OFDM waveform design based on mutual information for cognitive radar applications

Abstract

We propose a novel optimization method based on wideband orthogonal frequency division multiplexing (OFDM) signals to detect random extended targets with known covariance matrix in the presence of additive white Gaussian noise. Mutual information is used as our criterion for waveform design under transmitted power constraint. We utilize the advantage of OFDM signal to intelligently design the complex weights of the transmitted waveform. For making complete use of the transmission power, a novel iterative algorithm is introduced based on maximizing mutual information criterion between the target impulse response and the received echoes. We have derived the optimal Neyman–Pearson detector for the corresponding hypothesis testing problem and provided different numerical experiments to demonstrate the achieved performance improvement when the proposed method is applied.

Appendices

Appendix A: Optimal Neyman–Pearson detector

The PDF of OFDM received signal can be computed as [52]:

$$\begin{aligned} P_0(Y_{\mathrm{OFDM}})&=\frac{1}{\pi ^{LN}\sigma ^{2N}}\exp \left\{ -\frac{tr[Y_{\mathrm{OFDM}}Y_{\mathrm{OFDM}}^H]}{\sigma ^2}\right\} \nonumber \\ P_1(Y_{\mathrm{OFDM}})&=\frac{1}{\pi ^{LN}\mathrm{det}^N((N+1)AC_xA^H+\sigma ^2)}\nonumber \\&\quad \times \, \exp \{-tr[((N+1)AC_xA^H+\sigma ^2))^{-1}Y_{\mathrm{OFDM}}Y_{\mathrm{OFDM}}^H]\} \end{aligned}$$

(27)

The log-likelihood function can be simplified by Eq. (28).

$$\begin{aligned} l(Y_{\mathrm{OFDM}})=&\log \frac{P_1(Y_{\mathrm{OFDM}})}{P_0(Y_{\mathrm{OFDM}})} \nonumber \\ =&\frac{1}{\sigma ^2}tr[Y_{\mathrm{OFDM}}Y_{\mathrm{OFDM}}^H] \nonumber \\&-tr[((N+1)AC_xA^H+\sigma ^2))^{-1}Y_{\mathrm{OFDM}}Y_{\mathrm{OFDM}}^H]+K \end{aligned}$$

(28)

where $K=\pi ^{LN}\sigma ^{2N}-\pi ^{LN}\mathrm{det}((N+1)AC_xA^H)$ is a constant term which is not dependent on $Y_{\mathrm{OFDM}}$. Therefore, the optimal Neyman–Pearson detection statistics is stated by Eq. (29).

$$\begin{aligned} T(Y_{\mathrm{OFDM}})&=\frac{1}{\sigma ^2}tr[Y_{\mathrm{OFDM}}Y_{\mathrm{OFDM}}^H] \nonumber \\&\quad -tr[((N+1)AC_xA^H+\sigma ^2))^{-1}Y_{\mathrm{OFDM}}Y_{\mathrm{OFDM}}^H] \end{aligned}$$

(29)

If T(Y) exceeds a given threshold, we say a target exists.

Appendix B: Computing the diagonal elements of matrix B

We suppose $B=AR_{\tilde{\Phi }}A^H+R_n$ and now we want to compute diagonal elements of B as $d_l(B)$. Assuming AWGN noise with known power $P_n=\sigma ^2$, we can write matrix B by equation (30).

$$\begin{aligned} B&=AR_{\tilde{\Phi }}A^H+\sigma ^2 I \nonumber \\&=U_s\Lambda _s V_s R_{\tilde{\Phi }} (U_s\Lambda _s V_s)^H+\sigma ^2 I \nonumber \\&=\Lambda _s V_s^H R_{\tilde{\Phi }} V_s \Lambda _s^H+\sigma ^2 I \end{aligned}$$

(30)

Since $V_s$ is a unitary matrix, it must be equal to the eigen-matrix (matrix of eigenvectors) of $R_{\tilde{\Phi }}$. Now, since $R_{\tilde{\Phi }}$ is a diagonal matrix, its eigen-matrix is $I_L$. Thus, we can write $V_s=P$, where P is a permutation matrix, and we have:

$$\begin{aligned} B=\Lambda _s R_{\tilde{\Phi }} \Lambda _s^H+\sigma ^2 I \end{aligned}$$

(31)

Finally, the diagonal elements of matrix B can be written as Eq. (32).

$$\begin{aligned} d(B)=\sigma ^2 +\Lambda _s^2(l,l) R_{\tilde{\Phi }} (l,l) \end{aligned}$$

(32)