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Unformatted text preview: ECE 562 Fall 2011
December 1, 2011
Final (Max score = 130 pts) Assigned: December 1, 2011
Due:
December 9, 2011 at 1:00 PM in room 106 CSL Note: This is an openbook, takehome exam. You can use your class notes, the text book,
any of the references that I’ve put on reserve in the library, and the HW sets and solutions.
The contents of the exam are conﬁdential until 1 PM on December 9, 2011. Until that time,
you must not show this examination to, or discuss its contents with, any person other than
Prof. Veeravalli. This also applies to your solutions to the exam of course. You may use
Matlab or similar software packages to do your numerical computations. 1. (15 pts) Signal Space and Performance in AWGN. Consider the signal set shown below:
s1(t)
√ s2(t)
√ Es
0 0.5 1 1.5 2 0 0.5 t s3(t)
√ 1 1.5 2 t 1 1.5 2 t s4(t)
√ Es
0 0.5 Es 1 1.5 2 t Es
0 0.5 (a) Find an orthonormal basis for this signal set and express the signals in terms of
this basis.
(b) Find the measure of goodness ζ for the signal constellation.
(c) Suppose this signal set is used for communication over an AWGN channel with
noise PSD N0 . Assuming equally likely symbols, ﬁnd the IUB and NNA approximations for the average probability of symbol error Pe in terms of γs = Es /N0 .
(d) Now ﬁnd an exact expression for Pe in terms of γs . c V.V. Veeravalli, 2011 1 2. (25 pts) V.29 Modem. The V.29 standard for 9.6 Kbit/s modems uses the following
16ary signal constellation: 5 3 1 −5 −3 −1 1 3 5 −1 −3 −5 Assume equally likely symbols.
(a) Carefully draw the MPE decision regions for this constellation
(b) Find the IUB for Pe as a function of γs .
(c) Find the NNA for Pe as a function of γs .
(d) Find the NNA to Pe for a (square) 16QAM constellation as a function of γs .
(e) Comparing (c) and (d), which constellation is more power eﬃcient at high SNR?
(f) Determine whether you can label the signal points using three bits so that nearest
neighbors diﬀer by at most one bit (Gray coding). If so, ﬁnd such a labeling. If
not, state why not and ﬁnd a labeling that minimizes the maximum number of
bit transitions between neighbors.
(g) For the labelings found in part (a), compute the nearest neighbor approximation
for the average bit error probability Pb as a function of γb . c V.V. Veeravalli, 2011 2 3. (15 pts) QPSK with Errors and Erasures Decoding. Consider signaling using the QPSK
constellation shown in the ﬁgure below. Instead of making a decision in favor of one
of the symbols for every value of the demodulator output R, suppose the detector has
the option of declaring an erasure for certain values of R. The decision regions for one
such detector are shown, with the shaded region corresponding to erasure. (Note that
the shaded regions extend out to inﬁnity in all four directions.) 1111111111111111111111111111111
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a
d Assume equally likely symbols. Let α = a/d. Note that 0 ≤ α ≤ 1.
(a) Find exact expressions for the average probability of error (Pe ) and the average
probability of erasure (Ps ) as a function of γs and α.
(b) Find good high SNR approximations for Pe and Ps .
(c) Use the approximations of part (b) to ﬁnd the value of α such that Ps = 2Pe , at a
symbol SNR of 7 dB. You may use Matlab to solve for the value of α numerically.
Remark: The motivation for setting Ps = 2Pe in part (c) is that a typical error correcting code can correct twice as many erasures as errors. c V.V. Veeravalli, 2011 3 4. (15 pts) Demodulation with phase uncertainty. Consider the 4ary constellation shown
below:
Q
s2
d
s1
0
s4 s3
2d I 3d Suppose this signal set is used on an AWGN channel with phase oﬀset φ that is unknown
to the receiver. After demodulation we obtain the suﬃcient statistic
R = sm ejφ + W
when symbol m is sent, m = 1, 2, 3, 4, where W ∼ CN (0, N0 ).
(a) Assuming that φ ∈ [0, 2π ], ﬁnd mJML (r).
ˆ
(b) Carefully draw the decision regions for the JML detector.
(c) Find a closedform expression for maxm Pe,m for the JML detector.
5. (15 pts) Equalization. Consider the problem of BPSK communication on an ISI channel
with:
√
√
√
Es = 2, and h0 = 0.5, h1 = 0.3, h2 = − 0.2
Suppose the received sequence (z1 , z2 , . . . , z6 ) is given by (2,0.5,1,0,2,3).
(a) Find the MLSE estimate of the transmitted symbols.
(b) Find the estimate of the transmitted symbols produced by the matched ﬁlter
equalizer.
(c) Find the ZF equalizer (e.g., using Matlab), and ﬁnd the estimate of the transmitted symbols produced by the ZF equalizer.
6. (20 pts) Selection diversity for fading channels. Consider BPSK signaling on an Lth
order diversity channel, where the received signal at the output of channel is given
by
r (t) = ±α ejφ
Eb g (t) + w (t)
= 1, . . . , L.
where {g (t)} are unit energy pulse shaping functions, and the processes {w (t)} are
independent WGN processes with PSD N0 .
We know from class that maximalratio combining (MRC) across the L channels is
optimum. However, it may not always be practical to combine the signals in this
optimal fashion. In some cases (e.g., the reverse link in the wireless cellular system),
the receiver may base its decision only on the strongest of the L channels. This is
c V.V. Veeravalli, 2011 4 known as selection diversity. Speciﬁcally, the receiver ﬁrst determines the channel
with the largest amplitude, i.e.,
λ = arg max α
∈{1,...,L} and then uses the statistic
RSD = rλ (t), gλ (t)ejφλ
to make a decision. A decision in favor of +1 (symbol 1) is made if Re(RSD ) > 0, and
−1 (symbol 2) otherwise.
(a) Assuming that the fading across the L channels is i.i.d. Rayleigh, show that
distribution function of αλ is given by:
Fαλ (x) = P{αλ < x} = 1 − e−x 2 L 1 { x≥ 0 } .
1 (b) For ﬁxed {α }L=1 , show that the bit error probability for the selection diversity
receiver is given by:
Pb = Q(αλ 2γ b )
where γ b = Eb /N0 .
(c) Use integration by parts to show that the average bit error probability with fading
for the selection diversity receiver is given by
√
∞
γ
2
Fαλ (x) √ b e−γ b x dx
Pb =
π
0
(d) Show that the expression for Pb of part (c) can be written in closedform as:
1
Pb =
2 L L
k (−1)k
k=0 γb
,
k + γb Hint: You may want to use the binomial expansion for Fαλ (x) as given in part
(a), and the fact that a zeromean Gaussian pdf integrates to 1/2 over [0, ∞).
7. (25 pts) Bitbybit noncoherent demodulation of orthogonal signals. Consider M ary
orthogonal signaling on a slow, ﬂat fading channel. Assume that M = 2ν for some
positive integer ν . The received signal corresponding to one symbol interval is given
by
√
r(t) = αejφ E gm (t) + w(t)
when symbol m is sent on the channel. The functions {gm (t)} are orthonormal.
From class we know that the statistics Rk = r(t), gk (t) , k = 1, . . . , M , are suﬃcient
for optimum detection. The MPE decision rule for noncoherent detection is given by
mMPE = arg max Rk  = arg max Rk 2 = arg max Yk ,
ˆ
k c V.V. Veeravalli, 2011 k k (1)
5 where Yk = 1
Rk 2 .
N0 Each M ary symbol corresponds to a sequence of ν bits. Suppose these bits come
from the output of an error control encoder. Then the orthogonal demodulator will
be followed by a decoder. The MPE decision rule of (1) can be used to produce hard
decisions for each bit in the ν bit sequence. However, it is desirable to produce soft
outputs for each bit to enable soft decision decoding.
(a) Let Y = [Y1 , . . . , YM ] . Assuming that the fading is Rayleigh, show that the
likelihood function for symbol m, based on y , is given by:
ym e− γ +1
pm (y ) =
γ+1 1
e−yk 1 {y≥0}
k =m where 1 {y≥0} equals 1 if all the components of y are nonnegative, and 0 otherwise,
1
and γ = E /N0 .
(b) Based on Y , we would like to form a soft decision statistic for the th bit in the
ν bit sequence that represents the M ary symbol, = 1, 2, . . . , ν . Assume equal
()
priors on the symbols. Let pi (y ) denote the conditional pdf of Y , given that the
th bit equals i, i = 0, 1. The loglikelihood ratio for the th bit is given by:
() () L (y ) = ln p1 (y )
() . p0 (y )
Show that
m∈S L (y ) = ln exp ym γ
1+γ ¯
m∈S () exp ym γ
1+γ ¯
where S = {m : th bit of symbol m is 1}, and S is its complement.
(c) The metric L( ) (y ) is the desired soft decision statistic for the th bit. Argue that
L( ) (y ) can be approximated as
L( ) (y ) ≈ γ
Λ( ) (y )
1+γ where
Λ( ) (y ) = max ym − max ym .
m∈S ¯
m∈S (d) The approximate metric Λ( ) (y ), known as the dualmax metric, is independent
of γ , and is used for softdecision decoding in IS95 based CDMA systems. Now,
suppose instead that we use Λ( ) (y ) for making a hard decision on the th bit as
follows:
()
ˆ ,DM = 1 if Λ (y ) > 0
b
0 otherwise
Show that ˆ ,DM is the same as the hard decision for the th bit based on the
b
MPE decision rule of (1).
c V.V. Veeravalli, 2011 6 ...
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