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UniversitY of Minnesota
Department of Electrical and Computer Engineering
EE 454L: Digital Signal Processing, Fall 2008
Midterm #2 (Maximum Points: 100)
Your Name:
you must show oll work on these sheets. [Jse reverse side of each sheet if necessory.
problem 1- corries 20 points, Problem 2 15 points, Problem 3 30 points, Problem 4 15
points, and Problem 5 20 Points.
Page 1 of 18
University of Minnesota
Department of Electrical and Computer Engineering
EE 454L2 Digital Signal Processing, Fall 2008
Midterm #2 (Maximum Points: 100)
Your Name:
you must show oll work on these sheets. tJse reverse side of eoch sheet if necessory'
problem L corries 20 points, Problem 2 15 points, Problem 3 30 points, Problem 4 L5
points, ond Problem 5 20 Points.
Page 1 of 18
1-. (20 points) Consider the block diagram and the spectrum below:
xlnlI a
v,r' J --? vIn]
-r -nirl xl1
(a) (8 points) Derive an expression for W(ej*) and plot W(e:w;.
Lutnl= th . Lthl.-E
- gi?tt. T.gnl
^ 'q,
€ q;1eloy= x(ei(b'*T))
{ii,-
X{el "i
Page 2 of 18
(b) (12 points) Derive an expression for Y(ejw; and plot Y(elw;.
r r' '-tn *b-b * \N'-4'!t -l
yreu,.l* + | \ru(elT )+ t$(ei'- ) + \lot€'-T- )-J1L
:: 6 :4q:
= .+ [xr*r*t*-*)) *
x(e*F€)-t+) + xt*F(*E) dit )l' v .*€ ^*-2*) +xtJ#>l=t ltdT ) +xt€)"-r
Ysq C"uk need +c )se
ct'tv' -tv'\se q$ES '
Page 3 of 18
2. (15 points) Let xIn]={1, 2,3,4} and y={1, L,'J','J"}.
(a) (6 points) Find w[n] = x[n] * y[n]
rubr-l= ! i. :. e . lo. 1" -T . 4 \.
(b) (9 points) Find v[n] = *[n]@y[n] using wIn]. Do not compute vIn] from definition of
circular convolution.
( 3 6 io 1 T' +qT+' $d'- '/ (rrug$i- s[/'1+-*------*---"''
to [s lo to
-v8\kltortorto. to\
Page 4 of 18
3. (30 points) Let x[n] be a real sequence {a, b, c, d, e}, and its DFT is given by X[k]={A, B, C, D, E}.
(a) (5 poi,nts) What is the DFT of {A, B, C, D, E}?
!=
Esto9rtii : ')(ml <-9 N1l(-h')hl
.'. bF-f "€ 1* , E. c .b.E lse . se . 5*.. se " sb}
(b) (5 points) what is the DFT of the ffence
{a+d, b+e, c+a, d+b, e+c}?
lo-t. c.d- "] L!n.B-c..D.EibFT *.3o&
ld. e. a..t..] - xt(n->))sf < ) e)-E- Vclf
.i:r, ^gK' ' l?F - iCT' \
lA.B,a'b"E! + 1l',*i€'s' u'i$ c' eiJg b'e-'-E El
= [r*, (l+eiF> ts . cr+e >S) e -(t+e')F) u " (t+ei€")E 1
Page 5 of 18
(c) (5 points)What is*the DFTof {a, b, c, !2b,99?Xr.
q *8,Xc.ckll= i xocn:e 1F'e-n
!" --.j#h +**.n-s:s{*e'= f icfft1thF€ T ;Pu i'ck
= + t'rrnl *i€* ' eJh:!0 w-/
t ttJt/s^" f e\}o'an
-t t.:lvo-rr.. L "d$
f 2Xr+f . he'^e-v'=( to " k-od&
"'. [email protected]. b.2c. (l' 2b. e\ 2e . "3(d) (5 points) What is the DFT of {A, 0, B, O, C, 0, D, 0, E, 0,*€r?
DFT 6,€ 1 4. . b. c. b-E 3 = tso.'se,. sd'.. 5c. Stl
Up**f\ by {Uds}n 2 [e*dr rc -tr*e ceireS '
Page 5 of 18
(e) (5 points) What is the DFT of {a, e, d, c, b}?
)tnbrt= rt (t-rr))5J
:rlrc+'D5: +----' x*cbl
x."r c(- n l)sl
tsx.t (t*n'))sl
s Xetkf=SXerkf
= IEA* . EBl* 5C - Sbt' 5EI \
funtl- "r-tr1] ii r'wO.
x.cP3= lnt - B* \
(f) (5 points) What is the DFT[DFTlDFTlxln]lll?
i --'-,i YcLl 1
Lt(-h))g) =
C . 'b*. E*
€ xtp:
Page 7 of 18
4. (15 points) Consider the block diagram below:
wlnl
xlnl a
W(2.\(a) (3 points) rind #.
^\.)
n:-Uil + ' vlnl
XBr: r +?-r
Page 8 of 18
tt\ t- ' - \-, ,,V(Z)(b) (b pornts, Ftno *' X(Z)
| - z?++- B*a
(t+':'*i)>
=Q*?-f: i -:?{*?-?
(c) (3 points) Find H(z) :#
tN tt) \'cr)ft cb)=
xL}) *
x*>
= 2 *z* +?*z
vr3)
-=xl:)
Page 9 of 18
(d) (3 points) Find hlnl.
hrnt=t=.*,. '!
Page 10 of 18
5. (20 points) Consider X(e;w) and Y(elw).
X{*t*J
1
>rE
T
,l'it-4n19 -n/J -lr/rg lrils rr/l 4ri/q
25c
1(a) (8 points) Express Y(elw; in terms of X(elw).
uqg[: €*S : tr^tssw-?L'1 [V ]ItrJl
.'. Ytd* )= x(d=t**+))\Q***) ..- x(€' )
Page 11 of 18
(b) (8 points) Express y[n] in terms of x[n].
r13ir*rt) \yCdtul= X(e,., )ec .-f (;tr'{) h
= Z 'LCYr-le I
It=-€cs .11 --Tte '3n
= i nrv'f '(--t)'' e'f\;:S
eo 'nry' -'{uM= i a-L+f ' (-{):i'e '\Yr=-6't )
rn:vuutr5StZsei
*r+l^ (-tf/'d
o
l{r\ir:i-)
:. Sthk
xrh:l ----->s;-_-+ fFl* SLnl
tGrf
rwwtttle- b€ ]
br vJ.
(c) (a points) Draw a block diagram to obtain y[n] from x[n].
Page 12 of 18
