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Unformatted text preview: Question 1: [Multiple Choices, 20%] Consider a LT I system with impulse response h(t) =
Ll(t) — u(t — 2). Consider an input signal = cos(27rt) + sin(7rt) and denote the
corresponding output as y(t). For convenience, we let :zzc(t) = cos(27rt) and $80) = sin(7rt)
and X (to), Xc(w), Xs(w), H (w), and Y(w) are the corresponding the Fourier transforms
ofa:(t), watt), x505), h(t), and y(t) respectively. 1. [Outcome 4, 4%] What is the value of XC(7r)? (a) 0, (b) 0.5, (c) 0.500. 2. [Outcome 4, 4%] What is the value of Xs(7r)? (a) 0, (b) ——0.5j, (c) —0.500j.
3. [Outcomes 2 and 4, 4%] What is the value of H(0)? (a) 0, (b) 1, (c) 2. 4. [Outcomes 2, 4, and 5, 4%] What is the value of Y(1 ? (a) 0, (b) 1, (c) 2. 5. [Outcomes 4, and 5, 4%] How would you name the system? (a) a. low—pass ﬁlter, (b)
a highpass ﬁlter. Question 2: [Short—Answer, 15%] Please provide a short, onesentence explanation of the
following terms / theorems. 1. [Outcomes 1, 4, and 5, 3%] Fourier series / transformations convert the original
signal 27(23) (or to another representation with different “basis signals.” What
type of Signals are the “basis signals” of the Fourier series / transformation repre—
sentation? 2. [Outcome 4, 3%] Suppose X (to) is the Fourier transformation of What is the
physical meaning of X (0)? What is the physical meaning of X (100)? 3. [Outcome 4, 3%] What is the physical meaning of the Parseval’s theorem / relation—
ship? 4. [Outcomes 4, and 5, 3%] An important feature of converting signals to their Fourier
representations is that the response y(t) = h(t) >I< :r(t) of a LTI system becomes
Y(w) = H(w)X What is the physical meaning of the last equation? 5. [Outcomes 1, 4, and 5, 3%] A LTI system of impulse response h(t) is invertible if and only if its corresponding Fourier transform H (w) 7é 0 for all m. Why is it so?
Hint: Y(w) = H(w)X(w). / Sinuyolﬂfnl tum/63‘ (Campley ego/redial fig/14(5)
2. = DC ODmPone/Etw X0013) " WDMZ'mLe/fig/ [170 Campone/ft~ 3‘ fetal energy Conserve st 41 The output of a smorde Wax/e hp at F: §fAtLSonA[ wlﬁk Mpliztwfe C/tMge HCW)’ M
Phase climate 4m») 5‘ Because We, can OOAS'THACt ﬁlé flu/else
Pin/«26. resf arise as H10») 2 PIE{:5
:5 HifW)*YLW)= er) Question 3: [Workout Question 25%] Consider a discrete signal of period 5 such
that within one period, the values of the signal is 0 ifn=0
l ifn=1
:c[n[= 2 ifn=2.
0 ifn=3
0 ifn=4 1. [Outcome 4, 7%] Find out the Fourier series representation of Note that this
question is asking a complete representation so you not only have to specify the
coefﬁcient ak but also have to specify the corresponding frequency. 2. [Outcome 4, 5%] Find out the value of 2:20 ak. 3. [Outcome 4, 6%] Find out the value of 2:20 [ak2. Note: (1k may be a complex
number and  ~  is the corresponding absolute value of a complex number. 4. [Outcomes 4, and 5, 7%] Let y[n] = — :r[n — 1]. Find out the Fourier series
representation of Again, one needs to specify both the coefﬁcients Bk and the
corresponding frequency. 2K :9 ﬁle Corrapondfrér 'R/Q/ﬁuehC/é)” [ix «Qt—oil .— We, {l _ Question 4: [Workout Question 20+5%] Consider a serial concatenation of two LTI
systems as follows, h2(t) 3/“) where h1(t) = e“3tI/{(t) and h2(t) = 63—57105). 1. [Outcomes 2, 3, 4, and 5, 10%] Let h(t) = h1(t) * h2(t) denote the impulse response
of this serially concatenated system. Find H 2. [Outcomes 4, and 5, 10%] Suppose x(t) = em, and we know that the output y(t) = Aej(2t+0). Find out the gain factor A and the phase shift 6. Hint: Let C = A610 and
rewrite y(t) = Ceﬂt, Now consider input—output relationship in the Fourier domain
and ﬁnd the value of C. 3. [Bonus: Outcomes 4, and 5, 5%] Find h(t). Hint: One can obtain the result from
H (w) or one can directly compute the convolution. H‘ (to): hictye'wale Question 5: [Work—out Question 25%] Consider a discrete moving average system of
Window size 3: h[n] = 1/3(6[n] + 6[n ~— 1] + 6[n — 1. [Outcomes 4, and 5, 10%] Find out the Fourier transform H (w) of Note: we
are considering the discrete Fourier transform. 2. [Outcome 4, 10%] = cos(3n + 7r/2). Find out the Fourier transform X(w) of 3. [Outcomes 2, 3, 4, and 5, 5%] What is the output y[n] and its Fourier transform Y(w)?
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 Fall '06
 V."Ragu"Balakrishnan
 Digital Signal Processing, Fourier Series, 1K

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