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 , a
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
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 0
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.
Q
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
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=
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
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
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 choos
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 tha
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 applic
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 choos