Distributions of test statistics are essential in order to find analytically the probability of false alarm p
fa
and the detection probability p
d
. T
p
and T
a
have the same statistical distribution since W(k) is i.i.d. and S(k) is deterministic as s(n) is assumed to be deterministic. In the following, we develop the distribution of T
p
under H
0 and H
1 over a Gaussian channel, where the channel effect h is assumed to be constant. Similarly, the distribution of T
a
can be found. T
or
made its decision by applying the logical OR on the decisions of T
p
and T
n
which have the same distribution.
4.1 False alarm and detection probabilities of T
p
The distribution of T
p
depends on \({\sum \nolimits }_{k=1}^{\frac {N}{2}} \Gamma _{p}(k)\) as presented in Eq. (20). A simplification of the term \({\sum \nolimits }_{k=1}^{\frac {N}{2}} \Gamma _{p}(k)\) can be obtained as follows (see Appendix 1):
$$\begin{array}{*{20}l} \sum\limits_{k=1}^{\frac{N}{2}} \Gamma_{p}(k)&=\frac{2}{N^{2}\sigma_{w}^{2}}\sum\limits_{k=1}^{\frac{N}{2}} \sum\limits_{u=1}^{k}|Y(u)|^{2} \\ &=\frac{2}{N^{2}\sigma_{w}^{2}}\sum\limits_{k=1}^{\frac{N}{2}} \left(\frac{N}{2}-k+1\right)|Y(k)|^{2} \end{array} $$
(29)
4.1.1 False alarm probability of T
p
Under H
0, the test statistic T
p
is only related to the noise w(n). Using Eq. (29), T
p
can be written as follows:
$$\begin{array}{*{20}l} T_{p}&=\sum\limits_{k=1}^{\frac{N}{2}} \left[\frac{2}{N^{2}\sigma_{w}^{2}}\left(\frac{N}{2}-k+1\right)\left|W(k)\right|^{2}-R\left(1,\frac{N}{2};k\right)\right]\\ &=\sum\limits_{k=1}^{\frac{N}{2}}\left[\Gamma_{p}(k)-R\left(1,\frac{N}{2};k\right)\right] \end{array} $$
(30)
Being the discrete Fourier transform of a white noise w(n), W(k) asymptotically follows a normal distribution since it is the sum of independent terms. It is known that two Gaussian variables are independent if they are uncorrelated [22]. As E[W(k)W
∗(k−m)]=E[W(k)]E[W
∗(k−m)]=0 (see Appendix (2)), then W(k) becomes i.i.d.
Being the sum of independent terms and according to the central limit theorem, the distribution of T
p
tends towards \(\mathcal {N}(\mu _{0},V_{0})\) under H
0. In this case, the probability of false alarm \(p_{fa}^{p}\) of T
p
can be found as follows:
$$ p_{fa}^{p}=Q\left(\frac{\lambda-\mu_{0}}{\sqrt{V_{0}}}\right) $$
(31)
where Q(.) is the Q-function 4, and λ is the threshold of comparison. Since \(E\left [|W(k)|^{2}\right ]\ =N\sigma _{w}^{2}\) and based on Eq. (30), we can evaluate μ
0 as follows:
$$\begin{array}{*{20}l} \mu_{0}&=E[T_{p} ] \\ &=\frac{2}{N^{2}\sigma_{w}^{2}}\sum_{k=1}^{\frac{N}{2}} \left(\frac{N}{2}-k+1\right)E\left[|W(k)|^{2}\right]-\frac{N+2}{4} \\ &=0 \end{array} $$
(32)
In this case, the variance, V
0, of T
p
becomes (see Appendix 4)
$$\begin{array}{*{20}l} V_{0}&=E\left[T_{p}^{2}\right]=\frac{(N+2)(N+1)}{6N} \end{array} $$
(33)
4.1.2 Probability of detection of T
p
Under H
1, Y(k)=hS(k)+W(k), then Eq. (29) becomes as follows:
$$ \Gamma_{p}(k)=\frac{2}{N^{2}\sigma_{w}^{2}}\sum\limits_{k=1}^{\frac{N}{2}} \left(\frac{N}{2}-k+1\right)\left|hS(k)+W(k)\right|^{2} $$
(34)
Since S(k) is deterministic and the terms of W(k) are independent, the distribution of T
p
under H
1 tends also towards \(\mathcal {N}(\mu _{1},V_{1})\). In this case, the probability of detection \(p_{d}^{p}\), of T
p
can be found as follows:
$$ p_{d}^{p}=Q\left(\frac{\lambda-\mu_{1}}{\sqrt{V_{1}}}\right) $$
(35)
μ
1 and V
1 should be evaluated in order to find \(p_{d}^{p}\). Under H
1, |Y(k)|2 becomes
$$\begin{array}{*{20}l} |Y(k)|^{2}&=Y(k)Y^{*}(k) \\ &=|hS(k)|^{2}+|W(k)|^{2}+2Re\{hS(k)W^{*}(k)\} \end{array} $$
(36)
where hS(k)=DFT{hs(n)} and Re{X} is the real part of X. The mean value of T
p
under H
1 can be found as follows (see Appendix 3):
$$ \mu_{1}=b\gamma $$
(37)
where \(b=\frac {2}{N^{2}}\sum _{k=1}^{\frac {N}{2}}(\frac {N}{2}-k+1)|S(k)|^{2}\), and γ is the SNR as defined by Eq. (4).
Under H
1, the variance V
1 is given by the following equation (see Appendix 5):
$$\begin{array}{*{20}l} V_{1}&\,=\,V_{0}+\frac{8\gamma}{N^{3}}\sum_{k=1}^{\frac{N}{2}} \left(\frac{N}{2}-k+1\right)^{2}|S(k)|^{2} &=V_{0}+c\gamma \end{array} $$
(38)
where \(c=\frac {8}{N^{3}}{\sum \nolimits }_{k=1}^{\frac {N}{2}} \left (\frac {N}{2}-k+1\right)^{2}|S(k)|^{2}\).
T
a
is based on similar idea to T
p
, but it covers the N frequency points instead of just positive frequency points (\(\frac {N}{2}\) points). Since W(k) is i.i.d. and S(k) is deterministic, then by following the same process for \(p_{fa}^{p}\) and \(p_{d}^{p}\), the probability of false alarm \(p_{fa}^{a}\) and the probability of detection \(p_{d}^{a}\) of the detector T
a
can be found as follows:
$$\begin{array}{*{20}l} p_{fa}^{a}=Q\left(\frac{\lambda-\mu_{0}^{a}}{\sqrt{V_{0}^{a}}}\right) \end{array} $$
(39)
$$\begin{array}{*{20}l} p_{d}^{a}=Q\left(\frac{\lambda-\mu_{1}^{a}}{\sqrt{V_{1}^{a}}}\right) \end{array} $$
(40)
where \(\mu _{0}^{a}=0\); \( V_{0}^{a}=\frac {N}{3}+\frac {1}{2}+\frac {1}{6N}\); \(\mu _{1}^{a}=b_{a}\gamma \), where \(b_{a}=\frac {1}{N^{2}}\sum _{k=1}^{N}(N-k+1)|S(k)|^{2}\), and \(V_{1}^{a}=V_{0}^{a}+c_{a}\gamma \), where \(c_{a}=\frac {2}{N^{3}}\sum _{k=1}^{N} (N-k+1)^{2}|S(k)|^{2}\).
4.2 Probabilities of T
or
T
or
applies the OR rule between the decisions of T
p
and T
n
, then T
or
can be considered as a hard cooperative detector of these two detectors. Since T
p
and T
n
are independent and have the same statistics as we defined previously, the probability of false alarm \(p_{fa}^{or}\) and the probability of detection \(p_{d}^{or}\) of T
or
can be found as follows [2]:
$$\begin{array}{*{20}l} p_{fa}^{or}&=1-\left(1-p_{fa}^{p} \right)^{2} \end{array} $$
(41)
$$\begin{array}{*{20}l} p_{d}^{or}&=1-\left(1-p_{d}^{p} \right)^{2} \end{array} $$
(42)
4.3 Probabilities of T
av
T
av
can be developed following similar steps as Eq. (30).
$$\begin{array}{*{20}l} T_{av}&=\frac{2}{N^{2}\sigma_{w}^{2}}\sum\limits_{k=1}^{\frac{N}{2}}\left(\frac{N}{2}-k+1\right)\frac{|Y(k)|^{2}+|Y(-k+1)|^{2}}{2}\\ &\quad-\frac{N+2}{4} \end{array} $$
(43)
Under H
0, Y(k)=W(k), then T
av
becomes the sum of independent terms. Based on CLT, T
av
asymptotically follows \(\mathcal {N}(\mu _{0}^{av},V_{0}^{av})\) under H
0.
Under H
1, Y(k)=hS(k)+W(k) and S(k) is deterministic, so T
av
is still following a normal distribution: \(\mathcal {N}(\mu _{1}^{av},V_{1}^{av})\) under H
1.
The probability of false alarm, \(p_{fa}^{av}\), and detection, \(p_{d}^{av}\), of T
av
are expressed as follows:
$$\begin{array}{*{20}l} p_{fa}^{av}=Q\left(\frac{\lambda-\mu_{0}^{av}}{\sqrt{V_{0}^{av}}}\right) \end{array} $$
(44)
$$\begin{array}{*{20}l} p_{d}^{av}=Q\left(\frac{\lambda-\mu_{1}^{av}}{\sqrt{V_{1}^{av}}}\right) \end{array} $$
(45)
Since W(k) is i.i.d. and P
s
(k) is even, \(\mu _{0}^{av},V_{0}^{av}\), \(\mu _{1}^{av}\), \(V_{1}^{av}\) can be found by following similar steps to μ
0, V
0, μ
1 and V
1, we can find that \(\mu _{0}^{av}=\mu _{0}\), \(V_{0}^{av}=\frac {V_{0}}{2}\), \(\mu _{1}^{av}=\mu _{1}\), and \(V_{1}^{av}=\frac {V_{1}}{2}\).
The theoretical and the simulated ROC curves of proposed detectors are with good agreement as shown in Figs. 3
a, b. Simulations were done under following conditions: 16-QAM modulation, γ=−12 dB, N=1000 samples, and N
s
=4 sps.
As shown in Fig. 3
a, T
av
is the most efficient detector for both considered shaping filters. For the simulations of Section 6 under Gaussian channel, only T
av
and T
or
are compared to other well-known detectors. The rest of simulations in this paper are done with a rectangular pulse-shaping filter.