ECE 614 – Principles of Digital Communications
Homework 1
Assigned on: 01/26/2016
Due by: 02/04/2016
1.1
(a) According to Claude Shannon, what is the minimum SNR required to achieve 56.6 kb/s
over an ideal channel with AWGN and a bandwidth of 4 kHz?
(b) According to Claude Shannon, what is the maximum bit rate that can be achieved by a
transmitter with power P=80 mW over an ideal channel with 4 kHz of bandwidth and
AWGN with one-sided noise power density N0=0.1 W/Hz?
(c) Which of the above two systems has a larger spectral efficiency?
1.2
Let st   k  g t  kT  be a pulse train with period T and the following pulse shape:

g t   4
sin 10t / T 
10t / T
(a) Sketch s(t) (using MATLAB)
(b) Find the energy of g(t)
(c) Find the power of g(t)
(d) Find the energy of s(t)
(e) Find the power of s(t).
1.3

(a) Find the Hilbert transform of st   cos 440t  
8


5

5
(b) Find the in-phase and quadrature components of st    2 cos 90t  
 with
respect to the carrier frequency f 0  55 Hz
(c) Find the in-phase and quadrature components of st    2 cos 90t  
 with
respect to the carrier frequency f 0  60 Hz
(d) Find the in-phase and quadrature components of st    2 cos500t  sin t   with
t 

respect to the carrier frequency f 0  250 Hz
(e) Find the in-phase and quadrature components of st    2 cos500t  sin t   with
t 

respect to the carrier frequency f 0  251 Hz
1.4
Let st   sin 1024t 
. Specify a particular carrier frequency f 0 (in Hz) such that the
1024t
quadrature component of st  with respect to f 0 is zero.
1.5
Let xt  denote the in-phase component of a real signal st  with respect to the particular
carrier frequency of f 0  300 Hz
(a) Does there exist a signal st  such that xt   st  ?
(b) If yes, give an example of such an st  .
If no, explain why it is not possible for a signal to be equal to its in-phase component.
1.6
Consider a real-valued signal r t  with Hilbert transform r̂ t  . Let rI t  and rQ t  denote
the in-phase and quadrature components of r t  with respect to a carrier frequency f 0 , so
that r t   rI t  2 cos2f 0 t   rQ t  2 sin 2f 0 t  , and let xt  and yt  denote the inphase and quadrature components of r̂ t  with respect to f 0 , so that
rˆt   xt  2 cos2f 0 t   yt  2 sin 2f 0 t  .
Express xt  and yt  as a function of rI t  and rQ t  .
1.7
(a) Prove that a real signal st  has the same energy as its Hilbert transform ŝ t  .
(b) Prove that the energy of a real st  is equal to the energy of its complex envelope ~
s t  .