ECE 532 Homework 1 Solutions
1. X is a random variable with pdf () = 2 , with -1x1.
a. What is the value of k?
b. If = 2 , find [].
Answer:
1
a. () = 1 2 = 1
3
1
3
1
1
1
= 3 + 3 =
2
3
3
= 1 = 2
b. = 2 . If = (), [] = [()] = ()().
1
3
3 5
[] = 1 2 2 2 =
ECE 532
Final project guidelines
The final project includes two sections: an investigation section and a coding section.
In the investigation section, describe the feature you selected, its properties, how it is used, why it is
used. This section has a wi
Channel coding
ECE 532 Advanced Digital Communications
Parameters
Input: k bits Output: n bits (n>k)
Code rate: = ( <1)
Assume constellation of size M is used in modulator (M=power of 2)
Number of transmitted symbols: =
Symbol duration:
Time to transmit
Repeaters and
link budget
ECE 532 Advanced Digital Communications
Analog and regenerative repeaters
Analog repeaters
Periodically boost signal strength, but also boost the noise
Example: binary PAM =
Regenerative repeaters
2
Demodulate/detect the signal,
KL expansion
ECE 532 Advanced Digital Communications
Karhunen-Loeve expansion
Set of functions cfw_1(t), 2(t), , n(t) is orthonormal if
1
=
m(t)n(t)dt = =
0
A wide sense stationary random process x(t) can be expanded as:
= n(t), with random variable
Signal demodulation
ECE 532 Advanced Digital Communications
Receiver
Signal demodulator: converts r(t) into an N-dimensional vector r=[r1 r2 rN]
Correlator demodulator
Matched filter demodulator
Detector: decides which of the M waveforms was transmitted (
Digital modulation
schemes
ECE 532 Advanced Digital Communications
Modulation
Definition: Mapping between the digital sequence (bits) and the transmitted sequence
MEMORYLESS MODULATION: Each group of k bits is mapped independently of previous bits
MODULAT
Optimum detector
ECE 532 Advanced Digital Communications
Maximum a posteriori (MAP) criterion
and Maximum Likelihood (ML) criterion
The optimum detector makes a decision based on vector such that the
probability of correct decision is maximized
Posterior
Random processes
ECE 532 Advanced Digital Communications
Definitions and stationarity
A function X(,t) is a random process if for each t, P(X(,t)x) is defined for
all x.
t integer discrete parameter random process
t continuous continuous parameter random
Random variables
ECE 532 Advanced Digital Communications
Discrete Random Variables
Random variable: random function X(), with P(X()x) defined for all x
, = sample space
Cumulative distribution function: P(X()x)
Probability distribution function (or proba
ECE 532 Home Work 5
Submitted by
Moaaz Mohammed Mumtaz
UIN 671175056
As per the given instructions I first generated a sequence of Binary bits of length
Nbits=1280 and then mapped them to a 16 QAM Constellation using Gray Mapping.
As a result the below pl