document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166278>Question 1166278</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.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852767 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=852767\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Yes, the random process $Y(t) = X(at)$ is wide-sense stationary (WSS) if $X(t)$ is WSS, provided $a$ is a non-zero constant.\r<BR>\n" );
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document.write( "To prove that $Y(t)$ is WSS, we must verify the two conditions for wide-sense stationarity:\r<BR>\n" );
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document.write( "1.  The **mean** of $Y(t)$, $E[Y(t)]$, is a constant (independent of $t$).<BR>\n" );
document.write( "2.  The **autocorrelation function** of $Y(t)$, $R_Y(t_1, t_2)$, depends only on the time difference $\tau = t_2 - t_1$.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 1. Mean of $Y(t)$ is Constant\r<BR>\n" );
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document.write( "Since $X(t)$ is WSS, its mean, $E[X(t)]$, is a constant, which we'll call $\mu_X$.<BR>\n" );
document.write( "$$E[X(t)] = \mu_X$$\r<BR>\n" );
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document.write( "Now we find the mean of $Y(t)$:<BR>\n" );
document.write( "$$E[Y(t)] = E[X(at)]$$<BR>\n" );
document.write( "Since $X(t)$ is WSS, the mean of $X$ evaluated at *any* time (including the time $at$) is the constant $\mu_X$.<BR>\n" );
document.write( "$$E[Y(t)] = \mu_X$$\r<BR>\n" );
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document.write( "Since $\mu_X$ is a constant (independent of $t$), the first condition is satisfied.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 2. Autocorrelation of $Y(t)$ Depends Only on Time Difference ($\tau$)\r<BR>\n" );
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document.write( "Since $X(t)$ is WSS, its autocorrelation function, $R_X(t_1, t_2)$, depends only on the time difference $\tau = t_2 - t_1$. We denote this as $R_X(\tau)$.<BR>\n" );
document.write( "$$R_X(t_1, t_2) = E[X(t_1)X(t_2)] = R_X(t_2 - t_1)$$\r<BR>\n" );
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document.write( "Now we find the autocorrelation function of $Y(t)$:<BR>\n" );
document.write( "$$R_Y(t_1, t_2) = E[Y(t_1)Y(t_2)]$$<BR>\n" );
document.write( "Substitute the definition $Y(t) = X(at)$:<BR>\n" );
document.write( "$$R_Y(t_1, t_2) = E[X(at_1)X(at_2)]$$\r<BR>\n" );
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document.write( "The term inside the expectation is the autocorrelation of $X(t)$ evaluated at times $t' = at_1$ and $t'' = at_2$. Since $X(t)$ is WSS, this expectation depends only on the difference between these times, $t'' - t'$:<BR>\n" );
document.write( "$$R_Y(t_1, t_2) = R_X(t'' - t') = R_X(at_2 - at_1)$$<BR>\n" );
document.write( "Factor out the constant $a$:<BR>\n" );
document.write( "$$R_Y(t_1, t_2) = R_X(a(t_2 - t_1))$$\r<BR>\n" );
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document.write( "Let $\tau = t_2 - t_1$.<BR>\n" );
document.write( "$$R_Y(t_1, t_2) = R_X(a\tau)$$\r<BR>\n" );
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document.write( "Since $R_X(a\tau)$ is a function that depends only on the time difference $\tau$ (and the constant $a$), the second condition is satisfied.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "Since both the mean of $Y(t)$ is constant and the autocorrelation of $Y(t)$ depends only on the time difference, the process $Y(t) = X(at)$ is **wide-sense stationary**. </SPAN>\n" );
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