Abstract
In this article, we study the performance of intelligent reflecting surfaces (IRS) with adaptive transmit power (ATP). The power of secondary source is adapted to have a low interference at primary destination. IRS with ATP offers 14, 20, 26, 32, and 38 dB gains versus the absence of IRS for reflectors. Rayleigh channels are studied with arbitrary positions of primary and secondary nodes.
1. Introduction
In underlay cognitive radio networks, secondary users adapt their power to minimize the interference at primary nodes [1, 2]. Power adaptation allows joint transmission of primary and secondary nodes over the same band [2–5]. The system throughput is high in UCRN since it is the sum of primary and secondary throughputs. In non-UCRN, the throughput is equal to that of primary network and is therefore small than that of UCRN. In order to enhance data rates in UCRN, we suggest the use of reflecting surfaces [6–8]. The phase of each reflector is wisely chosen so that received signals have a zero phase at the destination. The phase shifts of IRS can be quantized over bits [9–12]. The number of reflectors is . The diversity order is high since it is equal to leading to significant performance enhancement. The throughput of IRS with constant transmit power was derived in [13–19]. Channel estimation for millimeter wave communications using IRS was studied in [20]. Joint power allocation and user association for millimeter wave communications using IRS were suggested in [21]. In all previous studies, IRS were studied with fixed transmit power [6–21]. In this paper, we derive the performance of IRS with adaptive transmit power. In fact, the secondary source adapts its power so that the interference at the primary destination is lower than . The secondary source adapts its power, and its signal is reflected by IRS so that all reflections have a zero phase at secondary destination.
When intelligent reflecting surfaces (IRS) are used in wireless systems with fixed transmit power, the signal-to-noise ratio (SNR) follows a chi-square distribution with one degree of freedom. IRS can be used in the secondary network of underlay CRN where the secondary source adapts its power to generate a low interference at primary destination. In this case, the SNR is the product of a chi-square random variable (r.v.) with one degree of freedom and the random adaptive transmit power. The statistics of SNR are not available when IRS are used with adaptive transmit power which is the first motivation of this research article. Besides, IRS were previously studied without interference. The second motivation of the paper is to derive the throughput of CRN using IRS and adaptive transmit power in the presence of interference from primary source.
The main innovations of the article are as follows: (i)We compute the throughput of IRS and adaptive transmit power. IRS is between secondary source and destination to improve the throughput. IRS with fixed constant power was studied in [6–22].(ii)IRS with ATP offers 14, 20, 26, 32, and 38 dB gains versus the absence of IRS for reflectors(iii)We derive the CDF of SNR and SINR while taking account of primary interference. Our results are confirmed with computer simulations
Sections 2 and 3 derive the throughput when IRS is a reflector or a transmitter. Section 4 describes the results. Conclusions are given in Section 5.
2. SNR and SINR Analysis when IRS Is a Reflector
Figure 1 depicts the network model containing a primary destination and source (, ) and a secondary destination and source (, ). has an ATP. Besides, an IRS is placed between and in order to improve the secondary throughput.
2.1. SNR Analysis
Secondary source adapts its symbol energy so that the interference at is small: where is the channel of the - link.
The adaptive transmit energy of a symbol is defined as where is the maximum energy of a symbol.
The adaptive transmit power (ATP) is equal to where is the symbol period.
Let be the channel of --th reflector link. where is the distance from to , and is the path loss exponent (PLE). We can write where .
Let be the channel from -th reflector to . . We can write where .
The phase of shift of -th reflector is [1]:
The received signal at is where is the -th symbol and is a zero-mean Gaussian r.v. of variance .
We replace and by their expressions and use (4) to obtain where
Using the central limit theorem, can be approximated by a Gaussian random variable with mean and variance denoted by and . This approximation is valid for a large number of reflectors .
The SNR at is expressed as
Equation (8) shows that the SNR is the product of a chi-square r.v. and which is also random as given in (2) depend on channel gain . In case of fixed transmit power, is deterministic, and the SNR follows a chi-square distribution as it is proportional to .
The CDF of SNR is
We have where and is the channel from to .
When the adaptive transmit symbol energy is larger than , , and the SNR is expressed as
When , we have where is the generalized Marcum -function.
When , , and the first term of (8) is written as
We deduce
has a Gaussian distribution, and we have
Let , so that , and we deduce
We use ([22], Equation (34)): with , , , and .
Using (10), (11), (13), (17), and (18), we finally obtain the CDF of SNR of CRN using IRS and adaptive transmit power:
2.2. SINR Analysis
The SINR at is equal to where and is symbol energy.
The SINR can be expressed in terms of SNR (8) as follows:
The CDF of SINR can be evaluated numerically as follows: where is provided in (19) and where .
The integral (21) always converges as it computes the CDF of SINR defined in (20). The numerical evaluation of (21) was done by truncating the integral as follows: while is
At the end, return the value of the integral as
2.3. Throughput Analysis
The packet error probability (PEP) is given by [23]. where
is packet length and for -QAM.
For -PSK, we have
The throughput at is deduced from the PEP as follows:
3. SNR and SINR Analysis when IRS Is a Transmitter
Figure 2 shows the network model when IRS are used as transmitter at . Let be the channel coefficient of -th reflector to . We can write where .
Let be -th RIS phase [1]. where is the phase of symbol.
The received signal at is equal to where
The SNR at is given by
When IRS are used as a transmitter, the SNR is the product of a chi-square r.v. , and where given in (2) is also random as it depends on channel gain . When IRS is used as a transmitter, the SNR is similar to the case where IRS is a reflector given in (8) where we have to replace by and by .
The SINR at is expressed as:
The CDF of SNR and SINR are computed as Equations (17) and (20). We have only to replace by and , by , . The throughput can be evaluated in Section 2.
4. Numerical Results
We used the MATLAB software to make the simulations and plot the numerical results. In this paper, Rayleigh channels were studied so that channel gains are complex random variables. We generated using the command where is a constant to set the value of fix . Channel gains such as , ... were generated similarly. The other variables such as and were generated using and . The parameters of the simulations are , , , , , , and
Figures 3–5 show the throughput at for QPSK, 16QAM, and 64QAM when IRS is a reflector for . Figure 5 shows that IRS allows 14, 20, 26, 32, and 38 dB gains versus the absence of IRS for . When the number of reflectors is doubled, we obtained 6 dB enhancement in throughput. The simulations results give a better throughput than theoretical curves at low SNR because we used the upper bound (26). At medium and large SNR, the theoretical curves are very close to simulation results.
Figure 6 compares the throughput at for different values of interference threshold and for QPSK and . The throughput at increases as increases due to less constraints on the interference level. In fact, the secondary source can increase its power when increases resulting in a larger throughput. The adaptive transmit power (3) increases when increases so that the throughput improves.
Figure 7 compares the throughput at for 16PSK modulation when IRS is a reflector and a transmitter for . We observe that IRS used as a transmitter offers the best performance. IRS used as a transmitter offers 1 dB gain with respect to IRS used as a reflector.
Figure 8 shows that the secondary throughput decreases as is near where . This degradation in performance is expected due to interference term. In fact, when is close to , the interference term in Equations (20) and (34) increases so that the SINR and throughput decrease.
5. Conclusions
In this article, we derived the secondary throughput of underlay CRN using IRS with adaptive power. IRS allows 14, 20, 26, 32, and 38 dB gains versus the absence of IRS for reflectors. We derived the throughput at the packet level as well as the CDF of SNR and SINR. IRS with fixed power was studied in previous papers while our results deal with adaptive power.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was funded by Saudi Electronic University.