Lecture 5
Analysis of Error Probability
P(e) = 1 P(C)
Analysis of P(e) for the case where M > 2 is not straightforward. We shall start with the case of M = 2
signals, s0 (t ) and s1 (t ) , 0 t < T , and assume the messages are equally likely, i.e., P0 = P
Lecture 4
The Optimum Receiver
Given the received signal
r (t ) = s (t ) + n(t ) ,
0t <T ,
we would like to decide which signal s (t ) S = cfw_s i (t ),0 t < T iM 0 1 has been sent.
=
We can construct an orthonormal basis cfw_i (t ), 0 t < T i =0 such tha
Lecture 6
Analysis of Digital Modulation Techniques
Binary phase-shift-keying (BPSK)
An example of an antipodal set is binary phase shift-keying (BPSK). The signals in the set are
s0 (t ) =
2 Es
cos 0t , 0 t <T , 0-bit
T
s1 (t ) =
2 Es
cos(0t + )
T
=
2 Es
Lecture 7
Communication Theory
By now, we know how the error probability P(e) depends on Es / N 0 (SNR), and that P(e) decreases if
Es / N 0 increases. We would like to explore the following issues.
Question 1: Can P(e) be decreased by other means such as
Lecture 11
Coded Systems (continued)
Convolutional codes
We use the running example of the rate-, constraint length-3 code with generators g
and g
(2)
(1)
= [1 0 1]
= [1 1 1] .
00
00
01
00
00
11
11
00
01
10
10
11
11
11
01
10
10
11
0
t
1
2
3
For a message
Lecture 10
Channel capacity (continued)
Question: What is the capacity C if W ?
P
C = lim CW = lim W log 2 1 + s .
W
W
WN 0
Using the definition of e given by
1
1
= lim (1 + ) x ,
i ! X
x
i =0
e=
we can write
1
W
P
C = lim log 2 1 + s
W
WN 0
WN
Lecture 9
Communication Theory (continued)
The ensemble average codeword error probability taken over the set of all codes,
P (e ) <
1 N ( R0 RN )
,
2
2
also known as a random coding bound, implies P (e) 0 as code block length N provided that
the code rat
Lecture 8
Communication Theory (continued)
Coded systems
The requirement of a very large bandwidth as well as a very stringent timing synchronization accuracy
makes it very difficult to implement the block-orthogonal signaling system in practice. A practi
Lecture 1
The study of digital communications starts with the following fundamental problem.
Given a set of messages cfw_si (t ), 0 t < T iM0 1 , where T is the message duration, the transmitter chooses
=
a message s (t ) = s i (t ) for transmission in ea
Lecture 2
Vector Representation (continued)
Given a set of signals S = cfw_si (t ),0 t < T iM 0 1 for transmission, we can construct an orthonormal basis
=
cfw_i (t ), 0 t < T iN 1 where N M, such that every si (t) S can be represented as
=0
N 1
si (t ) =
Lecture 3
Random Processes (continued)
Statisitical stationarity statistical properties of the random process are time-invariant in some sense.
Such stationarity is required in many engineering applications.
Strict-Sense Stationarity of order N states tha
Lecture 1
The study of digital communications starts with the following fundamental problem.
Given a set of messages cfw_si (t ), 0 t < T iM0 1 , where T is the message duration, the transmitter chooses
=
a message s (t ) = s i (t ) for transmission in ea