TUTORIAL #4-2013
1
T UTORIAL 4
Question 1
Let vp denote a real passband signal, with Fourier transform Vp (f ) specied as follows for negative
frequencies:
f + 101,
Vp (f ) =
0,
101 f 99
(1)
f < 101 or 99 < f < 0.
(a) Sketch Vp (f ) for both positive and

TUTORIAL #1-2013
1
T UTORIAL 1
Question 1:
2
Consider the vector subspace L2 (R C) = cfw_u : u
2
|u(t)| dt
< . Let v and u be two
vectors in this space represented as v (t) and u(t). Let the inner product be dened by
v, u =
v (t)u (t)dt.
(1)
(a) Assume t

MIDTERM PRACTICE PROBLEM SOLUTIONS-2013
1
Solution 1
Since CN is an N -dimensional vector space over the eld F, then S is a vector subspace of CN iff
1) The zero vector, 0 S;
2) x1 + x2 S;
where x1 , x2 S and , F. Show that property 1) holds:
A0 = 0, A CM

1
PRACTICE PROBLEMS
Problem 1. Let A be an M N matrix and
S = cfw_x : Ax = 0, x CN ,
(1)
where each x denotes an N 1 column vector. Please check whether or not S is a linear subspace of
CN and justify your answer.
Problem 2. Consider the following modulat

TUTORIAL #2-2013
1
T UTORIAL 2
Question 1:
Consider the following set of functions cfw_um (t) for integer m 0:
u0 (t) =
1, 0 t 1
0, otherwise
.
.
.
um (t) =
1, 0 t 2m
(1)
0, otherwise
.
.
.
Consider these functions as vectors cfw_u0 , u1 , . . . over the

TUTORIAL #7-2013
1
T UTORIAL 7
Question 1:
The signal constellation shown in Fig.1 is obtained by moving the outer corner points in rectangular
16-QAM to the I and Q axes.
Fig. 1.
Signal Constellation for Question 1
(a) Sketch the ML decision regions.
(b)

TUTORIAL #6-2013
1
T UTORIAL 6
Question 1:
The 8-ary signal constellations are shown in Fig.1.
Fig. 1.
Signal Constellations for Question 1
(2)
(1)
(a) Express R and dmin in terms of dmin so that all three constellations have the same Eb .
(b) For a given

TUTORIAL #9-2013
1
T UTORIAL 9
Question 1:
(a) For L = 4 and K = 4, explicitly represent the channel matrix H1 in equation (1), in terms of the
entries of h.
y1 = H1 s + 1 .
(1)
(b) If we let y1 denote the remaining vector of length 8 after we have delete

TUTORIAL #8-2013
1
T UTORIAL 8
Question 1:
Consider 2-user BPSK signaling in AWGN, with received signal
y = b1 u1 + b2 u2 + n,
(1)
where u1 = (1, 1)T , u2 = (2, 1)T , b1 , b2 take values 1 with equal probability and n is AWGN of
variance 2 per dimension.

TUTORIAL #3-2013
1
T UTORIAL 3
Question 1:
Consider a general function, p(t), which satises the following constraints:
p(t) = p2 (t), t;
p(t) = 0, for |t| > 1;
p(t) = p(t), t;
p(t) = 1 p(t 1), for 0 t 1.
(1)
(a) Show that p(t) is orthogonal to p(t 1).
(b)

TUTORIAL #5-2013
1
T UTORIAL 5
Question 1:
Consider two real-valued passband pulses of the form
s0 (t) = cos(2f0 t + 0 ), 0 t T,
s1 (t) = cos(2f1 t + 1 ), 0 t T,
where f1 f0
1/T . The pulses are said to be orthogonal if s0 , s1 =
(1)
T
0
s0 (t)s1 (t)dt =