Efﬁcient Deep Unfolding for SISO-OFDM Channel Estimation Baptiste Chatelieryz Luc Le Magoarouyz Getachew Redieteabxz

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Efﬁcient Deep Unfolding for SISO-OFDM

Channel Estimation

Baptiste Chatelier†‡, Luc Le Magoarou†‡, Getachew Redieteab§‡

†Univ Rennes, INSA Rennes, CNRS, IETR-UMR 6164, Rennes, France

‡b<>com, Rennes, France

§Orange Innovation, Rennes, France

baptiste.chatelier@insa-rennes.fr

Abstract—In modern communication systems, channel state

information is of paramount importance to achieve capacity. It is

then crucial to accurately estimate the channel. It is possible to

perform SISO-OFDM channel estimation using sparse recovery

techniques. However, this approach relies on the use of a physical

wave propagation model to build a dictionary, which requires

perfect knowledge of the system’s parameters. In this paper,

an unfolded neural network is used to lighten this constraint.

Its architecture, based on a sparse recovery algorithm, allows

SISO-OFDM channel estimation even if the system’s parameters

are not perfectly known. Indeed, its unsupervised online learning

allows to learn the system’s imperfections in order to enhance

the estimation performance. The practicality of the proposed

method is improved with respect to the state of the art in

two aspects: constrained dictionaries are introduced in order

to reduce sample complexity and hierarchical search within

dictionaries is proposed in order to reduce time complexity.

Finally, the performance of the proposed unfolded network

is evaluated and compared to several baselines using realistic

channel data, showing the great potential of the approach.

Index Terms—Deep Unfolding, Frugal AI, SISO-OFDM

channel estimation, Sparse recovery

I. INTRODUCTION

Recently, machine learning techniques have emerged as

a promising solution in many wireless communications

areas [1], [2] such as beamforming [3], [4], channel chart-

ing [5]–[7] or channel mapping [8], [9]. The Deep Unfolding

machine learning approach [10]–[13] is very promising as it

proﬁts from both the controlled complexity of classical signal

processing approaches and the ﬂexibility of machine learning

techniques. Indeed, this approach considers that iterative algo-

rithms can be unfolded as neural networks, that can be trained,

where each layer represents one iteration of the algorithm.

In the ﬁeld of channel estimation, classical statistical

estimation techniques such as the Least Squares (LS) or

the Minimum Mean Squared Error (MMSE) have been

extensively used in the past. However, those techniques suffer

from several drawbacks: on one hand, for the LS estimator,

there is a huge Mean Squared Error (MSE), on the other

hand, for the MMSE estimator, there is a huge computational

complexity. Other approaches have been envisioned to counter

those drawbacks: one of them is the usage of sparse recovery

algorithms. It is well known that propagation channels are

dominated by a few strong propagation paths: the channel is

then said to be sparse. It has been shown that one can use

that sparsity notion to propose channel estimation strategies

relying on sparse recovery algorithms such as Matching

Pursuit (MP) [14]–[17]. However, those techniques rely on

the knowledge of a physical wave propagation model, and it

has been shown in [17] that a small uncertainty on system

parameters could lead to high estimation performance losses.

Contributions and related work. In [17], the Deep

Unfolding approach was used to solve this performance loss

issue, with the unfolded network mpNet, allowing to learn

the physical imperfections of the system so as to enhance

the estimation performance. However, this approach yields a

high number of parameters to learn, leading to both a high

sample complexity and a high time complexity. In this paper,

mpNet is adapted to constrained dictionaries, which allows to

drastically reduce sample complexity. Moreover, hierarchical

search over dictionaries is introduced in order to reduce time

complexity. Furthermore, the proposed method is illustrated

on a Single Input Single Output Orthogonal Frequency

Division Multiplexing (SISO-OFDM) system here instead of

a Multi-User Multiple Input Multiple Output (MU-MIMO)

system as in [17]. However, note that the contributions apply

equally to both types of systems.

The rest of the paper is organized as follows, Section II

presents the SISO-OFDM channel estimation problem, the

used physical wave propagation model, and the sparse

recovery approach. Section III presents the unfolded neural

network and the two contributions of this paper: the sample

complexity reduction strategy, and the time complexity

reduction strategy. Section IV presents simulation results.

Finally, Section V concludes the paper and gives perspectives.

II. PROBLEM FORMULATION

A. System model

In this paper, a SISO-OFDM scenario with Nsubcarriers

in the uplink is considered. Let g∈CNdenote the antenna

gain vector representing the Base Station (BS) antenna gain

at each subcarrier frequency, and f∈RNdenote a frequency

vector containing the different subcarrier frequencies. Finally,

a multipath channel with Lppropagation paths is considered.

Let x∈CNbe an LS channel estimate of the channel

vector h∈CN. This estimate is noisy as it possesses a

arXiv:2210.06588v1 [cs.IT] 11 Oct 2022

residual estimation error:

x=h+n,(1)

where n∼ CN 0, σ2IN. Its Signal to Noise Ratio (SNR)

can be computed as:

SNRin =khk2

Nσ2.(2)

The goal of this paper is to denoise this LS channel estimate

using sparse recovery techniques and a physical model.

B. Physical model

For a given subcarrier fk, the channel coefﬁcient can be

written as:

hk=

l=1

αlgke−j2πfkτl,(3)

where αland τlrespectively represent the complex attenuation

coefﬁcient and the propagation delay for the lth path.

Under this model, the channel vector h∈CNcan be

expressed as:

l=1

αl





g1e−j2πf1τl

gNe−j2πfNτl





=

l=1

αlψ(τl),(4)

where ψ(τl)is the frequency response vector (FRV) for the

lth path. We can see that the channel vector can be deﬁned

as a linear combination of FRVs. The linear combination is

said to be sparse when Lpis small.

C. Sparse recovery approach

As the channel can be written as a linear combination of

FRVs, it is possible to construct a ﬁxed matrix Ψ∈CN×A

with AFRV columns, which amounts to the discretization

of the delays, and a projection vector u∈CAsuch that the

channel could be estimated as:

h=Ψu.(5)

For the rest of this paper, the FRV matrix Ψwill be called

dictionary and its columns will be called atoms, i.e. each

atom will represent a FRV corresponding to a speciﬁc delay.

As the channel vector can be seen as a linear combination

of a few FRVs, the channel vector is said to be sparse in

a dictionary of FRVs. For a ﬁxed dictionary Ψ, the sparse

recovery optimization problem is:

min

ukh−Ψuk2

s.t.kuk0≤A

(6)

The previous optimization problem makes sense only if the

dictionary is correct, i.e. a sparse linear combination of

dictionary columns can represent a channel vector. If this is

not the case, the channel estimate won’t be trustworthy. It is

possible to ﬁnd the optimal dictionary as:

Ψ?= arg min

Ehmin

u,kuk0≤Akh−Ψuk2

2.(7)

Solving Eq. (6) for a ﬁxed FRV dictionary is possible through

sparse recovery algorithm such as the MP algorithm [18]: this

is the approach followed in this paper, as in [17]. The goal of

this algorithm is to ﬁnd the most correlated FRVs, i.e. atoms,

of the dictionary with the noisy channel estimate and subtract

their projections. After Lpiterations of the algorithm, in a

perfect scenario, as all the corresponding FRVs will have

been subtracted to the residual, the residual will only be

composed of noise.

Algorithm 1 MP algorithm for channel denoising

Input: Dictionary Ψ, Noisy LS estimate x, Tolerance level 

1: Initialize the residual: r←x

2: while krk2

2>  do

3: Find the most correlated atom: i?←arg maxi



ψH

ir



4: Update the residual: r←r−ψi?ψH

i?r

5: end while

Output: ˆ

h←x−r(Denoised LS estimate)

D. Hardware impairments

In reality, the FRVs are not exactly known as there

exists a lot of hardware imperfections related to frequency

generation/acquisition. More speciﬁcally, Carrier Frequency

Offset (CFO) and Sampling Clock Offset (SCO) can offset

the subcarrier frequencies. Let ˜

fibe the nominal ith subcarrier

frequency, i.e. the non-offseted subcarrier frequency, and

fibe the real ith subcarrier frequency, i.e. the potentially

offseted subcarrier frequency. For the CFO, the frequency

offset δf is common to all subcarriers:

∀i∈t−N

2,N

2|, fi=˜

fi+δf. (8)

For the SCO, it has been shown in [19], [20] that the

frequency offset is dependent on the subcarrier index, on

the oscillator part per million (ppm) value ξ, and on the

subcarrier spacing ∆f:

∀i∈t−N

2,N

2|, fi=˜

fi+iδf =˜

fi+iξ∆f. (9)

Moreover, we consider an antenna gain imperfection. Let ˜gi

be the nominal antenna gain, and gibe the real antenna gain

for the ith subcarrier. We obtain:

∀i∈t−N

2,N

2|, gi= ˜gi+ngi,(10)

with ngi∼ N 0, σ2

g.

For the rest of this paper, we will only consider SCO and

antenna gain imperfections. We can then deﬁne the notions

of nominal and real dictionaries. The nominal dictionary ˜

will represent the dictionary constructed from the nominal

knowledge of the system parameters, which is unaware

of impairments. On the other hand, the real dictionary

will represent the dictionary constructed from the perfect

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EfcientDeepUnfoldingforSISO-OFDMChannelEstimationBaptisteChatelieryz,LucLeMagoarouyz,GetachewRedieteabxzyUnivRennes,INSARennes,CNRS,IETR-UMR6164,Rennes,Francezbcom,Rennes,FrancexOrangeInnovation,Rennes,Francebaptiste.chatelier@insa-rennes.frAbstractInmoderncommunicationsystems,channelstateinformat...

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Efﬁcient Deep Unfolding for SISO-OFDM Channel Estimation Baptiste Chatelieryz Luc Le Magoarouyz Getachew Redieteabxz

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