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Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2010
Homework #9
Due 12/10/10 (in class) #$" !©§
¥ ¡¨
¦¥¤£¢ ¡ be a widesense stationary Gaussian random process with mean zero and autocorrelation
be a white Gaussian noise process with power spectral density ¦¥¤£¢ ¡ ¥
&¤£¡ % . Let ¦¥¤£¢ ¡
2
31 (a) Find , the power in (b) Is . strictsense stationary?
. ¥4 7 ¥ ¡ I¥ ¡
6`YTSRQ¢XHG¢¤ ¡ 1
¥¡U
WVHB
¥ ¡ I¥ ¡ I¥ ¡ 658¢QPTSR¢QPHG¢9 F
¦¥¤£¡ %
¥
C5B¡ E &¤£¡ D
¥
¦¥¤£¢ C5BA¨ @
¡ ¥¡
) that has input
. Find (g) Find , the pdf of . . is a zeromean Gaussian random process with autocorrelation function through an ideal lowpass ﬁlter with unity gain and bandwidth
and zero otherwise) as shown below: Ideal LPF
BW edcp
bG edcG hf B f
bg
¥¡
&¤£Q ¥
¡ 0 (a) Suppose that I run
is,
= 1 for Hz &i¥ ¤£rq
¡ ¥¡
&¤£Q 2. Suppose that
sinc
. and output with power spectral denisty ¥
C5B¡ E ¥¡
&¤£Q i Find: ¥¡
&£¤Q ¥ ¥¡
68`vaHGuq¡ 1 s
t
31 s
¦¥¤£¡uq
¥t
C5B¡ )@ s
, the power in . ¦¥¤£uq
¡ , the power spectral density of . . through an ideal differentiator as shown below: ¥¡
&¤£rq i ¦xw w
i (b) Suppose that I run edcG
b (f) Let or . . ¥¡¨
a§ (e) Find a ﬁlter (give of F (d) Find the power spectral density ¥¡
C5BA¨ @ ¥8 7 ¥ ¡
69!654¢¤ ¡ 1 (c) Find . )0( ' 1. Let Hz (that ¥¡
&¤£Q Find: ¥ ¥¡
68`vaHGuq¡ 1 s
t
31 s
¥¦¤£¡uq
¥t
C5B¡ )@ s
, the power in . ¦¥¤£uq
¡ , the power spectral density of . . G ¦¥¤£¨
¡ ¥¡
&¤£Q 3. Let
be a widesense stationary Gaussian random process with mean
and autocorrelation function
, where
is the Dirac delta function. The random process
is
run through a ﬁlter combination as shown below, where the ﬁrst ﬁlter has frequency response: ¥ ¤£h¡
¡ g
4 cf B f R § ¨
¨
© G ¥
0 pC5B¡ 0 E
¥ ¡¡ ¥ ¡
W¢G R ¨ § C5B¡ ¦E
¥¥ . ¥ ¥ ¥¡
C5B¡ E i &¤£Q ¥¡
&¤£rq ¥
&¤£¡ F i C5B¡ 0 E i
¥
¥
&¤£¡ F ¦¥¤£¡ 0 F
¦¥¤£uq &¤£¡ 0 q
¡¥ (b) Find the power ¥¡
&¤£Q and the second ﬁlter has frequency response (a) Find the power otherwise in . in . Be sure to simplify your answer as much as possible. 4. Per class, a main utility of widesense stationarity (WSS) is that it allows us to deﬁne the power
spectral density for a given random process. However, there is a large class of random processes that
are not widesense stationary for which we are still able to derive the power spectral density. ¥
&¤£¡ t ¥¡
&¤£uq A random process
is deﬁned to be cyclostationary if both its mean function
and the
autocorrelation function
are periodic (in ) at some period ; that
is,
and
for some . #
&" is given by the Fourier trans ¥ BB A @86¥
&S£©C©58¡ ¢97&¤£¡
3 ¡
¦¥¤£¢ is some constant. Use the deﬁnitions above to derive the power spectral density for ¥¡
&¤£Q be a zeromean widesense stationary random process with autocorrelation function ¥4
¡ 5§ ¦¥¤£¡ 3
¥
¡ t ' §
0
¥¡
&£¤rq
¥t
C5B¡ P@
(
0(
1)# "
£ ¨ I
H¤2¦¥¦£¢P¤£¡ t § ( R ¡ t ' §
¥
¥
&¤£¡rq
#%"
#
I # I § I ¨
)¦£¨ &" C¤£¡ t p¥ QP¦£!¤£¡ t § t ¥ %" P¤£¡ t ¦¥¤£¡ t I
C¤¡ q &¤¡ q
¥ I £ ¥ £ P¦£!¤£¡ §
¥ I¨
¥
¢I B B where £ Let
and let of a cyclostationary process #
$" and the power spectral density
form of
. is given by: The “average autocorrelation” of a cyclostationary process . , ...
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This note was uploaded on 09/16/2011 for the course ECE ECE603 taught by Professor Dennisgoeckel during the Fall '10 term at UMass (Amherst).
 Fall '10
 DennisGoeckel

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