WU et al.: IRS-AIDED WIRELESS COMMUNICATIONS: A TUTORIAL 3319
From IRS element n to the receiver, the reflected signal
undergoes a similar equivalent narrow-band frequency-flat
channel given by α
2,n
e
−ξ
2,n
. Then the passband signal arriv-
ing at the receiver via IRS element n’s reflection is expressed
as
y
r,n
(t)=Re
α
1,n
e
−ξ
1,n
β
n
e
θ
n
α
2,n
e
−ξ
2,n
x(t)
e
2πf
c
t
.
(4)
Thus, the cascaded channel from the transmitter to the receiver
via IRS element n has been modeled. Let h
∗
r,n
α
1,n
e
−ξ
1,n
and g
n
α
2,n
e
−ξ
2,n
. The corresponding baseband signal
model of (4) is given by
y
n
(t)=β
n
e
θ
n
h
∗
r,n
g
n
x(t). (5)
From (5), it is observed that the IRS reflected channel is a
multiplication of three terms, namely, transmitter-to-element
n channel, IRS reflection, and element n-to-receiver channel.
For simplicity, we assume that there is no signal coupling
in the reflection by neighbouring IRS elements, i.e., all IRS
elements reflect the incident signals independently. Due to the
substantial path loss, we only consider signals reflected by
the IRS for the first time and ignore those reflected by it
two or more times. As such, the received signal from all IRS
elements can be modeled as a superposition of their respective
reflected signals; thus, the baseband signal model accounting
for all the N IRS elements is given by
y(t)=
N
n=1
β
n
e
θ
n
h
∗
r,n
g
n
x(t)=h
H
r
Θgx(t), (6)
where h
H
r
=[h
∗
r,1
, ··· ,h
∗
r,N
], g =[g
1
, ··· ,g
N
]
T
,and
Θ = diag(β
1
e
θ
1
, ··· ,β
N
e
θ
N
). Note that the IRS with
N elements essentially performs a linear mapping from the
incident (input) signal vector to a reflected (output) signal
vector by an N × N diagonal complex reflecting matrix
Θ, which is diagonal because each IRS element reflects the
signal independently and there is no signal coupling or joint
processing over the IRS elements [8]–[10], [43].
Note that the channel coefficients in h
H
r
and g generally
depend on distance-related path loss, large-scale shadowing,
and small-scale multipath fading. In particular, the path loss
of IRS-reflected channel captures its average power and is
thus essential to the link budget analysis and performance
evaluation o f IRS-aided communications. Without loss of
generality, con sider IRS elem ent n, which is assumed to be
located sufficiently far from both the transmitter an d receiver,
with the distances fr om them given by d
1,n
and d
2,n
, respec-
tively, as shown in Fig. 4 (a). Under the far-field prorogation
condition, we can assume that d
1,n
= d
1
and d
2,n
= d
2
, ∀n.
Then, it follows that E(|h
r,n
|
2
) ∝ c
1
(
d
1
d
0
)
−a
1
and E(|g
n
|
2
) ∝
c
2
(
d
2
d
0
)
−a
2
, ∀n,wherec
1
(c
2
) denotes the corresponding path
loss at the reference distance d
0
, while a
1
(a
2
) denotes the
corresponding path loss exponent with typical values from 2
(in free-space propagation) to 6 [7]. From (5), it then follows
that the average received signal power via the reflection b y
IRS element n, denoted by P
r,n
, is inversely proportional to
d
a
1
1
d
a
2
2
, i.e.,
P
r,n
∝
1
d
a
1
1
d
a
2
2
. (7)
In other words, the IRS reflected channel via element n suffers
from double path loss, which is thus referred to as the product-
distance path loss model. As such, a large number of IRS
reflecting elements are needed in practice to compensate for
the severe power loss due to double attenuation, by jointly
designing their reflection amplitudes and/or phases to achieve
high passive beamforming gains, as will be detailed later in
Section III.
Remark 1: In Fig. 4 (b), the IRS is replaced by an infinitely
large perfect electric conductor (PEC) (or metallic plate).
Assuming free-space propagation and applying the image the-
ory [44], it can be shown that the signal power received at the
receiver via the PEC’s reflection, denoted by P
r
,isinversely
proportional to the square of the sum distance of the two-hop
links, i.e.,
P
r
∝
1
(d
1
+ d
2
)
2
. (8)
This model is usually referred to as the sum-distance path
loss model. Intuitively, due to the reflection of the infinitely
large PEC, the received signal at the receiver were as if from
an equivalent transmitter located at the image point of the
original transmitter as shown in Fig. 1 (b), with the same link
distance d
1
+ d
2
, which is also known as specular reflection.
Note that this model is valid for the free-space propaga tion
with an infinitely large PEC, but in ge neral inapplicable to
the IRS-reflected channel modeled from the element level as
given in (5)–(7). In particular, it is inappropriate to apply
the sum-distance model to the scenario with one or more
finite-size tunable PECs and conclude that the received signal
power scales with the number of PECs by exploiting their
multiplicative passive beamform ing gains and at the same time
following the more favorable (as compared to the product-
distance model) sum-distance based path loss, even under the
free-space propagation. More theoretical and/or experimental
studies on this issue can be found in [45]–[49].
B. IRS Architecture, Hardware, and Practical Constraints
The highly controllable reflection of IRS can be practi-
cally achieved b y leveraging the existing digitally reconfig-
urable/programmable metasurface [50]. Specifically, metasur-
face is a planar array composed of massive properly d esigned
reflecting elements/meta-atoms whose electrical thickness is
typically in the order of subwavelength of the signal of interest.
By designing their geometry shape (e.g., square or split-ring),
size/dimension, orientation, arrangement, and so on, desired
signal response (e.g., reflection amplitude and/or phase shift)
of each element/atom can be realized. However, in wireless
communication, the channel is g enerally time-varying due to
the mobility of the transmitter/receiver as well as the surround-
ing objects, thus calling for real-time tunable response of IRS
based on the channel variation. To this end, IRS elements need
to be manufactured with dynamically adjustable reflection
coefficients and IRS is required to connect to the wireless
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