TUTORIAL #9-2014
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
Summary II: Modulation and
Demodulation
Instructor : Jun Chen
Department of Electrical and Computer Engineering,
McMaster University
Room: ITB A221, ext. 20163
Email: junchen@mail.ece.mcmaster.ca
Website: http:/www.ece.mcmaster.ca/junchen/
EE 4TM4
Digital
Summary I: Signal Space
Instructor : Jun Chen
Department of Electrical and Computer Engineering,
McMaster University
Room: ITB A221, ext. 20163
Email: junchen@ece.mcmaster.ca
Website: http:/www.ece.mcmaster.ca/junchen/
EE 4TM4
Digital Communications
Vecto
S OLUTION
OF THE
M IDTERM T EST
We start the solution manual with some reminders about the Fourier transform. Note that sinc(t) in
the time domain is equivalent to rect(f ) in the frequency domain, where
0 |f | > 1
2
.
rect(f ) =
1 |f | 1
2
(1)
It is also
S OLUTION
OF THE
S AMPLE M IDTERM T EST
We start the solution manual with some reminders about the Fourier transform. Note that sinc(t) in
the time domain is equivalent to rect(f ) in the frequency domain, where
0 |f | > 1
2
.
rect(f ) =
1 |f | 1
2
(1)
It
S OLUTION
OF
Q UIZ 2A
Question 1
for part (a) note that convex envelope of an arbitrary passband function xp is x where xp (t) =
2Re(x(t)ej2f0 t ). Note that f0 is the reference frequency. Hence,
u(t)
1
1
up (t) = 2Re( ej(200t+/6) ) = 2Re( ej/6 ej200t )
2
Solution of Sample Final
Question 1
Omitted.
Question 2
If two functions u and v are orthogonal, it means that their inner product is
equal to zero, i.e., u, v = uv dt = 0. For part (a)
, =
dt =
(t)ej2fc t (t)ej2fc t dt =
(t) (t)dt = 0
where the last equ
TUTORIAL #1-2014
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 tha
TUTORIAL #3-2014
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 #4-2014
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 #5-2014
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 =
TUTORIAL #2-2014
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)
1, 0 t 2m
0, otherwise
.
.
.
Consider these functions as vectors cfw_u0 , u1 , . . . over the
TUTORIAL #7-2014
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 #8-2014
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 #6-2014
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
Summary III: MIMO Communications
Instructor : Jun Chen
Department of Electrical and Computer Engineering,
McMaster University
Room: ITB A221, ext. 20163
Email: junchen@mail.ece.mcmaster.ca
Website: http:/www.ece.mcmaster.ca/junchen/
EE 4TM4
Digital Commun