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Unformatted text preview: (22 pts) 1. Let and y(t) be the input and the output of a continuous
time system, respectively. Answer each of the questions below with either
yes or no (no justification needed). Yes No
If y(t) : :I:(2t), is the system causal? If y(t) = (if + 2):):(t), is the system causal? If y(t) : 3:(~t2), is the system causal? >5
If y(t) = + t v 1, is the system memoryless? R><
If y(t) : 00(9), is the system memoryless?
If y(t) = 113(25/3), is the system stable? l><
If y(t) : 7555(25/3), is the system stable? If y(t) = [:00 513(7‘)d7’, is the system stable? If 3/(25) : sin is the system time invariant? >5
If y(t) 2: u(t) * 513(t), is the system LTI? If y(t) : (tu(t)) * ﬁt), is the system linear? x“? (15 pts) 2. ‘ An LTl system has unit impulse response Mt) : u(t + 2).
Compute the system’s response to the input $(t) : e‘tu(t). (Simplify your
answer until all 2 signs disappear.) «d ,
Q ulﬂu liwde”
W0 (15 pts) 3. Compute the energy and the power of the Signal 512(t) : Mm 3*??me
Emt 3 MM 3‘: 33%.] gm W «.W «(33%
E e
w Er‘M‘V g «L i» Z? :rmim
a” " %:eé? :2 A? 22'7"“ e § 3W
“T” o W} i" :54??? m g (15 pts) 4. Computethe coefﬁcients ak of the Fourier series of the signal periodic with period T : 4 defined by / _ sinbﬂ), O§t§2
$(t)_{ 0, 2<t§4' (Simplify your answer as much as possible.) 5. A discrete—time system is such that when the input is one of the signals
in the left column, then the output is the corresponding signal in the right
column: input output
$0 n] = —> yaw] = (5[n m l], :6[n1] —+ y1[nl:46[n~2],
TQM] : 6[n — 2] a yin] : 96in * 3],
$3M :6[n~3] —> y3[n] :166[n—4], = (5[n ~ k3] —+ 2 (Is +1)26[n ~ (k: + 1)] for any integer k.  “M (10 PtS) 8) Can this system be time—invariant? Explain. _ V m v.1 E
r r ’ lmv. ' i , {2W ' if is xiii} ‘ie 53 i its W 13> it let U i» (10 pts) b) Assuming that this system is linear, What input would yield
the output y[n] 2 Mn — 1]? Wei» xiii}: my {33%wa in 2 3 Facts and Formulas CT Signal Energy and Power E00 —/ WNW:
T P00 : lim T——>oo 2T VT : Z akejk(27ﬂ)t
kzioo
1 T W
ak : :I:(t)e’ﬂ”(2T Properties of CT Fourier Series x(t)[2dt Fourier Series of CT Periodic Signals with period T Let 56(25) be a periodic signal with fundamental period T and fundamental frequency
wo. Let y(t) be another periodic signal with the same fundamental period T and
fundamental frequency we. Denote by ak and bk the Fourier series coﬂicients of X(t) and y(t) respectively. Linearity:
Time Shifting: Conjugation: Parseval’s Relation Signal 019005) + 6W) :z:(t e to) 93W) real and even
‘ Mt) real and odd FT
flak: + 53% (5)
e—jkwotoak (6)
air (7)
ak real and even (8)
ak pure imaginary and odd
(9)
(10) 4: DT Signal Energy and Power E0O : Z (11)
P ~ 1' 1 i lrrrlnllg <12)
00 ‘ NEEO2N+1WW 5 Fourier Series of DT Periodic Signals with period N V N~1 ‘ 2 : Z akeJMWﬂ)” (13)
TL _ 2W
ak : N a:[n]e’]k(W)n (14)
71:0 6 Properties of DT Fourier Series Let be a periodic signal with fundamental period N and fundamental fre—
quency we. Let y[n] be another periodic signal with the same fundamental period
N and fundamental frequency we. Denote by ak and bk the Fourier series cofﬁcients
of X(t) and y(t) respectively. Signal FT
Linearity: arm] + ﬂy 04a;C + ﬁbk (15)
Time Shifting: ~ no] e_jk‘“0”0ak (16)
Conjugation: 56* afik (17) real and even ak real and even (18) real and odd ak pure imaginary and odd
(19) 7 . 1 N—l 2 N—1 2 Parseval 8 Relation. N r; dt — 1;) akl (20) Properties of LTI systems LTI systems commute. The response of an LTI system with unit impulse response h to a signal 36
is the same as the response of an LT i system with unit impulse response 1: to the signal h. An LTI system consisting of a cascade of k LT I systems with unit impulse
responses in, fig, . A . , hk respectively, is the same as an LTI system with unit impulse response hl >k hg * . . . * his. The response of a CT LTI system with unit impulse response h(t) to the
signal eSt is H(s)e‘9t where H(s) : ff; h(T)e’STdT. The response of a DT LTI system with unit impulse response h[n] to the
signal 2“ is Hm" where He) 2 22:.“ hlklzw ...
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 Spring '06
 V."Ragu"Balakrishnan
 Digital Signal Processing, Fourier Series, LTI system theory, Impulse response, DT LTI

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