document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166371>Question 1166371</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose the random variable X(t)is a wide-sense stationary random process.<BR>\n" );
document.write( "Prove that the random process Y(t)=X(at) is wide-sense stationary, where is a constant. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852760 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2030)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852760\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a proof in the theory of random processes. To prove that $Y(t) = X(at)$ is a **Wide-Sense Stationary (WSS)** random process, we must show that it satisfies the two conditions for WSS:\r<BR>\n" );
document.write( "\n" );
document.write( "1.  Its mean is constant (independent of time $t$).<BR>\n" );
document.write( "2.  Its autocorrelation function depends only on the time difference, $\tau = t_2 - t_1$.\r<BR>\n" );
document.write( "\n" );
document.write( "We are given that $X(t)$ is WSS. Therefore, $X(t)$ satisfies:<BR>\n" );
document.write( "$$E[X(t)] = \mu_X \quad (\text{a constant})$$<BR>\n" );
document.write( "$$R_X(t_1, t_2) = E[X(t_1)X(t_2)] = R_X(t_2 - t_1) = R_X(\tau)$$\r<BR>\n" );
document.write( "\n" );
document.write( "Let's check the two WSS conditions for $Y(t) = X(at)$.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 1. Mean of $Y(t)$\r<BR>\n" );
document.write( "\n" );
document.write( "The mean of $Y(t)$ is $E[Y(t)]$.\r<BR>\n" );
document.write( "\n" );
document.write( "$$E[Y(t)] = E[X(at)]$$\r<BR>\n" );
document.write( "\n" );
document.write( "Since $X(t)$ is WSS, its mean is constant, $\mu_X$, regardless of the time index used (whether it is $t$ or $at$).\r<BR>\n" );
document.write( "\n" );
document.write( "$$E[X(at)] = \mu_X$$\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the mean of $Y(t)$ is:<BR>\n" );
document.write( "$$\mu_Y = E[Y(t)] = \mu_X$$\r<BR>\n" );
document.write( "\n" );
document.write( "Since $\mu_X$ is a **constant**, $\mu_Y$ is **independent of time $t$**. The first condition for WSS is met.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 2. Autocorrelation of $Y(t)$\r<BR>\n" );
document.write( "\n" );
document.write( "The autocorrelation function of $Y(t)$ is $R_Y(t_1, t_2) = E[Y(t_1)Y(t_2)]$.\r<BR>\n" );
document.write( "\n" );
document.write( "$$R_Y(t_1, t_2) = E[X(at_1)X(at_2)]$$\r<BR>\n" );
document.write( "\n" );
document.write( "Since $X(t)$ is WSS, its autocorrelation $R_X$ depends only on the time difference, $\tau_X = at_2 - at_1$.\r<BR>\n" );
document.write( "\n" );
document.write( "$$R_Y(t_1, t_2) = R_X(at_2 - at_1)$$<BR>\n" );
document.write( "$$R_Y(t_1, t_2) = R_X(a(t_2 - t_1))$$\r<BR>\n" );
document.write( "\n" );
document.write( "Let $\tau = t_2 - t_1$ be the time difference for $Y(t)$.\r<BR>\n" );
document.write( "\n" );
document.write( "$$R_Y(t_1, t_2) = R_X(a\tau)$$\r<BR>\n" );
document.write( "\n" );
document.write( "Let $g(\tau) = R_X(a\tau)$. Since $\mu_X$ and $a$ are constants, $g(\tau)$ is a function that depends **only on the time difference $\tau$**, and not on $t_1$ or $t_2$ individually.\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the autocorrelation of $Y(t)$ is:<BR>\n" );
document.write( "$$R_Y(t_1, t_2) = R_Y(\tau)$$\r<BR>\n" );
document.write( "\n" );
document.write( "The second condition for WSS is met.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## Conclusion\r<BR>\n" );
document.write( "\n" );
document.write( "Since $Y(t)$ has a constant mean and its autocorrelation function depends only on the time difference $\tau$, the random process $Y(t)=X(at)$ is **Wide-Sense Stationary**. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );