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Multiband Spectrum Sensing Using Modified Daniell Windowing Technique in Full-Duplex Cognitive Radio Networks: A Performance Study

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Abstract

Full-duplex cognitive radio (CR) is a promising technology for upcoming 5G wireless communication systems. This paper presents a robust fast Fourier transform (FFT) based multiband spectrum sensing using two-dimensional averaging algorithms in orthogonal frequency division multiplexing systems in full-duplex CR networks with residual self interference under Rayleigh flat fading scenario. In the proposed algorithm, we have used modified Daniell windowing technique both in time and frequency dimensions to smoothen the FFT spectrum under full-duplex scenario. The analytical expressions for the performance metrics are derived for the aforementioned algorithm. The simulated and analytical results, obtained for the proposed algorithm using modified Daniell windowing technique are found in good agreement. Finally, the comparison studies between the proposed scheme using modified Daniel windowing technique and the conventional rectangular windowing scheme clearly depict that the proposed scheme gives optimal performance even at low SNR using fewer filter lengths in time and frequency dimensions as the modified Daniell windowing algorithm is less influenced due to spectral leakage.

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Authors and Affiliations

  1. Electronics and Communication Engineering Department, National Institute of Technology Patna, Patna, India

    Tripta & Seemanti Saha

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  1. Tripta
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  2. Seemanti Saha
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Appendix

Appendix

The mean of the text statistic T(Y[n, k]) normalized with respect to \(\sigma _{H_0}^2\) under \(H_0\) is calculated as:

$$\begin{aligned} E\left[ \frac{T(Y[n,k])}{\sigma _{H_0}^2}\right]= & {} \sum _{k=k_0,N_f}\frac{1}{2(N_f-1)}\left[ \sum _{n=n_0,N_t}\frac{1}{2(N_t-1)}E\left[ \frac{\vert Y(n,k)\vert ^2/H_0}{\sigma _{H_0^2}}\right] \right. \\&+\,\left. \sum _{n=n_0+1}^{N_t-1}\frac{1}{N_t-1}E\left[ \frac{\vert Y(n,k)\vert ^2/H_0}{\sigma _{H_1^2}}\right] \right. \\&+\,\sum _{k=k_0+1}^{N_f-1}\frac{1}{(N_f-1)}\left[ \sum _{n=n_0,N_t}\frac{1}{2(N_t-1)}E\left[ \frac{\vert Y(n,k)\vert ^2/H_0}{\sigma _{H_0^2}}\right] \right. \\&+\,\left. \sum _{n=n_0+1}^{N_t-1}\frac{1}{N_t-1}E\left[ \frac{\vert Y(n,k)\vert ^2/H_0}{\sigma _{H_0^2}}\right] \right. . \end{aligned}$$
(31)

Considering Eq. (7), the above equation is simplified as:

$$\begin{aligned} E\left[ \frac{T(Y[n,k])}{\sigma _{H_1}^2}\right]= & {} \sum _{k=k_0,N_f}\frac{1}{2(N_f-1)}\left[ \frac{1}{2(N_t-1)}(1+1)+\frac{1}{N_t-1}(1+1+\cdots N_t-2)\right] \\&\quad +\,\sum _{k=k_0+1}^{N_f-1}\frac{1}{(N_f-1)}\left[ \frac{1}{2(N_t-1)}(1+1)+\frac{1}{N_t-1}(1+1+\cdots N_t-2)\right] =1. \end{aligned}$$
(32)

Similarly, the mean of the text statistic T(Y[n, k]) normalized with respect to \(\sigma _{H_1}^2\) under \(H_1\) is calculated as:

$$\begin{aligned} E\left[ \frac{T(Y[n,k])}{\sigma _{H_1}^2}\right]= & {} \sum _{k=k_0,N_f}\frac{1}{2(N_f-1)}\left[ \sum _{n=n_0,N_t}\frac{1}{2(N_t-1)}E\left[ \frac{\vert Y(n,k)\vert ^2/H_1}{\sigma _{H_1^2}}\right] \right. \\&+\,\left. \sum _{n=n_0+1}^{N_t-1}\frac{1}{N_t-1}E\left[ \frac{\vert Y(n,k)\vert ^2/H_1}{\sigma _{H_1^2}}\right] \right. \\&+\,\sum _{k=k_0+1}^{N_f-1}\frac{1}{(N_f-1)}\left[ \sum _{n=n_0,N_t}\frac{1}{2(N_t-1)}E\left[ \frac{\vert Y(n,k)\vert ^2/H_1}{\sigma _{H_1^2}}\right] \right. \\&+\,\left. \sum _{n=n_0+1}^{N_t-1}\frac{1}{N_t-1}E\left[ \frac{\vert Y(n,k)\vert ^2/H_1}{\sigma _{H_1^2}}\right] \right. . \end{aligned}$$
(33)

Using Eq. (11), the above equation is written as:

$$\begin{aligned} E\left[ \frac{T(Y[n,k])}{\sigma _{H_1}^2}\right]= & {} \sum _{k=k_0,N_f}\frac{1}{2(N_f-1)}\left[ \frac{1}{2(N_t-1)}(1+1)+\frac{1}{N_t-1}(1+1+\cdots N_t-2)\right] \\&+\,\sum _{k=k_0+1}^{N_f-1}\frac{1}{(N_f-1)}\left[ \frac{1}{2(N_t-1)}(1+1)+\frac{1}{N_t-1}(1+1+\cdots N_t-2)\right] =1. \end{aligned}$$
(34)

Further, under \(H_0\), the variance of the text statistic T(Y[n, k]) normalized with respect to \(\sigma _{H_0}^2\) is calculated as:

$$\begin{aligned} var\left[ \frac{T(Y[n,k])}{\sigma _{H_0}^2}\right]= & {} \frac{1}{4(N_f-1)^2}\frac{1}{4(N_t-1)^2}\sum _{k=k_0,N_f}\sum _{n=n_0,N_t}\left[ \frac{1}{\sigma _{H_0^4}}var(\vert Y(n,k)\vert ^2/H_0)\right] \\&+\,\frac{1}{4(N_f-1)^2}\frac{1}{(N_t-1)^2}\sum _{k=k_0,N_f}\sum _{n=n_0+1}^{N_t-1}\left[ \frac{1}{\sigma _{H_0^4}}var(\vert Y(n,k)\vert ^2/H_0)\right] \\&+\,\frac{1}{(N_f-1)^2}\frac{1}{4(N_t-1)^2}\sum _{k=k_0+1}^{N_f-1}\sum _{n=n_0,N_t}\left[ \frac{1}{\sigma _{H_0^4}}var(\vert Y(n,k)\vert ^2/H_0)\right] \\&+\,\frac{1}{(N_f-1)^2}\frac{1}{(N_t-1)^2}\sum _{k=k_0+1}^{N_f-1}\sum _{n=n_0+1}^{N_t-1}\left[ \frac{1}{\sigma _{H_0^4}}var(\vert Y(n,k)\vert ^2/H_0)\right] , \end{aligned}$$
(35)

where, \(var[\vert Y[n,k]\vert ^2/H_0]=var[\vert Z[n,k]+W[n,k]\vert ^2]=2\sigma _{z,k}^4+2\sigma _{w,k}^4.\) Thus, Eq. (35) is further simplified as:

$$\begin{aligned}&var\left[ \frac{T(Y[n,k])}{\sigma _{H_0}^2}\right] =\frac{8}{16(N_f-1)^2(N_t-1)^2}+\frac{4N_t-8)}{4(N_f-1)^2(N_t-1)^2} \\&\quad +\,\frac{4N_f-8}{4(N_f-1)^2(N_t-1)^2}+\frac{2(N_t-2)(N_f-2)}{(N_f-1)^2(N_t-1)^2}=\frac{4N_tN_f-6N_t-6N_f+9}{2(N_f-1)^2(N_t-1)^2}. \end{aligned}$$
(36)

Similarly, the calculation of varT(Y[n, k]) under the hypothesis \(H_1\) is summarized as:

$$\begin{aligned} var\left[ \frac{T(Y[n,k])}{\sigma _{H_1}^2}\right]= & {} \frac{1}{4(N_f-1)^2}\frac{1}{4(N_t-1)^2}\sum _{k=k_0,N_f}\sum _{n=n_0,N_t}\left[ \frac{1}{\sigma _{H_1^4}}var(\vert Y(n,k)\vert ^2/H_1)\right] \\&+\,\frac{1}{4(N_f-1)^2}\frac{1}{(N_t-1)^2}\sum _{k=k_0,N_f}\sum _{n=n_0+1}^{N_t-1}\left[ \frac{1}{\sigma _{H_1^4}}var(\vert Y(n,k)\vert ^2/H_1)\right] \\&+\,\frac{1}{(N_f-1)^2}\frac{1}{4(N_t-1)^2}\sum _{k=k_0+1}^{N_f-1}\sum _{n=n_0,N_t}\left[ \frac{1}{\sigma _{H_1^4}}var(\vert Y(n,k)\vert ^2/H_1)\right] \\&+\,\frac{1}{(N_f-1)^2}\frac{1}{(N_t-1)^2}\sum _{k=k_0+1}^{N_f-1}\sum _{n=n_0+1}^{N_t-1}\left[ \frac{1}{\sigma _{H_1^4}}var(\vert Y(n,k)\vert ^2/H_1)\right] \\= & {} \frac{4N_tN_f-6N_t-6N_f+9}{2(N_f-1)^2(N_t-1)^2}. \end{aligned}$$
(37)

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Tripta, Saha, S. Multiband Spectrum Sensing Using Modified Daniell Windowing Technique in Full-Duplex Cognitive Radio Networks: A Performance Study. Wireless Pers Commun 117, 293–309 (2021). https://doi.org/10.1007/s11277-020-07869-z

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