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Unformatted text preview: If ECElOl Final Exam Fall 2008 Please remember to put your name on all answer pages. 1. (7 points) A wireless channel has a bandwidth of 480 kHz. a) If a signaltonoise ratio of this channel is 27 dB, what is the information capacity of the
channel?
b) If information is transmitted through this channel at a rate of 18.4 kb/s, what is the minimum signaltonoise ratio required to support this transmission rate? 2. (10 points, total) Given the following Huffman code for a symbol source: a) (3 pts) What is the average code word length?
b) (3 pts) Decode the bit string: 11010011110110010110
c) (4 pts) If the original source symbols were encoded in a 2bit code, ﬁnd the compression ration for the given compressed string (of part b)? (show calculations / work for this part) 3. (6 points) Write a paritycheck matrix of a binary (15,11) Hamming code.
4. (5 points) How many polynomials of degree 2 with coefﬁcients in the binary ﬁeld are irreducible? 5. (16 points, total) A stream of independent information bits, each equally likely to be 0 or 1, are grouped to form row vectors 2 of length 3, which are used by the channel encoder to form code words by
1_1G, where: CD
ll
00"“ O O
1 O
O 1 Ol—‘H
j—‘p—lﬂ 0 1
1 0
1 1 (Hint: You may m need to construct a standard array in order to complete this problem.) a)
i.) (1 pt) What is the rate, r, of this code?
ii.) (2 pts) Give the set of code words, C.
iii.) (2 pts) Give the parity check matrix, H, of the code.
iv.) (1 pt) How many syndromes are there?
v.) (1 pt) Is this code cyclic?
b) (2 pts) Does this code have the largest minimum (Hamming) distance of any (7,3) linear
block code?
c) (4 pts) List the syndromes, with the coset leaders corresponding to each syndrome.
d) (3 pts) Consider a binary sequence x of length 7 that is chosen so as to maximize the Hamming distance between 3 and the nearest codeword (i.e. x is chosen as far away from
the code as possible). How far (in Hamming distance) is x from the nearest codeword? Page 1 of 3 ECE l 01 Final Exam Fall 2008 6. (10 points) The signals ¢1(t)= %cos(27;fct) and (152 (t) = %sin(27_zfct) for (k — l)T < t S kT , where k is an integer and T is the symbol duration, are used as basis functions for
band pass digital modulation. A QPSK modulated signal can be written as: s,(t)=\/§cos(2zygr+e,), (k—1)T<tskT where, . 7T
6,. = (1—1)? i=1,2,3,4
a) Express si(t) in terms of the basis ﬁinctions (M0 and (Mt). (Hint: use a trigonometric
identity.)
b) Represent the signals si(t), i=1,2,3,4, as vectors in the ¢1¢2plane (or IQ plane). Draw and
label axes, points and all relevant information. c) Sketch the decision regions, labeled 2,, i=l,2,3,4, in the (twinplane. 7. (8 points, total) Let C be the binary linear [n,k,d]code with parity check matrix:
1 1 l l O 0 0
H = l 1 0 0 1 1 0
1 0 l 0 1 0 1
a) (2 pts) Find n.
b) (2 pts) Find k.
c) (4 pts) Does d=3? Explain why or why not. 8. (8 points) A binary PAM system employs rectangular pulses of duration Tb and amplitudes :A to
transmit digital information at a rate Rb=100 kbits/s. If the power spectral density of the AWGN is N0/2,
where N0=10'2w/Hz, ﬁnd the value of A that is required to achieve a probability of error Pb=10*5. Give your answer numerically. 2E,
N 0 (Hint: For binary PAM, P(e) = : and bit energy E b = Asz .) Page 2 of 3 ECElOl 9. (30 points, 2 points each) Final Exam Fall 2008 Answer each question with True or False, and make a short notation as to why you chose an answer. a)
b) c)
d) e) g)
h) j) k)
1) If C is the capacity of a channel in bits per second, then it is not possible to transmit data
at a higher rate than R=C bits per second. If w(t) is white Gaussian noise, then w(tl) and w(tz) are independent random variables for
any t1¢t2. The MAP decision rule is a Special case of the ML rule. Let C(e) denote the capacity of a Binary Symmetric Channel (BSC) with crossover
probability a. It holds that C(1)<C(0.5). A convolutional code is described by: g1=(100), g2=(101), g3=(111) The corresponding statetransition diagram for this code is depicted below. Antipodal signals require a factor of two more energy to achieve the same error
probability as orthogonal signals. If X and Y are independent Gaussian variables, then E[XY]=0, always. The mutual information I(X;Y) between two discrete random variables X and Y is
maximized when X and Y are independent. A Hamming code always has minimum distance 3. At least three orthonormal waveforms are required to represent these four signals: 81(t) 33 (t)
t I
52 (t) 5‘ (t)
t t
PSK is a special case of QAM. Let {Xn} be a memoryless, stationary, binary (X n 6 {0,1}) random process, with P,(X,,=l) = P,(Xn=0) = '/2. Applying lossless source coding (e.g. Hamming) to this
process cannot result in any compression of the source. A modulation format cannot be both memoryless and linear. Consider transmitting a stream of binary data with rate 16 Mb/s using 8PSK modulation.
Then the distance between two nearest constellation points is d=7.42x104, when the
transmitted power is 0.05 Watts. When an AM signal is overmodulated, i.e. when 100k0m(t) > 100 , there wil be
distortion at the output of the envelope detector for any choice of modulating signal m(t). Page 3 of 3 Some potentially useful information cos(0) % (6” + 6—39) sin(0) — (ej‘9 — e'jg) sin(a :i: b)
cos(a :l: b) sin(a) cos(b) :l: cos(a) sin(b)
cos(a) cos(b) IF sin(a) sin(b) cos(a) cos(b) % [cos(a — b) + cos(a + b)]
sin(a) sin(b) % [cos(a — b) — cos(a + b)]
% [sin(a — b) + sin(a + b)] z(at + b) 1%[X 3327%]?
t _ 1 t _  p()— 0 It P<f)—%<;—ﬂ=smcm
2ij6(ffo)2ij5(f+fo) Parseval’s Relation: If X (f) is the Fourier Transform of $(t), sin(a) cos(b) /°° Izwdt = /°° X(f)2df —oo —oo TABLE OF Q(x) VALUES x Q(X) I Q(X) x Q00
0 5.000000c—01 2.4 8.1975346—03 4.8 7.933274e—07
0.1 4.601722e01 2.5 6.209665e—03 4.9 4.791830e07
02 4.207403e01 2.6 4.6611890—03 5.0 2.866516é—07
0.3 3.8208860—01 2.7 3.466973e—03 5.1 1.6982686—07
0.4 3.445783e—01 2.8 2.555131e—03 5.2 9.964437e—OG
0.5 3.085375e—01 2.9 1.865812e—03 5.3 5.7901280—08
0.6 2742531301 3.0 1.349898e—03 5.4 3.3320439—08
0.7 2.4196376—01 3.1 9.676035e—04 5.5 1.898956e—08
0.8 2.118554e01 3.2 6.8713780—04 5.6 1.071760c—08
0.9 1.840601001 3.3 4.8342429—04 5.7 5 .990378e—09
1.0 1.5865539—01 3.4 3.3692910—04 5.8 3.315742609
1.1 1.356661e01 3.5 2.3262919—04 5.9 1.8175075—09
1.2 1.150697e—01 3.6 1.591086e04 6.0 9.8658769—10
1.3 9.680049e02 3.7 1.07 7997e—O4 6.1 5.303426910
1.4 8.075666b02 3.8 7.2348063—05 6.2 2.823 1618] 0
1.5 6.680720e02 3.9 4.8096330—05 6.3 1.488226010
1.6 5.479929002 4.0 3.167124e05 6.4 7.768843e—11
1.7 4.456546902 4.1 2.065752e—05 6.5 4.016001611
1.8 3.593032c—02 4.2 1.3345760—05 6.6 2.055790er11
1.9 2.871656e—02 4.3 8.539898c—06 6.7 1.0420995—11
2.0 2.2750139—02 4.4 5.412542e06 6.8 5 .23095 1012
2.1 1.786442e—02 4.5 3.3976730—06 6.9 2.600125e—12
2.2 1.390345502 4.6 2.1124563—06 7 .0 1.279813e12
23 1.0724110—02 4.7 1.3008090—06 Bounds on Qfunction. ...
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This note was uploaded on 02/27/2012 for the course CHEMISTRY/ CH/ECE/PH/ taught by Professor Faculty during the Spring '08 term at Cooper Union.
 Spring '08
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