Performance Analysis of a Cognitive Phased Array Radar with Online Tracking Capability

Abstract

This paper presents the performance analysis of a cognitive phased array radar with online tracking capability. The versatile model has the ability to sense the environment for adaptive selection of active number of array elements in the transmitter, as well as, in receiver arrays. An online extended Kalman filter with a kernel recursive least squares observer has been used to predict the future direction of target. The proposed radar avoids continuous scanning of the surveillance region. Instead, the transmitter, based on the feedback from the receiver, ensures maximum power in the target future direction. It saves a lot of power. Likewise, the intelligent selection of transmitting and receiving array elements, at each scan, results in less computational complexity and improved detection probability along with signal to interference plus noise ratio than that of a conventional phased array. Monte-Carlo based simulations have been conducted to validate the performance of the design.

Appendix

In order to formulate KRLS algorithm, we proceed by taking the recorded sequence

$$\varvec{d}\left( k \right) = \left[ {\left( {y_{1} } \right), \ldots ,\left( {y_{k} } \right)} \right]$$

KRLS algorithm assumes a functional form given as,

$${\varvec{\Phi}}\left( k \right) = \left[ {\varphi \left( 1 \right), \ldots ,\varphi \left( k \right)} \right]$$

In order to find the weight vector w(k) at each iteration we need to minimize the following cost function

$$\sum\limits_{j = 1}^{i} {\left| {d\left( j \right) - w^{T} \varphi \left( j \right)} \right|^{2} } + \lambda \left| {\left| w \right|} \right|^{2}$$

Differentiate the above mentioned cost function and by putting equal to zero we get

$$\varvec{w}\left( k \right) = \left[ {\lambda {\mathbf{I}} + {\varvec{\Phi}}\left( k \right){\varvec{\Phi}}\left( k \right)^{\varvec{T}} } \right]^{ - 1} {\varvec{\Phi}}\left( k \right)\varvec{d}\left( k \right)$$

By using the matrix inversion lemma with the following identification

$$\lambda {\mathbf{I}} \to {\mathbf{A}},\quad {\varvec{\Phi}}\left( i \right) \to {\mathbf{B}},\quad {\mathbf{I}} \to {\mathbf{C}},\quad {\varvec{\Phi}}\left( i \right)^{\varvec{T}} \to \varvec{D}$$

$$\left[ {\lambda {\mathbf{I}} + {\varvec{\Phi}}\left( k \right){\varvec{\Phi}}\left( k \right)^{\varvec{T}} } \right]^{ - 1} {\varvec{\Phi}}\left( k \right) = {\varvec{\Phi}}\left( k \right)\left[ {\lambda {\mathbf{I}} + {\varvec{\Phi}}\left( k \right)^{\varvec{T}} {\varvec{\Phi}}\left( k \right)} \right]^{ - 1}$$

So the weight vector is defined as

$$\varvec{w}\left( k \right) = {\varvec{\Phi}}\left( k \right)\left[ {\lambda {\mathbf{I}} + {\varvec{\Phi}}\left( k \right)^{\varvec{T}} {\varvec{\Phi}}\left( k \right)} \right]^{ - 1} \varvec{d}\left( k \right)$$

$$\varvec{w}\left( k \right) = {\varvec{\Phi}}\left( k \right)\varvec{a}\left( k \right)$$

$$\varvec{a}\left( k \right) = \varvec{Q}\left( k \right)\varvec{d}\left( k \right)$$

$$\varvec{Q}\left( k \right) = \left[ {\lambda {\mathbf{I}} + {\varvec{\Phi}}\left( k \right){\varvec{\Phi}}\left( k \right)^{\varvec{T}} } \right]^{ - 1}$$

By using the matrix partition, we partition the matrix Q as

$$\varvec{Q}\left( k \right) = \left[ {\begin{array}{*{20}c} {\lambda {\mathbf{I}} + {\varvec{\Phi}}\left( {k - 1} \right)^{\varvec{T}} {\varvec{\Phi}}\left( {k - 1} \right)} & {\varvec{h}\left( k \right)} \\ {\varvec{h}\left( k \right)^{\varvec{T}} } & {\lambda + \varphi \left( k \right)^{T} \varphi \left( k \right)} \\ \end{array} } \right]^{ - 1}$$

here $\varvec{h}\left( k \right) = {\varvec{\Phi}}\left( {k - 1} \right)^{\varvec{T}} \varphi \left( k \right)$.

By using the Block matrix inversion lemma

$$\left[ {\begin{array}{*{20}l} \varvec{A} \hfill & \varvec{B} \hfill \\ \varvec{C} \hfill & \varvec{D} \hfill \\ \end{array} } \right]^{ - 1} = \left[ {\begin{array}{*{20}l} {\left( {A - BD^{ - 1} C} \right)^{ - 1} } \hfill & { - A^{ - 1} B\left( {D - CA^{ - 1} B} \right)^{ - 1} } \hfill \\ { - D^{ - 1} C\left( {A - BD^{ - 1} C} \right)^{ - 1} } \hfill & {\left( {D - CA^{ - 1} B} \right)^{ - 1} } \hfill \\ \end{array} } \right]$$

$$\varvec{Q}\left( k \right) = \left[ {\begin{array}{*{20}c} {\varvec{Q}\left( {k - 1} \right) + \varvec{z}\left( k \right)\varvec{z}\left( k \right)^{\varvec{T}} r\left( k \right)^{ - 1} } & { - \varvec{z}\left( k \right)r\left( k \right)^{ - 1} } \\ { - \varvec{z}\left( k \right)^{\varvec{T}} r\left( k \right)^{ - 1} } & {r\left( k \right)^{ - 1} } \\ \end{array} } \right]$$

As $\varvec{a}\left( k \right) = \varvec{Q}\left( k \right)\varvec{d}\left( k \right)$

$$\varvec{a}\left( k \right) = \left[ {\begin{array}{*{20}c} {\varvec{Q}\left( {k - 1} \right) + {\mathbf{z}}\left( k \right){\mathbf{z}}\left( k \right)^{\varvec{T}} r\left( k \right)^{ - 1} } & { - {\mathbf{z}}\left( k \right)r\left( k \right)^{ - 1} } \\ { - {\mathbf{z}}\left( k \right)^{\varvec{T}} r\left( k \right)^{ - 1} } & {r\left( k \right)^{ - 1} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varvec{d}\left( {k - 1} \right)} \\ {d\left( k \right)} \\ \end{array} } \right]$$

$$\left[ {\begin{array}{*{20}c} {\varvec{a}\left( {k - 1} \right)} \\ {a\left( k \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\varvec{a}\left( {k - 1} \right) - {\mathbf{z}}\left( k \right)r\left( k \right)^{ - 1} e\left( k \right)} \\ {r\left( k \right)^{ - 1} e\left( k \right)} \\ \end{array} } \right]$$

here ${\mathbf{z}}\left( k \right) = \varvec{Q}\left( {k - 1} \right)\varvec{h}\left( k \right)$, $r\left( k \right) = \lambda + \varphi \left( k \right)^{T} \varphi \left( k \right) - {\mathbf{z}}\left( k \right)^{\varvec{T}} \varvec{h}\left( k \right)$, and $e\left( k \right) = d\left( k \right) - \varvec{h}\left( k \right)\varvec{a}\left( {k - 1} \right)$.

As we know that the output of the filter is

$$y\left( k \right) = \varvec{w}\left( {k - 1} \right)^{\varvec{T}} \varvec{\varphi }\left( k \right)$$

We also know that

$$\varvec{w}\left( {k - 1} \right)^{\varvec{T}} = \varvec{a}\left( {k - 1} \right)^{\varvec{T}} {\varvec{\Phi}}\left( {k - 1} \right)^{\varvec{T}}$$

So the output is

$$y\left( k \right) = \varvec{a}\left( {k - 1} \right)^{\varvec{T}} {\varvec{\Phi}}\left( {k - 1} \right)^{\varvec{T}} \varvec{\varphi }\left( k \right)$$

$$y\left( k \right) = \varvec{a}\left( {k - 1} \right)^{\varvec{T}} \varvec{h}\left( k \right)$$

where $\varvec{h}\left( k \right) = {\varvec{\Phi}}\left( {k - 1} \right)^{\varvec{T}} \varvec{\varphi }\left( k \right)$.