Processing of Auto and Cross-correlations of Two Jointly Wss Random Processes through Various Types of Windows in Time Domain

PROCESSING OF AUTO AND CROSS-CORRELATIONS OF TWO JOINTLY WSS RANDOM PROCESSES THROUGH VARIOUS TYPES OF WINDOWS IN TIME DOMAIN Ejaz A. Ansari, Saleem Akhtar, Muhammad Nadeem Department of Electrical Engineering, COMSATS Institute of Information Technology 1.5 km Defense Road, off Raiwind Road, 53700, Lahore, Pakistan {dransari, sakhtar, munadeem{@ciitlahore.edu.pk} corresponding author’s email: {[email protected]} ABSTRACT: In many situations, transform techniques are quite useful for analyzing the wide sense stationary (WSS) processes through LTIsystems. However, if the input process is not a stationary white noise and is further passed through a(n) exponential, rectangular, triangular and trapezoidal shaped window, then, it becomes difficult to compute auto-correlation and cross-correlation of the output process using transform techniques. Thus, in this paper, we present a very simple and efficient technique in time domain for the computation of auto and cross correlations of the output WSS process through the above windows. First, we develop the relationship of auto and cross correlation of the output signal in terms of auto-correlation of the input deterministic signal and the impulse response of the filter without using the integral definition of the convolution explicitly. Then, we repeat the analysis in terms of two jointly WSS processes, X(t) and Y(t) denote input and output process respectively. Subsequently, we apply these results and derive the algorithm first for an exponential shaped window and further use it intelligently while processing X(t) through a rectangular (also known as moving average), triangular and trapezoidal shaped windows. Furthermore, we validate our proposed method in case of a moving average window through comparison with the results already reported in the literature. Finally, our proposed technique in time domain can easily be extended to discrete-time signals and discrete-time WSS random processes.