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Unformatted text preview: ECE 5670 Prelim
Spring 2011 Instructor: Salmau Avesﬁmehr Qﬂice: 325 Rhodes Hall
Tel: 59915. Email: avestimehr©ecacomel£edu March 17, 2011 Do not open this exam until you are instructed to do so. You have 90 minutes to complete this exam. You are permitted one letter—size crib sheet (oneside only). Other
wise, the exam is closedbook and closednote. No collaboration is permitted. Name Going)” Marconi
Problem 1 (18 pts)
Problem 2 (20 pts)
Problem 3 (30 pts) I Problem 4 (32 pts) I II Total (100 pts) H Marja: #0
871) ‘. If l. (18 points) Short questions. (a) (3 points) What is the minimum energy per bit required for communicating over a
discrete AWGN channel with noise variance 2‘? {byltrzlgwl :7} a.” 7Qﬁt~9~ (b) (5 points) The additional SNR required to send one extra bit in PAM modulation
scheme is about 6 dB. Explain precisely what this statement means and why it is
true. I‘) (arro') it: e,
212%» +0 la“? 41“: reliqLLL'j (em ?"I’“L5tfj) He Same Owl/f 36A]
amt ouJJJL‘na] L}; we: RCeJ ‘I‘o quaJraﬂ/g #5 SNR, ' ‘ in; I, S {Q I?
#10133“ :1 {0'5 3‘) W; “QEJ 4; agonMJej’ mCé’ 6 ﬁle, N .j 5% (c) (5 points) State the inverse of the sampling Theorem. What is its role in a digital
communication system? f .— 1‘3 ﬁre.» +fn$ 6‘) “£48 max;,.4m (unite, 5'1 D/A (opt/E’ia;aa /’( (6M“““(" é‘rq
war a chnAet amid gm 14/. (d) (5 points) What is the largest possible rate of reliable communication over a contin
uous time AWGN channel with no bandwidth constraints, but with power constraint
P and noise spectral density Sw( f) = 329, f E (—001 00)? What is the connection
between this rate and your answer to part (a)? 2. (20 points) Consider two received symbols over an ISI channel ylll = $111+ mm] + will
y[2] 3x[1] + 2$[0] + sc[2] + w[2] Here x[0], $[1],1:[2] are independent and equally likely to be ix/E. The additive noises
w[1] and w[2] are i.i.d. N (0, 1) random variables independent of the transmit symbols.
We would like to use y[1] and y[2] to detect the symbol :c[1] using a linear receiver. Derive expressions (in terms of E) for (a) (5 points) Matched ﬁlter, <3” :' thjrﬁjfz] (b) (15 points) MMSE equalizer. A .. I (a) MC 3 UL'Z] dxmfl]dtkfa)_¢u(l
We. fIrslr Cfea‘l'e j 5"): 3013 0‘30] .. 31C 342.!» 4 1 + . ’ (3‘ 3*)5‘03 + (9.51.0).wnf2') 4—w(13_dm(e]¢u(
a M 2ch Suck “mi" £013.}... Lufolegfﬂ) )] , 5 [clans.0132] ,, d. efucn‘} =(2—ME ¢'~ : o ECécﬂ (xcolwm :37 ‘7‘ 5 #9:?—
6H
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We. Hm. crude g0]: C303” C3 2) ELM/‘56 : 30“ (3,006. 5 J 4 z . @615“ 3 Lemma} 5ch
: (\  “343C?" ) 'jCI'J 4. C3416 351319;: Egéeﬂ 3. (30 points) Consider an AWGN channel
yin] = min] + win] where {w[n]} is an i.i.d. sequence of N (0, 02) noise. In class we consider detection
based directly on the analog outputs y[n]’s. In many communication systems, however,
only a quantized version of the analog output is available, generated by the AID. In
that case, detection is based on T(y[n]) instead of y[n], where T(.) is a quantizer. For
simplicity, we consider the 1bit quantizer T(y) = sign(y)a, where sign(y) is the sign
of y and a is a constant. (a) (8 points) Suppose x[n] is equally likely to be +\/E or —\/E. What is the optimal
rule for detecting $[n] from T(y[n])? What is the detection error probability of the
optimal detector? Is there any loss in performance in using T(y[n]) instead of y[n] ' in the detection of ﬁn]? In His 01 a.
E’W T (303) 15 “amid" Sik‘i'l's‘h'cs L? claiccizj 10.} iron '30:) J— TC (“1% +0.
3’37 &.C"37 ‘V 5 j k Assn“: ‘1)u)
are thm]; _o\ ﬁniteL is also m. ’05) 1‘“ 9&1.ng ”‘7 '1) (av O. (Ema)? QC Jaye) (b; (8 points) Redo (a) if Pr(..~:[n] = x/E) > Pr(:z:[n] = —\/E). w . 3,632+»)?
ﬁ>r (T(jCﬂ])SC\“C’J=*I—é)? >C Qr£TLanjrc
afnjzJ—é m. w irtTtveWmlrWHR) =?r(wcnl>J'£)s W Iacﬂ/r)
no pvt1* tanl ): {ml v.01): are) .— §>r (“01371.J2): u CL (fa—5:] MT tutu): 41am. Mz) = Pr (“0‘3 “’63 .— MQLJEIT)
l‘LTC‘JCﬂ‘J‘i “CnlsJ'elsf" (“C"JWE): l—Q(f€;¢) 5
angina, ﬁt; ,f’hml decision a 1e is ; )‘ﬁl anr‘fé) CH5:
=> PRCJHJ‘G
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H? #00363 (PCerror ) 5 P(U"" [11(0)=.Jg
= 1?( B000} )5; + 1P(C“°*lMCn]:—JE)U10) ’JLC53:+Jg)f* ?(‘jC“)>o IMCn]:J—é )(i—P) [email protected]{ﬁmhf1MQ(‘E/r)7 QUE/w) ”6% :0 H25 (use. ‘Hlel'ﬂ. 35 Ox 1555 L“ F‘jﬂﬂwen beta“5'3. Mﬁ? W"?— ‘omeo’ 0,. Wm) mu» a. W w an m. (c) (14 points) Again suppose that $[n] is equally likely to be +x/E or —\/E, but we
now use a repetition code, where each symbol is repeated 3 times. Is there any loss in
performance in using the quantized outputs instead of the analog outputs themselves
to perform the detection? If not, explain. If so, compute approximately the additional
energy per bit (in dB) required to achieve the same target error probability because
we are using quantized outputs instead of the outputs themselves? (You can assume
the target error probability is small and use appropriate approximations.) 11 ‘ ol 3
P5(Ty=ﬂn—E=O‘HT3:OK3 l anlHE‘ ); (l—Gltfgl) (10%) Where, '0‘ c # 0L 0‘55 309!“ ““14 ﬁll2H1 Hte auﬂu‘dea‘! Sl‘GLJ‘Bl'iLS i5 :7 n‘
._  \r‘ (1‘. EMS;
Sim‘kﬁeb: ?('ﬁfqt, lg:0L,T 2“; lﬂcnj:FK/E*) :C2(—§O:) ({*Q(%€')J
A ' its m2:
:7 ML WIC : QKC"): . a: He major/17 ”’6 4. (32 points) Consider the communication over two channel uses. The transmitter chooses
with equal probability a signal 33’ E {51, :32, 553, 5134} shown in Figure 1. The receiver receives
3?: hi I” f+rﬁ
—h2 hr where 13 = (whwgf and ml and 'UJQ are i.i.d. Gaussian noises with mean zero and
variance a2, and hl and h2 are ﬁxed known real constants. Figure 1: Signal constellation. (a) (10 points) Assume that bl = 1 and kg = 0 (i.e. ﬂ = f + 13). Draw the decision
region for the ML detector and ﬁnd the detection error probability. 2
'k
Psuccessz (LOX"0:”
l
:‘7 par?!“ : l'Ll'QLl&)) (b) (10 points) Assume that h1 = kg = 1. Find the ML detection rule and the detection
error probability.  SMe. P usmuerh‘nle, 4L3 Problem I3 Hm.
3?
Same; as dgjfecijrj wt Bram
Jr 2:7 .2,
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2.
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— mm (magma) .5 a 12W. $10 «a: (c) (12 points) Assume that hf + h2 = 1. Find the ML detection error probability. Is
your answer different from part (a)? N 9 .— H :5 3915'” Rte {95%.‘1'55 mal'rfx.
, 8m. 4!». Jabw.» c! 63"“ 0‘“; M4 alum/jg WILL m1?» QM , Hﬂe egror {Valletl'a‘}
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 Spring '11
 SCAGLIONE

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