# Application of a Channel Estimation Algorithm to Spectrum Sensing in a Cognitive Radio Context

Authors: Vincent Savaux; Moïse Djoko-Kouam; Yves Louët; Alexandre Skrzypczak
Journal: International Journal of Antennas and Propagation
 (2014)

---

## Abstract

This paper deals with spectrum sensing in an orthogonal frequency division multiplexing (OFDM) context, allowing an opportunistic user to detect a vacant spectrum resource in a licensed band. The proposed method is based on an iterative algorithm used for the joint estimation of noise variance and frequency selective channel. It can be seen as a second-order detector, since it is performed by means of the minimum mean square error criterion. The main advantage of the proposed algorithm is its capability to perform spectrum sensing, noise variance estimation, and channel estimation in the presence of a signal. Furthermore, the sensing duration is limited to only one OFDM symbol. We theoretically show the convergence of the algorithm, and we derive its analytical detection and false alarm probabilities. Furthermore, we show that the detector is very efficient, even for low SNR values, and is robust against a channel uncertainty.


---

## Body

## 1. Introduction

Wireless communications are facing a constant increase of data-rate-consuming transmissions, due to the multiplications of the applications and services, while the available spectrum resource is naturally limited. Furthermore, most of the frequency bands are already allocated to specific licenses. However, some of these licensed bands are not used at their full capacity, which results in spectrum holes along the time and frequency axes [1], whereas they could be exploited in order to achieve the requirements of data rate. Away from the usual paradigm in which the channels are allocated only for licensed users, Mitola and Maguire Jr. defined the cognitive radio [2], allowing an opportunistic access by unlicensed users to the unused frequency bands. In such network, the opportunistic users, called secondary users (SUs), can use licensed bands when primary users (PUs) are not active. The main condition for the SUs to use the licensed bands is to minimize the interferences with PUs. Thus, they must be able to sense the presence of the PUs, even if the PU’s signal is attenuated compared to the noise level. Figure 1 depicts the concept of spectrum sensing: a PU transmitter (PU-Tx) is transmitting to a PU receiver (PU-Rx) while a SU transmitter intends to transmit in the same band. In order to avoid the interferences with the PU, the SU has to perform spectrum sensing. In order to lighten the drawing, only one PU-Rx and two SU-Rxs are depicted, but the network can obviously be more complex. The process set up by the SUs to sense the presence of the PUs is called spectrum sensing. The authors of [3–5] propose detailed reviews of the different techniques of spectrum sensing. The different methods are usually classified into two main categories: the cooperative detection and the noncooperative detection. In this paper, we take an interest in the latter.Figure 1
Illustration of the principle of the spectrum sensing.The noncooperative detection concerns a sole SU who tries to detect the presence of the PU alone. Among the wide range of methods [3–6], one can describe the main ones: the energy detector measures the energy of the received signal and compares it to a threshold. It has a low complexity of implementation and does not require any knowledge on the PU’s signal features. However, the choice of the threshold value depends on the noise variance, and the uncertainties on the noise level may cause important degradations of the detector performance [7, 8]. The matched-filter correlates the received signal with the one transmitted by the PU, which is supposed to be known at the receiver [9, 10]. This is the optimal detector when the signal is transmitted with AWGN only and supposed to be known at the receiver. Due to these hypotheses, this method is generally not applicable in practice, and its performance is degraded when the knowledge of the signal is erroneous [11]. Less binding than the matched-filter, the feature detectors only use several characteristics of the signal to detect the PU. Thus, the waveform-based sensing uses the preamble of the PU’s signal (used for the synchronization, the estimation, etc.) to perform a correlation with the received signal [12]. However, the performance of the waveform-based sensing is degraded in the presence of selective channels. In the same way, when a CP is used in an OFDM signal, its autocorrelation function becomes time-varying, so a second-order-based method consists of detecting the peaks in the autocorrelation function at time-lag T
s. More generally, the cyclostationarity detector exploits the periodic redundancy of all telecommunications signals to differentiate it from a pure Gaussian noise [13, 14]. As indicated in [5], the redundancy can occur due to periodic patterns such as the CP, the symbol rate, or the channel code. However, these second-order detectors require large sensing time, that is, a large number of symbols to be performed. In [15], a hybrid architecture composed of both energy and cyclostationarity is proposed. It allows the energy detector to compensate its limitation due to the noise uncertainty thanks to a cyclostationarity detection stage whose computation time is reduced. Another attractive technique called eigenvalue-based detection uses the characteristics of the covariance matrix of large-sized random matrices (e.g., containing noise samples only) [16–18]. Indeed, the random matrix theory proved that the distribution of the eigenvalues of such matrix tends to a deterministic function. In [16], if the noise variance is known, the signal is detected if a peak appears outside of the domain of the function. Using the same theory, the authors of [17] propose the maximum-minimum eigenvalue (MME) detection, whose principle consists of comparing the ratio between the maximum and minimum eigenvalues with a threshold to take the decision. Based on the same theory, both techniques have the same asymptotic performance, but the latter does not require the noise level to be performed. However, these two methods require matrices with very large sizes, hence, a large number of sensors and a long sensing time. In order to use MME detection with a single sensor, the authors of [19] propose to artificially create a large matrix by stacking the shifted vectors of the received sampled signal. However, this method is limited since the rows of the created matrix are correlated.In this paper, we propose to perform spectrum sensing by means of a minimum mean square error (MMSE-)based iterative algorithm developed in [20] for the joint estimation of noise variance and frequency selective channel. Since we consider a sole receiver, the context of the next sections will be the noncooperative detection of a PU transmitting an OFDM signal by a single SU in a given narrow band. In the presence or absence of signal, the algorithm converges after a few iterations and performs the estimation of the noise variance. We then add a metric to the method presented in [20] in order to turn it into a spectrum sensing algorithm. The metric is defined by the difference between the second-order moment M
2 of the received signal and the estimated noise variance. If the PU is present (resp. absent), the metric is equal to the power of the transmitted signal (resp. equal to zero). As the algorithm is based on MMSE criterion, it requires the estimation of the channel covariance matrix, so the detector can be classified as a second-order statistics detector. Compared to usual second-order statistics detectors such as MME, the proposed one only needs the time duration of one OFDM symbol to be performed. It is also robust in frequency selective channels context. Furthermore, when the PU’s signal is present, it achieves a joint estimation of noise variance and channel. When the PU is absent, it also performs the noise variance estimation, and it is proved that it reaches the exact noise power value. In this paper, a theoretical expression of the detection and the false alarm probabilities are derived, and we show that they are very close to the simulations.In this paper, the normal fontx is used for scalar variables, the boldface x is used for vectors, and the underlined boldface x
_ is used for matrices. Furthermore, small letters x point out the variables in the time domain and capital letters X in the frequency domain.This paper is organized as follows: Section2 presents the system model and the algorithm developed in [20]. In Section 3, we prove the convergence of the algorithm in the absence of the signal, and we characterize the detector in Section 4. In Section 5, we give the theoretical expressions of the false alarm and detection probabilities. Simulations results are depicted in Section 6, and finally we draw our conclusions in Section 7.


## 2. Background

### 2.1. System Model

We consider the problem of the detection of an OFDM pilot preamble over a Rayleigh fading channel with additive white Gaussian noise (AWGN) in a given band. After theM-points discrete Fourier transform (DFT), the received signal is noted U. According to the presence or the absence of the primary user (PU) in the band, the usual hypothesis test is given by
(1)
ℍ
0
:
U
=
W
,
ℍ
1
:
U
=
C
_
H
+
W
,
where ℍ
0 and ℍ
1 denote the absence and the presence of the PU hypotheses, respectively. According to the model given in [20], the variable H is the M
×
1 complex vector of the frequency response of the channel, composed of the frequency response samples H
m, m
=
0,1
,
…
,
M
-
1:
(2)
H
m
=
∑
l
=
0
L
-
1
h
l
exp
⁡
(
-
2
j
π
m
τ
l
M
)
,
where L is the number of paths of the channel and h
l and τ
l are the zero-mean Gaussian path coefficients and the sampled path delays, respectively. We assume that the channel H follows a wide sense stationary uncorrelated scattering (WSSUS) model [21]. Consequently, |
H
m
| follows a Rayleigh distribution. The variable C
_ is the M
×
M diagonal matrix composed of the pilots C
m such that |
C
m
|
=
1, without loss of generality, and W is the M
×
1 vector of AWGN with variance σ
2. Let us assume that, under hypothesis ℍ
1, the receiver is synchronized on the position of the preamble C
_.


### 2.2. Iterative Algorithm for the Channel and Noise Variance Estimation

We now briefly recall the steps of the algorithm for the joint and iterative estimation of the channel and the noise variance as given in [20], that is, under the hypothesis ℍ
1. Basically, it is an MMSE-based iterative algorithm in which, at each step, the noise variance estimation feeds the channel estimation and vice versa. In addition to the noise variance, the linear-MMSE (LMMSE) channel estimation [22] requires the channel covariance matrix that has to be estimated at the receiver. This estimated channel covariance matrix is noted R
~
_
H. The algorithm from [20] is described by Figure 2 and its steps are detailed in the following.Figure 2
Block diagram of the iterative algorithm in the realistic scenario.(1) At the beginning, only the LS channel estimationH
^
LS performed on a pilot preamble is available at the receiver, so the only way to estimate the covariance matrix denoted by R
~
_
H
LS is
(3)
R
~
_
H
LS
=
H
^
LS
(
H
^
LS
)
H
,
where (
·
)
H is the Hermitian transposition. Furthermore, a stopping criterion e
σ is fixed. Let us denote i to be the index of the iteration.(2) At the first step(
i
=
1
) of the algorithm, the LMMSE channel estimation [22, 23] is performed with R
~
_
H
LS:
(4)
H
^
(
i
=
1
)
LMMSE
=
R
~
_
H
LS
(
R
~
_
H
LS
+
σ
^
(
i
=
0
)
2
C
_
C
_
H
)
-
1
H
^
LS
,
where σ
^
(
i
=
0
)
2 points out the initialization value of the noise variance and C
_
C
_
H is equal to the identity matrix I
_.(3) The noise variance is estimated by means of the MMSE criterion [24] σ
^
(
i
=
1
)
2
=
(
1
/
M
)
E
{
∥
H
^
LS
-
H
^
∥
2
}, with H
^
=
H
^
(
i
=
1
)
LMMSE:
(5)
σ
^
(
i
=
1
)
2
=
1
M
E
{
∥
H
^
LS
-
H
^
(
i
=
1
)
LMMSE
∥
2
}
,
with ∥
·
∥ being the Frobenius matrix norm, defined by ∥
A
_
∥
=
tr
⁡
(
A
_
A
_
H
). If the algorithm keeps on computing with R
~
_
H
LS, it is proved in [20] that (
σ
^
(
i
)
2
) converges to zero. Under this condition, the algorithm enters into an endless loop. This is due to the fact that R
~
_
H
LS is sensitive to the noise and then it is a rough approximation of the exact covariance matrix. In order to obtain a more accurate channel covariance matrix, it is now possible to use H
^
(
i
=
1
)
LMMSE, such that
(6)
R
~
_
H
LMMSE
=
H
^
(
i
=
1
)
LMMSE
(
H
^
(
i
=
1
)
LMMSE
)
H
.(4) Fori
≥
2, the iterative estimation steps (4) and (6) are performed by using (6):
(7)
H
^
(
i
)
LMMSE
=
R
~
_
H
LMMSE
(
R
~
_
H
LMMSE
+
σ
^
(
i
-
1
)
2
I
_
)
-
1
H
^
LS
,
(8)
σ
^
(
i
)
2
=
1
M
E
{
∥
H
^
LS
-
H
^
(
i
)
LMMSE
∥
2
}
.It will be shown afterwards that the characterization of the initializationσ
^
(
i
=
0
)
2 remains the same in the presence or absence of the PU. However, it is already obvious that σ
^
(
i
=
0
)
2 must be strictly positive; otherwise, H
^
(
i
)
LMMSE
=
H
^
LS in (7). In that case, σ
^
(
i
)
2
=
0, and the algorithm enters into an endless loop.(5) While|
σ
^
(
i
)
2
-
σ
^
(
i
-
1
)
2
|
>
e
σ, go back to Step 4 with i
←
i
+
1; otherwise, go to Step 6.(6) End of the algorithm. We notei
f to be the index of the last iteration.It is proved in [20] that this algorithm converges if the initialization value of the noise variance is chosen such that σ
^
(
i
=
0
)
2
≫
M
2, where M
2 is the second-order moment of the received signal U. Moreover, the algorithm converges to limits (
H
^
(
i
f
)
,
σ
^
(
i
f
)
2
) that are close to the exact values (
H
,
σ
2
). From (4) and (7) we can deduce the complexity of the algorithm. The LMMSE channel estimation requires M
3 scalar multiplications for the matrix inversion and multiplication and the noise variance M
2. The covariance matrices estimation also requires M
2 operations. Finally, we then evaluate the complexity of the proposed algorithm by 𝒪
(
i
f
M
3
).Unlike the presented model, the next section investigates the behavior of the algorithm when the PU is absent, that is, under hypothesisℍ
0.


## 2.1. System Model

We consider the problem of the detection of an OFDM pilot preamble over a Rayleigh fading channel with additive white Gaussian noise (AWGN) in a given band. After theM-points discrete Fourier transform (DFT), the received signal is noted U. According to the presence or the absence of the primary user (PU) in the band, the usual hypothesis test is given by
(1)
ℍ
0
:
U
=
W
,
ℍ
1
:
U
=
C
_
H
+
W
,
where ℍ
0 and ℍ
1 denote the absence and the presence of the PU hypotheses, respectively. According to the model given in [20], the variable H is the M
×
1 complex vector of the frequency response of the channel, composed of the frequency response samples H
m, m
=
0,1
,
…
,
M
-
1:
(2)
H
m
=
∑
l
=
0
L
-
1
h
l
exp
⁡
(
-
2
j
π
m
τ
l
M
)
,
where L is the number of paths of the channel and h
l and τ
l are the zero-mean Gaussian path coefficients and the sampled path delays, respectively. We assume that the channel H follows a wide sense stationary uncorrelated scattering (WSSUS) model [21]. Consequently, |
H
m
| follows a Rayleigh distribution. The variable C
_ is the M
×
M diagonal matrix composed of the pilots C
m such that |
C
m
|
=
1, without loss of generality, and W is the M
×
1 vector of AWGN with variance σ
2. Let us assume that, under hypothesis ℍ
1, the receiver is synchronized on the position of the preamble C
_.


## 2.2. Iterative Algorithm for the Channel and Noise Variance Estimation

We now briefly recall the steps of the algorithm for the joint and iterative estimation of the channel and the noise variance as given in [20], that is, under the hypothesis ℍ
1. Basically, it is an MMSE-based iterative algorithm in which, at each step, the noise variance estimation feeds the channel estimation and vice versa. In addition to the noise variance, the linear-MMSE (LMMSE) channel estimation [22] requires the channel covariance matrix that has to be estimated at the receiver. This estimated channel covariance matrix is noted R
~
_
H. The algorithm from [20] is described by Figure 2 and its steps are detailed in the following.Figure 2
Block diagram of the iterative algorithm in the realistic scenario.(1) At the beginning, only the LS channel estimationH
^
LS performed on a pilot preamble is available at the receiver, so the only way to estimate the covariance matrix denoted by R
~
_
H
LS is
(3)
R
~
_
H
LS
=
H
^
LS
(
H
^
LS
)
H
,
where (
·
)
H is the Hermitian transposition. Furthermore, a stopping criterion e
σ is fixed. Let us denote i to be the index of the iteration.(2) At the first step(
i
=
1
) of the algorithm, the LMMSE channel estimation [22, 23] is performed with R
~
_
H
LS:
(4)
H
^
(
i
=
1
)
LMMSE
=
R
~
_
H
LS
(
R
~
_
H
LS
+
σ
^
(
i
=
0
)
2
C
_
C
_
H
)
-
1
H
^
LS
,
where σ
^
(
i
=
0
)
2 points out the initialization value of the noise variance and C
_
C
_
H is equal to the identity matrix I
_.(3) The noise variance is estimated by means of the MMSE criterion [24] σ
^
(
i
=
1
)
2
=
(
1
/
M
)
E
{
∥
H
^
LS
-
H
^
∥
2
}, with H
^
=
H
^
(
i
=
1
)
LMMSE:
(5)
σ
^
(
i
=
1
)
2
=
1
M
E
{
∥
H
^
LS
-
H
^
(
i
=
1
)
LMMSE
∥
2
}
,
with ∥
·
∥ being the Frobenius matrix norm, defined by ∥
A
_
∥
=
tr
⁡
(
A
_
A
_
H
). If the algorithm keeps on computing with R
~
_
H
LS, it is proved in [20] that (
σ
^
(
i
)
2
) converges to zero. Under this condition, the algorithm enters into an endless loop. This is due to the fact that R
~
_
H
LS is sensitive to the noise and then it is a rough approximation of the exact covariance matrix. In order to obtain a more accurate channel covariance matrix, it is now possible to use H
^
(
i
=
1
)
LMMSE, such that
(6)
R
~
_
H
LMMSE
=
H
^
(
i
=
1
)
LMMSE
(
H
^
(
i
=
1
)
LMMSE
)
H
.(4) Fori
≥
2, the iterative estimation steps (4) and (6) are performed by using (6):
(7)
H
^
(
i
)
LMMSE
=
R
~
_
H
LMMSE
(
R
~
_
H
LMMSE
+
σ
^
(
i
-
1
)
2
I
_
)
-
1
H
^
LS
,
(8)
σ
^
(
i
)
2
=
1
M
E
{
∥
H
^
LS
-
H
^
(
i
)
LMMSE
∥
2
}
.It will be shown afterwards that the characterization of the initializationσ
^
(
i
=
0
)
2 remains the same in the presence or absence of the PU. However, it is already obvious that σ
^
(
i
=
0
)
2 must be strictly positive; otherwise, H
^
(
i
)
LMMSE
=
H
^
LS in (7). In that case, σ
^
(
i
)
2
=
0, and the algorithm enters into an endless loop.(5) While|
σ
^
(
i
)
2
-
σ
^
(
i
-
1
)
2
|
>
e
σ, go back to Step 4 with i
←
i
+
1; otherwise, go to Step 6.(6) End of the algorithm. We notei
f to be the index of the last iteration.It is proved in [20] that this algorithm converges if the initialization value of the noise variance is chosen such that σ
^
(
i
=
0
)
2
≫
M
2, where M
2 is the second-order moment of the received signal U. Moreover, the algorithm converges to limits (
H
^
(
i
f
)
,
σ
^
(
i
f
)
2
) that are close to the exact values (
H
,
σ
2
). From (4) and (7) we can deduce the complexity of the algorithm. The LMMSE channel estimation requires M
3 scalar multiplications for the matrix inversion and multiplication and the noise variance M
2. The covariance matrices estimation also requires M
2 operations. Finally, we then evaluate the complexity of the proposed algorithm by 𝒪
(
i
f
M
3
).Unlike the presented model, the next section investigates the behavior of the algorithm when the PU is absent, that is, under hypothesisℍ
0.


## 3.  Convergence of the Iterative Algorithm under Hypothesisℍ
0

The signalC
_ is now supposed to be absent, so the received signal is U
=
W. The convergence of the proposed algorithm in the case of absence of signal is going to be proved. Furthermore, it will be proved afterwards that the nonnull solution allows to turn the MMSE-based algorithm into a free band detector. To this end, the first four steps of the iterative algorithm presented in Section 2.2 are expressed under hypothesis ℍ
0.


### 3.1. Expression of the Algorithm underℍ
0

Let us consider that the receiver does not know if the signal is present or absent, so the same formalism as in Section2.2 is used, and the steps of the algorithm are recalled by considering noise only. At the beginning of the process, the LS channel estimation has been performed, H
^
LS
=
C
_
-
1
U
=
C
_
-
1
W. The following steps are as follows.(1) FromH
^
LS, the channel covariance matrix is estimated by
(9)
R
~
_
H
LS
=
H
^
LS
(
H
^
LS
)
H
=
W
W
H
.Additionally, a stopping criterione
σ and an initialization σ
^
(
i
=
0
)
2 are set.(2) At iterationi
=
1 of the algorithm, the LMMSE channel estimation H
^
(
i
=
1
)
LMMSE is performed by using R
~
_
H
LS:
(10)
H
^
(
i
=
1
)
LMMSE
=
R
_
H
LS
(
R
_
H
LS
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
H
^
LS
.(3) The MMSE noise variance estimationσ
^
(
i
=
1
)
2 is performed with H
^
=
H
^
(
i
=
1
)
LMMSE:
(11)
σ
^
(
i
=
1
)
2
=
1
M
E
{
∥
H
^
LS
-
H
^
(
i
=
1
)
LMMSE
∥
2
}
,
and a new covariance matrix is computed by
(12)
R
~
_
H
LMMSE
=
H
^
(
i
=
1
)
LMMSE
(
H
^
(
i
=
1
)
LMMSE
)
H
.Indeed, it is proved in the Appendix that if the algorithm keeps on computing withR
~
_
H
LS
=
W
W
H, then the sequence σ
^
(
i
)
2 necessarily converges to zero. When R
~
_
H
LS is used, and in spite of its inputs being different, the algorithm has exactly the same response whatever the hypothesis, ℍ
0 or ℍ
1.(4) Then, fori
≥
2, iteratively perform the channel and the noise variance estimation:
(13)
H
^
(
i
)
LMMSE
=
R
~
_
H
LMMSE
(
R
~
_
H
LMMSE
+
σ
^
(
i
-
1
)
2
I
_
)
-
1
H
^
LS
,
(14)
σ
^
(
i
)
2
=
1
M
E
{
∥
H
^
LS
-
H
^
(
i
)
LMMSE
∥
2
}
.From these first four steps of the algorithm, it is now possible to prove that the algorithm converges to a nonnull solution underℍ
0.


### 3.2. Scalar Expression of the Sequence(
σ
^
(
i
)
2
) under ℍ
0

The convergence of the algorithm is now going to be proved, and its limit characterized. To this end, we will first obtain a scalar expression of the sequence(
σ
^
(
i
)
2
). We use the Hermitian property of R
~
_
H
LS
=
(
R
~
_
H
LS
)
H, and we develop (12) with (10) to get
(15)
R
~
_
H
LMMSE
=
H
^
(
i
=
1
)
LMMSE
(
H
^
(
i
=
1
)
LMMSE
)
H
=
R
~
_
H
LS
(
R
~
_
H
LS
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
H
^
LS
×
(
R
~
_
H
LS
(
R
~
_
H
LS
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
H
^
LS
)
H
=
R
~
_
H
LS
(
R
~
_
H
LS
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
H
^
LS
(
H
^
LS
)
H
×
(
R
~
_
H
LS
(
R
~
_
H
LS
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
)
.Let us assume thatM is large enough to justify the approximation tr
⁡
(
W
W
H
)
=
tr
⁡
(
σ
2
I
_
). Since the estimation of the noise variance is calculated by means of the trace in (14), we make the assumption that as a first approximation R
~
_
H
LS
≈
σ
2
I
_, and then it is possible to replace R
_
H
LMMSE by
(16)
R
~
_
H
LMMSE
=
σ
2
I
_
(
σ
2
I
_
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
R
~
_
H
LS
(
σ
2
I
_
(
σ
2
I
_
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
)
=
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
I
_
in (14). Thus, by reinjecting (16) in (13) and (14), it yields
(17)
σ
^
(
i
+
1
)
2
=
1
M
E
{
∥
H
^
LS
-
H
^
(
i
+
1
)
LMMSE
∥
2
}
=
1
M
E
{
∥
×
(
R
~
_
H
LMMSE
+
σ
^
(
i
)
2
I
_
)
-
1
H
^
LS
-
R
~
_
H
LMMSE
m
h
h
h
h
h
l
l
×
(
R
~
_
H
LMMSE
+
σ
^
(
i
)
2
I
_
)
-
1
H
^
LS
∥
2
}
=
1
M
E
{
×
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
I
_
+
σ
^
(
i
)
2
I
_
)
-
1
W
∥
2
∥
h
h
h
h
h
×
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
I
_
+
σ
^
(
i
)
2
I
_
)
-
1
W
-
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
I
_
m
h
h
h
h
h
h
l
l
×
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
I
_
+
σ
^
(
i
)
2
I
_
)
-
1
W
∥
2
}
=
1
M
E
{
+
σ
^
(
i
)
2
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
)
I
_
)
-
1
)
W
(
σ
^
(
i
)
2
I
_
(
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
∥
2
∥
+
σ
^
(
i
)
2
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
)
I
_
)
-
1
)
W
(
σ
^
(
i
)
2
I
_
(
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
∥
2
(
+
σ
^
(
i
)
2
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
)
I
_
)
-
1
σ
^
(
i
)
2
I
_
(
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
m
m
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
+
σ
^
(
i
)
2
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
)
I
_
)
-
1
)
W
(
σ
^
(
i
)
2
I
_
(
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
∥
2
}
=
σ
2
σ
^
(
i
)
4
(
σ
2
+
σ
^
(
i
=
0
)
2
)
4
(
σ
6
+
σ
^
(
i
)
2
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
)
2
.For a better readability, we note the following mathematical developments:(18)
A
=
σ
2
+
σ
^
(
i
=
0
)
2
.


### 3.3. Convergence of the Sequence(
σ
^
(
i
)
2
) to a Nonnull Solution

One can observe that the sequence(
σ
^
(
i
+
1
)
2
) is built from a function f
s
1 such that if we note x
=
σ
^
(
i
)
2, we have
(19)
f
s
1
(
x
)
=
σ
2
A
4
x
2
(
σ
6
+
A
2
x
)
2
.The sequence converges iff
s
1 has at least one fixed point. Zero is an obvious fixed point, but it has been proved in the Appendix that the algorithm enters into an endless loop if (
σ
^
(
i
)
2
) converges to zero. We then solve the equation f
s
1
(
x
)
=
x to find the other fixed points:
(20)
f
s
1
(
x
)
=
x
⟺
σ
2
A
4
x
2
(
σ
6
+
A
2
x
)
2
=
x
⟺
σ
2
A
4
x
2
=
x
(
σ
6
+
A
2
x
)
2
.Since we exclude zero as a solution, the previous expressions can be simplified byx, and the problem amounts to look for real roots of the polynomial A
4
x
2
+
x
(
2
A
2
σ
6
-
σ
2
A
4
)
+
σ
12. Since it is a second order polynomial, in order to find real solutions, the first condition on the initialization σ
^
(
i
=
0
)
2 is to obtain the discriminant Δ
=
(
2
A
2
σ
6
-
σ
2
A
4
)
2
-
4
A
4
σ
12 positive; that is,
(21)
Δ
≥
0
⟺
(
2
A
2
σ
6
-
σ
2
A
4
)
2
≥
4
A
4
σ
12
⟺
A
2
≥
4
σ
4
⟺
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
≥
4
σ
4
⟺
σ
^
(
i
=
0
)
2
≥
3
σ
2
.Asσ
2 is absolutely unknown, one can find a stronger condition on σ
^
(
i
=
0
)
2, conditionally to Δ
>
0. We then find the roots r
s
+ and r
s
- of the polynomial under the condition Δ
>
0:
(22)
r
s
-
+
=
(
σ
2
A
4
-
2
A
2
σ
6
)
±
(
2
A
2
σ
6
-
σ
2
A
4
)
2
-
4
A
4
σ
12
2
A
4
⟺
r
s
-
+
=
(
σ
2
A
2
-
2
σ
6
)
±
σ
4
A
4
-
4
σ
8
A
2
2
A
2
.If we notice that whenσ
^
(
i
=
0
)
2 tends to +
∞, then A
=
σ
2
+
σ
^
(
i
=
0
)
2 also tends to +
∞, we get
(23)
lim
⁡
A
→
∞
r
s
+
=
σ
2
A
2
+
σ
2
A
2
2
A
2
=
σ
2
,
lim
⁡
A
→
∞
r
s
-
=
σ
2
A
2
-
σ
2
A
2
2
A
2
=
0
.It can be seen that by choosing a sufficiently large initialization valueσ
^
(
i
=
0
)
2, the sequence (
σ
^
(
i
)
2
) converges to a value as close as possible to the exact value of the noise variance σ
2. This characterization of the initialization value σ
^
(
i
=
0
)
2 perfectly tallies with the one made for the sufficient condition in [20]; that is, σ
^
(
i
=
0
)
2
≫
M
2. Moreover, it will be further shown that this condition allows to differentiate ℍ
0 from ℍ
1. Thus, choosing σ
^
(
i
=
0
)
2 with a large value is the condition for the algorithm to converge to a nonnull solution for both hypotheses ℍ
0 and ℍ
1. Besides that, since it converges, the stopping criterion |
σ
^
(
i
)
2
-
σ
^
(
i
-
1
)
2
|
<
e
σ can also be the same under ℍ
0. Finally, the MMSE-based algorithm can be used as a free band detector.Figure3 displays the function f
s
1 for different values of (
σ
^
(
i
=
0
)
2
), compared with y
=
x and for a fixed value σ
2
=
1. By comparing the curves of f
s
1 for different initializations values, we verify that the larger the value of σ
^
(
i
=
0
)
2, the closer σ
^
(
i
f
)
2 to the real value of σ
2.Figure 3
Aspect off
s
1 for different values of σ
(
0
)
2, σ
2
=
1 compared with y
=
x.


## 3.1. Expression of the Algorithm underℍ
0

Let us consider that the receiver does not know if the signal is present or absent, so the same formalism as in Section2.2 is used, and the steps of the algorithm are recalled by considering noise only. At the beginning of the process, the LS channel estimation has been performed, H
^
LS
=
C
_
-
1
U
=
C
_
-
1
W. The following steps are as follows.(1) FromH
^
LS, the channel covariance matrix is estimated by
(9)
R
~
_
H
LS
=
H
^
LS
(
H
^
LS
)
H
=
W
W
H
.Additionally, a stopping criterione
σ and an initialization σ
^
(
i
=
0
)
2 are set.(2) At iterationi
=
1 of the algorithm, the LMMSE channel estimation H
^
(
i
=
1
)
LMMSE is performed by using R
~
_
H
LS:
(10)
H
^
(
i
=
1
)
LMMSE
=
R
_
H
LS
(
R
_
H
LS
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
H
^
LS
.(3) The MMSE noise variance estimationσ
^
(
i
=
1
)
2 is performed with H
^
=
H
^
(
i
=
1
)
LMMSE:
(11)
σ
^
(
i
=
1
)
2
=
1
M
E
{
∥
H
^
LS
-
H
^
(
i
=
1
)
LMMSE
∥
2
}
,
and a new covariance matrix is computed by
(12)
R
~
_
H
LMMSE
=
H
^
(
i
=
1
)
LMMSE
(
H
^
(
i
=
1
)
LMMSE
)
H
.Indeed, it is proved in the Appendix that if the algorithm keeps on computing withR
~
_
H
LS
=
W
W
H, then the sequence σ
^
(
i
)
2 necessarily converges to zero. When R
~
_
H
LS is used, and in spite of its inputs being different, the algorithm has exactly the same response whatever the hypothesis, ℍ
0 or ℍ
1.(4) Then, fori
≥
2, iteratively perform the channel and the noise variance estimation:
(13)
H
^
(
i
)
LMMSE
=
R
~
_
H
LMMSE
(
R
~
_
H
LMMSE
+
σ
^
(
i
-
1
)
2
I
_
)
-
1
H
^
LS
,
(14)
σ
^
(
i
)
2
=
1
M
E
{
∥
H
^
LS
-
H
^
(
i
)
LMMSE
∥
2
}
.From these first four steps of the algorithm, it is now possible to prove that the algorithm converges to a nonnull solution underℍ
0.


## 3.2. Scalar Expression of the Sequence(
σ
^
(
i
)
2
) under ℍ
0

The convergence of the algorithm is now going to be proved, and its limit characterized. To this end, we will first obtain a scalar expression of the sequence(
σ
^
(
i
)
2
). We use the Hermitian property of R
~
_
H
LS
=
(
R
~
_
H
LS
)
H, and we develop (12) with (10) to get
(15)
R
~
_
H
LMMSE
=
H
^
(
i
=
1
)
LMMSE
(
H
^
(
i
=
1
)
LMMSE
)
H
=
R
~
_
H
LS
(
R
~
_
H
LS
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
H
^
LS
×
(
R
~
_
H
LS
(
R
~
_
H
LS
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
H
^
LS
)
H
=
R
~
_
H
LS
(
R
~
_
H
LS
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
H
^
LS
(
H
^
LS
)
H
×
(
R
~
_
H
LS
(
R
~
_
H
LS
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
)
.Let us assume thatM is large enough to justify the approximation tr
⁡
(
W
W
H
)
=
tr
⁡
(
σ
2
I
_
). Since the estimation of the noise variance is calculated by means of the trace in (14), we make the assumption that as a first approximation R
~
_
H
LS
≈
σ
2
I
_, and then it is possible to replace R
_
H
LMMSE by
(16)
R
~
_
H
LMMSE
=
σ
2
I
_
(
σ
2
I
_
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
R
~
_
H
LS
(
σ
2
I
_
(
σ
2
I
_
+
σ
^
(
i
=
0
)
2
I
_
)
-
1
)
=
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
I
_
in (14). Thus, by reinjecting (16) in (13) and (14), it yields
(17)
σ
^
(
i
+
1
)
2
=
1
M
E
{
∥
H
^
LS
-
H
^
(
i
+
1
)
LMMSE
∥
2
}
=
1
M
E
{
∥
×
(
R
~
_
H
LMMSE
+
σ
^
(
i
)
2
I
_
)
-
1
H
^
LS
-
R
~
_
H
LMMSE
m
h
h
h
h
h
l
l
×
(
R
~
_
H
LMMSE
+
σ
^
(
i
)
2
I
_
)
-
1
H
^
LS
∥
2
}
=
1
M
E
{
×
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
I
_
+
σ
^
(
i
)
2
I
_
)
-
1
W
∥
2
∥
h
h
h
h
h
×
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
I
_
+
σ
^
(
i
)
2
I
_
)
-
1
W
-
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
I
_
m
h
h
h
h
h
h
l
l
×
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
I
_
+
σ
^
(
i
)
2
I
_
)
-
1
W
∥
2
}
=
1
M
E
{
+
σ
^
(
i
)
2
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
)
I
_
)
-
1
)
W
(
σ
^
(
i
)
2
I
_
(
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
∥
2
∥
+
σ
^
(
i
)
2
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
)
I
_
)
-
1
)
W
(
σ
^
(
i
)
2
I
_
(
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
∥
2
(
+
σ
^
(
i
)
2
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
)
I
_
)
-
1
σ
^
(
i
)
2
I
_
(
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
m
m
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
+
σ
^
(
i
)
2
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
)
I
_
)
-
1
)
W
(
σ
^
(
i
)
2
I
_
(
(
σ
6
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
∥
2
}
=
σ
2
σ
^
(
i
)
4
(
σ
2
+
σ
^
(
i
=
0
)
2
)
4
(
σ
6
+
σ
^
(
i
)
2
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
)
2
.For a better readability, we note the following mathematical developments:(18)
A
=
σ
2
+
σ
^
(
i
=
0
)
2
.


## 3.3. Convergence of the Sequence(
σ
^
(
i
)
2
) to a Nonnull Solution

One can observe that the sequence(
σ
^
(
i
+
1
)
2
) is built from a function f
s
1 such that if we note x
=
σ
^
(
i
)
2, we have
(19)
f
s
1
(
x
)
=
σ
2
A
4
x
2
(
σ
6
+
A
2
x
)
2
.The sequence converges iff
s
1 has at least one fixed point. Zero is an obvious fixed point, but it has been proved in the Appendix that the algorithm enters into an endless loop if (
σ
^
(
i
)
2
) converges to zero. We then solve the equation f
s
1
(
x
)
=
x to find the other fixed points:
(20)
f
s
1
(
x
)
=
x
⟺
σ
2
A
4
x
2
(
σ
6
+
A
2
x
)
2
=
x
⟺
σ
2
A
4
x
2
=
x
(
σ
6
+
A
2
x
)
2
.Since we exclude zero as a solution, the previous expressions can be simplified byx, and the problem amounts to look for real roots of the polynomial A
4
x
2
+
x
(
2
A
2
σ
6
-
σ
2
A
4
)
+
σ
12. Since it is a second order polynomial, in order to find real solutions, the first condition on the initialization σ
^
(
i
=
0
)
2 is to obtain the discriminant Δ
=
(
2
A
2
σ
6
-
σ
2
A
4
)
2
-
4
A
4
σ
12 positive; that is,
(21)
Δ
≥
0
⟺
(
2
A
2
σ
6
-
σ
2
A
4
)
2
≥
4
A
4
σ
12
⟺
A
2
≥
4
σ
4
⟺
(
σ
2
+
σ
^
(
i
=
0
)
2
)
2
≥
4
σ
4
⟺
σ
^
(
i
=
0
)
2
≥
3
σ
2
.Asσ
2 is absolutely unknown, one can find a stronger condition on σ
^
(
i
=
0
)
2, conditionally to Δ
>
0. We then find the roots r
s
+ and r
s
- of the polynomial under the condition Δ
>
0:
(22)
r
s
-
+
=
(
σ
2
A
4
-
2
A
2
σ
6
)
±
(
2
A
2
σ
6
-
σ
2
A
4
)
2
-
4
A
4
σ
12
2
A
4
⟺
r
s
-
+
=
(
σ
2
A
2
-
2
σ
6
)
±
σ
4
A
4
-
4
σ
8
A
2
2
A
2
.If we notice that whenσ
^
(
i
=
0
)
2 tends to +
∞, then A
=
σ
2
+
σ
^
(
i
=
0
)
2 also tends to +
∞, we get
(23)
lim
⁡
A
→
∞
r
s
+
=
σ
2
A
2
+
σ
2
A
2
2
A
2
=
σ
2
,
lim
⁡
A
→
∞
r
s
-
=
σ
2
A
2
-
σ
2
A
2
2
A
2
=
0
.It can be seen that by choosing a sufficiently large initialization valueσ
^
(
i
=
0
)
2, the sequence (
σ
^
(
i
)
2
) converges to a value as close as possible to the exact value of the noise variance σ
2. This characterization of the initialization value σ
^
(
i
=
0
)
2 perfectly tallies with the one made for the sufficient condition in [20]; that is, σ
^
(
i
=
0
)
2
≫
M
2. Moreover, it will be further shown that this condition allows to differentiate ℍ
0 from ℍ
1. Thus, choosing σ
^
(
i
=
0
)
2 with a large value is the condition for the algorithm to converge to a nonnull solution for both hypotheses ℍ
0 and ℍ
1. Besides that, since it converges, the stopping criterion |
σ
^
(
i
)
2
-
σ
^
(
i
-
1
)
2
|
<
e
σ can also be the same under ℍ
0. Finally, the MMSE-based algorithm can be used as a free band detector.Figure3 displays the function f
s
1 for different values of (
σ
^
(
i
=
0
)
2
), compared with y
=
x and for a fixed value σ
2
=
1. By comparing the curves of f
s
1 for different initializations values, we verify that the larger the value of σ
^
(
i
=
0
)
2, the closer σ
^
(
i
f
)
2 to the real value of σ
2.Figure 3
Aspect off
s
1 for different values of σ
(
0
)
2, σ
2
=
1 compared with y
=
x.


## 4. Proposed Detector

### 4.1. Decision Rule for the Proposed Detector

In this section, a decision rule for the detector is proposed. To this end, whateverℍ
0 or ℍ
1, it is supposed that the algorithm has converged; that is, the condition |
σ
^
(
i
)
2
-
σ
^
(
i
-
1
)
2
|
<
e
σ is reached and then i
=
i
f. The second-order moment M
2
=
(
1
/
M
)
∑
m
=
0
M
-
1
‍
|
U
m
|
2 of the received signal is expressed under the hypotheses ℍ
0 and ℍ
1:
(24)
M
2
=
{
1
M
∑
m
=
0
M
-
1
|
W
m
|
2
,
if
ℍ
0
1
M
∑
m
=
0
M
-
1
|
C
m
H
m
+
W
m
|
2
,
if
ℍ
1
.
The second-order moment M
2 is the decision metric used for the energy detector. Here, a different metric noted ℳ is proposed and defined by
(25)
ℳ
=
|
M
2
-
σ
^
2
|
,
where σ
^
2
=
σ
^
(
i
f
)
2 is the noise variance estimation performed by the proposed iterative algorithm. From (24), the metric (25) is rewritten according to the hypotheses ℍ
0 and ℍ
1:
(26)
ℳ
=
{
|
1
M
∑
m
=
0
M
-
1
|
W
m
|
2
-
σ
^
2
|
,
under
ℍ
0
|
1
M
∑
m
=
0
M
-
1
|
C
m
H
m
+
W
m
|
2
-
σ
^
2
|
,
under
ℍ
1
.
By fixing a threshold ς, the decision criterion is now
(27)
ℍ
0
,
if
ℳ
<
ς
,
ℍ
1
,
else
.
The detection and false alarm probabilities are defined by
(28)
P
d
=
P
(
ℳ
>
ς
∣
ℍ
1
)
,
P
fa
=
P
(
ℳ
>
ς
∣
ℍ
0
)
.The detection probabilityP
d is the probability to decide ℍ
1 while the PU is present, and the false alarm probability P
fa is the probability to decide ℍ
1 while the PU is absent. As mentioned in [6, 25], the sensibility of the detector (the expected value of P
fa and P
d) depends on the application. In a cognitive radio context, the SU has to minimize the interference with the PU, so the probability of detection has to be maximized, whereas if the false alarm probability is not optimized, it only implies that the SU misses white spaces. On the contrary, in a radar application, a false alarm could have serious consequences, especially in a military context.


### 4.2. Expression of the Proposed Detector

By taking into account the previous decision rule, it is possible to extend the practical algorithm proposed in the scenario of the joint estimation of the SNR and the channel for free band detections, as it is summed up in Algorithm1.Algorithm 1:Application of the MMSE-based algorithm to free band detection.
begin
Initialization:R
_
~
H
LS, e
σ
>
0, σ
^
(
i
=
0
)
2 and ς;
i
←
1;
while |
σ
^
(
i
)
2
-
σ
^
(
i
-
1
)
2
|
>
e
σ do
if  i
=
1 then
Perform LMMSE channel estimation;
Perform the noise variance estimation;
Calculate the matrix R
_
~
H
LMMSE;
else
Perform an LMMSE channel estimation with R
_
~
H
LMMSE;
Perform the noise variance estimation;
end
i
←
i
+
1;
end
Calculate the metricℳ;
if ℳ
<
ς then
return  ℍ
0;
else
return  ℍ
1;
end
endIt can be seen that the structure of Algorithm1 is similar to the one of [20] and summarized in Section 2.2, but with a detection part. Thus, compared to the methods of the literature, the proposed one not only returns the decision ℍ
0 and ℍ
1 but also provides(i)
the noise variance estimation, ifℍ
0;
(ii)
the channel and SNR estimations, ifℍ
1.An a priori qualitative analysis of the detector can be done. Indeed, from (26), one can deduce that by supposing a good estimation of σ
^
2, ℳ tends to a value close to zero under ℍ
0 and to a value close to P
s under ℍ
1. By supposing a normalized signal power, one can suppose that choosing a value ς between zero and one allows getting a viable detector. Concerning the value of the threshold e
σ, since it ensures the convergence of the algorithm, it has no effect on the detector performance. This property will be shown by simulations afterward.In the context of cognitive radio, the SUs have to target a given detection probability, notedP
d
t. Thus, according to the Neyman-Pearson criterion [26], the best value of the threshold ς can be analytically derived (when it is possible) by solving P
(
ℳ
>
ς
∣
ℍ
1
)
≥
P
d
t and by maximizing the likelihood ratio test (LRT)
(29)
Λ
(
x
)
=
p
(
x
∣
ℍ
1
)
p
(
x
∣
ℍ
0
)
≷
ℍ
0
ℍ
1
ς
.
To this end, the probability density functions (pdfs) of ℳ have to be expressed, which is proposed in the next section.


## 4.1. Decision Rule for the Proposed Detector

In this section, a decision rule for the detector is proposed. To this end, whateverℍ
0 or ℍ
1, it is supposed that the algorithm has converged; that is, the condition |
σ
^
(
i
)
2
-
σ
^
(
i
-
1
)
2
|
<
e
σ is reached and then i
=
i
f. The second-order moment M
2
=
(
1
/
M
)
∑
m
=
0
M
-
1
‍
|
U
m
|
2 of the received signal is expressed under the hypotheses ℍ
0 and ℍ
1:
(24)
M
2
=
{
1
M
∑
m
=
0
M
-
1
|
W
m
|
2
,
if
ℍ
0
1
M
∑
m
=
0
M
-
1
|
C
m
H
m
+
W
m
|
2
,
if
ℍ
1
.
The second-order moment M
2 is the decision metric used for the energy detector. Here, a different metric noted ℳ is proposed and defined by
(25)
ℳ
=
|
M
2
-
σ
^
2
|
,
where σ
^
2
=
σ
^
(
i
f
)
2 is the noise variance estimation performed by the proposed iterative algorithm. From (24), the metric (25) is rewritten according to the hypotheses ℍ
0 and ℍ
1:
(26)
ℳ
=
{
|
1
M
∑
m
=
0
M
-
1
|
W
m
|
2
-
σ
^
2
|
,
under
ℍ
0
|
1
M
∑
m
=
0
M
-
1
|
C
m
H
m
+
W
m
|
2
-
σ
^
2
|
,
under
ℍ
1
.
By fixing a threshold ς, the decision criterion is now
(27)
ℍ
0
,
if
ℳ
<
ς
,
ℍ
1
,
else
.
The detection and false alarm probabilities are defined by
(28)
P
d
=
P
(
ℳ
>
ς
∣
ℍ
1
)
,
P
fa
=
P
(
ℳ
>
ς
∣
ℍ
0
)
.The detection probabilityP
d is the probability to decide ℍ
1 while the PU is present, and the false alarm probability P
fa is the probability to decide ℍ
1 while the PU is absent. As mentioned in [6, 25], the sensibility of the detector (the expected value of P
fa and P
d) depends on the application. In a cognitive radio context, the SU has to minimize the interference with the PU, so the probability of detection has to be maximized, whereas if the false alarm probability is not optimized, it only implies that the SU misses white spaces. On the contrary, in a radar application, a false alarm could have serious consequences, especially in a military context.


## 4.2. Expression of the Proposed Detector

By taking into account the previous decision rule, it is possible to extend the practical algorithm proposed in the scenario of the joint estimation of the SNR and the channel for free band detections, as it is summed up in Algorithm1.Algorithm 1:Application of the MMSE-based algorithm to free band detection.
begin
Initialization:R
_
~
H
LS, e
σ
>
0, σ
^
(
i
=
0
)
2 and ς;
i
←
1;
while |
σ
^
(
i
)
2
-
σ
^
(
i
-
1
)
2
|
>
e
σ do
if  i
=
1 then
Perform LMMSE channel estimation;
Perform the noise variance estimation;
Calculate the matrix R
_
~
H
LMMSE;
else
Perform an LMMSE channel estimation with R
_
~
H
LMMSE;
Perform the noise variance estimation;
end
i
←
i
+
1;
end
Calculate the metricℳ;
if ℳ
<
ς then
return  ℍ
0;
else
return  ℍ
1;
end
endIt can be seen that the structure of Algorithm1 is similar to the one of [20] and summarized in Section 2.2, but with a detection part. Thus, compared to the methods of the literature, the proposed one not only returns the decision ℍ
0 and ℍ
1 but also provides(i)
the noise variance estimation, ifℍ
0;
(ii)
the channel and SNR estimations, ifℍ
1.An a priori qualitative analysis of the detector can be done. Indeed, from (26), one can deduce that by supposing a good estimation of σ
^
2, ℳ tends to a value close to zero under ℍ
0 and to a value close to P
s under ℍ
1. By supposing a normalized signal power, one can suppose that choosing a value ς between zero and one allows getting a viable detector. Concerning the value of the threshold e
σ, since it ensures the convergence of the algorithm, it has no effect on the detector performance. This property will be shown by simulations afterward.In the context of cognitive radio, the SUs have to target a given detection probability, notedP
d
t. Thus, according to the Neyman-Pearson criterion [26], the best value of the threshold ς can be analytically derived (when it is possible) by solving P
(
ℳ
>
ς
∣
ℍ
1
)
≥
P
d
t and by maximizing the likelihood ratio test (LRT)
(29)
Λ
(
x
)
=
p
(
x
∣
ℍ
1
)
p
(
x
∣
ℍ
0
)
≷
ℍ
0
ℍ
1
ς
.
To this end, the probability density functions (pdfs) of ℳ have to be expressed, which is proposed in the next section.


## 5. Detection and False Alarm Probabilities

### 5.1.  Probability Density Function ofℳ under ℍ
1

Under the hypothesisℍ
1, since it is proved in [20] that the noise variance estimation is very accurate, it is reasonable to suppose that the noise variance estimation is good enough to consider that σ
^
2
≈
(
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2, so the contribution of C
m
H
m is prevailing in ℳ (26) so that
(30)
ℳ
=
|
1
M
∑
m
=
0
M
-
1
(
|
C
m
H
m
+
W
m
|
2
)
-
σ
^
2
|
=
|
1
M
∑
m
=
0
M
-
1
(
|
C
m
H
m
|
2
+
|
W
m
|
2
+
CF
m
)
-
σ
^
2
|
,
where ∀
m
=
0
,
…
,
M
-
1, CF
m are the cross-factors (
C
m
H
m
W
m
*
) + (
C
m
H
m
W
m
*
)
*, whose means (for a sufficiently large value of M) are equal to zero, since H
m and W
m are zero-mean uncorrelated Gaussian processes. The development of (30) then simply yields
(31)
ℳ
=
|
1
M
∑
m
=
0
M
-
1
|
C
m
H
m
|
2
+
|
W
m
|
2
-
σ
^
2
|
=
1
M
∑
m
=
0
M
-
1
|
H
m
C
m
|
2
.The result in (31) obtained with the approximation σ
^
2
≈
(
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2 may be debated, since it has been proved in [20] that the noise estimation under hypothesis ℍ
1 is slightly biased. However, it will be shown in Section 6 that this approximation is accurate for low values of σ
2. From the channel frequency response expression (2) and remembering that C
m
C
m
*
=
1, the metric (31) can be rewritten by
(32)
ℳ
=
1
M
∑
m
=
0
M
-
1
|
∑
l
=
0
L
-
1
h
l
exp
⁡
(
-
2
j
π
m
β
l
M
)
C
m
|
2
=
1
M
∑
m
=
0
M
-
1
(
∑
l
1
=
0
L
-
1
h
l
1
exp
⁡
(
-
2
j
π
m
β
l
1
M
)
C
m
)
hhhhhhhh
×
(
∑
l
2
=
0
L
-
1
h
l
2
exp
⁡
(
-
2
j
π
m
β
l
2
M
)
C
m
)
*
=
∑
l
=
0
L
-
1
|
h
l
|
2
+
1
M
∑
m
=
0
M
-
1
∑
l
1
=
0
L
-
1
∑
l
2
≠
l
1
l
2
=
0
L
-
1
h
l
1
h
l
2
*
exp
⁡
(
-
2
j
π
m
(
β
l
1
-
β
l
2
)
M
)
.According to the Rayleigh distributed WSSUS channel model, whateverl
=
0
,
…
,
L
-
1, the gains h
l are uncorrelated zero-mean Gaussian processes. For a large enough value M, let us assume that the mean of the cross-factors of the right side in (32) are equal to zero. Finally, the metric ℳ is simply written as follows:
(33)
ℳ
=
∑
l
=
0
L
-
1
|
h
l
|
2
,
under
ℍ
1
.
ℳ then follows a chi-square distribution with 2
L degrees of freedom. The probability density function (pdf) noted p
ℳ
(
x
) of the decision statistic under ℍ
1 is then expressed by
(34)
p
ℳ
(
x
)
=
1
2
L
P
s
L
Γ
(
L
)
x
L
-
1
exp
⁡
(
-
x
2
P
s
)
,
under
ℍ
1
,
where Γ
(
·
) is the gamma function [27].


### 5.2. Probability Density Function ofℳ under ℍ
0

The theoretical probability density function (pdf) expression of the metric under the hypothesisℍ
0
(35)
ℳ
=
|
1
M
∑
m
=
0
M
-
1
|
W
m
|
2
-
σ
^
(
i
)
2
|
is now developed. To this end, let us assume that the initialization value of the algorithm is chosen large enough to state σ
^
2
≈
σ
2, in accordance with the previously formulated hypotheses in Section 3.3. Whatever m
=
0
,
…
,
M
-
1, each sample W
m follows a zero-mean Gaussian process with variance σ
2; |
W
m
|
2 has a chi-square distribution χ
2
2 with a degree of liberty equal to 2:
(36)
χ
2
2
(
x
)
=
1
σ
2
e
-
x
/
σ
2
.The mean and the variance of this distribution are equal toσ
2 and σ
4, respectively. In an OFDM context, we reasonably suppose that M is large enough (e.g., M
>
100) to consider that from the central limit theorem (
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2 follows a normal distribution 𝒩
~
(
σ
2
,
σ
4
/
M
), and then (
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2
-
σ
^
2 follows a centered normal distribution 𝒩
~
(
0
,
σ
4
/
M
). Consequently, the metric ℳ
=
|
(
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2
-
σ
^
(
i
)
2
| has a chi distribution χ
1 with one degree of liberty:
(37)
p
ℳ
(
x
)
=
2
Γ
(
1
/
2
)
σ
4
/
M
exp
⁡
(
-
1
2
(
x
σ
4
/
M
)
2
)
,
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
l
l
l
l
under
ℍ
0
.As a conclusion, the probability density functions of the metricℳ, according to ℍ
0 and ℍ
1, are given by
(38)
p
ℳ
(
x
)
=
{
2
Γ
(
1
/
2
)
σ
4
/
M
exp
⁡
(
-
1
2
(
x
σ
4
/
M
)
2
)
,
under
ℍ
0
1
2
L
P
s
L
Γ
(
L
)
x
L
-
1
exp
⁡
(
-
x
2
P
s
)
,
under
ℍ
1
.


### 5.3. Analytical Expressions ofP
d and P
fa

The detection and false alarm probabilitiesP
d and P
fa are obtained by integrating (38) between the fixed threshold ς and +
∞. For the calculation of P
d, the solution is derived in [28, 29]:
(39)
P
d
=
P
(
ℳ
>
ς
∣
ℍ
1
)
=
∫
ς
+
∞
x
L
-
1
2
L
P
s
L
Γ
(
L
)
exp
⁡
(
-
x
2
P
s
)
d
x
=
Γ
(
L
,
ς
/
2
P
s
)
Γ
(
L
)
,
where Γ
(
·
,
·
) is the incomplete gamma function [27]. In the case ℍ
0, we have
(40)
P
fa
=
∫
ς
+
∞
p
ℳ
(
x
)
d
x
=
∫
ς
+
∞
2
Γ
(
1
/
2
)
σ
4
/
M
e
-
(
1
/
2
)
(
x
/
σ
4
/
M
)
2
d
x
.By using the variable changeX
=
x
/
2
σ
4
/
M and knowing that Γ
(
1
/
2
)
=
π, one can recognize the complementary error function e
r
f
c
(
x
)
=
1
-
e
r
f
(
x
):
(41)
P
fa
=
∫
ς
/
2
σ
4
/
M
+
∞
2
π
e
-
X
2
d
X
=
e
r
f
c
(
ς
M
2
σ
2
)
.Since the incomplete gamma function is not directly invertible in (39), it is not possible to derive an analytical expression of the threshold ς in function of the targeted detection probability P
d
t. However, an approximation by means of a computer calculation or a series expansion of the invert of (39) or a simple characterization of ς by simulations can be done. We will consider this third solution thereafter. Furthermore, the next section aims to characterize the performance of the proposed detection algorithm and the validity of the proposed analytical developments.


## 5.1.  Probability Density Function ofℳ under ℍ
1

Under the hypothesisℍ
1, since it is proved in [20] that the noise variance estimation is very accurate, it is reasonable to suppose that the noise variance estimation is good enough to consider that σ
^
2
≈
(
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2, so the contribution of C
m
H
m is prevailing in ℳ (26) so that
(30)
ℳ
=
|
1
M
∑
m
=
0
M
-
1
(
|
C
m
H
m
+
W
m
|
2
)
-
σ
^
2
|
=
|
1
M
∑
m
=
0
M
-
1
(
|
C
m
H
m
|
2
+
|
W
m
|
2
+
CF
m
)
-
σ
^
2
|
,
where ∀
m
=
0
,
…
,
M
-
1, CF
m are the cross-factors (
C
m
H
m
W
m
*
) + (
C
m
H
m
W
m
*
)
*, whose means (for a sufficiently large value of M) are equal to zero, since H
m and W
m are zero-mean uncorrelated Gaussian processes. The development of (30) then simply yields
(31)
ℳ
=
|
1
M
∑
m
=
0
M
-
1
|
C
m
H
m
|
2
+
|
W
m
|
2
-
σ
^
2
|
=
1
M
∑
m
=
0
M
-
1
|
H
m
C
m
|
2
.The result in (31) obtained with the approximation σ
^
2
≈
(
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2 may be debated, since it has been proved in [20] that the noise estimation under hypothesis ℍ
1 is slightly biased. However, it will be shown in Section 6 that this approximation is accurate for low values of σ
2. From the channel frequency response expression (2) and remembering that C
m
C
m
*
=
1, the metric (31) can be rewritten by
(32)
ℳ
=
1
M
∑
m
=
0
M
-
1
|
∑
l
=
0
L
-
1
h
l
exp
⁡
(
-
2
j
π
m
β
l
M
)
C
m
|
2
=
1
M
∑
m
=
0
M
-
1
(
∑
l
1
=
0
L
-
1
h
l
1
exp
⁡
(
-
2
j
π
m
β
l
1
M
)
C
m
)
hhhhhhhh
×
(
∑
l
2
=
0
L
-
1
h
l
2
exp
⁡
(
-
2
j
π
m
β
l
2
M
)
C
m
)
*
=
∑
l
=
0
L
-
1
|
h
l
|
2
+
1
M
∑
m
=
0
M
-
1
∑
l
1
=
0
L
-
1
∑
l
2
≠
l
1
l
2
=
0
L
-
1
h
l
1
h
l
2
*
exp
⁡
(
-
2
j
π
m
(
β
l
1
-
β
l
2
)
M
)
.According to the Rayleigh distributed WSSUS channel model, whateverl
=
0
,
…
,
L
-
1, the gains h
l are uncorrelated zero-mean Gaussian processes. For a large enough value M, let us assume that the mean of the cross-factors of the right side in (32) are equal to zero. Finally, the metric ℳ is simply written as follows:
(33)
ℳ
=
∑
l
=
0
L
-
1
|
h
l
|
2
,
under
ℍ
1
.
ℳ then follows a chi-square distribution with 2
L degrees of freedom. The probability density function (pdf) noted p
ℳ
(
x
) of the decision statistic under ℍ
1 is then expressed by
(34)
p
ℳ
(
x
)
=
1
2
L
P
s
L
Γ
(
L
)
x
L
-
1
exp
⁡
(
-
x
2
P
s
)
,
under
ℍ
1
,
where Γ
(
·
) is the gamma function [27].


## 5.2. Probability Density Function ofℳ under ℍ
0

The theoretical probability density function (pdf) expression of the metric under the hypothesisℍ
0
(35)
ℳ
=
|
1
M
∑
m
=
0
M
-
1
|
W
m
|
2
-
σ
^
(
i
)
2
|
is now developed. To this end, let us assume that the initialization value of the algorithm is chosen large enough to state σ
^
2
≈
σ
2, in accordance with the previously formulated hypotheses in Section 3.3. Whatever m
=
0
,
…
,
M
-
1, each sample W
m follows a zero-mean Gaussian process with variance σ
2; |
W
m
|
2 has a chi-square distribution χ
2
2 with a degree of liberty equal to 2:
(36)
χ
2
2
(
x
)
=
1
σ
2
e
-
x
/
σ
2
.The mean and the variance of this distribution are equal toσ
2 and σ
4, respectively. In an OFDM context, we reasonably suppose that M is large enough (e.g., M
>
100) to consider that from the central limit theorem (
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2 follows a normal distribution 𝒩
~
(
σ
2
,
σ
4
/
M
), and then (
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2
-
σ
^
2 follows a centered normal distribution 𝒩
~
(
0
,
σ
4
/
M
). Consequently, the metric ℳ
=
|
(
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2
-
σ
^
(
i
)
2
| has a chi distribution χ
1 with one degree of liberty:
(37)
p
ℳ
(
x
)
=
2
Γ
(
1
/
2
)
σ
4
/
M
exp
⁡
(
-
1
2
(
x
σ
4
/
M
)
2
)
,
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
l
l
l
l
under
ℍ
0
.As a conclusion, the probability density functions of the metricℳ, according to ℍ
0 and ℍ
1, are given by
(38)
p
ℳ
(
x
)
=
{
2
Γ
(
1
/
2
)
σ
4
/
M
exp
⁡
(
-
1
2
(
x
σ
4
/
M
)
2
)
,
under
ℍ
0
1
2
L
P
s
L
Γ
(
L
)
x
L
-
1
exp
⁡
(
-
x
2
P
s
)
,
under
ℍ
1
.


## 5.3. Analytical Expressions ofP
d and P
fa

The detection and false alarm probabilitiesP
d and P
fa are obtained by integrating (38) between the fixed threshold ς and +
∞. For the calculation of P
d, the solution is derived in [28, 29]:
(39)
P
d
=
P
(
ℳ
>
ς
∣
ℍ
1
)
=
∫
ς
+
∞
x
L
-
1
2
L
P
s
L
Γ
(
L
)
exp
⁡
(
-
x
2
P
s
)
d
x
=
Γ
(
L
,
ς
/
2
P
s
)
Γ
(
L
)
,
where Γ
(
·
,
·
) is the incomplete gamma function [27]. In the case ℍ
0, we have
(40)
P
fa
=
∫
ς
+
∞
p
ℳ
(
x
)
d
x
=
∫
ς
+
∞
2
Γ
(
1
/
2
)
σ
4
/
M
e
-
(
1
/
2
)
(
x
/
σ
4
/
M
)
2
d
x
.By using the variable changeX
=
x
/
2
σ
4
/
M and knowing that Γ
(
1
/
2
)
=
π, one can recognize the complementary error function e
r
f
c
(
x
)
=
1
-
e
r
f
(
x
):
(41)
P
fa
=
∫
ς
/
2
σ
4
/
M
+
∞
2
π
e
-
X
2
d
X
=
e
r
f
c
(
ς
M
2
σ
2
)
.Since the incomplete gamma function is not directly invertible in (39), it is not possible to derive an analytical expression of the threshold ς in function of the targeted detection probability P
d
t. However, an approximation by means of a computer calculation or a series expansion of the invert of (39) or a simple characterization of ς by simulations can be done. We will consider this third solution thereafter. Furthermore, the next section aims to characterize the performance of the proposed detection algorithm and the validity of the proposed analytical developments.


## 6. Simulations Results

### 6.1. Simulations Parameters

The signal parameters used for the simulations are based on those of the digital radio mondiale (DRM/DRM+) standard [30]. This standard designs the digital radio broadcasting over the current AM/FM bands. When it is transmitted, the signal is composed of 148 independent carriers. The symbol and the cyclic prefix durations are 14.66 ms and 5.33 ms, respectively. Although the DRM standard recommends a pilot distribution in staggered rows, we consider a block-type pilot arrangement, according to the model used in [20]. The channel used in the presence of a PU is based on the U
S
C
o
n
s
o
r
t
i
u
m model of the DRM/DRM+ standard, whose path gains are normalized. The channel parameters are summed up in Table 1.Table 1
Table of parameters of the channel model.
Channel model
Path 1
Path 2
Path 3
Path 4
Delay
0 ms
0.7 ms
1.5 ms
2.2 ms
Gain
0.7448
0.5214
0.3724
0.1862


### 6.2. Choice of the Thresholdς

Figure4 depicts the metric ℳ
=
|
M
2
-
σ
^
2
| versus the number of iterations, under the hypotheses ℍ
0 and ℍ
1. The SNR is fixed equal to 0 dB. In presence of signal, the average signal power P
s is equal to 1. The simulation is performed by means of 4000 simulation runs.Figure 4
|
M
2
-
σ
^
2
| versus the iteration number under ℍ
0 and ℍ
1, for SNR = 0 dB.It can be seen that the a priori qualitative analysis is verified. Indeed, for a sufficient number of iterations (according to the valuee
σ, as shown thereafter), ℳ converges to P
s under ℍ
1 and converges toward zero under ℍ
0. It has been noticed that it is not possible to find an exact value of ς according to P
(
ℳ
>
ς
∣
ℍ
1
)
=
P
d
t. However, it is observable on Figure 4 that the choice of the threshold is not restrictive. Indeed, choosing ς as small as expected ensures a probability P
d close to one, and, for a sufficient number of iterations, it also ensures a low value for P
fa. Nevertheless, reducing the value of e
σ increases the number of required iterations, as shown in the following. Hence, for an expected detection probability, a tradeoff between the complexity and the acceptable level of false alarm probability has to be taken into account, since each iteration requires 𝒪
(
i
f
M
3
) operations.


### 6.3. Effect of the Choice ofe
σ on the Detector Performance

It is shown in this section that the choice of the thresholde
σ value does not have any effect on the detection performance of the proposed method but only impacts the convergence speed of the algorithm. Figure 5 depicts the curves of detection and false alarm probabilities P
d and P
fa versus the SNR from −15 dB to 10 dB. In order to ensure the convergence of the algorithm, e
σ must have a low value. The subfigures (a) and (b) then depict the curves P
d and P
fa for e
σ
=
0.01 and e
σ
=
0.0001, respectively. According to the previous recommendations, the initialization σ
(
i
=
0
)
2 is equal to 40
×
M
2. We also arbitrarily fix the threshold ς
=
0.01, its effect on the detection performance being further studied. The figure is obtained thanks to 2000 simulation runs.Detection and false alarm probabilities versus SNR, for two valuese
σ and for a fixed value ς
=
0.01.
(a)
P
d and P
fa versus SNR, for e
σ
=
0.01
(b)
P
d and P
fa versus SNR, for e
σ
=
0.0001We observe that the curves ofP
d and P
fa match from Figures 5(a) to 5(b). P
fa is equal to zero or nearly for all SNR values and P
d reaches one from SNR = −5 dB. The detector can then reach the perfect one from SNR ≥ −5 dB, that is, in low SNR environment. We conclude that, assuming a value of e
σ low enough to ensure the convergence of the algorithm, this threshold does not have any effect on the detection performance of the proposed method.Figure6 displays the iterations number the algorithm needs before it stops versus the SNR from −10 to 10 dB. We consider three different values for the threshold: e
σ = 0.01, 0.001, and 0.0001. The simulations conditions remain the same.Figure 6
Means of the number of iterations needed by the algorithm to stop versus SNR (in dB), for three values of thresholde
σ.Although Figures5(a) and 5(b) display almost the same probabilities whatever the threshold e
σ, they differ from each other according to the number of iterations the algorithm requires before stopping. Indeed, remembering that we compare |
σ
^
(
i
)
2
-
σ
^
(
i
-
1
)
2
| with e
σ, the lower e
σ, the larger the number i of iterations needed to reach e
σ. However, Figure 6 shows that the maximum mean of iterations is less than 7 for SNR = −10 dB and shows the maximum mean of iterations is less than 5 for SNR = −10 dB and e
σ = 0.0001, which is a reasonable number of iterations. We conclude that the choice of e
σ has no effect on the detector efficiency, while it allows the convergence of the algorithm. Besides this result, the number of required iterations reasonably increases when e
σ and the SNR have low values. The detector then remains usable in practice under these conditions.


### 6.4. Detector Performance with Channel Uncertainty

In this part, we study the behavior of the proposed detector when a non-WSS channel is considered. To this end, we artificially correlate the different paths by inserting the gainh
0 into the other path gains. Thus, from the originally created channel impulse response [
h
0
,
…
,
h
L
-
1
] with independent paths, we build a new correlated vector [
h
0
,
…
,
h
~
l
,
…
,
h
~
L
-
1
] such that, for l
=
1
,
…
,
L
-
1, we define a correlation coefficient ρ
h by
(42)
ρ
h
=
E
{
h
0
h
~
l
*
}
σ
0
σ
~
l
,
where h
~
l
=
h
l
+
α
l
h
0, α
l being a coefficient that is calculated in function of the expected ρ
h, and σ
0
2 and σ
~
l
2 are the variances of h
0 and h
~
l, respectively. Figure 7 displays the detection probability P
d versus the SNR for the proposed detection under a channel correlation condition. Three curves are considered: the reference (ρ
h
=
0) and two correlated channels with ρ
h
=
0.1 and ρ
h
=
0.5. We observe a limited gap of 1 dB between the reference curve and the two others. We conclude that the proposed detector is robust against the channel uncertainty.Figure 7
Detection probability versus SNR for a non-WSSUS channel.


### 6.5. Receiver Operating Characteristic of the Detector

The performance of a detector is usually evaluated by means of the receiver operating characteristic (ROC) curves, depicting the detection probabilityP
d in function of the false alarm probability P
fa. The optimal detector is logically reached at the point (
P
fa
=
0
,
P
d
=
1
). The curve P
fa
=
P
d is called line of chance and corresponds to a detector that makes as much good decisions as false alarms. If the ROC curve is above the first bisector, the detector is efficient, since P
d
>
P
fa.Figure8 shows the ROC curves of the proposed detector for low SNR values (−10 dB and SNR = 0 dB). It is compared to the energy detector and the second-order moment-based MME [31]. The simulations conditions remain the same, and we fix the threshold e
σ
=
0.01. In Figure 8(a), the proposed detector is compared to the usual energy detector, whose metric ℳ is equal to the second order-moment of the received signal M
2. This metric is compared to a threshold ς
e to obtain the following decision rule:
(43)
ℍ
0
,
if
ℳ
<
ς
e
ℍ
1
,
else
.Receiver operating characteristic (ROC) curves of the proposed method compared to the energy and MME detectors.
(a)
Proposed detector compared with the energy detector
(b)
Proposed detector compared with the MME detectorIn Figure8(b), the proposed detector is also compared to the usual MME detector, whose metric ℳ is equal to the ratio of the maximum and the nonzero minimum eigenvalues of the received signal covariance matrix R
_
˘; that is, ℳ
=
λ
max
⁡
/
λ
min
⁡. The same aforementioned decision rule is used. Since a SISO system is assumed, R
_
˘ is obtained by concatenating N consecutive OFDM symbols so that U
_
N
=
[
U
1
,
…
,
U
N
] and then R
_
˘
=
U
_
N
U
_
N
T. In that way, U
_
N is equivalent to a system with N sensors. However, due to the nature of the channel, the different received OFDM symbols are correlated. In Figure 8(b), the ROC curves of MME are obtained for N
=
4, 10, and 20 symbols, and the SNR is equal to −10 dB. Each point of the curves is obtained by means of 2000 simulation runs.We observe in Figure8(a) that the proposed detector outperforms the energy detector, whatever the SNR. Indeed, as we consider the detection of a preamble transmitted over a Rayleigh channel, the power of the received signal P
S in (30) is not constant and follows a chi-square distribution. Consequently, for simulations made at a fixed SNR, the noise variance is also a varying process, which deteriorates the detector performance. For additional details about the theoretical development of the energy detection of signals with random amplitude, please refer to [28, 29]. We also may explain the performance of our detector by the fact that we use the same sensing time to compare the energy detector and the proposed algorithm, that is, only one OFDM symbol length. The 148 samples of one OFDM symbol are not enough to obtain an accurate energy detector. Figure 8(a) also confirms that the proposed detector is very efficient, since it is able to reach the perfect detector for ς
=
0.01. Indeed, for SNR = 0 dB, we observe that the ROC curve reaches the point (
P
fa
=
0
,
P
d
=
1
), as we remarked in Figures 5(a) and 5(b) for SNR ≥ −5 dB. In Figure 8(b), we observe that MME requires N
=
20 symbols to reach the performance of the proposed method, because MME is efficient for a very large size of U
_
N, and the vectors of the latter matrix are correlated. Thus, for a given performance, the complexity of MME is 𝒪
(
N
M
2
) (for the computation and the diagonalization of R
_
˘) and the one of the proposed algorithm is 𝒪
(
i
f
M
3
). Since we reasonably have N
<
i
f
M, we conclude that the iterative method is more complex than usual second-order moment-based techniques. However, the proposed algorithm also performs the noise variance estimation if ℍ
0 and the SNR and channel estimation if ℍ
1, which is an advantage by comparison with the techniques of the literature.Figure9 compares the ROC curves of the proposed detector given by simulation with the theoretical ones P
d and P
fa given by (39) and (41), respectively. We notice that the theoretical curve for SNR = 0 dB is very close to the one obtained by simulation, whereas for SNR = −10 dB, the difference is more noticeable. This observation tallies with the discussion on the approximation σ
^
2
≈
(
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2 in the calculation of the metric ℳ under the hypothesis ℍ
1. Indeed, this approximation is justified for high values of SNR but becomes wrong for the very low SNR values. However, the theoretical curves give an idea of the detector performance for a given SNR, even for low SNR values.Figure 9
Comparison of the receiver operating characteristic (ROC) curves obtained by simulation and in theory.


## 6.1. Simulations Parameters

The signal parameters used for the simulations are based on those of the digital radio mondiale (DRM/DRM+) standard [30]. This standard designs the digital radio broadcasting over the current AM/FM bands. When it is transmitted, the signal is composed of 148 independent carriers. The symbol and the cyclic prefix durations are 14.66 ms and 5.33 ms, respectively. Although the DRM standard recommends a pilot distribution in staggered rows, we consider a block-type pilot arrangement, according to the model used in [20]. The channel used in the presence of a PU is based on the U
S
C
o
n
s
o
r
t
i
u
m model of the DRM/DRM+ standard, whose path gains are normalized. The channel parameters are summed up in Table 1.Table 1
Table of parameters of the channel model.
Channel model
Path 1
Path 2
Path 3
Path 4
Delay
0 ms
0.7 ms
1.5 ms
2.2 ms
Gain
0.7448
0.5214
0.3724
0.1862


## 6.2. Choice of the Thresholdς

Figure4 depicts the metric ℳ
=
|
M
2
-
σ
^
2
| versus the number of iterations, under the hypotheses ℍ
0 and ℍ
1. The SNR is fixed equal to 0 dB. In presence of signal, the average signal power P
s is equal to 1. The simulation is performed by means of 4000 simulation runs.Figure 4
|
M
2
-
σ
^
2
| versus the iteration number under ℍ
0 and ℍ
1, for SNR = 0 dB.It can be seen that the a priori qualitative analysis is verified. Indeed, for a sufficient number of iterations (according to the valuee
σ, as shown thereafter), ℳ converges to P
s under ℍ
1 and converges toward zero under ℍ
0. It has been noticed that it is not possible to find an exact value of ς according to P
(
ℳ
>
ς
∣
ℍ
1
)
=
P
d
t. However, it is observable on Figure 4 that the choice of the threshold is not restrictive. Indeed, choosing ς as small as expected ensures a probability P
d close to one, and, for a sufficient number of iterations, it also ensures a low value for P
fa. Nevertheless, reducing the value of e
σ increases the number of required iterations, as shown in the following. Hence, for an expected detection probability, a tradeoff between the complexity and the acceptable level of false alarm probability has to be taken into account, since each iteration requires 𝒪
(
i
f
M
3
) operations.


## 6.3. Effect of the Choice ofe
σ on the Detector Performance

It is shown in this section that the choice of the thresholde
σ value does not have any effect on the detection performance of the proposed method but only impacts the convergence speed of the algorithm. Figure 5 depicts the curves of detection and false alarm probabilities P
d and P
fa versus the SNR from −15 dB to 10 dB. In order to ensure the convergence of the algorithm, e
σ must have a low value. The subfigures (a) and (b) then depict the curves P
d and P
fa for e
σ
=
0.01 and e
σ
=
0.0001, respectively. According to the previous recommendations, the initialization σ
(
i
=
0
)
2 is equal to 40
×
M
2. We also arbitrarily fix the threshold ς
=
0.01, its effect on the detection performance being further studied. The figure is obtained thanks to 2000 simulation runs.Detection and false alarm probabilities versus SNR, for two valuese
σ and for a fixed value ς
=
0.01.
(a)
P
d and P
fa versus SNR, for e
σ
=
0.01
(b)
P
d and P
fa versus SNR, for e
σ
=
0.0001We observe that the curves ofP
d and P
fa match from Figures 5(a) to 5(b). P
fa is equal to zero or nearly for all SNR values and P
d reaches one from SNR = −5 dB. The detector can then reach the perfect one from SNR ≥ −5 dB, that is, in low SNR environment. We conclude that, assuming a value of e
σ low enough to ensure the convergence of the algorithm, this threshold does not have any effect on the detection performance of the proposed method.Figure6 displays the iterations number the algorithm needs before it stops versus the SNR from −10 to 10 dB. We consider three different values for the threshold: e
σ = 0.01, 0.001, and 0.0001. The simulations conditions remain the same.Figure 6
Means of the number of iterations needed by the algorithm to stop versus SNR (in dB), for three values of thresholde
σ.Although Figures5(a) and 5(b) display almost the same probabilities whatever the threshold e
σ, they differ from each other according to the number of iterations the algorithm requires before stopping. Indeed, remembering that we compare |
σ
^
(
i
)
2
-
σ
^
(
i
-
1
)
2
| with e
σ, the lower e
σ, the larger the number i of iterations needed to reach e
σ. However, Figure 6 shows that the maximum mean of iterations is less than 7 for SNR = −10 dB and shows the maximum mean of iterations is less than 5 for SNR = −10 dB and e
σ = 0.0001, which is a reasonable number of iterations. We conclude that the choice of e
σ has no effect on the detector efficiency, while it allows the convergence of the algorithm. Besides this result, the number of required iterations reasonably increases when e
σ and the SNR have low values. The detector then remains usable in practice under these conditions.


## 6.4. Detector Performance with Channel Uncertainty

In this part, we study the behavior of the proposed detector when a non-WSS channel is considered. To this end, we artificially correlate the different paths by inserting the gainh
0 into the other path gains. Thus, from the originally created channel impulse response [
h
0
,
…
,
h
L
-
1
] with independent paths, we build a new correlated vector [
h
0
,
…
,
h
~
l
,
…
,
h
~
L
-
1
] such that, for l
=
1
,
…
,
L
-
1, we define a correlation coefficient ρ
h by
(42)
ρ
h
=
E
{
h
0
h
~
l
*
}
σ
0
σ
~
l
,
where h
~
l
=
h
l
+
α
l
h
0, α
l being a coefficient that is calculated in function of the expected ρ
h, and σ
0
2 and σ
~
l
2 are the variances of h
0 and h
~
l, respectively. Figure 7 displays the detection probability P
d versus the SNR for the proposed detection under a channel correlation condition. Three curves are considered: the reference (ρ
h
=
0) and two correlated channels with ρ
h
=
0.1 and ρ
h
=
0.5. We observe a limited gap of 1 dB between the reference curve and the two others. We conclude that the proposed detector is robust against the channel uncertainty.Figure 7
Detection probability versus SNR for a non-WSSUS channel.


## 6.5. Receiver Operating Characteristic of the Detector

The performance of a detector is usually evaluated by means of the receiver operating characteristic (ROC) curves, depicting the detection probabilityP
d in function of the false alarm probability P
fa. The optimal detector is logically reached at the point (
P
fa
=
0
,
P
d
=
1
). The curve P
fa
=
P
d is called line of chance and corresponds to a detector that makes as much good decisions as false alarms. If the ROC curve is above the first bisector, the detector is efficient, since P
d
>
P
fa.Figure8 shows the ROC curves of the proposed detector for low SNR values (−10 dB and SNR = 0 dB). It is compared to the energy detector and the second-order moment-based MME [31]. The simulations conditions remain the same, and we fix the threshold e
σ
=
0.01. In Figure 8(a), the proposed detector is compared to the usual energy detector, whose metric ℳ is equal to the second order-moment of the received signal M
2. This metric is compared to a threshold ς
e to obtain the following decision rule:
(43)
ℍ
0
,
if
ℳ
<
ς
e
ℍ
1
,
else
.Receiver operating characteristic (ROC) curves of the proposed method compared to the energy and MME detectors.
(a)
Proposed detector compared with the energy detector
(b)
Proposed detector compared with the MME detectorIn Figure8(b), the proposed detector is also compared to the usual MME detector, whose metric ℳ is equal to the ratio of the maximum and the nonzero minimum eigenvalues of the received signal covariance matrix R
_
˘; that is, ℳ
=
λ
max
⁡
/
λ
min
⁡. The same aforementioned decision rule is used. Since a SISO system is assumed, R
_
˘ is obtained by concatenating N consecutive OFDM symbols so that U
_
N
=
[
U
1
,
…
,
U
N
] and then R
_
˘
=
U
_
N
U
_
N
T. In that way, U
_
N is equivalent to a system with N sensors. However, due to the nature of the channel, the different received OFDM symbols are correlated. In Figure 8(b), the ROC curves of MME are obtained for N
=
4, 10, and 20 symbols, and the SNR is equal to −10 dB. Each point of the curves is obtained by means of 2000 simulation runs.We observe in Figure8(a) that the proposed detector outperforms the energy detector, whatever the SNR. Indeed, as we consider the detection of a preamble transmitted over a Rayleigh channel, the power of the received signal P
S in (30) is not constant and follows a chi-square distribution. Consequently, for simulations made at a fixed SNR, the noise variance is also a varying process, which deteriorates the detector performance. For additional details about the theoretical development of the energy detection of signals with random amplitude, please refer to [28, 29]. We also may explain the performance of our detector by the fact that we use the same sensing time to compare the energy detector and the proposed algorithm, that is, only one OFDM symbol length. The 148 samples of one OFDM symbol are not enough to obtain an accurate energy detector. Figure 8(a) also confirms that the proposed detector is very efficient, since it is able to reach the perfect detector for ς
=
0.01. Indeed, for SNR = 0 dB, we observe that the ROC curve reaches the point (
P
fa
=
0
,
P
d
=
1
), as we remarked in Figures 5(a) and 5(b) for SNR ≥ −5 dB. In Figure 8(b), we observe that MME requires N
=
20 symbols to reach the performance of the proposed method, because MME is efficient for a very large size of U
_
N, and the vectors of the latter matrix are correlated. Thus, for a given performance, the complexity of MME is 𝒪
(
N
M
2
) (for the computation and the diagonalization of R
_
˘) and the one of the proposed algorithm is 𝒪
(
i
f
M
3
). Since we reasonably have N
<
i
f
M, we conclude that the iterative method is more complex than usual second-order moment-based techniques. However, the proposed algorithm also performs the noise variance estimation if ℍ
0 and the SNR and channel estimation if ℍ
1, which is an advantage by comparison with the techniques of the literature.Figure9 compares the ROC curves of the proposed detector given by simulation with the theoretical ones P
d and P
fa given by (39) and (41), respectively. We notice that the theoretical curve for SNR = 0 dB is very close to the one obtained by simulation, whereas for SNR = −10 dB, the difference is more noticeable. This observation tallies with the discussion on the approximation σ
^
2
≈
(
1
/
M
)
∑
m
=
0
M
-
1
‍
|
W
m
|
2 in the calculation of the metric ℳ under the hypothesis ℍ
1. Indeed, this approximation is justified for high values of SNR but becomes wrong for the very low SNR values. However, the theoretical curves give an idea of the detector performance for a given SNR, even for low SNR values.Figure 9
Comparison of the receiver operating characteristic (ROC) curves obtained by simulation and in theory.


## 7. Conclusion

In this paper, an iterative algorithm for spectrum sensing in a cognitive radio context has been presented. Originally proposed in [20] for the joint estimation of the noise and the channel, this method is based on the second-order moment of the received signal. In the presence of a primary user (PU), the algorithm estimates the channel and the noise variance. If the PU is not active, the algorithm returns a very accurate estimation of the noise level. By comparing the noise variance to the second moment of the received signal estimation (useful signal with noise or only noise), it is then possible to determine if the PU is present or absent. From that an analytical expression of the detection and false alarm probabilities have been proposed, and it is shown that they are very close to the simulations. It is also shown that the detector reaches the perfect one from very low SNR values. The algorithm offers numerous advantages as it performs a PU detection, the noise variance, and the channel estimation if the PU is active and it returns the noise level in the frequency band when the PU is absent, without changing the structure proposed in [20]. The future work concerning the detector will focus on the synchronization of the SU on the PU’s signal.



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*Source: 506457-2014-03-26.xml*