# Wideband Channel Estimation for IRS-Aided Systems in the face of Beam Squint

Siqi Ma, Wenqian Shen, Jianping An, Lajos Hanzo,  S. Ma, W. Shen, J. An are
with the School of Information and Electronics, Beijing Institute of
Technology, Beijing 100081, China (E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>an@bit.edu.cn). L. Hanzo is with the Department of
Electronics and Computer Science, University of Southampton, Southampton SO17
1BJ, UK (e-mail: lh@ecs.soton.ac.uk).

###### Abstract

Intelligent reflecting surfaces (IRSs) improve both the spectral and energy
efficiency of wideband communication systems by using low-cost passive
elements for reflecting the impinging signals with adjustable phase shifts. To
fulfill the potentials of IRS-aided communication systems, accurate channel
state information (CSI) is indispensable, but it is challenging to acquire,
since these passive devices cannot carry out transmit/receive signal
processing. The existing channel estimation methods conceived for wideband
IRS-aided communication systems only consider the channel’s frequency
selectivity, but ignore the effect of beam squint, despite its severe
performance degradation. Hence we fill this gap and conceive the wideband
channel estimation for IRS-aided communication systems by taking the effect of
beam squint into consideration. We demonstrate that the mutual correlation
function between the spatial steering vectors and the two-hop channel
reflected by the IRS has two peaks, which leads to two angles estimated for a
single propagation path, due to the effect of beam squint. One of the two
estimated angles is frequency-independent ‘actual angle’, while the other one
is the frequency-dependent ‘false angle’. To reduce the influence of false
angles on channel estimation, we propose a twice-stage orthogonal matching
pursuit (TS-OMP) algorithm, where the path angles of the two-hop channel
reflected by the IRS are obtained in the first stage, while the propagation
gains and delays are obtained in the second stage. Moreover, we propose the
corresponding bespoke pilot design by exploiting the specific the
characteristics of the mutual correlation function and cross-entropy theory,
for achieving an improved channel estimation performance. Our simulation
results demonstrate the superiority of the proposed channel estimation
algorithm and pilot design, compared to their conventional counterparts.

###### Index Terms:

Wideband channel estimation, intelligent reflecting surface (IRS), beam
squint.

##  I Introduction

Recently, intelligent reflecting surface (IRS) aided communication systems
have emerged as a promising solution for next generation systems [1, 2, 3],
which are capable of achieving improved spectral and energy efficiency at a
low cost. In contrast to traditional amplify-and-forward (AF) relaying, IRS-
aided systems use low-cost passive elements for reflecting the incident
signals with adjustable phase shifts [4]. By appropriately designing the phase
shifts of IRS elements and the two-hop channel reflected through IRS, we can
improve the communications between transceivers [5, 6]. There are numerous
studies about beamforming design, including active beamforming at the base
station (BS) and passive beamforming at the IRS for the narrow-band [7, 8, 9,
10, 11]. Specifically, Wu et al.[7] proposed a semidefinite relaxation (SDR)
based method for maximizing the spectral efficiency, while Huang et al.[8]
advocated a gradient descent approach and sequential fractional programming
method for maximizing the energy efficiency. As a further development, Feng et
al. [10] conceived a deep reinforcement learning based framework for solving
the non-convex optimization problem of passive beamforming design at the IRS.
Moreover, a geometric mean decomposition-based method [12], a convex
optimization-based technique [13] and block coordinate descent iterative
algorithms [14] were proposed for the joint passive and active beamforming
design in wideband IRS-aided systems. All these contributions relied on the
idealized simplifying assumption that the channel state information (CSI) is
perfectly known. However, in reality channel estimation for IRS-aided
communication systems is very challenging. That is because the large number of
reflective elements are passive devices, which cannot perform active
transmit/receive signal processing. We fill this gap in the literature by
conceiving an efficient channel estimation method for the IRS-aided
communication systems.

Although, there is paucity of related solutions, some researchers have
designed channel estimation for IRS-aided narrow-band communication systems
[15, 16, 17, 18]. Specifically, Mishra et al. [15] proposed an ‘on-off’ state
control based channel estimation method, where only a single element of
reflecting surface was switched on, while all other elements remained off at
each time slot. In this way, the channel reflected through the activated
element can be estimated without interference from the signals reflected by
all other elements of the IRS. Wang et al. [16] proposed a three-phase channel
estimation method for the uplink of IRS-aided multiuser systems, which
exploited the fact that the IRS elements reflect the signals arriving from
different users to the BS via the same IRS to BS channel. Lin [17] et al.
proposed a Lagrange optimation based channel estimation strategy for
minimizing the mean-squared error of channel estimation. However, the above
methods require a large number of measurements for distinguishing the large
number of reflective elements at the IRS. To circulate this problem, Wang [18]
et al. formulated the channel estimation of IRS-aided systems as a sparse
signal recovery problem by exploiting the channel sparsity in the angular
domain. Then, they applied popular compressed sensing (CS) algorithms, such as
the basis pursuit and simultaneous orthogonal matching pursuit techniques for
recovering the channel at a much reduced number of measurements.

To expound a little further, some authors have also extended channel
estimation from the narrow-band regime to wideband scenarios[19, 20, 21, 22].
Specifically, Zheng et al. [19, 20] formulated a wideband channel estimation
problem for the IRS-aided orthogonal frequency-division multiplexing (OFDM)
systems and proposed a least-square (LS) based method for estimating
frequency-selective fading channels. To reduce the overhead of channel
training, Yang et al.[22] proposed an IRS elements grouping method, where each
group consists of a set of adjacent IRS elements that share a common
reflecting coefficient. Wan [21] et al. designed a CS-based wideband channel
estimation method by assuming that the path-angle of BS to IRS channel is
known. Then, they utilized a distributed orthogonal matching pursuit algorithm
for estimating the IRS to user channel. However, these existing wideband
channel estimation methods[19, 20, 21, 22] only consider the frequency-
selectivity of the wideband channel, but ignore the effect of beam squint.
Beam squint of wideband systems will lead two distinct angles of two-hop
channel reflected through the IRS for the same propagation path, which makes
traditional channel estimation methods ineffective. To the best of our
knowledge, this is the first paper studying channel estimation for IRS-aided
communication systems considering the effect of beam squint. The main
contributions of this paper are summarized as follows:

  * •

We formulate the estimation problem of the reflected channel, which is defined
as the cascaded BS-to-IRS and IRS-to-user channel, in the face of beam squint.
Specifically, the cascaded channel is characterized by the equivalent angles,
gains and delays of the associated propagation paths. In contrast to the
conventional channel model, where the spatial steering vectors are frequency-
independent, we consider the steering vectors to be frequency-dependent.
Therefore, the angular range of the steering vectors of the reflected channel
is no longer equivalent to $[-1/2,1/2)$ [23]. Instead, we should consider the
extended angular range of steering vectors spanning $(-1,1)$.

  * •

We study the effect of beam squint on the channel estimation of IRS-aided
communication systems and propose a twice-stage OMP (TS-OMP) algorithm, which
is robust to beam squint. A popular technique is to search for the peak of
correlation function between the spatial steering vectors and the reflected
channel, which determines the equivalent angles. However, due to the extended
angular range of steering vector, two angles will be returned for a single
propagation path. One of the two estimated angles is the frequency-independent
actual angle, while the other one is the frequency-dependent false angle,
which may substantially erode the performance of channel estimation. To
eliminate the influence of false angles, we propose a TS-OMP algorithm.
Specifically, in the first stage, we exploit the fact that frequency-dependent
false angles vary across the subcarriers, while the actual angles remain
constant across the subcarriers. Then, we propose a block-sparse method for
estimating the actual equivalent angles, while suppressing the influence of
the frequency-dependent false angles. In the second stage, based on the
equivalent angles estimated in the first stage, we propose an OMP based method
for calculating the gains and delays of the equivalent paths having different
equivalent angles.

  * •

Finally, we propose the corresponding pilot design, where a pair of pilot
design requirement are considered. Firstly, to eliminate the interference
imposed by false angles in the first stage, we have to reduce the accumulated
values corresponding to the false angles of the mutual correlation function
across the different subcarriers. This means that the false angles at the
pilot-subcarriers corresponding to the certain path should be separated from
each other. Secondly, a high grade of orthogonality is preferred among the
columns of the measurement matrix for the estimation of channel gains and
delays in the second stage. Based on the above mentioned pair of requirements,
we propose a cross-entropy based pilot design method, which improves the
cascaded BS-IRS-user channel estimation performance. Finally, our simulation
results demonstrate that the proposed channel estimation algorithm and pilot
design combination outperforms its conventional counterpart.

The rest of this paper is organized as follows. In Section II, we present the
system model of IRS-aided wideband systems considering the effect of beam
squint. In Section III, we analyze the effect of beam squint on estimation of
BS-IRS-user channel. In Section IV, we propose a TS-OMP method for estimating
the channel, which is robust to beam squint, while in Section V, we propose a
pilot design based on cross-entropy. In Section VI, our simulation results are
provided, followed by our conclusion in Section VII.

_Notations_ : We use the following notations throughout the paper. We let a,
$\mathbf{a}$, $\mathbf{A}$ represent the scalar, vector, and matrix
respectively; $\rm vec\\{\cdot\\}$ denotes the vectorization of a matrix and
$\rm ivec\\{\cdot\\}$ denotes the invectorization of a vector;
$(\cdot)^{\rm{T}}$, $(\cdot)^{\text{H}}$, and $(\cdot)^{-1}$ denote the
transpose, conjugate transpose, and inverse of a matrix, respectively;
$\left[\mathbf{a}\right]_{m:n}$ denotes the $m$-th to $n$-th element of vector
$\mathbf{a}$; $\left[\mathbf{A}\right]_{m,n}$ denotes the $(m,n)$-th element
of matrix $\mathbf{A}$; $\left[\mathbf{A}\right]_{:,n}$ denotes the $n$-th
column of matrix $\mathbf{A}$; The operator $\circ$, $\otimes$ and $*$
represent the Hadamard-product, Kronecker product and convolution,
respectively; Finally, $\mathbf{0}$ denotes the zero matrix, $\mathbf{I}$
denotes identity matrix, and ${\mathbf{1}}_{m,n}$ denotes all 1 matrix of size
$m\times n$.

##  II System Model With Beam Squint

Figure 1: IRS-aided communication system

As depicted in Fig. 1, we consider the IRS-aided wideband OFDM system, where
both the BS and the user have a single antenna- 111 The channel estimation
method proposed in this paper can be readily extended to multiple-antennas BS
and multiple received single-antenna users by using orthogonal pilots., while
the IRS is a ($M\times 1$)-element uniform linear array (ULA) [24]. Let
$\mathbf{g}(t)\in\mathbb{C}^{M\times 1}$ denote the channel impulse response
(CIR) spanning from the BS to the IRS, and
$\mathbf{u}(t)\in\mathbb{C}^{M\times 1}$ denote the CIR of the link from the
IRS to the user. In this paper, we neglect the direct path between the BS and
the user. In fact, if the direct path exists, we can estimate it by turning
off the IRS and using traditional channel estimation methods [18]. Based on
this assumption, the direct channel can be cancelled from the IRS-aided
channel model. OFDM with $N_{p}$ subcarriers is adopted for combating the
multipath effects. We define the transmission bandwidth as $W$, yielding the
subcarrier of $W/{N_{p}}$. We assume that the cyclic prefix (CP) is longer
than the maximum multipath delay and all the complex-valued channels remain
approximately constant within the channel’s coherence time [20].

Assume furthermore that there are $L_{1}$ propagation paths between the BS and
the IRS, where $\tau_{{l_{1}},m}^{\rm{TR}}$ denotes the time delay of the
$l_{1}$-th path from the BS to the $m$-th IRS elements, where
$l_{1}\in\\{1,2,\cdots,L_{1}\\}$ and $m\in\\{1,2,\cdots,M\\}$. Thus, the CIR
between the BS and the $m$-th IRS element can be expressed as [25]

$\displaystyle{g_{m}}(t)=\sum\limits_{{l_{1}}=1}^{{L_{1}}}{{{\overline{\alpha}}_{{l_{1}}}}}{e^{-j2\pi{f_{c}}\tau_{{l_{1}},m}^{\rm{TR}}}}\delta(t-\tau_{{l_{1}},m}^{\rm{TR}}),$
(1)

where $f_{c}$ is the carrier frequency and ${\overline{\alpha}}_{{l_{1}}}$ is
the complex path gain of the $l_{1}$-th path. We define the channel vector
${\mathbf{g}}(t)=[{g_{1}}(t),\cdots,{g_{M}}(t)]^{\rm T}\in\mathbb{C}^{M\times
1}$. Similar to the definition of ${g_{m}}(t)$, the CIR between the $m$-th IRS
element and the user can be expressed as

$\displaystyle{u_{m}}(t)=\sum\limits_{{l_{2}}=1}^{{L_{2}}}{{{\overline{\beta}}_{l_{2}}}}{e^{-j2\pi{f_{c}}\tau_{l2,m}^{\rm{RR}}}}\delta(t-\tau_{{l_{2}},m}^{\rm{RR}}),$
(2)

where $L_{2}$ and ${\overline{\beta}}_{l_{2}}$ denote the number of paths
between the IRS as well as the user and the complex gain of the $l_{2}$-th
path, $l_{2}\in\\{1,\cdots,L_{2}\\}$. We denote the delay of the $l_{2}$-th
path from the $m$-th IRS element to the user by $\tau_{l2,m}^{\rm{RR}}$ and
define the channel vector by
${\mathbf{u}}(t)=[{u_{1}}(t),\cdots,{u_{M}}(t)]^{\rm T}\in\mathbb{C}^{M\times
1}$.

(a) Illustrate AOA ${\chi_{{l_{1}}}}$. (b) Illustrate DOA
${\vartheta_{{l_{2}}}}$.

Figure 2: The illustration of AOA and DOA

Let us express the signal reflected by the $m$-th IRS element as

$\displaystyle{r_{m}}(t)$ $\displaystyle={\theta_{m}}{g_{m}}(t)*s(t)$
$\displaystyle={\theta_{m}}\sum\limits_{{l_{1}}=1}^{{L_{1}}}{{{\overline{\alpha}}_{{l_{1}}}}{e^{-j2\pi{f_{c}}\tau_{{l_{1,m}}}^{\rm{TR}}}}s(t-\tau_{{l_{1,m}}}^{\rm{TR}})},$
(3)

where
${\boldsymbol{\theta}}={[{\theta_{1}},\cdots,{\theta_{m}},\cdots,{\theta_{M}}]^{T}}={[{\varepsilon_{1}}{e^{j{\phi_{1}}}},\cdots,{\varepsilon_{m}}{e^{j{\phi_{m}}}},\cdots,{\varepsilon_{M}}{e^{j{\phi_{M}}}}]^{T}}$
represent the reflection coefficients of the IRS, where
${\phi_{m}}\in[0,2\pi)$ and ${\varepsilon_{m}}\in[0,1]$ are the phase shift
and amplitude reflection coefficient of the $m$-th IRS element, respectively.
To maximize the signal power reflected by the IRS, we set
${\varepsilon_{m}}=1,{\forall_{m}}\in M$ [7]. Moreover, $s(t)$ is the signal
transmitted by the BS.

Then, the signal $r_{m}(t)$ is reflected by the IRS to the user and the
received signal is expressed as

$\displaystyle{y_{m}}(t)$ $\displaystyle={u_{m}}(t)*{r_{m}}(t)+n_{m}(t)$
$\displaystyle=\sum\limits_{{l_{2}}=1}^{{L_{2}}}{{{\overline{\beta}}_{{l_{2}}}}{e^{-j2\pi{f_{c}}\tau_{{l_{2},m}}^{\rm{RR}}}}}{r_{m}}(t-\tau_{{l_{2},m}}^{\rm{RR}})+n_{m}(t),$
(4)

where $n_{m}(t)$ is the additive complex Gaussian noise. By substituting (3)
into (4), we have

$\displaystyle{y_{m}}(t)={\theta_{m}}$
$\displaystyle\sum\limits_{{l_{1}}=1}^{{L_{1}}}{\sum\limits_{{l_{2}}=1}^{{L_{2}}}{{{\overline{\alpha}}_{{l_{1}}}}{{\overline{\beta}}_{{l_{2}}}}}}{e^{-j2\pi{f_{c}}\tau_{l_{1},m}^{\rm{TR}}}}{e^{-j2\pi{f_{c}}\tau_{{l_{2},m}}^{\rm{RR}}}}\cdot
s(t-\tau_{l_{1},m}^{\rm{TR}}-\tau_{{l_{2},m}}^{\rm{RR}})+n_{m}(t)$
$\displaystyle={\theta_{m}}$ $\displaystyle{h_{m}}(t)*s(t)+n_{m}(t),$ (5)

where CIR of the $m$-th cascaded BS-IRS-user element channel is written as

$\displaystyle{h_{m}}(t)=$
$\displaystyle\sum\limits_{{l_{1}}=1}^{{L_{1}}}{\sum\limits_{{l_{2}}=1}^{{L_{2}}}{{{\overline{\alpha}}_{{l_{1}}}}{{\overline{\beta}}_{{l_{2}}}}}}{e^{-j2\pi{f_{c}}\tau_{{l_{1}},m}^{\rm{TR}}}}{e^{-j2\pi{f_{c}}\tau_{{l_{2}},m}^{\rm{RR}}}}\cdot\delta(t-\tau_{{l_{1}},m}^{\rm{TR}}-\tau_{{l_{2}},m}^{\rm{RR}}).$
(6)

For the sake of simplicity, we define $\tau_{{l_{1}}}^{\rm
TR}=\tau_{{l_{1},1}}^{\rm TR}$ and $\tau_{{l_{2}}}^{\rm
RR}=\tau_{{l_{2},1}}^{\rm RR}$. Based on the assumption that IRS array size is
much smaller than the distance between the BS and the IRS as well as the IRS-
user distance, the path delay $\tau_{l_{1},m}^{\rm TR}$ and
$\tau_{l_{2},m}^{\rm RR}$ can be expressed as

$\displaystyle\tau_{{l_{1}},m}^{\rm TR}=\tau_{{l_{1}}}^{\rm
TR}+(m-1)\frac{{d\sin{\chi_{{l_{1}}}}}}{c}=\tau_{{l_{1}}}^{\rm
TR}+(m-1)\frac{{\varphi_{{l_{1}}}^{\rm TR}}}{{{f_{c}}}},$ (7)

$\displaystyle\tau_{{l_{2}},m}^{\rm RR}=\tau_{{l_{2}}}^{\rm
RR}-(m-1)\frac{{d\sin{\vartheta_{{l_{2}}}}}}{c}=\tau_{{l_{2}}}^{\rm
RR}-(m-1)\frac{{\varphi_{{l_{2}}}^{\rm RR}}}{{{f_{c}}}},$ (8)

where ${\chi_{{l_{1}}}}$ denotes the angle of arrival (AOA) of the $l_{1}$-th
path from the BS to the IRS, as shown in Fig. 2 (a). Similarly,
${\vartheta_{{l_{2}}}}$ denotes the angle of departure (AOD) of the $l_{2}$-th
path from the IRS to the user, as shown in Fig. 2 (b). We define
${\varphi_{{l_{1}}}^{\rm TR}}=\frac{{d\sin{\chi_{{l_{1}}}}}}{{{\lambda_{c}}}}$
and ${\varphi_{{l_{2}}}^{\rm
RR}}=\frac{{d\sin{\vartheta_{{l_{2}}}}}}{{{\lambda_{c}}}}$ as the normalized
AOA and AOD, respectively, where $\lambda_{c}$ represents the carrier
wavelength. For the typical half-wavelength element spacing of
$d=\lambda_{c}/2$, we have ${\varphi_{{l_{1}}}^{\rm TR}}\in[-1/2,1/2)$,
${\varphi_{{l_{2}}}^{\rm RR}}\in[-1/2,1/2)$.

By substituting (7) and (8) into (6), we have

$\displaystyle{h_{m}}(t)=$
$\displaystyle\sum\limits_{{l_{1}}=1}^{{L_{1}}}{\sum\limits_{{l_{2}}=1}^{{L_{2}}}{\underbrace{{{\overline{\alpha}}_{{l_{1}}}}{e^{-j2\pi{f_{c}}\tau_{{l_{1}}}^{TR}}}}_{{\alpha_{{l_{1}}}}}}}\underbrace{{{\overline{\beta}}_{{l_{2}}}}{e^{-j2\pi{f_{c}}\tau_{{l_{2}}}^{RR}}}}_{{\beta_{{l_{2}}}}}{e^{-j2\pi(m-1)\varphi_{{l_{1}}}^{TR}}}{e^{j2\pi(m-1)\varphi_{{l_{2}}}^{RR}}}\delta(t-\tau_{{l_{1}},m}^{TR}-\tau_{{l_{2}},m}^{RR}),$
(9)

where
$\alpha_{l_{1}}={{{\overline{\alpha}}_{{l_{1}}}}{e^{-j2\pi{f_{c}}\tau_{{l_{1}}}^{TR}}}}$
and
$\beta_{l_{2}}={{{\overline{\beta}}_{{l_{2}}}}{e^{-j2\pi{f_{c}}\tau_{{l_{2}}}^{RR}}}}$.
By applying the continuous time Fourier transform to (9), the frequency-domain
(FD) cascaded channel corresponding to the $m$-th element of IRS can be
written as

$\displaystyle{h_{m}}(f)=\int\limits_{-\infty}^{+\infty}{{h_{m}}(t){e^{-j2\pi
ft}}dt}$
$\displaystyle=\sum\limits_{{l_{1}}=1}^{{L_{1}}}{\sum\limits_{{l_{2}}=1}^{{L_{2}}}}{{{{{\alpha_{{l_{1}}}}}}{{\beta_{{l_{2}}}}}}}{e^{-j2\pi(m-1)\left(\varphi_{{l_{1}}}^{\rm
TR}-\varphi_{{l_{2}}}^{\rm RR}\right)}}{e^{-j2\pi f\left(\tau_{{l_{1}},m}^{\rm
TR}+\tau_{{l_{2}},m}^{\rm RR}\right)}}$
$\displaystyle={\sum\limits_{{l_{3}}=1}^{{L_{1}L_{2}}}{c_{{l_{3}}}^{\rm{C}}{e^{-j2\pi(m-1)\varphi_{{l_{3}}}^{\rm{C}}\left({1+\frac{f}{{{f_{c}}}}}\right)}}{e^{-j2\pi
f\tau_{{l_{3}}}^{\rm{C}}}}}},$ (10)

where the equivalent delay ${\tau_{{l_{3}}}^{\rm{C}}}$, angle
$\varphi_{{l_{3}}}^{\rm C}$ and the complex gain ${c_{{l_{3}}}^{\rm{C}}}$ of
the cascaded BS-IRS-user channel can be defined as

$\displaystyle\tau_{{l_{3}}}^{\rm C}$ $\displaystyle=\tau_{{l_{1}}}^{\rm
TR}+\tau_{{l_{2}}}^{\rm RR},$ (11) $\displaystyle\varphi_{{l_{3}}}^{\rm C}$
$\displaystyle=\varphi_{{l_{1}}}^{\rm TR}-\varphi_{{l_{2}}}^{\rm RR},$ (12)
$\displaystyle c_{{l_{3}}}^{\rm{C}}$
$\displaystyle={\alpha_{{l_{1}}}}{\beta_{{l_{2}}}},$ (13)

where $l_{3}\in\\{1,2,\cdots,L_{1}L_{2}\\}$ and the range of
$\varphi_{{l_{3}}}^{\rm C}$ is $(-1,1)$. Let us discuss the range of
$\varphi_{{l_{3}}}^{\rm C}$ in the next section in detail. Note that if we can
obtain the parameters of $\tau_{{l_{3}}}^{\rm C},\varphi_{{l_{3}}}^{\rm C}$
and $c_{{l_{3}}}^{\rm{C}}$, the cascaded FD channel can be recovered according
to (10).

Based on the convolution theorem and equation (5), the FD signal received by
the user through the $m$-th IRS element can be expressed as

$\displaystyle{y_{m}}(f)={\theta_{m}}{h_{m}}(f)s(f)+n_{m}(f),$ (14)

where $n_{m}(f)$ is the additive white Gaussian noise (AWGN). The FD signal
$s(f)$ denotes the pilot symbol transmitted by the BS. Without loss of
generality, we let $s(f)=1$. Thus, the FD signal received by the user can be
expressed as

$\displaystyle{y}(f)$ $\displaystyle=\sum\limits_{m=1}^{M}{{y_{m}}(f)}$
$\displaystyle={\boldsymbol{\theta}}^{\rm{T}}{\mathbf{h}}(f)+n(f),$ (15)

where $n(f)\sim{CN}(0,{\delta}^{2})$ and ${\delta}^{2}$ is the noise power.
The cascaded channel vector
${\mathbf{h}}(f)=[h_{1}(f),h_{2}(f),\cdots,h_{M}(f)]^{\rm T}$ can be rewritten
as

$\displaystyle{\bf{h}}(f)=\sum\limits_{{l_{3}}=1}^{{L_{1}}{L_{2}}}{c_{{l_{3}}}^{\rm{C}}}{\bf{a}}\left(\left(1+\frac{f}{{{f_{c}}}}\right)\varphi_{{l_{3}}}^{\rm
C}\right){e^{-j2\pi f\tau_{{l_{3}}}^{\rm C}}},$ (16)

where

$\displaystyle{\bf{a}}\left(\left(1+\frac{f}{{{f_{c}}}}\right)\varphi_{{l_{3}}}^{\rm
C}\right)=$
$\displaystyle\left[1,{e^{-j2\pi\left(1+\frac{f}{{{f_{c}}}}\right)\varphi_{{l_{3}}}^{\rm
C}}},\cdots,{e^{-j2\pi
m\left(1+\frac{f}{{{f_{c}}}}\right)\varphi_{{l_{3}}}^{\rm
C}}},\cdots,{e^{-j2\pi(M-1)\left(1+\frac{f}{{{f_{c}}}}\right)\varphi_{{l_{3}}}^{\rm
C}}}\right]^{\rm T}$ (17)

is the spatial steering vector. In contrast to the conventional channel model
[26, 27], the spatial steering vector is dependent on frequency and has the
following property:

${\lim_{M\to\infty}}\frac{1}{M}{{\bf{a}}^{\rm
H}}\left(\left(1+\frac{f}{{{f_{c}}}}\right)\varphi_{1}\right){\bf{a}}\left(\left(1+\frac{f}{{{f_{c}}}}\right)\varphi_{2}\right)=\delta\left(\varphi_{1}-\varphi_{2}\right)$,
which is termed as the angular orthogonality property. The proof is similar to
that in [28].

The channel model derived in (10) can be directly extended to the uniform
planar array (UPA) scenario. For an IRS of $M_{x}\times M_{y}$ elements, we
have

$\displaystyle{h_{{m_{x}},{m_{y}}}}(f)$
$\displaystyle=\sum\limits_{{l_{1}}=1}^{{L_{1}}}{\sum\limits_{{l_{2}}=1}^{{L_{2}}}{{\alpha_{{l_{1}}}}{\beta_{{l_{2}}}}{e^{-j2\pi({m_{x}}-1)\left(u_{{l_{1}}}^{\rm
TR}-u_{{l_{2}}}^{\rm RR}\right)\left(1+\frac{f}{{{f_{c}}}}\right)}}}}$
$\displaystyle\times{{e^{-j2\pi({m_{y}}-1)\left(v_{{l_{1}}}^{\rm
TR}-v_{{l_{2}}}^{RR}\right)\left(1+\frac{f}{{{f_{c}}}}\right)}}{e^{-j2\pi
f\left(\tau_{{l_{1}}}^{TR}+\tau_{{l_{2}}}^{\rm RR}\right)}}},$ (18)

where ${h_{{m_{x}},{m_{y}}}}(f)$ denotes the cascaded FD channel corresponding
to the $(m_{x},m_{y})$-th IRS element. We have $u_{l}^{\rm
x}=\frac{{d\cos(\overline{u}_{{}_{l}}^{\rm x})}}{{{\lambda_{c}}}}$ and
$v_{l}^{\rm x}=\frac{{d\cos(\overline{u}_{{}_{l}}^{\rm
x})\sin(\overline{v}_{{}_{l}}^{\rm x})}}{{{\lambda_{c}}}}$, where
${\overline{u}_{{}_{l}}^{\rm x}}$ and ${\overline{v}_{{}_{l}}^{\rm x}}$ denote
the elevational angle and azimuth angle, respectively,
$l\in\\{l_{1},l_{2}\\}$, ${\rm x}\in\\{{\rm RR,TR}\\}$.

##  III Effect Of Beam Squint on channel estimation of IRS-Aided systems

In contrast to the conventional wideband channel model [26, 27], where the
steering vector is independent of frequency, in this paper, we consider the
frequency-dependent steering vectors of (16) for wideband IRS-aided systems.
We observe from (16) that the spatial angle
$\left(1+\frac{f}{f_{c}}\right)\varphi_{{l_{3}}}^{\rm C}$ has different
angular spread over different subcarriers, which is referred to as “beam
squint” [28] [29]. The effect of beam squint will change the equivalent angle
range $\varphi_{{l_{3}}}^{\rm C}$ of the cascaded channel, which will be
discussed as follows.

According to (16) and the angular orthogonality property of steering vectors,
one can get the estimation of the equivalent angle $\varphi_{{l_{3}}}^{\rm C}$
by finding the peak of mutual correlation function $\Gamma^{f}({x})$ between
steering vectors and the cascaded channel, which can be expressed as

$\displaystyle\Gamma^{f}(x)$ $\displaystyle={{\bf{a}}^{\rm
H}}\left(\left(1+\frac{f}{{{f_{c}}}}\right)x\right){\bf{h}}(f)$
$\displaystyle=\sum\limits_{{l_{3}}=1}^{{L_{1}}{L_{2}}}{\sum\limits_{m=1}^{M}{{c_{{l_{3}}}}{e^{-j2\pi
f\tau_{{l_{3}}}^{\rm{C}}}}{e^{-j2\pi(m-1)\left(1+\frac{f}{{{f_{c}}}}\right)(x-\varphi_{{l_{3}}}^{\rm{C}})}}}}$
$\displaystyle=\sum\limits_{{l_{3}}=1}^{{L_{1}}{L_{2}}}{{c_{{l_{3}}}}{e^{-j2\pi
f\tau_{{l_{3}}}^{\rm{C}}}}\frac{{\sin\left(\pi
M\left(1+\frac{f}{{{f_{c}}}}\right)(x-\varphi_{{l_{3}}}^{\rm{C}})\right)}}{{\sin\left(\pi\left(1+\frac{f}{{{f_{c}}}}\right)(x-\varphi_{{l_{3}}}^{\rm{C}})\right)}}},$
(19)

where the search range is $x\in(-1,1)$, since
$\varphi_{{l_{3}}}^{\rm{C}}\in(-1,1)$. We consider a single path of the
channel $\mathbf{h}(f)$ for better elaborating on the effect of beam squint on
the range of the equivalent angle $\varphi_{{l_{3}}}^{\rm{C}}$. Without loss
of generality, we consider the path of $l_{3}=1$. Then equation of (19) can be
simplified as

$\displaystyle\Gamma^{f}(x)={c_{{1}}}{e^{-j2\pi
f\tau_{{1}}^{\rm{C}}}}\frac{{\sin\left(\pi
M\left(1+\frac{f}{{{f_{c}}}}\right)(x-\varphi_{{1}}^{\rm{C}})\right)}}{{\sin\left(\pi\left(1+\frac{f}{{{f_{c}}}}\right)(x-\varphi_{{1}}^{\rm{C}})\right)}},$
(20)

where the range of $\varphi_{{1}}^{\rm{C}}$ is $(-1,1)$.

For the conventional wideband channel model disregarding the effect of beam
squint, the term of $\frac{f}{f_{c}}$ is ignored in (20). Hence, the steering
vector is expressed as
$\mathbf{a}(\varphi_{{l_{3}}}^{\rm{C}})=\left[1,{e^{-j2\pi\varphi_{{l_{3}}}^{\rm
C}}}\right.$ $\left.\cdots,{e^{-j2\pi(M-1)\varphi_{{l_{3}}}^{\rm
C}}}\right]^{\rm T}$ and for the single path of $l_{3}=1$,
$\Gamma^{f}(x)={c_{{1}}}{e^{-j2\pi f\tau_{{1}}^{\rm{C}}}}\frac{{\sin\left(\pi
M\left(x-\varphi_{{1}}^{\rm{C}}\right)\right)}}{{\sin\left(\pi\left(x-\varphi_{{1}}^{\rm{C}}\right)\right)}}$.
Since the value of M is usually large, the function
$\frac{1}{M}\left|{\frac{{\sin(\pi M\xi)}}{{\sin(\pi\xi)}}}\right|$ only has
non-zero value when $\xi\in\mathbb{Z}$. Thus, we can estimate the equivalent
angle $\varphi_{{l_{3}}}^{\rm{C}}$ of the cascaded channel $\mathbf{h}(f)$ by
finding the peak of the function $\Gamma^{f}({x})$. When we search for $x$
between $-1$ and $1$, the function $\Gamma^{f}(x)$ reaches its peak at
$x=\varphi_{{1}}^{\rm{C}}$, while the other peak at
$x=\varphi_{{1}}^{\rm{C}}+1\;(\varphi_{{1}}^{\rm{C}}<0)$ or
$x=\varphi_{{1}}^{\rm{C}}-1\;(\varphi_{{1}}^{\rm{C}}>0)$. Then, we will get
two estimated angles, one of which is the actual angle
$\varphi_{{1}}^{\rm{C}}$ and the other one is separated from the actual angle
by $1$, which is termed as false angle. The false angle is frequency-
independent, and the steering vector ${\bf{a}}(\varphi_{{1}}^{\rm{C}})$ of the
actual angle and steering vector ${\bf{a}}(\varphi_{{1}}^{\rm{C}}\pm 1)$ of
false angle are equivalent. Thus, it is easy to understand that searching
across the range of $x\in(-1,1)$ is equivalent to
$x\in[-\frac{1}{2},\frac{1}{2})$, and the equivalent angle estimation is
formulated as

$\displaystyle x=\left\\{\begin{array}[]{l}\varphi_{{1}}^{\rm
C}\quad\;\;\;\varphi_{{1}}^{\rm C}\in[-\frac{1}{2},\frac{1}{2})\\\
\varphi_{{1}}^{\rm C}-1\;\;\varphi_{{1}}^{\rm C}\in[\frac{1}{2},1)\\\
\varphi_{{1}}^{\rm C}+1\;\;\varphi_{{1}}^{\rm
C}\in(-1,-\frac{1}{2}],\end{array}\right.$ (24)

where $x\in[-\frac{1}{2},\frac{1}{2})$. Explicitly, we only have to consider
the range of equivalent angles in the steering vector as
$[-\frac{1}{2},\frac{1}{2})$ instead of $(-1,1)$. Similar conclusions can be
found in [23].

However, when we consider the effect of beam squint, the range of equivalent
angles in the steering vector
${\mathbf{a}}\left(\left(1+\frac{f}{{{f_{c}}}}\right)x\right)$ can no longer
be equivalent to $[-\frac{1}{2},\frac{1}{2})$. That is because the false
angles become frequency-dependent. Specifically, when we search for $x$
between $-1$ and $1$, we will get the actual peak at
$x=\varphi_{{1}}^{\rm{C}}$ and the false peak at
$x=\varphi_{{1}}^{\rm{C}}+\frac{f_{c}}{f+f_{c}}\;(-1<\varphi_{{1}}^{\rm{C}}\leq
0)$ or
$x=\varphi_{{1}}^{\rm{C}}-\frac{f_{c}}{f+f_{c}}\;(0<\varphi_{{1}}^{\rm{C}}<1)$,
which is frequency-dependent. Thus, we will get two estimated angles, one of
which is the actual angle of $\varphi_{{1}}^{\rm{C}}$ and the other is the
false angle of $\varphi_{{1}}^{\rm{C}}\pm\frac{f_{c}}{f+f_{c}}$. The false
angle is separated from the actual angle by $\frac{f_{c}}{f+f_{c}}$. Thus, the
squint of false angle over all subcarriers is
$\frac{f_{c}}{0+f_{c}}-\frac{f_{c}}{W+f_{c}}$, which is independent of the
specific equivalent angle $\varphi_{{1}}^{\rm{C}}$.

The false angle has a grave impact on channel estimation. We will elaborate by
considering an example, where there is a single path with equivalent angle of
${\varphi_{{1}}^{\rm C}}=-1/6$ in cascaded FD channel $\mathbf{h}(f)$. The
angular-domain index $x$ is selected from $-1$ to $1$, the carrier frequency
is $f_{c}=10$GHz, the number of subcarrier is $N_{p}=128$ and the system
bandwidth is $W=500$MHz. Thus, the subcarrier frequency is
$f=\frac{n_{p}}{N_{p}}W$, where $n_{p}\in\\{0,1,\cdots,N_{p}-1\\}$. The
function $\Gamma^{f}({x})$ at subcarriers $30,60,90,120$ is shown in Fig. 3.
We observe that there are two peaks over the angular range of $(-1,1)$, which
correspond to the index of the actual angle and the index of false angle. The
index of actual angle remains $-\frac{1}{6}$ for these four subcarriers, as
well as for all other subcarriers. By contrast, the index of false angle
varies from about $0.78$ to $0.82$ between subcarriers $30$ and $120$.
Furthermore, we note that the peak of false angle is quite comparable to that
of actual angle, which will seriously interfere with the estimation of the
actual angle, since we cannot readily distinguish the actual angle and false
angle by searching for the peak of mutual correlation function. This motivates
us to propose an efficient estimation method of the cascaded FD channel
$\mathbf{h}(f)$, which is robust to beam squint.

Figure 3: The mutual correlation function $\Gamma^{f}({x})$ between steering
vectors ${\bf{a}}\left((1+\frac{f}{f_{c}})x\right)$ and the cascaded FD
channel $h(f)$ at different subcarriers, for ${\varphi_{{1}}^{\rm C}}=-1/6$ ,
$M=256$, $N_{p}=128$, $f_{c}=$10GHz, $W=500$MHz.

##  IV IRS-Aided Channel Estimation

###  IV-A Overview of the Proposed Channel Estimation Method

In this section, we propose a TS-OMP based method for estimating the
parameters of equivalent angles, delays and gains of cascaded channel. The
philosophy of proposed method is as follows:

  * •

In the first stage, we propose a block-sparse processing based method for
estimating the equivalent angles of the cascaded channel. As shown in Fig. 3,
we find that the index of the actual angle is invariant for all subcarriers,
while the index of the false angle varies across all the subcarriers. Thus, if
we accumulate the function $\Gamma^{f}({\varphi_{{1}}^{\rm C}})$ across the
different subcarriers, where ${\varphi_{{1}}^{\rm C}}$ denotes the actual
angle of path $l_{3}=1$, the accumulated value $\Gamma^{f}({\varphi_{{1}}^{\rm
C}})$ corresponding to the actual angel will be quite high. By contrast, when
we accumulate the function $\Gamma^{f}({\varphi_{{1}}^{\rm F}})$ across the
subcarriers, where ${\varphi_{{1}}^{\rm F}}={\varphi_{{1}}^{\rm
C}}\pm\frac{f_{c}}{f+f_{c}}$ denotes the false angle corresponding to the path
of $l_{3}=1$, the value $\Gamma^{f}({\varphi_{{1}}^{\rm F}})$ for the angle
${\varphi_{{1}}^{\rm F}}$ accumulated over the subcarriers does not increase
as fast as the accumulated value of $\Gamma^{f}({\varphi_{{1}}^{\rm C}})$.
Therefore, we propose to accumulate the mutual correlation function
$\Gamma^{f}(x)$ over the subcarriers for eliminating the interference of the
false angle.

  * •

In the second stage, based on the estimated equivalent angles, we propose an
OMP based method for estimating the equivalent path delays and gains of
cascaded channel. Since there may be multiple paths having different delays
for the same equivalent angle, an appropriate stopping condition is needed.

###  IV-B Equivalent Angle Estimation for the Cascaded Channel

Observe from (16), that there may be multiple paths having the same equivalent
angle among the $L_{1}L_{2}$ number of cascaded channel paths. Thus, (II) can
be rewritten as

$\displaystyle y(f)$
$\displaystyle=\sum\limits_{{l_{3}}=1}^{{L_{1}}{L_{2}}}{{{\boldsymbol{\theta}}^{\rm
T}}{\bf{a}}\left({\left({1+\frac{f}{{{f_{c}}}}}\right)\varphi_{{l_{3}}}^{\rm
C}}\right){c_{{l_{3}}}}{e^{-j2\pi f\tau_{{l_{3}}}^{\rm{C}}}}}+n(f)$
$\displaystyle=\sum\limits_{i=1}^{{N_{a}}}{{\boldsymbol{\theta}}^{\rm
T}{\bf{a}}\left({\left({1+\frac{f}{{{f_{c}}}}}\right){{\overline{\varphi}}_{i}}}\right)\sum\limits_{{j_{i}}=1}^{{J_{i}}}{{{\overline{c}}_{{j_{i}}}}}}{e^{-j2\pi
f{{\overline{\tau}}_{{j_{i}}}}}}+n(f),$ (25)

where ${\overline{c}_{{j_{i}}}}$ and ${\overline{\tau}_{{j_{i}}}}$ denote the
equivalent gain and delay of the $j_{i}$-th cascaded channel path,
$j_{i}=\\{1,2,\cdots,J_{i}\\}$ and $J_{i}$ denotes the number of cascaded
channel paths having the same equivalent angle ${{{\overline{\varphi}}_{i}}}$,
where $i=\\{1,2,\cdots,N_{a}\\}$ and $N_{a}$ denotes the number of different
values of equivalent angles ${\varphi_{{l_{3}}}^{\rm C}}$ in the cascaded
channel. Thus, we have $\sum_{i}^{N_{a}}J_{i}=L_{1}L_{2}$.

In the $n_{p}$-th subcarrier, i.e., $f=\frac{n_{p}W}{N_{p}}$, the signal
received by the user can be expressed as

$\displaystyle{{y}}({n_{p}})$
$\displaystyle={{{\bf{{\boldsymbol{\theta}}}}}^{\rm
T}{\bf{A}}({n_{p}})}{\bf{z}}({n_{p}})+{{n}(n_{p})},$ (26)

where we denote $y(\frac{n_{p}W}{N_{p}})$ and $n(\frac{n_{p}W}{N_{p}})$ as
$y(n_{p})$ and $n(n_{p})$ for simplicity. The sparse vector
$\mathbf{z}(n_{p})\in{\mathbb{C}}^{N_{d}\times 1}$ has $N_{a}$ none-zero
elements. The matrix $\mathbf{A}(n_{p})\in\mathbb{C}^{M\times N_{d}}$ is the
dictionary matrix composed of $N_{d}$ steering vectors, which can be expressed
as

$\displaystyle{\bf{A}}(n_{p})=$
$\displaystyle\left[{\bf{a}}\left({\left({-1}\right)\left({1+\frac{{{n_{p}}W}}{{{N_{p}}f_{c}}}}\right)}\right),{\bf{a}}\left({\left({-1+\frac{2}{{{N_{d}}}}}\right)\left({1+\frac{{{n_{p}}W}}{{{N_{p}}f_{c}}}}\right)}\right),\right.$
$\displaystyle\cdots,\left.{\bf{a}}\left({\left({1-\frac{2}{{{N_{d}}}}}\right)\left({1+\frac{{{n_{p}}W}}{{{N_{p}}f_{c}}}}\right)}\right)\right].$
(27)

In order to estimate the equivalent angles
$\\{{{\overline{\varphi}}_{i}}\\}_{i=1}^{N_{a}}$ of the cascaded channel, we
need multiple sets of observations at the user. We consider $N_{s}$ OFDM
symbols and assume that the IRS reflection coefficients are reconfigured
during different OFDM symbols. Thus, the signal received by the user in the
$n_{p}$-th subcarrier after $N_{s}$ OFDM symbols can be expressed as

$\displaystyle{\mathbf{y}}({n_{p}})$
$\displaystyle={{{\bf{{\bf{\Theta}}}}}{\bf{A}}({n_{p}})}{\bf{z}}({n_{p}})+{\mathbf{n}(n_{p})}$
$\displaystyle={{\bf{F}}({n_{p}})}{\bf{z}}({n_{p}})+{\mathbf{n}(n_{p})},$ (28)

where
$\mathbf{y}(n_{p})=[y^{1}(n_{p}),y^{2}(n_{p}),\cdots,y^{N_{s}}(n_{p})]^{\rm
T}$ and the element $y^{N_{s}}(n_{p})$ denotes the signal received during the
$N_{s}$-th OFDM symbol. The matrix
${\bf{\Theta}}=[{\boldsymbol{\theta}}^{1},{\boldsymbol{\theta}}^{2},\cdots,$
${\boldsymbol{\theta}}^{N_{s}})]^{\rm T}\in{\mathbb{C}}^{N_{s}\times M}$ and
the vector ${\boldsymbol{\theta}}^{N_{s}}\in\mathbb{C}^{M\times 1}$ denotes
the reflection coefficient vector during the $N_{s}$-th OFDM symbol. The noise
vector in (IV-B) is
${\mathbf{n}(n_{p})}=[n^{1}(n_{p}),n^{2}(n_{p}),\cdots,n^{N_{s}}(n_{p})]^{\rm
T}$ and the element $n^{N_{s}}(n_{p})$ represents the Gaussian noise during
the $N_{s}$-th OFDM symbol. While we have matrix
${{\bf{F}}({n_{p}})}={{{\bf{{\bf{\Theta}}}}}{\bf{A}}({n_{p}})}\in{\mathbb{C}}^{N_{s}\times
N_{d}}$. To elaborate, equation (IV-B) represents a sparse signal recovery
problem, where $\mathbf{y}(n_{p})$ is the observation vector,
$\mathbf{F}(n_{p})$ is the measurement matrix and $\mathbf{z}(n_{p})$ is the
sparse vector to be recovered. By using classic OMP algorithm, we can get the
non-zero elements’ indices in the vector $\mathbf{z}(n_{p})$, which are
expressed as ${{{\cal I}^{a}}}=\\{{\mathcal{I}}_{i}^{\rm
a}\\}_{i=1}^{i=N_{a}}$. The equivalent angles ${{{\overline{\varphi}}_{i}}}$
can be estimated as
${{{\overline{\varphi}}_{i}}}={-1+\frac{2({\mathcal{I}}_{i}^{\rm
a}-1)}{{{N_{d}}}}}$.

However, as discussed in Section III, we have to combine several subcarriers
for suppressing the deleterious influence of false angles. Collecting the
signal $\mathbf{y}(n_{p})$ received in ${N_{{P1}}}$ subcarriers where $N_{P1}$
is the number of pilot subcarriers, we have

$\displaystyle\overline{\bf{y}}=\overline{\bf{F}}\overline{\bf{z}}+\overline{\bf{n}},$
(29)

where

$\displaystyle\overline{\bf{F}}=\left[{\begin{array}[]{*{20}{c}}{{\bf{F}}({{n}}_{1})}&{\bf{0}}&\cdots&{\bf{0}}\\\
{\bf{0}}&{{\bf{F}}({{n}}_{2})}&\cdots&{\bf{0}}\\\
\vdots&\vdots&\ddots&\vdots\\\
{\bf{0}}&{\bf{0}}&\cdots&{{\bf{F}}({{n}}_{N_{P1}})}\end{array}}\right]\in{{\mathbb{C}}^{{N_{s}}{N_{{P1}}}\times{N_{{P1}}}{N_{d}}}},$
(34)

and $\overline{\bf{y}}=[{\bf{y}}({{n}}_{1})^{\rm T},{\bf{y}}({{n}}_{2})^{\rm
T},\cdots,{\bf{y}}({{n}}_{N_{P1}})^{\rm T}]^{\rm
T}\in\mathbb{C}^{N_{s}N_{P1}\times 1}$,
$\overline{\bf{z}}=[{\bf{z}}({{n}}_{1})^{\rm T},{\bf{z}}({{n}}_{2})^{\rm
T},\cdots,{\bf{z}}({{n}}_{N_{P1}})^{\rm T}]^{\rm
T}\in\mathbb{C}^{N_{P1}N_{a}\times 1}$,
$\overline{\bf{n}}=[{\bf{n}}({{n}}_{1})^{\rm T},{\bf{n}}({{n}}_{2})^{\rm
T},\cdots,{\bf{n}}({{n}}_{N_{P1}})^{\rm T}]^{\rm
T}\in\mathbb{C}^{N_{s}N_{P1}\times 1}$. Equation (29) also represents a sparse
signal recovery problem. Since the actual angle is the same for all
subcarriers, the vector $\overline{\bf{z}}$ exhibits inherent block sparsity.
By exploiting the block-sparsity, we propose a TS-OMP algorithm for
eliminating the effect of false angles and get an accurate estimate of the
cascaded channel angles, which is summarized in Algorithm 1.

In Step 1 of Algorithm 1, we apply an elementary transformation to (29) for
locating those particular elements of vector $\overline{\bf{z}}$ that
represent the same equivalent angle in different subcarriers, which are then
lumped together.. Specifically, this operation can be expressed as
$[\widetilde{\bf{z}}]_{(n_{d}-1){N_{{P1}}}+{{n}}_{p}}=[\overline{\bf{z}}]_{({{n}}_{p}-1){N_{d}}+n_{d}}$,
where $n_{p}=\\{1,2,\cdots,N_{P1}\\}$ and $n_{d}=\\{1,2,\cdots,N_{d}\\}$. To
ensure that equation (29) holds, we apply a similar transformation to the
columns of matrix $\overline{\bf{F}}$, which can be expressed as
$[\widetilde{\bf{F}}]_{:,(n_{d}-1){N_{{P1}}}+{{n}}_{p}}=[\overline{\bf{F}}]_{:,({{n}}_{p}-1){N_{d}}+n_{d}}$.
Thus, (29) can be rewritten as
$\overline{\bf{y}}=\widetilde{\bf{F}}\widetilde{\bf{z}}+\overline{\bf{n}}$. In
this way, the $N_{P1}$ adjacent elements of vector $\widetilde{\bf{z}}$ are
either all zero elements or all non-zero elements. In Step 3, we estimate the
indices of non-zero elements of vector $\widetilde{\bf{z}}$. Traditionally, we
can obtain the indices by finding the maximum value of objective function
(OF), which can be expressed as ${\cal
I}_{i}^{a}=\mathop{\arg\max}\limits_{{\cal
I}_{i}^{a}\in\\{1,2,\cdots,{N_{d}}\\}}\sum\limits_{k=1}^{{N_{P1}}}{\left\|{{{({{[{{\widetilde{\bf{F}}}_{{b_{1}}}}]}_{:,t}})}^{\rm{H}}}{\bf{r}}_{a}^{i-1}}\right\|_{2}^{2}}$,
where $t=({\cal I}_{i}^{a}-1)N_{P1}+k$ and the expression ${\bf{r}}_{a}^{i-1}$
can be found in Step 8, while ${{\widetilde{\bf{F}}}_{{b_{1}}}}$ denotes the
buff matrix processing in Step 6, where we set
${{\widetilde{\bf{F}}}_{{b_{1}}}}={{\widetilde{\bf{F}}}}$ in Step 1 of
Algorithm 1. We find that there are many zeros elements in each column of
matrix ${\widetilde{\bf{F}}}$. To reduce the complexity, we remove those items
of ${\bf{r}}_{a}^{i-1}$ that interact with the zero elements of
${\bf{F}}_{{b_{1}}}$. The OF can be rewritten as ${\cal
I}_{i}^{a}=\mathop{\arg\max}\limits_{{\cal
I}_{i}^{a}\in\\{1,2,\cdots,{N_{d}}\\}}\sum\limits_{k=1}^{{N_{P1}}}{\left\|{{{({{[{{\widetilde{\bf{F}}}_{{b_{1}}}}]}_{j:q,t}})}^{\rm{H}}}{{[{\bf{r}}_{a}^{i-1}]}_{j:q}}}\right\|_{2}^{2}}$,
where $j=(k-1)N_{s}+1$, $q=kN_{s}$ and $t=({\cal I}_{i}^{a}-1)N_{\rm P1}+k$.
Repeat Steps 3 to 9 of Algorithm 1 until the stop criterion is met, when we
can get the index set ${{{\cal I}^{a}}}=\\{{\cal I}^{a}\\}_{i=1}^{N_{d}}$.

Then, we can use least squares (LS) estimator to estimate the non-zero
elements of vector $\mathbf{z}(n_{p})$, which can be expressed as

$\displaystyle{\rm LS}:\quad{[{\bf{z}}({{{n}}}_{p})]_{{{\cal
I}^{a}}}}={\left({{{[{\bf{F}}({{{n}}}_{p})]}_{:,{{\cal
I}^{a}}}}}\right)^{\dagger}}{\bf{y}}({{{n}}}_{p}),$ (35)

where

$\displaystyle{({[{\bf{F}}({{{n}}}_{p})]_{:,{{\cal
I}^{a}}}})^{\dagger}}={\left[{({[{\bf{F}}({{{n}}}_{p})]_{:,{{\cal
I}^{a}}}})^{\rm{H}}}{[{\bf{F}}({{{n}}}_{p})]_{:,{{\cal
I}^{a}}}}\right]^{-1}}{({[{\bf{F}}({{{n}}}_{p})]_{:,{{\cal
I}^{a}}}})^{\rm{H}}}$ (36)

denotes the pseudo-inverse of ${[{\bf{F}}({n}_{p})]_{:,{{\cal I}^{a}}}}$.

###  IV-C Delay- and Gain- Estimation of the Cascaded Channel

According to (25) and (26), the $i$-th non-zero element of vector
$\mathbf{z}(n_{p})$ can be expressed as

$\displaystyle{[{\bf{z}}({n_{p}})]_{{{\cal
I}^{a}_{i}}}}=\sum\limits_{{j_{i}}=1}^{{J_{i}}}{{{\overline{c}}_{{j_{i}}}}{e^{-j2\pi\frac{{{n_{p}}W}}{{{N_{P}}}}{{\overline{\tau}}_{{j_{i}}}}}}}.$
(37)

Upon collecting ${[{\bf{z}}({n_{p}})]_{{{\cal I}^{a}_{i}}}}$ of pilot-
subcarriers, we have

$\displaystyle{{\bf{z}}^{{{\cal
I}^{a}_{i}}}}=\sum\limits_{{j_{i}}=1}^{{J_{i}}}{{{\overline{c}}_{{j_{i}}}}\overline{\bf{b}}}\left({{{\overline{\tau}}_{{j_{i}}}}}\right),$
(38)

where ${{\bf{z}}^{{{\cal
I}^{a}_{i}}}}={\left[{{{[{\bf{z}}({{{n}}}_{1})]}_{{\cal
I}_{i}^{a}}},{{[{\bf{z}}({{{n}}}_{2})]}_{{\cal
I}_{i}^{a}}},\cdots,{{[{\bf{z}}({{{n}}}_{{N_{{P1}}}})]}_{{\cal
I}_{i}^{a}}}}\right]^{\rm{T}}}$ and
$\overline{\bf{b}}\left({{{\overline{\tau}}_{{j_{i}},i}}}\right)$ can be
expressed as

$\displaystyle\overline{\bf{b}}\left({{{\overline{\tau}}_{{j_{i}}}}}\right)={\left[{{e^{-j2\pi\frac{{{n_{1}}W}}{{{N_{P}}}}{{\overline{\tau}}_{{j_{i}}}}}},{e^{-j2\pi\frac{{{n_{2}}W}}{{{N_{P}}}}{{\overline{\tau}}_{{j_{i}}}}}},\cdots,{e^{-j2\pi\frac{{{n_{{N_{P1}}}}W}}{{{N_{P}}}}{{\overline{\tau}}_{{j_{i}}}}}}}\right]^{{{\rm
T}}}}.$ (39)

Equation (38) can also be formulated as a sparse signal recovery problem,
which can be expressed as

$\displaystyle{\bf{z}}^{{\cal
I}_{i}^{a}}={\bf{B}}{{\bf{{\overline{c}}}}^{{\cal I}_{i}^{a}}},$ (40)

where the sparse vector ${{\bf{{\overline{c}}}}^{{\cal
I}_{i}^{a}}}\in{\mathbb{C}}^{N_{\tau}\times 1}$ has $J_{i}$ non-zero elements
and
$\mathbf{B}=[{\bf{b}}(0),{\bf{b}}(1),\cdots,{\bf{b}}({N_{\tau}}-1)]\in{\mathbb{C}}^{N_{P1}\times
N_{\tau}}$ is the dictionary matrix for the delay domain composed of
$N_{\tau}$ steering vectors. The vector $\mathbf{b}(k)$ of equation(40) can be
expressed as

$\displaystyle{\bf{b}}(k)={[{e^{-j2\pi\frac{W}{{{N_{p}}}}\frac{{k{T_{\tau}}}}{{{N_{\tau}}}}{{{n}}}_{1}}},{e^{-j2\pi\frac{W}{{{N_{p}}}}\frac{{k{T_{\tau}}}}{{{N_{\tau}}}}{{{n}}}_{2}}},\cdots,{e^{-j2\pi\frac{W}{{{N_{p}}}}\frac{{k{T_{\tau}}}}{{{N_{\tau}}}}{{{n}}}_{{N_{{P1}}}}}}]^{\rm{T}}},$
(41)

where $T_{\tau}$ is the maximum channel delay in the cascaded channel. In this
way, we can utilize our OMP-based method to estimate the vector
${{\bf{{\overline{c}}}}^{{\cal I}_{i}^{a}}}$. The non-zero elements of vector
${{\bf{{\overline{c}}}}^{{\cal I}_{i}^{a}}}$ correspond to the channel gain
${{{\overline{c}}_{{j_{i}}}}}$ and the index corresponds to the channel delay
${{{\overline{\tau}}_{{j_{i}}}}}$. The exact details are shown in Algorithm 1.

According to (25), the FD channel response of the ${\mathbf{h}}(n_{p})$ of the
$n_{p}$-th subcarrier in (16) can be expressed as

$\displaystyle{\mathbf{h}}(f)=\sum\limits_{i=1}^{{N_{d}}}{{\bf{a}}\left({\left({1+\frac{f}{{{f_{c}}}}}\right){{\overline{\varphi}}_{i}}}\right)\sum\limits_{{j_{i}}=1}^{{J_{i}}}{{{\overline{c}}_{{j_{i}}}}}}{e^{-j2\pi
f{{\overline{\tau}}_{{j_{i}}}}}}.$ (42)

Thus, after obtain the parameters of
$\\{{{{\overline{\varphi}}_{i}}},{{{\overline{c}}_{{j_{i}}}}},{{{\overline{\tau}}_{{j_{i}}}}}\\}$,
where $i\in\\{1,2,\cdots,{N_{a}}\\}$ and $j_{i}=\\{1,2,\cdots,J_{i}\\}$, the
cascaded FD channel $h(f)$ of all the subcarriers can be obtained according to
(42).

$\mathbf{Input}$: Received signal $\overline{\bf{y}}$. Measurement matrix
$\overline{\bf{F}}$ and dictionary matrix $\mathbf{B}$.

$\mathbf{Initialization}$: ${\bf{r}}_{a}^{0}=\overline{\bf{y}}$, ${{\cal
I}^{a}}=[\;]$, ${\widetilde{\bf{F}}_{{b_{2}}}}=[\;]$ and iteration counter
$i=1$.

1: Perform elementary transformation:
$[\widetilde{\bf{F}}]_{:,(n_{a}-1){N_{{P1}}}+{{n}}_{p}}=[\overline{\bf{F}}]_{:,({{n}}_{p}-1){N_{a}}+n_{a}}$
and
$[\widetilde{\bf{z}}]_{(n_{a}-1){N_{{P1}}}+{{n}}_{p}}=[\overline{\bf{z}}]_{({{n}}_{p}-1){N_{a}}+n_{a}}$.
We set ${\widetilde{\bf{F}}_{{b_{1}}}}=\widetilde{\bf{F}}$.

2: repeat

3: Estimate angular support: ${\cal I}_{i}^{a}=\mathop{\arg\max}\limits_{{\cal
I}_{i}^{a}\in\\{1,2,\cdots,{N_{d}}\\}}\sum\limits_{k=1}^{{N_{P1}}}\left\|{{{({{[{{\widetilde{\bf{F}}}_{{b_{1}}}}]}_{j:q,t}})}^{\rm{H}}}[{{\bf{r}}_{a}^{i-1}}]_{j:q}}\right\|_{2}^{2}$
, where $j=(k-1)N_{s}+1$, $q=kN_{s}$ and $t=({\cal I}_{i}^{a}-1)N_{\rm P1}+k$.

4: Update index set: ${{\cal I}^{a}}={{\cal I}^{a}}\cup\\{{\cal
I}_{i}^{a}\\}$.

5: Expand matrix:

${\widetilde{\bf{F}}_{b_{2}}}={\widetilde{\bf{F}}_{b_{2}}}\cup{[{\widetilde{\bf{F}}_{{b_{1}}}}]_{:,({\cal
I}_{i}^{a}-1){N_{{P1}}}+1:{\cal I}_{i}^{a}{N_{{P1}}}}}$.

6: Update matrix: ${[{\widetilde{\bf{F}}_{{b_{1}}}}]_{:,({\cal
I}_{i}^{a}-1){N_{{P_{1}}}}+1:{\cal I}_{i}^{a}{N_{{P_{1}}}}}}={\bf{0}}$.

7: Calculate the value $\mathbf{z}_{b}$:

${{\bf{z}}_{b}}={\left({{{({{\widetilde{\bf{F}}}_{b_{2}}})}^{\rm{H}}}{{\widetilde{\bf{F}}}_{b_{2}}}}\right)^{-1}}{({\widetilde{\bf{F}}_{b_{2}}})^{\rm{H}}}\overline{\bf{y}}$.

8: Update residual:
${\bf{r}}_{a}^{i}=\overline{\bf{y}}-{\widetilde{\bf{F}}_{{b_{2}}}}{{\bf{z}}_{b}}$.

9: Set $i=i+1$.

10: until
$\frac{{\left\|{{\bf{r}}_{a}^{i-2}-{\bf{r}}_{a}^{i-1}}\right\|_{2}^{2}}}{{\left\|{\overline{\bf{y}}}\right\|_{2}^{2}}}\leq\zeta$

11: Exploit LS estimator to calculate vector $\overline{\bf{z}}$ according to
(35).

12: for $1\leq i\leq N_{d}$ do

13: $\mathbf{Initialization}$: $\mathbf{B}_{b_{1}}=\mathbf{B}$,
$\mathbf{B}_{b_{2}}=[\;]$, $\mathbf{R}_{\tau}^{0}={\bf{z}}_{{P1}}^{{\cal
I}_{i}^{a}}$, $\bf{T}_{b}=[\;]$ and the counter $t=1$.

14: repeat

15: Estimate delay support:
$k=\mathop{\arg\max}\limits_{k\in\\{1,2,\cdots,{N_{\tau}}\\}}\left\|{{{({{[{{\bf{B}}_{{b_{1}}}}]}_{:,k}})}^{\rm{H}}}{\bf{r}}_{\tau}^{t-1}}\right\|_{2}^{2}$

16: Update index set: $\mathbf{T}_{b}=\mathbf{T}_{b}\cup\\{k\\}$.

17: Expand matrix:
${{\bf{B}}_{{b_{2}}}}={{\bf{B}}_{{b_{2}}}}\cup{[{{\bf{B}}_{{b_{1}}}}]_{:,k}}$.

18: Update matrix: ${[{{\bf{B}}_{{b_{1}}}}]_{:,k}}={\bf{0}}$.

19: Calculate the value ${{\bf{\overline{c}}}_{{\cal I}_{i}^{a}}}$:
${{\bf{\overline{c}}}_{{\cal
I}_{i}^{a}}}={\left({{{({{\bf{B}}_{{b_{2}}}})}^{\rm{H}}}{{\bf{B}}_{{b_{2}}}}}\right)^{-1}}{({{\bf{B}}_{{b_{2}}}})^{\rm{H}}}{\bf{z}}^{{\cal
I}_{i}^{a}}$.

20: Update residual: ${\bf{r}}_{\tau}^{t}={\bf{z}}^{{\cal
I}_{i}^{a}}-{{\bf{B}}_{{b_{2}}}}{{\bf{\overline{c}}}_{{\cal I}_{i}^{a}}}$.

21: Set $t=t+1$.

22: until
$\frac{{\left\|{{\bf{r}}_{\tau}^{t-2}-{\bf{r}}_{\tau}^{t-1}}\right\|_{2}^{2}}}{{\left\|{{\bf{z}}_{{P1}}^{{\cal
I}_{i}^{a}}}\right\|_{2}^{2}}}\leq\zeta$

23: $\mathbf{Output}$ the set ${\bf{T}}_{b}$ and vector
${{\bf{\overline{c}}}_{{\cal I}_{i}^{a}}}$.

24: We have the channel angle
${{{\overline{\varphi}}_{i}}}={-1+\frac{2({\mathcal{I}}_{i}^{\rm
a}-1)}{{{N_{a}}}}}$, the delay
${{{\overline{\tau}}_{{j_{i}}}}}=\\{\frac{{{{[{{\bf{T}}_{b}}]}_{{{j}}_{i}}}{T_{\tau}}}}{{{N_{\tau}}}}|{{j}}_{i}\in\\{1,2,\cdots,J_{i}\\}\\}$
and the channel gain
${{{\overline{c}}_{{j_{i}}}}}={[{{\bf{\overline{c}}}_{{\cal
I}_{i}^{a}}}]_{j_{i}}}$.

25: end for

Output:
${{{\overline{\varphi}}_{i}}},{{{\overline{c}}_{{j_{i}}}}},{{{\overline{\tau}}_{{j_{i}}}}}$

Algorithm 1 The proposed TS-OMP algorithm

##  V Pilot Design

In this section, we propose a cross-entropy based pilot design for improving
the estimation performance of the channel’s equivalent angles, delays and
gains. The pilot design guidelines have to consider both the first stage and
the second stage of the proposed TS-OMP algorithm, which are as follows:

  * •

In the first stage of the TS-OMP algorithm, we estimate the equivalent angles
of cascaded FD channel. We define
$v_{{n_{1}}}=\varphi_{1}^{\rm{C}}\pm\frac{{{f_{{n_{1}}}}}}{{{f_{{n_{1}}}}+{f_{c}}}}$
and
$v_{{n_{{}_{{N_{P1}}}}}}=\varphi_{1}^{\rm{C}}\pm\frac{{{f_{{n_{{N_{P1}}}}}}}}{{{f_{{n_{{N_{P1}}}}}}+{f_{c}}}}$
as the false angle corresponding to the actual angle $\varphi_{1}^{\rm{C}}$ at
the $n_{1}$-th and $n_{N_{P1}}$-th subcarrier, where
${f_{n_{1}}}=\frac{{{n_{1}}W}}{{{N_{P}}}}$ and
${f_{{n_{{N_{P1}}}}}}=\frac{{{n_{{N_{P1}}}}W}}{{{N_{P}}}}$ denotes the
frequency of the $n_{1}$-th and $n_{N_{P1}}$-th subcarrier, respectively. To
eliminate the interference imposed by the false angle, we should decrease the
accumulated values of function $\Gamma^{f}({\varphi_{{l_{3}}}^{\rm F}})$ over
the pilot- subcarriers, as mentioned in Section III. This means that the false
angle $\varphi_{l_{3}}^{\rm F}={\varphi_{{l_{3}}}^{\rm
C}}\pm\frac{f_{c}}{f+f_{c}}$ corresponding to a certain actual angle
$\varphi_{l_{3}}^{\rm C}$ at different subcarriers should be at a different
angular index in Fig. 3. We have defined the resolution of angular-domain
dictionary matric as $\frac{2}{N_{d}}$ in (IV-B). Thus, our first guideline is
formulated as $v_{{n_{1}}}-v_{{n_{{}_{{N_{P1}}}}}}>\frac{2}{N_{d}}$.

  * •

In the second stage of our TS-OMP algorithm, we formulate the problem of
estimating the equivalent channel delays and gains as a sparse signal recovery
problem. Naturally, a high grade of orthogonality of the measurement matrix
columns is preferred for sparse signal recovery [30][31]. Hence the second
guideline of pilot design should ensure that the measurement matrix $\bf{B}$
in (40), which depends on the pilot-subcarriers, has near-orthogonal columns.

As for our first guideline of pilot design, we have

$\displaystyle\frac{{{f_{c}}}}{{{f_{c}}+{f_{n_{1}}}}}-\frac{{{f_{c}}}}{{{f_{c}}+{f_{{{n_{{}_{{N_{P1}}}}}}}}}}\geq\frac{2}{N_{d}}.$
(43)

Thus, the span $D={f_{{n_{{N_{P1}}}}}}-{f_{{n_{1}}}}$ of pilots has to satisfy

$\displaystyle
D\geq\frac{{{f_{n_{1}}}({f_{c}}+\frac{N_{d}}{2}{f_{c}})+f_{c}^{2}-{f_{n_{1}}}(\frac{N_{d}}{2}-1)-f_{n_{1}}^{2}}}{{(\frac{N_{d}}{2}-1){f_{c}}+{f_{n_{1}}}}}.$
(44)

As for our second guideline of pilot design, we have to design ${\bf{B}}$ so
that ${{\bf{B}}^{\rm{H}}}{\bf{B}}$ becomes an approximately identity matrix,

$\displaystyle{{\bf{B}}^{\rm{H}}}{\bf{B}}\approx{N_{\rm P1}}{{\bf
I}_{{N_{\tau}}}},$ (45)

where ${{\bf I}_{{N_{\tau}}}}$ is the identity matrix of size $N_{\tau}\times
N_{\tau}$ . Thus, our pilot design problem can formulated as

$\displaystyle\mathop{\min}\limits_{\bf{B}}\mu({\bf B})$
$\displaystyle=\left\|{{{\bf{B}}^{\rm{H}}}{\bf{B}}-{N_{\rm
P1}}{{\bf{I}}_{{N_{\tau}}}}}\right\|_{2}^{2}$
$\displaystyle\rm{s.t.}\quad(\ref{irs10}),$ (46)

where the measurement matrix $\mathbf{B}$ is dependent of the subcarrier index
of pilots $\\{{n_{1}},{n_{2}},\cdots,{n_{{N_{P1}}}}\\}$. The optimization
problem (V) can then be solved by exhaustive search. However, the complexity
of exhaustive search is excessive. For example, if the number of subcarriers
is $128$ and the number of pilots is $6$, we need $5.4\times 10^{9}$ searches
to find the optimal pilots. Inspired-by the cross-entropy method of [32], we
hence propose the pilot design of Algorithm 2.

Input: The number of candidates $N_{c}$, the number of elites $N_{e}$ and the
number of iterations $N_{iter}$. The number of subcarriers $N_{\rm P}$ and the
number of pilots $N_{\rm P1}$.

$\mathbf{Initialization}$: $P_{\rm B}^{0}=\frac{N_{\rm P1}}{N_{\rm
P}}\times{\bf{1}}_{N_{\rm P},1}$ and iteration counter $i=0$.

for  _$0\leq i\leq N_{iter}-1$_ do

1\. Randomly generate $N_{c}$ candidate vectors
$\\{{\mathbf{d}_{n_{c}}}\\}_{n_{c}=1}^{N_{c}}$ according to
${\mathbf{P}_{\rm{B}}^{i}}$ and constraint of (44), where
$\mathbf{d}_{n_{c}}\\!\in\\!\\{0,1\\}^{N_{\rm{P}}\times 1}$. Generate the
$N_{c}$ candidate measurement matrix $\\{{\bf B}_{n_{c}}\\}_{n_{c}=1}^{N_{c}}$
according to $\mathbf{d}_{n_{c}}$.

2\. Calculate the objective function: $\mu({\bf B}_{n_{c}})=\left\|{{{\bf
B}_{n_{c}}^{\rm{H}}}{\bf B}_{n_{c}}-{N_{\rm
P1}}{{\bf{I}}_{{N_{\tau}}}}}\right\|_{2}^{2}$

3\. Sort the objective function $\mu({\bf B}_{n_{c}})$ in ascending order:
$\mu({\bf B}_{d_{e,1}})\leq\mu({\bf B}_{d_{e,2}})\leq\cdots\leq\mu({\bf
B}_{d_{e,N_{c}}})$;

4\.
${{\mathbf{P}_{\rm{B}}^{i+1}}\\!=\\!\frac{1}{{N_{e}}}({\mathbf{d}_{d_{e,1}}}\\!+\\!{\mathbf{d}_{d_{e,2}}}\\!+\\!\cdots\\!+\\!{\mathbf{d}_{d_{e,N_{e}}}})}$;

5\. $i=i+1$;

end for

Output: Designed pilot (the index of none-zero elements in
${\mathbf{d}_{d_{e,1}}}$).

Algorithm 2 Pilot design

In Step 1 of Algorithm 2, we define the probability vector ${\bf P}_{\rm
B}^{i}\in\mathbb{C}^{N_{P}\times 1}$ of the $i$-th iteration, where the
$n_{p}$-th element of ${\bf P}_{\rm B}^{i}$ denotes the probability that the
$n_{p}$-th element of ${\bf{d}}_{n_{c}}$ is equal to $1$. The $n_{p}$-th
element of ${\bf{d}}_{n_{c}}\in\mathbb{C}^{N_{\rm P}\times 1}$ indicates
whether the pilot found occupies the $n_{p}$-th subcarrier. Specifically, if
the $n_{p}$-th element of vector ${\bf{d}}_{n_{c}}$ is equal to $1$, then the
$n_{p}$-th subcarrier is a pilot. Otherwise, if the $n_{p}$-th element of
vector ${\bf{d}}_{n_{c}}$ is equal to $0$, the $n_{p}$-th subcarrier is not a
pilot. We utilize the probability matrix ${\bf P}_{\rm
B}^{i}\in\mathbb{C}^{N_{P}\times 1}$ to generate $N_{c}$ candidate vectors
expressed as $\mathbf{d}_{n_{c}}\\!\in\\!\\{0,1\\}^{N_{\rm{P}}\times 1}$.
Then, we check whether the candidate pilots satisfy (44). If not, we
regenerate the pilots until the number of candidates reaches $N_{c}$. we then
use the candidate pilots chosen to generate candidate measurement matrix
$\\{{\bf B}_{n_{c}}\\}_{n_{c}=1}^{N_{c}}$ according to (41). In Step 2 of
Algorithm 2, we calculate the OF: $\mu({\bf B}_{n_{c}})=\left\|{{{\bf
B}_{n_{c}}^{\rm{H}}}{\bf B}_{n_{c}}-{N_{\rm
P1}}{{\bf{I}}_{{N_{\tau}}}}}\right\|_{2}^{2}$. In Step 3 of Algorithm 2, we
sort the OF $\mu({\bf B}_{n_{c}})$ in ascending order and retain the first
$N_{e}$ elements. Then, we select the corresponding $N_{e}$ candidates
$\\{{{\bf{B}}_{{n_{c}}}}\\}_{{n_{c}}={d_{e,1}}}^{{n_{c}}={d_{e,{N_{e}}}}}$ as
elites and record their indices as
$\\{{d_{e,1}},{d_{e,2}},\cdots,{d_{e,N_{e}}}\\}$. In Step 4 of Algorithm 2, we
update the probability matrix as
${{\mathbf{P}_{\rm{B}}^{i+1}}\\!=\\!\frac{1}{{N_{e}}}({\mathbf{d}_{d_{e,1}}}\\!+\\!{\mathbf{d}_{d_{e,2}}}\\!+\\!\cdots\\!+\\!{\mathbf{d}_{d_{e,N_{e}}}})}$.
After $N_{iter}$ iterations, we arrive at the pilot subcarrier index set of
$\\{n_{1},\cdots,n_{p},\cdots,n_{N_{P1}}\\}$, where $n_{p}$ is the index of
non-zero elements in ${\mathbf{d}_{d_{e,1}}}$.

##  VI Simulation Results

In this section, we present our simulation results for characterizing the
performance of the proposed channel estimation and pilot design method. We
assume that the IRS elements of our ULA have a half-wavelength spacing. The
carrier frequency is $f_{c}=20$ GHz, The AOA is ${\varphi_{{l_{1}}}^{\rm
TR}}\in[-\frac{\pi}{2},\frac{\pi}{2})$ and the DOA is ${\varphi_{{l_{2}}}^{\rm
RR}}\in[-\frac{\pi}{2},\frac{\pi}{2})$. The maximum delay spread of the
cascaded channel is $200$ ns. The signal bandwidth is $W=510$ MHz and the
other parameteters are set to $M=256$, $N_{p}=128$, $N_{c}=100$, $N_{e}=20$
and $L_{1}L_{2}=6$. We define the normalized mean-squared error (NMSE) of the
cascaded channel at all subcarriers as

$\displaystyle{\rm
NMSE}{{}_{\bf{h}}}=\frac{{E\left({\sum\limits_{{n_{p}}=1}^{{N_{p}}}{\left\|{\widehat{\bf{h}}(\frac{{{n_{p}}W}}{{{N_{P}}}})-{\bf{h}}(\frac{{{n_{p}}W}}{{{N_{P}}}})}\right\|}_{2}^{2}}\right)}}{{E\left({\sum\limits_{{n_{p}}=1}^{{N_{p}}}{\left\|{{\bf{h}}(\frac{{{n_{p}}W}}{{{N_{P}}}})}\right\|}_{2}^{2}}\right)}},$
(47)

where $\widehat{\bf{h}}(\frac{{{n_{p}}W}}{{{N_{P}}}})$ denotes the estimate of
${\bf{h}}(\frac{{{n_{p}}W}}{{{N_{P}}}})$.Furthermore, we define the NMSE of
vector ${\bf z}(n_{p})$ in (IV-B) and ${{\bf{{\overline{c}}}}^{{\cal
I}_{i}^{a}}}$ in (40) as

$\displaystyle{\rm
NMSE}_{\bf{z}}=\frac{{E\left({\sum\limits_{{n_{p}}=1}^{{N_{P}}}{\left\|{\widehat{\bf{z}}({n_{p}})-{\bf{z}}({n_{p}})}\right\|_{2}^{2}}}\right)}}{{E\left({\sum\limits_{{n_{p}}=1}^{{N_{P}}}{\left\|{{\bf{z}}({n_{p}})}\right\|_{2}^{2}}}\right)}},$
(48) $\displaystyle{\rm
NMSE}_{\overline{\bf{c}}}=\frac{{E\left({\sum\limits_{i=1}^{{N_{d}}}{\left\|{{{\widehat{\bf{c}}}^{I_{i}^{a}}}-{{\overline{\bf{c}}}^{I_{i}^{a}}}}\right\|_{2}^{2}}}\right)}}{{E\left({\sum\limits_{i=1}^{{N_{d}}}{\left\|{{{\overline{\bf{c}}}^{I_{i}^{a}}}}\right\|_{2}^{2}}}\right)}},$
(49)

where ${{\widehat{\bf{c}}}^{I_{i}^{a}}}$ and $\widehat{\bf{z}}({n_{p}})$
denote the estimate of ${{\overline{\bf{c}}}^{I_{i}^{a}}}$ and
${\bf{z}}({n_{p}})$.

Figure 4: The NMSE performance of cascaded channel estimation against the
system bandwidth. The number of IRS elements $M=256$. The number of
subcarriers $N_{p}=128$. The frequency $f_{c}=20$ GHz and the pilot overhand
is $10\%$.

Fig. 4 shows the NMSE of the cascaded channel vs. the bandwidth. The curve
Method [23] denotes the NMSE of the conventional compressive sensing based
method proposed in [23], which assumes that the range of equivalent angles is
$[-1/2,1/2)$ and directly utilizes the OMP algorithm. We observe that when the
system bandwidth is small, the conventional method and the proposed method
have similar NMSE performance. However, as the bandwidth grows, the effect of
beam squint becomes severe, which degrades the performance of the traditional
channel estimation method.

Figure 5: The NMSE vs. SNR performance of cascaded channel gains and delays
estimation. The number of IRS elements $M=256$. The number of subcarriers
$N_{p}=128$. The frequency $f_{c}=20$ GHz and the signal bandwidth is $510$
MHz. Figure 6: The NMSE vs. SNR performance of cascaded channel angles
estimation. The number of IRS elements $M=256$. The number of subcarriers
$N_{p}=128$. The frequency $f_{c}=20$ GHz and the signal bandwidth is $510$
MHz.

Fig. 5 shows the impact of different pilot designs on the channel estimation
of delays and gains vs. SNR. The Y-axis represents the NMSE of vector
${{\bf{{\overline{c}}}}^{{\cal I}_{i}^{a}}}$. In Fig. 5, we estimate the
channel delays and gains with known equivalent angles. The curve Designed
pilots, $N_{P1}=6$ and Designed pilots, $N_{P1}=4$ rely on the pilots obtained
by Algorithm 2, whose subcarrier indices are $\\{2,20,26,43,67,91\\}$ and
$\\{21,45,68,86\\}$, respectively. The curve Random pilots, $N_{P1}=6$ and
Random pilots, $N_{P1}=4$ correspond to the randomly selected pilots, where
the number of pilots is $6$ and $4$, respectively. The curve Pilots without
constraint denotes the pilots whose subcarrier indices obtained by Algorithm 2
are $\\{2,4,5,6,10,12\\}$, except that our first pilot design guideline is
ignored. Fig. 5 shows that the proposed pilot design substantially improves
the channel gain and delay estimation performance, and the constraint of (44)
has a minor effect on the delay and gain estimation performance of the second
stage.

Fig. 6 shows the impact of different pilot designs on the equivalent angle
estimation vs. SNR. The Y-axis represents the NMSE of vector
${\mathbf{z}}(n_{p})$. We consider the range of equivalent angles as $(-1,1)$.
Fig. 6 shows that the proposed pilot design substantially improve the
algorithm’s equivalent angles estimation performance. However, since the curve
Pilots without constraint does not consider the first guideline, it represents
poor performance. Observe in Fig. 5 and Fig. 6, that the proposed pilot design
obtains both good equivalent angle as well as gain and delay estimation
performance, simultaneously.

Fig. 7 shows the impact of different pilot designs on the estimation of
cascaded channel against SNR. The definition of curves in Fig. 7 is the same
as those in Fig. 6. Fig. 7 shows that the proposed pilot design benefically
improves the algorithm’s cascaded channel estimation performance.

Fig. 8 shows the NMSE performance comparison of the proposed TS-OMP and the
conventional method of [23] vs. the SNR. The curve Proposed TS-OMP and
designed pilots represents the NMSE of the cascaded channel estimated by the
proposed TS-OMP algorithm based on our designed pilots, where the range of
equivalent angles is $(-1,1)$. The curve Method [23] and random pilots
correspond to the NMSE of the cascaded channel estimated by the conventional
OMP algorithm [23] based on random pilots, where the range of equivalent
angles is $[-1/2,1/2)$. The curve Ideal solution represents the NMSE of
channel estimation relying on known equivalent angles and delays of the
cascaded channel. We observe that the proposed TS-OMP algorithm and cross-
entropy based pilot design substantially improve the channel estimation
performance in IRS-aided wideband systems.

Figure 7: The NMSE vs. SNR performance of cascaded channel estimation. The
number of IRS elements $M=256$. The number of subcarriers $N_{p}=128$. The
frequency $f_{c}=20$ GHz and the signal bandwidth is $510$ MHz. Figure 8: The
NMSE vs. SNR performance of cascaded channel estimation. The number of IRS
elements $M=256$. The frequency $f_{c}=20$ GHz and the signal bandwidth is
$510$ MHz. The pilot overhand is $10\%$.

Fig.9 shows the NMSE performance of the proposed channel estimation method
against the number of measurements, which is defined as the number of OFDM
symbols required for channel estimation. The SNR is $10$dB. The curve Method
[23] and random pilots shows that we cannot achieve the accurate channel
estimation, not even for a high overhand of training OFDM symbols, due to the
effect of beam squint. Moreover, the curve Proposed method and designed pilots
indicates that the proposed method achieves good performance using a small
number of training OFDM symbols.

Figure 9: The NMSE performance of cascaded channel estimation against the
number of measurements. The number of IRS elements $M=256$. The frequency
$f_{c}=20$ GHz and the signal bandwidth is $510$ MHz. The SNR is 10dB. The
pilot overhand is $10\%$. Figure 10: The NMSE performance of estimated channel
against the number of pilots. The number of IRS elements $M=256$. The number
of subcarriers $N_{p}=128$. The frequency $f_{c}=20$ GHz and the signal
bandwidth is $510$ MHz. The SNR is 15dB. Figure 11: The SER performance
against SNR. The number of IRS elements $M=256$. The number of subcarriers
$N_{p}=128$. The frequency $f_{c}=20$ GHz and the signal bandwidth is $510$
MHz. The pilot overhand is $10\%$.

Fig.10 shows the NMSE performance of the proposed cascaded channel estimation
method against the number of pilots. The SNR is $15$dB and the number of
training OFDM symbols is $35$. We find that the channel estimation becomes
more and more accurate as the number of pilots increases.

Fig. 11 shows the symbol error rate (SER) against SNR. We use 16-QAM
transmitted symbols. The curve Random phase represents the SER performance,
where the IRS phases are chosen randomly. The curve Method [23] and designed
phase represents the SER performance when the CSI is estimated by the method
proposed in [23] and the phase is designed using the method of [14]. The curve
Designed phase and designed pilots represents the SER performance, where the
IRS phase is designed when the CSI is estimated by the proposed TS-OMP
algorithm based on our designed pilots. The curve Designed phase and random
pilots represents the SER performance, where the IRS phase is designed when
the CSI is estimated by the proposed TS-OMP algorithm based on random pilots.
The curve Designed phase and perfect CSI represents the SER performance, where
the IRS phase is based on the perfect CSI. The curve Random phase and Designed
phase and perfect CSI shows that the appropriate design of the IRS phase
substantially reduces the SER. The curve Designed phase and perfect CSI and
Designed phase and designed pilots shows that the proposed TS-OMP algorithm
based on our pilot design achieves similar SER to the perfect CSI scenario.

##  VII Conclusions

Cascaded channel estimation of IRS-aided systems have been investigated in
wideband scenarios considering the effect of beam squint. The cascaded channel
is characterized by the equivalent angles, gains and delays of propagation
paths. We demonstrated that it is hard to distinguish between the actual angle
and false angle, which is occurred due to the effect of beam squint. We
proposed a TS-OMP algorithm, which can estimate the cascaded channel with
small training overhand. Moreover, to further improve the channel estimation
performance, we propose a pilot design method based on cross-entropy theory.
Our simulation results confirm that the proposed TS-OMP algorithm and pilot
design method achieve better performance than their conventional counterparts.

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