System model
Typical IA in cognitive radio system is shown in Fig. 1 [18]. There are K s secondary transmitterreceiver pair and one SU with MIMOCR interface and one primary transmitter. All the SUs share the transmission resource with the PU at the same time. Each transmitter and receiver has M and N antennas, respectively. H^{[ij]} ∈ ℂ^{N × M} represents the channel between the jth transmitter and the ith receiver, where i, j ∈ {0, 1, …, K} and 0th user represents the PU. All the elements of H^{[ij]} are independent and identically distributed (i.i.d.) and follow complex Gaussian distribution with zero mean and unit variance \( \mathcal{CN}\left(0,1\right) \). Then, the received signal at the ith receiver is expressed as:
$$ {\mathbf{y}}^{\left[i\right]}={\mathbf{H}}^{\left[ ii\right]}{\mathbf{x}}^{\left[i\right]}+\sum \limits_{j=0,j\ne i}^K{\mathbf{H}}^{\left[ ij\right]}{\mathbf{x}}^{\left[j\right]}+{\mathbf{z}}^{\left[i\right]} $$
(1)
where x^{[i]} expresses the transmitted symbols of user i and z^{[i]} ∈ ℂ^{N × 1} represents the circularly symmetric additive white Gaussian noise vector with \( \mathcal{CN}\left(0,{\sigma}^2{\mathbf{I}}_N\right) \), in which σ^{2} is noise variance and I_{
N
} is an identity matrix.
In the IA system consisting of a pair of MIMObased transceivers, the transmitter controls the precoding matrix so that the transmitted signal is limited to the interference space at an undesired receiver, and the receiver controls the decoding matrix to remove undesired received signals and to recover the signal. This IA design conditions can be represented as follows:
$$ {\mathbf{U}}^{{\left[i\right]}^{\ast }}{\mathbf{H}}^{\left[ ij\right]}{\mathbf{V}}^{\left[j\right]}=0 $$
(2)
$$ \operatorname{rank}\left({\mathbf{U}}^{{\left[i\right]}^{\ast }}{\mathbf{H}}^{\left[ ii\right]}{\mathbf{V}}^{\left[i\right]}\right)={d}^k,\forall i\ne j $$
(3)
where d^{k} is the desired number of streams of user i. V^{[j]} ∈ ℂ^{M × d}and U^{[i]} ∈ ℂ^{N × d} denote precoding matrix of jth user and decoding matrix of ith user, respectively. \( {\mathbf{U}}^{{\left[i\right]}^{\ast }} \) is the conjugate transpose of U^{[i]}.
We can represent the received signal recovered by decoding matrix and adjusted by precoding matrix from the IA design conditions as follows.
$$ {\tilde{\mathbf{y}}}^{\left[i\right]}=\underset{\mathrm{desired}\kern0.17em \mathrm{signal}}{{\mathbf{U}}^{{\left[i\right]}^{\ast }}{\mathbf{H}}^{\left[ ii\right]}{\mathbf{V}}^{\left[i\right]}{\mathbf{s}}^{\left[i\right]}}+\underset{\mathrm{interference}\kern0.17em \mathrm{signal}\mathrm{s}}{\sum \limits_{j=0,j\ne i}^K{\mathbf{U}}^{{\left[i\right]}^{\ast }}{\mathbf{H}}^{\left[ ij\right]}{\mathbf{V}}^{\left[j\right]}{\mathbf{s}}^{\left[j\right]}}+\underset{\mathrm{noise}}{{\mathbf{U}}^{{\left[i\right]}^{\ast }}{\mathbf{z}}^{\left[i\right]}} $$
(4)
where s^{[i]} is the transmission signal of the ith transmitter. From the IA condition, interference signals term of (4) is eliminated.
In this paper, we designate one of the SUs as a sensor node only responsible for spectrum sensing, as shown in Fig. 2, in order to eliminate dependence on the PU information in the CR system for IA.
The designated sensor node should sense the primary signal continuously so that it cannot transmit or receive data during the sensing process. It may also consume more energy than other secondary nodes. Therefore, a new sensor node needs to be selected after certain given time. In wireless ad hoc networks or wireless sensor networks, there have been similar studies on selecting the cluster head (CH) [27, 32,33,34]. In order to select the CH, various parameters can be considered, such as the number of member nodes covered by the CH, the current residual energy level, and the history of CH nodes. In this paper, the sensor is selected by considering the residual energy and energy draining rate of every node as in [34]. In this paper, for the selection of spectrum sensing node, the node with the largest ratio of residual energy to draining rate of energy is selected.
$$ \mathrm{sensor}\left(t+{T}_{\mathrm{sel}}\right)=\underset{i}{\mathrm{argmax}}\left(\frac{E_i}{D_i}\right) $$
(5)
where T_{sel} is a cycle for selecting sensor node. E_{
i
} and D_{
i
} are the residual energy and the draining rate of energy of node i, respectively.
The use of the designated sensor node can reduce the sensing overhead of other secondary nodes, but it may bring sensing accuracy degradation in some wireless scenarios. When a cooperative sensing method is used in the conventional CR system, by combining each SU’s sensing result, it can increase the sensing accuracy and detect hide primary transmitters. Even though the sensor node in the proposed method is the only node that senses the primary signal, by performing consecutive spectrum sensing as a form of sequential chaining of a fixed sensing time, in which the fixed sensing time is determined to satisfy the required minimum primary detection probability, our proposed method also can combine timedomain multiple sensing results. It can compensate the lack of physical cooperative sensing.
As with the usual IA scheme, the sensor performs the IA process with other SUs to limit signals from other SUs to interference space and remove them through the decoding matrix. The remaining signal space can be used to sense the PU. As the sensing role is dedicated to a particular sensor, other SUs do not need to participate on spectrum sensing and also can transmit data without wasting of time for sensing.
To satisfy the required primary detection probability, a fixed sensing time slot (t_{
s
}) is determined as in (27). The sensor node performs spectrum sensing every consecutive sensing time slot. In this paper, to notify the sensing result by the sensor node to all CR SUs, we propose two mechanisms.

i.
Periodic notification (default mode): the sensor node broadcasts sensing report which includes primary detection information at every predetermined sensing reporting interval. When SUs receive the primary detection notification, they should not transmit data until the next report broadcasting time. The sensing reporting interval is represented as t_{
r
}.

ii.
Notification using dedicated control channel transceiver: every secondary nodes including sensor node have dual transceivers, in which one is for data transmission (or spectrum sensing) and the other one is control signal exchange. When the sensor node detects the primary signal, it transmits detection notification signal on the dedicated narrow band channel, and other SUs seize their data transmission. If the data channel returns to idle, then the sensor also send channel idle notification and then SUs can again utilize the data channel.
The spectrum sensing and primary detection comparison for the conventional CR system and proposed IAbased spectrum sensing system is represented in Fig. 3. Figure 3a represents the primary system activities as a form of busy with times. Figure 3b shows the conventional CR system which uses a fixed spectrum sensing time and interval. The conventional CR system can only sense the primary signal only during the short sensing time so that if the primary appears during the secondary data transmission time (i.e., between two consecutive sensing times), then the secondary system will give harmful interference to the primary system. As shown in Fig. 3c, the designated sensor node can continuously sense the primary signal. At time (1), the sensor node broadcasts the primary nondetection report to SUs so that SUs can utilize the data channel using interference alignment. At time (2) and time (3), the sensor node detects the primary signal so that it sends the primary detection report at (4). As we can see in Fig. 3c, SUs can seize their transmission until the primary signal is not detected. In conventional CR in Fig. 3b, since SUs are not able to detect the primary signal during the short sensing time, they send data and cause strong interference to the primary user. Figure 3d shows the case that dual transceivers are used. At time (5) when the sensor node detects the primary signal, it can immediately send the detection notification using the dedicated control channel. And when the data channel returns to idle at time (6), the sensor node notifies the nondetection notification to SUs and SUs utilize the data channel again. Therefore, secondary node’s data throughput is enhanced.
The sensing time t_{
s
} and sensing reporting interval t_{
r
} impact on not only primary protection performance but also secondary system data transmission opportunity. The longer the sensing time, the lower miss detection and false alarm probabilities are obtained. On the other hand, the shorter sensing time results in the higher miss detection and false alarm probabilities especially at low signaltonoise ratio (SNR) of the longer sensing reporting interval makes the more transmission opportunity for secondary users; however, it generates the higher possible interference to primary users. In CR wireless network, the primary activity and wireless channel condition vary dynamically so that it is very difficult to derive the optimal sensing time and reporting interval. Therefore, in this paper, we propose a new dynamic optimal parameter control using reinforcement learning. The multiobjective function of the secondary system is given as in (6), in which the multiobjective function consists of three reward functions: interference ratio reward, transmission opportunity loss ratio, and overhead for sensing. The reward functions will be explained in detail in Section 3.3. Therefore, the proposed method derives the optimal \( \left({t}_s^{\ast },{t}_r^{\ast}\right) \) value that maximizes the multiobjective function with subject to primary protection and secondary throughput requirements.
$$ {\displaystyle \begin{array}{l}\operatorname{Maximize}:{f}_{\mathrm{intf}}\left({R}_I\right)+{f}_{\mathrm{loss}}\left({R}_L\right)+{f}_{\mathrm{overhead}}\left({t}_s,{t}_r\right)\\ {}\mathrm{Find}:{t}_s^{\ast },{t}_r^{\ast}\\ {}\mathrm{Subject}\kern0.17em \mathrm{to}:{P}_d\ge {P}_d^{\mathrm{th}},{R}_I\le {R}_I^{\mathrm{th}},{R}_L\le {R}_L^{\mathrm{th}}\end{array}} $$
(6)
where f_{intf}(R_{
I
}) and f_{loss}(R_{
L
}) are the functions of interference and transmission opportunity loss ratio; f_{overhead}(t_{
s
}, t_{
r
}) is the function of the overhead related to sensing time t_{
s
} and reporting interval (integer multiple of t_{
s
}); \( {t}_s^{\ast },\kern0.5em {t}_r^{\ast } \) are the optimal spectrum sensing time and reporting interval, respectively; \( {P}_d^{\mathrm{th}} \) is the required primary detection probability P_{
d
}; \( {R}_I^{\mathrm{th}} \) and \( {R}_L^{\mathrm{th}} \) are the tolerable interference ratio and secondary transmission opportunity loss ratio, respectively.
The main novel features of the proposed system architecture are as follows:

1.
The dedicated sensor is responsible for the sensing function and can operate spectrum sensing by IA process when SUs transmit the signal so that the operation of the PU can be continuously observed.

2.
We specify the target detection probability to basically satisfy the detection probability and operate in the range that satisfies the interference ratio and the secondary transmission opportunity loss ratio.

3.
We use the Qlearning to determine sensing time and reporting interval dynamically and design the suitable reward function.
Interference alignment and degree of freedom in the proposed system
A minimum DoF must be ensured for each transceiver pair to communicate using IA process. We derive the condition of DoF that the proposed system can obtain. In addition, this section provides a theoretical basis for the sensor node to perform sensing while the SU is transmitting. Suppose there is a MIMOCR interference network with K SUs, one sensor and one PU in Fig. 4. It is assumed that SU’s IA network is consist of symmetric (i.e., same transmission antennas and receive antennas). The 0^{th} SU is a sensor, and each transmitter and receiver of the SU has M and N antennas. Then, received signals at the sensor and the ith SU receiver are as shown in (7) and (8):
$$ {\mathbf{y}}^{\left[0\right]}={\mathbf{H}}^{\left[0p\right]}{\mathbf{x}}^{\left[p\right]}+\sum \limits_{j=1}^K{\mathbf{H}}^{\left[0j\right]}{\mathbf{x}}^{\left[j\right]}+{\mathbf{z}}^{\left[0\right]} $$
(7)
$$ {\mathbf{y}}^{\left[i\right]}={\mathbf{H}}^{\left[ ii\right]}{\mathbf{x}}^{\left[i\right]}+\sum \limits_{j=0,j\ne i}^K{\mathbf{H}}^{\left[ ij\right]}{\mathbf{x}}^{\left[j\right]}+{\mathbf{z}}^{\left[i\right]} $$
(8)
where x^{[p]} and x^{[i]} are the transmission symbol of PU and SU i, z^{[0]}, z^{[i]} are circularly symmetric additive white Gaussian noise vectors, with \( \mathcal{CN}\left(0,{\sigma}^2{\mathbf{I}}_N\right) \). H^{[ij]} ∈ ℂ^{N × M} represents the channel between the jth transmitter and the ith receiver, where i, j ∈ {0, 1, …, K}, K represents the number of SUs, and the index p represents the PU. All elements of H^{[ij]} are i.i.d. distributed and follow \( \mathcal{CN}\left(0,1\right) \). We assumed a quasistatic channel, i.e., the channel realization remains fixed throughout the duration of transmission.
In order to eliminate interference in a sensor and each SU, we use a decoding matrix and the received signal with d data streams of the ith user is recovered as follows:
$$ {\tilde{\mathbf{y}}}^{\left[0\right]}={\mathbf{U}}^{{\left[0\right]}^{\ast }}{\mathbf{H}}^{\left[0p\right]}{\mathbf{V}}^{\left[p\right]}{\mathbf{s}}^{\left[p\right]}+\sum \limits_{j=1}^K{\mathbf{U}}^{{\left[0\right]}^{\ast }}{\mathbf{H}}^{\left[0j\right]}{\mathbf{V}}^{\left[j\right]}{\mathbf{s}}^{\left[j\right]}+{\mathbf{U}}^{{\left[0\right]}^{\ast }}{\mathbf{z}}^{\left[0\right]} $$
(9)
$$ {\tilde{\mathbf{y}}}^{\left[i\right]}={\mathbf{U}}^{{\left[i\right]}^{\ast }}{\mathbf{H}}^{\left[ ii\right]}{\mathbf{V}}^{\left[i\right]}{\mathbf{s}}^{\left[i\right]}+\sum \limits_{j=0,j\ne i}^K{\mathbf{U}}^{{\left[i\right]}^{\ast }}{\mathbf{H}}^{\left[ ij\right]}{\mathbf{V}}^{\left[j\right]}{\mathbf{s}}^{\left[j\right]}+{\mathbf{U}}^{{\left[i\right]}^{\ast }}{\mathbf{z}}^{\left[i\right]} $$
(10)
where s^{[p]}, s^{[j]} are transmission signals of the PU and the jth SU. V^{[p]} and V^{[j]} are the precoding matrix of the PU and jth SU, and U^{[0]}, U^{[i]} ∈ ℂ^{N × d} are the decoding matrix of the sensor and the ith user. To completely remove interference from the SUs to the sensor or between the SUs, V^{[j]}, U^{[0]}, and U^{[i]} must satisfy the following conditions:
$$ {\mathbf{U}}^{{\left[0\right]}^{\ast }}{\mathbf{H}}^{\left[0j\right]}{\mathbf{V}}^{\left[j\right]}={\mathbf{0}}_d $$
(11)
$$ {\mathbf{U}}^{{\left[i\right]}^{\ast }}{\mathbf{H}}^{\left[ ij\right]}{\mathbf{V}}^{\left[j\right]}={\mathbf{0}}_d $$
(12)
$$ \operatorname{rank}\left({\mathbf{U}}^{{\left[i\right]}^{\ast }}{\mathbf{H}}^{\left[ ii\right]}{\mathbf{V}}^{\left[i\right]}\right)={d}^{\left[i\right]} $$
(13)
$$ \operatorname{rank}\left({\mathbf{U}}^{{\left[0\right]}^{\ast }}{\mathbf{H}}^{\left[0p\right]}{\mathbf{V}}^{\left[p\right]}\right)={d}^{\left[0\right]},\forall i\ne j,i\ne 0,j\ne 0,\forall i,j\in \left\{1,2,\dots, K\right\} $$
(14)
where d^{[i]} is the desired number of stream of user i.
Equations (11) and (12) show that interference in receiving signal dimension of sensor and SU receivers should be zero. Equations (13) and (14) represent the number of signal stream that each SU transceiver pair and sensor node can acquire. From these constraints, the DoF condition is expressed by (15):
$$ d\le \frac{N}{K+1}+\frac{KM}{\left(P+K\right)\left(K+1\right)} $$
(15)
Proof. See the Appendix.
Therefore, in the network with satisfying the DoF condition from (11), the sensor node can remove the interference from the signals of other SU and sense the PU signal.
To fulfill the requirements in (11) and (12), the iterative IA algorithms in [35, 36] can be adopted with some modifications. The sensor should minimize the total leakage interference that remains after canceling the interference by decoding. Other SUs can obtain the precoding and decoding matrix by the maximum SINR algorithm considering the total leakage interference of the sensor. By fixing all V^{[i]}, we can solve U^{[j]} as
$$ {\mathbf{U}}^{\left[j\right]}={\nu}_{\mathrm{max}}\left(\frac{{\mathbf{H}}^{\left[ jj\right]}{\mathbf{V}}^{\left[j\right]}{\mathbf{V}}^{{\left[j\right]}^{\ast }}{\mathbf{H}}^{{\left[ jj\right]}^{\ast }}}{\sum_{i\ne j}{\mathbf{H}}^{\left[ ji\right]}{\mathbf{V}}^{\left[i\right]}{\mathbf{V}}^{{\left[i\right]}^{\ast }}{\mathbf{H}}^{{\left[ ji\right]}^{\ast }}+{\sigma}^2{\mathbf{I}}_N}\right) $$
(16)
where ν_{max}(∙) denotes the dominant eigenvector when the eigenvalues are real.
Reversely, by fixing all U^{[j]}, we can solve V^{[i]} as
$$ {\mathbf{V}}^{\left[i\right]}={\nu}_{\mathrm{max}}\left(\frac{{\mathbf{H}}^{{\left[ ii\right]}^{\ast }}{\mathbf{U}}^{{\left[i\right]}^{\ast }}{\mathbf{U}}^{\left[i\right]}{\mathbf{H}}^{\left[ ii\right]}}{\sum_{j\ne i}{\mathbf{H}}^{{\left[ ji\right]}^{\ast }}{\mathbf{U}}^{{\left[j\right]}^{\ast }}{\mathbf{U}}^{\left[j\right]}{\mathbf{H}}^{\left[ ji\right]}+{\mathbf{Q}}_0}\right) $$
(17)
where Q_{0} is the interference covariance matrix at the sensor.
The interference covariance matrix at the sensor is
$$ {\mathbf{Q}}_0=\sum \limits_{i\ne 0}^K{\mathbf{U}}^{{\left[0\right]}^{\ast }}{\mathbf{H}}^{\left[0i\right]}{\mathbf{H}}^{{\left[0i\right]}^{\ast }}{\mathbf{U}}^{\left[0\right]} $$
(18)
The decoder minimizing the total leakage interference at the sensor is
$$ {\mathbf{U}}_0={\nu}_{\mathrm{min}}\left({\mathbf{Q}}_0\right) $$
(19)
where ν_{min}(∙) is the least dominant eigenvector.
Summarizing the process, the transmitters choose the initial precoders randomly and receivers choose the decoders maximizing SINR. The sensor node calculates the interference covariance matrix and chooses the decoding matrix. The transmitters choose the precoders by maximizing SINR by considering the total interference leakage at the sensor node. Then, the choices of decoding matrix of the receivers are followed and this sequence of processes continues to convergence.
Energy detection with/without interference alignment
Spectrum sensing in the proposed system stops transmission of SUs when PU is detected and performs general MIMObased spectrum sensing. If the sensor determines that the PU is in an idle state, the SUs send the signal through IA and the sensor performs IAbased spectrum sensing that allows sensing during communication of the SUs. Therefore, in the proposed system, MIMObased or IAbased spectrum sensing is selected according to the detection result of the PU signal. Since the parameters to be used to set thresholds for energy detection depend on the choice of sensing method, this section focuses on MIMObased sensing and IAbased sensing.
If an SU does not transmit because the PU state is determined as busy state, the hypothesis from the received signal y_{
i
} of the sensor is expressed in (20):
$$ {\displaystyle \begin{array}{c}{H}_0:{y}_i(n)={z}_i(n)\\ {}{H}_1:{y}_i(n)={h}_is(n)+{z}_i(n),\mathrm{where}\;1\le i\le N\end{array}} $$
(20)
where H_{0} represents the hypothesis corresponding to “no signal transmitted,” and H_{1} represents “signal transmitted.” s(n) is the signal waveform, and z_{
i
}(n) is a zero mean additive white Gaussian noise (AWGN). The PU is assumed to phase shift keying (PSK) modulated signal. The channel coefficient h_{
i
} follows \( \mathcal{CN}\left(0,{\sigma}_h^2\right) \), and z_{
i
} follows \( \mathcal{CN}\left(0,{\sigma}_n^2\right) \); \( {\sigma}_h^2 \) and \( {\sigma}_n^2 \) are the variance in channel gain and Gaussian noise. N is the number of receiving antennas.
The test statistic for the energy detector is given by
$$ Y=\sum \limits_{i=1}^N\sum \limits_{n=0}^{n_s1}{\left{y}_i(n)\right}^2 $$
(21)
where n_{
s
} is the samples of spectrum sensing.
We assumed the test statistic follows a Gaussian distribution under the central limit theorem. Therefore, each pdf of (21) under H_{0} and H_{1} is given by
$$ \left.Y\right{H}_0\sim \mathcal{N}\left({\mu}_{0, non IA},{\sigma}_{0, non IA}^2\right),\left.Y\right{H}_1\sim \mathcal{N}\left({\mu}_{1, non IA},{\sigma}_{1, non IA}^2\right) $$
(22)
where \( {\mu}_{0,\mathrm{non}\mathrm{IA}}=N{n}_s{\sigma}_n^2 \), \( {\sigma}_{0,\mathrm{non}\mathrm{IA}}^2=N{n}_s{\sigma}_n^4 \), \( {\mu}_{1,\mathrm{non}\mathrm{IA}}=N{n}_s\left(P{\sigma}_h^2{\lambda}_m+{\sigma}_n^2\right) \), \( {\sigma}_1^2=N{n}_s{\left(P{\sigma}_h^2{\lambda}_m+{\sigma}_n^2\right)}^2 \). λ_{
m
} is eigenvalue of the correlation matrix, and P is transmission power of the PU.
False alarm and detection probability for the nonIA case are given by
$$ {\displaystyle \begin{array}{c}{P}_f^{non IA}=\mathit{\Pr}\left\{\left.Y>{\varepsilon}_{non IA}\right{H}_0\right\}=\mathcal{Q}\left(\frac{\varepsilon_{non IA}{\mu}_{0, non IA}}{\sigma_{0, non IA}}\right)\\ {}{P}_d^{non IA}=\mathit{\Pr}\left\{Y>\left.{\varepsilon}_{non IA}\right{H}_1\right\}=\mathcal{Q}\left(\frac{\varepsilon_{non IA}{\mu}_{1, non IA}}{\sigma_{1, non IA}}\right)\end{array}} $$
(23)
The hypothesis for the received signal of the spectrum sensor produced from the decoding matrix when a SU transmits because the PU state is determined as idle is expressed with (24):
$$ {\displaystyle \begin{array}{l}{H}_0:{\tilde{y}}_i(n)={\tilde{z}}_i(n)\\ {}{H}_1:{\tilde{y}}_i(n)=\sum \limits_{j=1}^{M_p}{\tilde{\mathbf{G}}}^{\left[ ij\right]}{s}_j(n)+{\tilde{z}}_i(n),\mathrm{where}\;1\le i\le N\end{array}} $$
(24)
where s_{
j
}(n) is the signal waveform from jth antenna of PU, and \( {\overset{\sim }{z}}_i(n) \) is an AWGN. \( {\overset{\sim }{\mathbf{G}}}^{\left[ ij\right]} \) is the compound channel gain between the PU transmitter and the sensor, and M_{
p
} is the number of PU’s transmit antenna. We assumed that the gain does not change for multiple CR frames and can be estimated blindly while the PU is known to be present. Each statistical pdf of (24) is given by
$$ \left.Y\right{H}_0,{\tilde{\mathbf{H}}}_p\sim \mathcal{N}\left({\mu}_{0, IA},{\sigma}_{0, IA}^2\right),\left.Y\right{H}_1,{\tilde{\mathbf{H}}}_p\sim \mathcal{N}\left({\mu}_{1, IA},{\sigma}_{1, IA}^2\right) $$
(25)
where \( {\mu}_{0,\mathrm{IA}}=N{n}_s{\sigma}_n^2 \), \( {\sigma}_{0,\mathrm{IA}}^2=N{n}_s{\sigma}_n^4 \), \( {\mu}_{1,\mathrm{IA}}=N{n}_s\left(P{\overset{\sim }{g}}_m^2{\sigma}_h^2{\lambda}_m+{\sigma}_n^2\right) \), \( {\sigma}_{1,\mathrm{IA}}^2=N{n}_s{\left(P{\overset{\sim }{g}}_m^2{\sigma}_h^2{\lambda}_m+{\sigma}_n^2\right)}^2 \). \( {\overset{\sim }{g}}_m^2 \) is the sum of \( {\overset{\sim }{\mathbf{G}}}^{\left[ ij\right]} \) on j indexes.
False alarm and detection probability for the IA process case are given by
$$ {\displaystyle \begin{array}{c}{P}_f^{IA}=\mathit{\Pr}\left\{Y>\left.{\varepsilon}_{IA}\right{H}_0\right\}=\mathcal{Q}\left(\frac{\varepsilon_{IA}{\mu}_{0, IA}}{\sigma_{0, IA}}\right)\\ {}{P}_d^{IA}=\mathit{\Pr}\left\{Y>\left.{\varepsilon}_{IA}\right{H}_1\right\}=\mathcal{Q}\left(\frac{\varepsilon_{IA}{\mu}_{1, IA}}{\sigma_{1, IA}}\right)\end{array}} $$
(26)
For a given pair of target probabilities \( \left({P}_d^{\mathrm{th}},{P}_f^{\mathrm{th}}\right) \), the number of required samples can be determined by
$$ {n}_s={\left(\frac{Q^{1}\left({P}_f^{\mathrm{th}}\right){Q}^{1}\left({P}_d^{\mathrm{th}}\right){\left(\zeta {\lambda}_m{\overset{\sim }{g}}_m^2+1\right)}^2}{\zeta {\lambda}_m{\overset{\sim }{g}}_m^2}\right)}^2/N $$
(27)
where ζ is SNR and \( {\overset{\sim }{g}}_m^2=1 \) when this represents about the nonIA case.