors perform worse than the ML detector in terms of BER
performance. Since CPSC-RIS can be regarded as a conventional CPSC scheme with
enhanced CIR, the behaviors of ZF, ML, and MMSE detectors for CPSC-RIS are
similar to those for conventional CPSC schemes. Particularly, the ML and MMSE
detectors are able to harvest some diversity gains, while the ZF detector
cannot extract any multipath diversity since the equalization severely
amplifies the noise at frequencies in deep fades [3].

##  III Performance Analysis

In this section, we concentrate on the performance analysis of the ML detector
for CPSC-RIS. An upper bound on the BER of CPSC-RIS in the absence of channel
estimation errors is provided after deriving the pairwise error probability
(PEP) in closed-form.

Let us first study the conditional PEP with channel estimation errors, namely
$\Pr(\mathbf{X}\to\hat{\mathbf{X}}|\hat{\mathbf{g}}_{eq})$, which is the
probability of detecting $\mathbf{X}$ as $\hat{\mathbf{X}}$ conditioned on
$\hat{\mathbf{g}}_{eq}$. From (7), (9), and (13), we have

$\displaystyle\Pr\left(\mathbf{X}\to\hat{\mathbf{X}}|\hat{\mathbf{g}}_{eq}\right)=\Pr\left(\left\|\mathbf{y}-\mathbf{X}\hat{\mathbf{g}}_{eq}\right\|^{2}>\left\|\mathbf{y}-\hat{\mathbf{X}}\hat{\mathbf{g}}_{eq}\right\|^{2}\right).$
(18)

Based on (11), $\mathbf{y}$ can be expressed as

$\displaystyle\mathbf{y}=\mathbf{X}\left(\hat{\mathbf{g}}_{eq}-\mathbf{g}_{e}\right)+\mathbf{w}=\mathbf{X}\hat{\mathbf{g}}_{eq}+\bar{\mathbf{w}},$
(19)

where $\bar{\mathbf{w}}=-\mathbf{X}\mathbf{g}_{e}+\mathbf{w}$. After putting
(19) into (18), we are led to

$\displaystyle\Pr\left(\mathbf{X}\to\hat{\mathbf{X}}|\hat{\mathbf{g}}_{eq}\right)$
$\displaystyle=\Pr\left(-\left\|\left(\mathbf{X}-\hat{\mathbf{X}}\right)\hat{\mathbf{g}}_{eq}\right\|^{2}-2\Re\left\\{\bar{\mathbf{w}}^{H}\left(\mathbf{X}-\hat{\mathbf{X}}\right)\hat{\mathbf{g}}_{eq}\right\\}>0\right)$
$\displaystyle=\Pr\left(V>0\right).$ (20)

We observe that $V$ is Gaussian distributed with

$\displaystyle E\left\\{V\right\\}$
$\displaystyle=-\left\|\left(\mathbf{X}-\hat{\mathbf{X}}\right)\hat{\mathbf{g}}_{eq}\right\|^{2},$
$\displaystyle Var\left\\{V\right\\}$
$\displaystyle=2N_{0}\left\|\left(\mathbf{X}-\hat{\mathbf{X}}\right)\hat{\mathbf{g}}_{eq}\right\|^{2}+2\sigma_{e}^{2}\left\|\mathbf{X}^{H}\left(\mathbf{X}-\hat{\mathbf{X}}\right)\hat{\mathbf{g}}_{eq}\right\|^{2}.$

Therefore, (III) can be calculated as

$\displaystyle\Pr\left(\mathbf{X}\to\hat{\mathbf{X}}|\hat{\mathbf{g}}_{eq}\right)=Q\left(\frac{\left\|\left(\mathbf{X}-\hat{\mathbf{X}}\right)\hat{\mathbf{g}}_{eq}\right\|^{2}}{\sqrt{2N_{0}\left\|\left(\mathbf{X}-\hat{\mathbf{X}}\right)\hat{\mathbf{g}}_{eq}\right\|^{2}+2\sigma_{e}^{2}\left\|\mathbf{X}^{H}\left(\mathbf{X}-\hat{\mathbf{X}}\right)\hat{\mathbf{g}}_{eq}\right\|^{2}}}\right).$
(21)

In order to obtain the unconditional PEP, (21) should be further averaged over
$\hat{\mathbf{g}}_{eq}$. However, it is very difficult to derive a closed-form
expression. Here, we consider the case of perfect channel estimation, i.e.,
$\sigma_{e}^{2}=0$.

In the absence of channel estimation errors, (21) reduces to

$\displaystyle\Pr\left(\mathbf{X}\to\hat{\mathbf{X}}|\mathbf{g}_{eq}\right)=Q\left(\sqrt{\frac{\left\|\left(\mathbf{X}-\hat{\mathbf{X}}\right)\mathbf{g}_{eq}\right\|^{2}}{2N_{0}}}\right).$
(22)

The squared norm $\|(\mathbf{X}-\hat{\mathbf{X}})\mathbf{g}_{eq}\|^{2}$ can
equivalently be developed as

$\displaystyle\left\|(\mathbf{X}-\hat{\mathbf{X}})_{[L_{s}]}\mathbf{g}_{eq}^{\prime}\right\|^{2}$
$\displaystyle=\mathbf{g}_{eq}^{\prime H}\mathbf{A}\mathbf{g}_{eq}^{\prime}$
$\displaystyle=\mathbf{g}_{eq}^{\prime
H}\mathbf{U}^{H}\mathbf{D}\mathbf{U}\mathbf{g}_{eq}^{\prime}$
$\displaystyle=\sum\limits_{l=1}^{L_{s}}d(l)|\tilde{g}_{eq}(l)|^{2},$ (23)

where $(\mathbf{X}-\hat{\mathbf{X}})_{[L_{s}]}$ comprises $L_{s}$ columns of
$(\mathbf{X}-\hat{\mathbf{X}})$ corresponding to $L_{s}$ non-zero entries of
$\mathbf{g}_{eq}$,
$\mathbf{A}=(\mathbf{X}-\hat{\mathbf{X}})_{[L_{s}]}^{H}(\mathbf{X}-\hat{\mathbf{X}})_{[L_{s}]}$
is decomposed as $\mathbf{A}=\mathbf{U}^{H}\mathbf{D}\mathbf{U}$ with
$\mathbf{U}$ being a unitary matrix and
$\mathbf{D}=\mathrm{diag}\\{d(1),\ldots,d(L_{s})\\}$, and
$\tilde{\mathbf{g}}_{eq}=[\tilde{g}_{eq}(1),\ldots,\tilde{g}_{eq}(L_{s})]^{T}=\mathbf{U}\mathbf{g}_{eq}^{\prime}$.
Hence, (22) can be rewritten as

$\displaystyle\Pr\left(\mathbf{X}\to\hat{\mathbf{X}}|\mathbf{g}_{eq}\right)=Q\left(\sqrt{\frac{\sum\limits_{l=1}^{L_{s}}d(l)|\tilde{g}_{eq}(l)|^{2}}{2N_{0}}}\right),$
(24)

where $\tilde{g}_{eq}(l)=\mathbf{u}_{l}^{T}\mathbf{g}_{eq}^{\prime}$ with
$\mathbf{u}_{l}=[u_{l1},\ldots,u_{lL_{s}}]^{T}$ and $u_{ll^{\prime}}$ being
the $(l,l^{\prime})$-th element of $\mathbf{U}$ for
$l,l^{\prime}=1,\ldots,L_{s}$. Next, the approximations in [45] are used to
derive the moment generating function (MGF) of $|\tilde{g}_{eq}(l)|^{2}$.
Specifically, the real and imaginary parts of a complex Nakagami-$m$ fading
gain can be approximated as $\mathcal{N}(\mu_{X},\Omega_{s}/2)$ and
$\mathcal{N}(\mu_{Y},\Omega_{s}/2)$ distributions, respectively, where
$\mu_{X}=\sqrt{\sqrt{1-1/m}}\sqrt{\Omega}\cos\phi$,
$\mu_{Y}=\sqrt{\sqrt{1-1/m}}\sqrt{\Omega}\sin\phi$, and
$\Omega_{s}=\Omega(1-\sqrt{1-1/m})$ with $m$ being the fading parameter,
$\Omega$ being the spreading parameter, and $\phi$ being the angle parameter
of the complex Nakagami-$m$ distribution. Hence, by defining $\mu_{X}(l)$,
$\mu_{Y}(l)$, and $\Omega_{s}(l)$ as the corresponding parameters that are
associated with $g_{eq}^{\prime}(l)$, we have
$\Re\\{\tilde{g}_{eq}(l)\\}\sim\mathcal{N}(\Re\\{\mathbf{u}_{l}^{T}\bm{\mu}\\},\mathbf{u}_{2l}^{T}\bm{\Omega}_{s}/2)$
and
$\Im\\{\tilde{g}_{eq}(l)\\}\sim\mathcal{N}(\Im\\{\mathbf{u}_{l}^{T}\bm{\mu}\\},\mathbf{u}_{2l}^{T}\bm{\Omega}_{s}/2)$,
where
$\bm{\mu}=[\mu_{X}(1)+j\mu_{Y}(1),\ldots,\mu_{X}(L_{s})+j\mu_{Y}(L_{s})]^{T}$,
$\mathbf{u}_{2l}=[|u_{l1}|^{2},\ldots,|u_{lL_{s}}|^{2}]^{T}$, and
$\bm{\Omega}_{s}=[\Omega_{s}(1),\ldots,\Omega_{s}(L_{s})]^{T}$. Hence, the MGF
of $|\tilde{g}_{eq}(l)|^{2}$ can be given by

$\displaystyle\mathcal{M}_{l}(t)=\frac{1}{1-t\mathbf{u}_{2l}^{T}\bm{\Omega}_{s}}\exp\left(\frac{t|\mathbf{u}_{l}^{T}\bm{\mu}|^{2}}{1-t\mathbf{u}_{2l}^{T}\bm{\Omega}_{s}}\right).$
(25)

It can be shown that $|\mathbf{u}_{l}^{T}\bm{\mu}|^{2}$ in (25) is irrelevant
to the value of $\phi$. By using the well-known approximation $Q(x)\approx
1/12\cdot e^{-x^{2}/2}+1/4\cdot e^{-2x^{2}/3}$ [48], the unconditional PEP can
be derived as

$\displaystyle\Pr\left(\mathbf{X}\to\hat{\mathbf{X}}\right)$
$\displaystyle\approx
E\left\\{\frac{1}{12}\prod_{l=1}^{L_{s}}\exp\left(-\frac{d(l)|\tilde{g}_{eq}(l)|^{2}}{4N_{0}}\right)+\frac{1}{4}\prod_{l=1}^{L_{s}}\exp\left(-\frac{d(l)|\tilde{g}_{eq}(l)|^{2}}{3N_{0}}\right)\right\\}$
$\displaystyle=\frac{1}{12}\prod_{l=1}^{L_{s}}\mathcal{M}_{l}\left(-\frac{d(l)}{4N_{0}}\right)+\frac{1}{4}\prod_{l=1}^{L_{s}}\mathcal{M}_{l}\left(-\frac{d(l)}{3N_{0}}\right).$
(26)

Finally, according to the union bounding technique, the BER of CPSC-RIS can be
upper bounded by

$\displaystyle
P_{e}\leq\frac{1}{{b{2^{b}}}}\sum\limits_{\bf{X}}{\sum\limits_{\hat{\mathbf{X}}}{\Pr\left({{\bf{X}}\to\hat{\mathbf{X}}}\right)\xi\left({{\bf{X}},\hat{\mathbf{X}}}\right)}},$
(27)

where $\xi({{\bf{X}},\hat{\mathbf{X}}})$ is the number of erroneous bits when
$\mathbf{X}$ is detected as $\hat{\mathbf{X}}$.

Remark 1: The contribution of $L_{s}$ independent modified channel taps of
$\tilde{\mathbf{g}}_{eq}$ to the unconditional PEP in (III) suggests that the
highest possible diversity order is equal to $L_{s}$. However, this maximum
diversity order is achieved if and only if $d(l)$ are non-null for all $l$,
$\mathbf{X}$, and $\hat{\mathbf{X}}$. In general, the diversity order achieved
by CPSC-RIS is given by $\min\mathrm{rank}(\mathbf{A})$. Since there exist
some error events, such as

$\displaystyle\mathbf{X}=\begin{bmatrix}1&1&\cdots&1\\\ 1&1&\cdots&1\\\
\vdots&\vdots&\cdots&\vdots\\\ 1&1&\cdots&1\\\ 1&1&\cdots&1\\\
\end{bmatrix},\quad\hat{\mathbf{X}}=\begin{bmatrix}-1&-1&\cdots&-1\\\
-1&-1&\cdots&-1\\\ \vdots&\vdots&\cdots&\vdots\\\ -1&-1&\cdots&-1\\\
-1&-1&\cdots&-1\\\ \end{bmatrix},$

which make $\mathrm{rank}(\mathbf{A})=1$, the asymptotical (i.e., at infinite
SNR) diversity order of CPSC-RIS is equal to one. Since this type of error
events rarely occurs, simulation results in Section V will show that the
multipath diversity can be extracted by CPSC-RIS for moderate BER values and
reasonably large values of $N$.

On the other hand, compared with conventional CPSC systems without RIS and
CDD, CPSC-RIS involves extra $R$ RIS’s reflecting links with
$\sum_{{\bar{r}}=1}^{R}L_{\bar{r}}$ channel taps that are collected through
CDD. Therefore, from (III) with $L_{s}>>L_{0}$, the BER performance of CPSC-
RIS is expected to be much better than that of conventional CPSC systems
without RIS and CDD.

##  IV IM-Empowered CPSC-RIS System

In this section, we extend CPSC-RIS to CPSC-RIS-IM, which improves the SE of
CPSC-RIS by employing the concept of CDD-IM [49]. In CPSC-RIS-IM, the
information bits are modulated into the cyclic delays at the RIS in addition
to conventional constellation information of $N$ PSK symbols. Specifically, a
total of $b$ information bits are divided into two parts. The first part,
consisting of $b_{1}=\lfloor\log_{2}(R!)\rfloor$ bits, is used for permuting
the cyclic delays at the RIS. Unlike the CPSC-RIS scheme in which the cyclic
delay for the $r$-th reflecting group is fixed to $\Delta_{r}=r\Delta$, the
cyclic delay for the $r$-th reflecting group in CPSC-RIS-IM is given by

$\displaystyle\Delta_{r}=k_{r}\Delta,$ (28)

where $r=1,\ldots,R$ and $\mathbf{k}=[k_{1},k_{2},\ldots,k_{R}]^{T}$ is a full
permutation of $\\{1,2,\ldots,R\\}$ determined by the $b_{1}$ bits. The
mapping from $b_{1}$ bits to $\mathbf{k}$ can be realized by either a look-up
table or the permutation method [50]. Table I presents an example of a mapping
table between the $b_{1}$ bits and $\mathbf{k}$ for $R=3$ and $b_{1}=2$, where
the last two permutations are unused.

TABLE I: An example of mapping table between $p_{1}$ bits and $\mathbf{k}$ for $R=3$. $b_{1}$ bits | [0 0] | [0 1] | [1 0] | [1 1] | – | –
---|---|---|---|---|---|---
$\mathbf{k}$ | $[1,2,3]^{T}$ | $[2,1,3]^{T}$ | $[1,3,2]^{T}$ | $[2,3,1]^{T}$ | $[3,2,1]^{T}$ | $[3,1,2]^{T}$

The second part, comprised of $b_{2}=N\log_{2}(M)$ bits, is mapped to $N$ PSK
symbols $\mathbf{x}=[x(1),\ldots,x(N)]^{T}$ via the regular $M$-PSK
constellation $\mathcal{X}$ that is also used in CPSC-RIS. Therefore, the SE
of CPSC-RIS-IM is given by

$\displaystyle\textsf{F}_{\text{CPSC-RIS-
IM}}=\frac{N\log_{2}(M)+\lfloor\log_{2}(R!)\rfloor}{N+L}\quad\text{bps/Hz},$
(29)

which shows that CPSC-RIS-IM can transmit $\lfloor\log_{2}(R!)\rfloor$ more
bits per block transmission than CPSC-RIS. Obviously, increasing the value of
$R$ leads to higher SE. However, a larger value of $R$ also incurs higher
complexity of implementing the mapping between the $b_{1}$ bits and
$\mathbf{k}$ at the transceiver, cyclic delays at the RIS, and signal
detection at the receiver.

Figure 4: Examples of $\mathbf{x}_{CP}$ and
$\mathbf{x}_{CP}(\Delta_{r}),r=1,\ldots,R$ for CPSC-RIS-IM with
$\mathbf{k}=[2,1]^{T}$ and $\mathbf{k}=[1,2]^{T}$, where $N=8,R=2$, and
$\Delta=2$.

In Fig. 4, we present an example of $\mathbf{x}_{CP}$ and
$\mathbf{x}_{CP}(\Delta_{r}),r=1,\ldots,R$ for CPSC-RIS-IM, where
$N=8,R=2,\Delta=2$, and both two possible realizations of $\mathbf{k}$ are
considered. It should be noted that there is ambiguity at the receiver to
detect $\mathbf{x}$ and $\mathbf{k}$. We exemplify this point with Fig. 4 and
letting $x(1)=x(2)=\cdots=x(8)$. Obviously, in this case, both
$\mathbf{x}_{CP}(\Delta_{1})$ and $\mathbf{x}_{CP}(\Delta_{2})$ for
$\mathbf{k}=[2,1]^{T}$ are completely the same as those for
$\mathbf{k}=[1,2]^{T}$. Hence, we take the first symbol of $\mathbf{x}$,
namely $x(1)$, as an anchor point to avoid ambiguity in signal detection.
Specifically, instead of $\mathbf{x}$, we transmit
$\tilde{\mathbf{x}}=\mathbf{x}\odot[\exp(j\pi/M),\mathbf{1}_{1\times(N-1)}]^{T}$
at the CPSC transmitter. In this manner, the Euclidean distance between $x(1)$
and any other symbols in $\tilde{\mathbf{x}}$ is maximized.

Similar to (7), the received signal for CPSC-RIS-IM can be expressed as

$\displaystyle\mathbf{y}=\tilde{\mathbf{X}}\mathbf{g}_{eq,\mathbf{k}}+\mathbf{w},$
(30)

where the notations are defined in (7) except that
$\tilde{\mathbf{X}}=\mathrm{cir}(\tilde{\mathbf{x}})$ and
$\mathbf{g}_{eq,\mathbf{k}}$ is revised as

$\displaystyle\mathbf{g}_{eq,\mathbf{k}}=\sum_{{\bar{r}}=0}^{R}\mathbf{g}_{\bar{r}}^{0}(k_{r}\Delta),$
(31)

where $k_{0}=0$. Based on (30), ML and low-complexity detectors can be
designed for CPSC-RIS-IM, which are discussed in the following.

####  IV-1 ML Detector

The optimal ML detector makes a joint decision on $\mathbf{k}$ and
$\mathbf{x}$ by searching all possible combinations of them, namely

$\displaystyle\left(\hat{\mathbf{x}},\hat{\mathbf{k}}\right)=\arg\mathop{\min}\limits_{\mathbf{x},\mathbf{k}}{\left\|{{\bf{y}}-\tilde{\mathbf{X}}{\mathbf{g}_{eq,\mathbf{k}}}}\right\|^{2}},$
(32)

where $\hat{\mathbf{x}}$ and $\hat{\mathbf{k}}$ are the estimates of
$\mathbf{x}$ and $\mathbf{k}$, respectively. Then, the corresponding $b$
information bits can be readily recovered from $\hat{\mathbf{x}}$ and
$\hat{\mathbf{k}}$. Obviously, the ML detector should search all $R!M^{N}$
possible combinations of $\mathbf{x}$ and $\mathbf{k}$ to find out the one
that minimizes the ML metric, which results in the computational complexity in
terms of complex multiplications of order $\sim\mathcal{O}(R!M^{N})$ per block
transmission. To lower the computational complexity, we next develop a low-
complexity detector.

####  IV-2 Low-Complexity Detector

For a realization of $\mathbf{k}$, denoted by
$\mathbf{k}_{i},i=1,\ldots,2^{b_{1}}$, we can employ the ZF/MMSE detectors
similar to (15)-(17) for obtaining the estimate of $\tilde{\mathbf{x}}$,
denoted by $\hat{\tilde{\mathbf{x}}}_{\mathbf{k}_{i}}$. Further, given
$\mathbf{k}_{i}$, $\mathbf{x}$ is estimated as
$\hat{\mathbf{x}}_{\mathbf{k}_{i}}=[\hat{x}_{\mathbf{k}_{i}}(1),\ldots,\hat{x}_{\mathbf{k}_{i}}(N)]^{T}=\hat{\tilde{\mathbf{x}}}_{\mathbf{k}_{i}}\odot[\exp(-j\pi/M),\mathbf{1}_{1\times(N-1)}]^{T}$.
Then, each symbol in $\hat{\mathbf{x}}_{\mathbf{k}_{i}}$ is demodulated
independently via $M$-PSK demodulation, namely

$\displaystyle\hat{s}_{\mathbf{k}_{i}}(n)=\arg\mathop{\min}\limits_{s\in\mathcal{X}}{\left\|\hat{x}_{\mathbf{k}_{i}}(n)-s\right\|^{2}},\quad
n=1,\ldots,N$ (33)

where $\hat{s}_{\mathbf{k}_{i}}(n)$ represents the hard decision on
$\hat{x}_{\mathbf{k}_{i}}(n)$. Based on
$\hat{\mathbf{s}}_{\mathbf{k}_{i}}=[\hat{s}_{\mathbf{k}_{i}}(1),\ldots,\hat{s}_{\mathbf{k}_{i}}(N)]^{T}$,
the metric that $\mathbf{k}_{i}$ is used at the transmitter can be given by

$\displaystyle
T(\mathbf{k}_{i})={\left\|{{\bf{y}}-\hat{\tilde{\mathbf{S}}}_{\mathbf{k}_{i}}{\mathbf{g}_{eq,\mathbf{k}_{i}}}}\right\|^{2}},$
(34)

where
$\hat{\tilde{\mathbf{S}}}_{\mathbf{k}_{i}}=\mathrm{cir}(\hat{\tilde{\mathbf{s}}}_{\mathbf{k}_{i}})$
with
$\hat{\tilde{\mathbf{s}}}_{\mathbf{k}_{i}}=\hat{\mathbf{s}}_{\mathbf{k}_{i}}\odot[\exp(j\pi/M),\mathbf{1}_{1\times(N-1)}]^{T}$.
Finally, the estimates of $\mathbf{k}$ and $\mathbf{x}$ can be expressed as

$\displaystyle\hat{\mathbf{k}}=\min\limits_{\mathbf{k}_{i}}T(\mathbf{k}_{i}),$
(35)

and $\hat{\mathbf{x}}=\hat{\mathbf{s}}_{\hat{\mathbf{k}}}$, respectively.

####  IV-3 Performance Analysis

Here, we analyze the BER performance of ML detection for CPSC-RIS-IM. Let us
first define a permutation matrix $\mathbf{P}_{\mathbf{k}}$, which is made up
of $N$ columns of $\mathbf{I}_{N\times N}$ and satisfies
$\mathbf{P}_{\mathbf{k}}\mathbf{g}_{eq}=\mathbf{g}_{eq,\mathbf{k}}$. Then,
(30) can be rewritten as

$\displaystyle\mathbf{y}=\tilde{\mathbf{X}}\mathbf{P}_{\mathbf{k}}\mathbf{g}_{eq}+\mathbf{w}.$
(36)

In the case of perfect channel estimation, similar to (22), the conditional
PEP can be written as

$\displaystyle\Pr\left((\tilde{\mathbf{X}},{\mathbf{k}})\to(\hat{\tilde{\mathbf{X}}},{\hat{{\mathbf{k}}}})|{\mathbf{g}}_{eq}\right)$
$\displaystyle=\Pr\left(\left\|\mathbf{y}-\tilde{\mathbf{X}}\mathbf{P}_{\mathbf{k}}{\mathbf{g}}_{eq}\right\|^{2}>\left\|\mathbf{y}-\hat{\tilde{\mathbf{X}}}\mathbf{P}_{\hat{{\mathbf{k}}}}{\mathbf{g}}_{eq}\right\|^{2}\right)$
$\displaystyle=Q\left(\sqrt{\frac{\left\|\left(\tilde{\mathbf{X}}\mathbf{P}_{\mathbf{k}}-\hat{\tilde{\mathbf{X}}}\mathbf{P}_{\hat{{\mathbf{k}}}}\right)\mathbf{g}_{eq}\right\|^{2}}{2N_{0}}}\right).$
(37)

The unconditional PEP, namely
$\Pr((\tilde{\mathbf{X}},{\mathbf{k}})\to(\hat{\tilde{\mathbf{X}}},{\hat{{\mathbf{k}}}}))$,
can be obtained by following the methods in (III)-(III), which is omitted to
avoid redundancy. Finally, an upper bound on the BER of CPSC-RIS-IM is given
by

$\displaystyle
P_{e}\leq\frac{1}{{b{2^{b}}}}\sum\limits_{\tilde{\mathbf{X}},{\mathbf{k}}}{\sum\limits_{\hat{\tilde{\mathbf{X}}},\hat{{\mathbf{k}}}}}{\Pr\left((\tilde{\mathbf{X}},{\mathbf{k}})\to(\hat{\tilde{\mathbf{X}}},{\hat{{\mathbf{k}}}})\right)\xi\left((\tilde{\mathbf{X}},{\mathbf{k}}),(\hat{\tilde{\mathbf{X}}},\hat{{\mathbf{k}}})\right)},$
(38)

where
$\xi((\tilde{\mathbf{X}},{\mathbf{k}}),(\hat{\tilde{\mathbf{X}}},\hat{{\mathbf{k}}}))$
is the number of erroneous bits when $(\tilde{\mathbf{X}},{\mathbf{k}})$ is
detected as $(\hat{\tilde{\mathbf{X}}},\hat{{\mathbf{k}}})$.

Remark 2: In CPSC-RIS and CPSC-RIS-IM, the amplitude coefficients of all
reflecting groups are fixed to 1, i.e., $a_{r}=1$ for $r=1,\ldots,R$.
Actually, a reflection-type power amplifier can be deployed for each RIS
element, such that $a_{r}$ can be drawn from a discrete set, say
$\\{a^{1},\ldots,a^{t}\\}$ with all entries equal to or greater than 1 [51].
In this manner, the SE of CPSC-RIS and CPSC-RIS-IM can be further improved by
encoding partial information (up to $R\log_{2}(t)$ bits) into the amplitude
coefficients, and a multi-ring PSK constellation is observed at the receiver.
Obviously, there is a trade-off between the SE and error performance.

##  V Simulation Results

In this section, we conduct Monte Carlo simulations to evaluate the uncoded
BER performance of CPSC-RIS(-IM) by taking CPSC and OFDM-RIS [41] as
benchmarks. In OFDM-RIS, the RIS is configured to maximize the channel gain
according to the method in [41] by assuming perfect CSI. In all simulations,
we plot the BER versus SNR = $E_{b}/N_{0}$. We use $N_{G}=8$, while increasing
it is expected to improve the performance for a given value of $R$ since the
signal power reflected from each group is enhanced. The distances between the
transmitter and the receiver, between the transmitter and the RIS, and between
the RIS and the receiver are $D_{0}=50$ m, $D_{1}=5$ m, and $D_{2}=50$ m,
respectively. The large-scale path loss of the transmitter-to-receiver,
transmitter-to-RIS, and RIS-to-receiver links are given by $D_{0}^{-2.5}$,
$D_{1}^{-2}$, and $D_{2}^{-2}$, respectively. All wireless channels are
modeled by the exponentially decaying PDP with the decaying factor of unity,
where each tap is generated according to the Nakagami-$m$ distribution. For
simplicity, we assume that $L_{0}=\cdots=L_{R}=2$,
$m_{0}(1)=\cdots=m_{0}(L_{0})=m_{1}(1)=\cdots=m_{1}(L_{1})=\cdots=m_{R}(L_{R})=2$,
and $\Delta=L$. Each BER point is obtained by averaging over at least $10^{5}$
channel realizations.

Figure 5: Performance comparison among CPSC-RIS, CPSC, and OFDM-RIS, where
$N=8,M=2,R=2$, and perfect CSI is assumed at the receiver. Figure 6:
Performance comparison between CPSC-RIS and CPSC, where $N=8,16,M=2,4,R=2,4$,
and perfect CSI is assumed at the receiver.

Fig. 5 depicts the performance comparison among CPSC-RIS, CPSC, and OFDM-RIS,
where $N=8,M=2,R=2$, and perfect CSI is assumed at the receiver. CPSC-RIS
employs the ML, MMSE, and ZF detectors, while CPSC and OFDM-RIS use the ML
detectors. To verify the analysis given in Section III, we also plot the BER
upper bound (27) as a theoretical bound for the ML detector of CPSC-RIS in
Fig. 5. As seen from Fig. 5, for CPSC-RIS with the optimal ML detection, the
BER upper bound agrees with the simulation results in the high SNR region. In
particular, for CPSC-RIS, the ZF detector cannot harvest any diversity and
performs the worst. In contrast, for CPSC-RIS, both the ML and MMSE detectors
achieve some diversity gains for practical BER values, while they fails to
extract diversity gains at infinite SNR. Moreover, CPSC-RIS even with low-
complexity MMSE detector significantly outperforms conventional CPSC and OFDM-
RIS with the optimal ML detectors throughout the considered SNR region. CPSC-
RIS with the ML and MMSE detectors obtain approximately 10 dB and 5 dB SNR
gains, respectively, over CPSC with the ML detector at a BER value of
$10^{-4}$. Since the MMSE detector is able to achieve acceptable performance
with low implementation complexity, the MMSE-based detectors are adopted for
CPSC-RIS(-IM) and CPSC in the remaining simulations.

Fig. 6 evaluates the BER performance of CPSC-RIS and CPSC, where
$N=8,16,M=2,4,R=2,4$, and perfect CSI is assumed at the receiver. It can be
observed from Fig. 6 that all considered CPSC-RIS schemes perform better than
all CPSC schemes for all SNR values thanks to the extra $R$ reflecting links
introduced by the RIS, and the gain is more prominent for a larger value of
$R$. For $N=16$ and $M=2$, CPSC-RIS with $R=4$ obtains about 3 dB SNR gain
over that with $R=2$, at a BER value of $10^{-4}$. Increasing the value of $N$
achieves SNR gains of 2 dB and 1 dB for CPSC-RIS and CPSC, respectively, at a
BER value of $10^{-4}$. By contrast, a larger constellation size deteriorates
the performance (about 2 dB SNR loss) for both CPSC-RIS and CPSC.

Figure 7: Performance comparison between CPSC-RIS and CPSC, where
$N=16,32,M=2$, and $R=6$. The cases of perfect channel estimation and
practical channel estimation in (11) are considered.

Fig. 7 presents the performance comparison between CPSC-RIS and CPSC with
perfect and estimated CSI at the receiver, where $N=16,32,M=2$, and $R=6$. The
estimated CSI is obtained via (11). As expected, all schemes with estimated
CSI perform worse than the corresponding schemes with perfect CSI, and
approximately 4 dB SNR loss is incurred by the imperfect CSI. We observe from
Fig. 7 that, in both cases, increasing the value of $N$ from 16 to 32 leads to
SNR gains for both CPSC-RIS and CPSC. Specifically, the SNR gain for CPSC-RIS
is about 1 dB at a BER value of $10^{-4}$, while that for CPSC is minor. With
both perfect and estimated CSI at the receiver, CPSC-RIS outperforms CPSC for
all considered SNR values. Moreover, for $N=16$ and 32, CPSC-RIS with
estimated CSI achieves about 9 dB and 10 dB SNR gains, respectively, over CPSC
with perfect CSI, at a BER value of $10^{-3}$.

Figure 8: Performance comparison between CPSC-RIS and CPSC-RIS-IM, where
$N=16,32,M=2,R=2,4$, and perfect CSI is assumed at the receiver.

In Fig. 8, we compare the BER performance of CPSC-RIS and CPSC-RIS-IM, where
$N=16,32,M=2,R=2,4$, and perfect CSI is assumed at the receiver. As seen from
Fig. 8, CPSC-RIS-IM outperforms CPSC-RIS at high SNR with the same parameter
settings. Particularly, the performance gain achieved by CPSC-RIS-IM over
CPSC-RIS for $R=4$ is larger than that for $R=2$. This can be explained as
follows. The cyclic delay constraints in Section II are met for both $R=2$ and
$R=4$. As seen from (8), increasing the value of $R$ from 2 to 4 enhances the
equivalent CIR. Hence, the performance of both CPSC-RIS-IM and CPSC-RIS
becomes better when $R$ increases from 2 to 4. Moreover, for CPSC-RIS-IM, a
larger value of $R$ results in more transmitted IM bits, and the transmit
power becomes higher for a given value of SNR=$E_{b}/N_{0}$. In addition, the
transmission of IM bits itself does not consume power. Therefore, the
performance improvement for CPSC-RIS-IM is more significant than that for
CPSC-RIS when $R$ increases from 2 to 4. Also, increasing the value of $N$
enhances the performance of both CPSC-RIS and CPSC-RIS-IM. In particular,
doubling the value of $R$ has a more significant impact than that of $N$ on
both CPSC-RIS and CPSC-RIS-IM. Specifically, about 3 dB and 1 dB SNR gains can
be achieved at a BER value of $10^{-4}$, by increasing the value of $R$ from 2
to 4 and the value of $N$ from 16 to 32, respectively.

Figure 9: Performance comparison between CPSC-RIS and CPSC-RIS-IM, where
$N=16,R=4,M=2,4,8$, and perfect CSI is assumed at the receiver.

Fig. 9 illustrates the performance comparison between CPSC-RIS and CPSC-RIS-
IM, where $N=16,R=4,M=2,4,8$, and perfect CSI is assumed at the receiver. As
shown in Fig. 9, with $M=2$, CPSC-RIS-IM performs better than CPSC-RIS for
practical BER values with SNR$>38$ dB. With increasing the value of $M$, the
superiority of CPSC-RIS-IM over CPSC-RIS is lost. This is because that the
proportion of the IM bits in CPSC-RIS-IM decreases with increasing the value
of $M$. However, from Fig. 6, it is expected that CPSC-RIS-IM still achieves
much better BER performance than conventional CPSC without RIS and CDD.

##  VI Conclusion

In this paper, we have proposed a CDD-enhancing CPSC transmission scheme for
RIS-aided broadband wireless systems. The channel estimation, signal
detection, and SE improvement have all been studied for CPSC-RIS.
Specifically, a practical and efficient time-domain channel estimator has been
developed for CPSC-RIS, which can be considered as a general channel estimator
for RIS-aided broadband communications. The optimal ML and low-complexity
ZF/MMSE detectors have been designed for CPSC-RIS, and the BER performance of
the ML detector over frequency-selective Nakagami-$m$ fading channels has been
analyzed with the theoretically derived upper bound. Further, CPSC-RIS has
been extended to CPSC-RIS-IM for improving the SE by using the concept of IM.
We conclude that the proposed two schemes can be considered as promising
candidates for RIS-empowered broadband wireless communications.

##  Appendix A Proof of Proposition 1

Since $\mathbf{X}_{p}$ is a circulant matrix,
$\mathbf{X}_{p}^{H}\mathbf{X}_{p}$ is also a circulant matrix and can be
expressed as $\mathbf{X}_{p}^{H}\mathbf{X}_{p}=\mathrm{cir}(\mathbf{x}_{pp})$,
where $\mathbf{x}_{pp}=[x_{pp}(1),x_{pp}(2),\ldots,x_{pp}(N)]^{T}$ with
$x_{pp}(n)=\mathbf{x}_{p}^{H}(n-1)\mathbf{x}_{p}$ and $\mathbf{x}_{p}(n-1)$
being the cyclically delayed version of $\mathbf{x}_{p}$ with the cyclic delay
$n-1$ for $n=1,\ldots,N$. Obviously, we have
$x_{pp}(1)=\mathbf{x}_{p}^{H}\mathbf{x}_{p}=N$.

Further, as a circulant matrix, $\mathbf{X}_{p}^{H}\mathbf{X}_{p}$ can be
decomposed as
$\mathbf{X}_{p}^{H}\mathbf{X}_{p}=\mathbf{F}^{-1}\mathbf{D}\mathbf{F}$, where
$\mathbf{F}$ is the unitary DFT matrix and
$\mathbf{D}_{p}=\mathrm{diag}([d_{p}(1),\ldots,d_{p}(N)]^{T})$ with

$\displaystyle
d_{p}\left(k\right)=\sum_{n=1}^{N}{x_{pp}\left(n\right)\exp\left(-j\frac{2\pi\left(n-1\right)\left(k-1\right)}{N}\right)},\quad
k=1,\dots,N,$ (39)

and
$\sum\nolimits_{k=1}^{N}{d_{p}\left(k\right)}=\mathrm{Tr}\left\\{\mathbf{X}_{p}^{H}\mathbf{X}_{p}\right\\}=N^{2}$.
From (II-B1), the MSE can be rewritten as

$\displaystyle\epsilon=N_{0}\mathrm{Tr}\left\\{\left(\mathbf{X}_{p}^{H}\mathbf{X}_{p}\right)^{-1}\right\\}=N_{0}\mathrm{Tr}\left\\{\mathbf{F}^{-1}\mathbf{D}_{p}^{-1}\mathbf{F}\right\\}=N_{0}\mathrm{Tr}\left\\{\mathbf{D}_{p}^{-1}\right\\}=N_{0}\sum_{k=1}^{N}{\frac{1}{d_{p}\left(k\right)}}.$
(40)

From (40), under the condition that
$\sum\nolimits_{k=1}^{N}{d_{p}\left(k\right)}=N^{2}$, to minimize the MSE, we
should have $d_{p}(1)=\ldots=d_{p}(N)=N$. Finally, we conclude that
$x_{pp}=[N,0,\ldots,0]^{T}$ and
$\mathbf{X}_{p}^{H}\mathbf{X}_{p}=N\mathbf{I}_{N\times N}$ when the MSE is
minimized, which completes the proof.

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