HW4
1- Two discrete-time random processes are defined as ??[??] = ??[??] and ??[??] =
!”
cos( # ) ??[??] for ?? < ?? < ?, where U[n] is white noise with variance ??$% . Are
the random processes X[n] and Y[n] jointly WSS? (20 points)
)
2- If the CCS is given as ??&,( [??] = (? #)|+,)| for ?? < ?? < ?, plot it and describe
its three properties that are different from an ACS. (20 points)
3- If ??[??] and ??[??] are jointly WSS with ACSs
2 |%+|
??& [?? ] = 5 5 8
+ ?? [?? ]
3
??( [?? ] = 5?? [??] + 3?? [?? ? 1] + 3?? [?? + 1] + ?? [?? ? 2] + ?? [?? + 2]
determine the maximum possible value of ??&,( [?? ]. (10 points)
)
4- For the two sinusoidal random processes ??[??] = % ??????(2????- ?? + ??) ) and ??[??] =
)
??????( 2????- ?? + ??% ), where ??) = ??% ~??(0,2??) find the CCS and CPSD and explain
#
your results versus the case when ??) and ??% are independent random variables. (20
points)
5- For the random processes ??[??] = ??[??] and ??[??] = ??[??] + ????[?? ? 1], where
U[n] is discrete white noise with variance ??$% = 4, find the CPSD and explain what
happens as ?? ? 0. (20 points)
6- If a continuous-time white noise process ??(??) with ACF ??& (??) = ??(??) is input to
an LTI system with impulse response ?(??) =
points)
. (“#$)
#
??(??), determine ??&,( (??). (10

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