Error Probability Performance of Binary
Transmission Systems
The theory of probability is an essential mathematical tool in the design of digital
communication, this is due to:
1) The statistical modeling of source that generates the information
2) The digitization of source output
3) The characterization of the channel through which the digital information is
transmitted
4) The design of the receiver that process the information-bearing signal from
the channel
5) The evaluation of the performance of the communication system.
In digital communications, at the most basic level, we talk about receiving bits of a
message after they pass through a channel with some probability of error. This
description doesn't even have meaning until we have a theory of probability.
In digital transmission, the number of bit errors is the number of received bits of
a data stream over a communication channel that have been altered due
to noise, interference, distortion or bit synchronization errors. The bit error
rate (BER) is the number of bit errors per unit time.
Example
As an example, assume this transmitted bit sequence:
0110001011
and the following received bit sequence:
0 0 1 0 1 0 1 0 0 1,
The number of bit errors (the underlined bits) is, in this case, 3. The BER is 3
incorrect bits divided by 10 transferred bits, resulting in a BER of 0.3 or 30%.
The bit error rate defined as
BER=
number of bit errors
total number of transferred bits
The BER depends on several factors
•
•
•
•
the modulation type, ASK FSK or PSK
the demodulation method
the noise in the system
the signal to noise ratio
In a noisy channel, the BER is often expressed as a function of the
normalized carrier-to-noise ratio measure denoted
Eb/N0, (energy per bit to noise power spectral density ratio),
or
Es/N0 (energy per modulation symbol to noise spectral density).
Gaussian Probability Distribution and White Noise
Gaussian processes play an important role in communication systems. The
fundamental reason for their importance is that thermal noise in electronic devices,
which is produced by random movement of electrons due to thermal agitation, can
be closely modeled by a Gaussian process.
1
f ( x  , ) 
e
 2
2
Where
 mean
 standard deviation
x random variable
 x- 2
2 2
The Error Function and Q-Function
The error function sometimes called as Gauss error function. The effect of noise
in digital communications is evaluated using the Q-function. The error function is
defined as:
The Q-function and complementary error function as related as follows:
Where the complementary error function erfc(x) is defined by
Error Probability Performance of Binary Transmission Systems
A) Amplitude Shift Keying
 A cos(ct ) ,for symbol 1
s (t )  
0
,for symbol 0

0t T
0t T
With T is an integer time1/f c . The probability of error Pe is
 A2T
Pe  Q 
 4 N0


 Eb 
  Q 


N
0 


Where
Eb  A2T 4 is the average signal energy per bit
N 0 Noise power spectral density
B) Frequency Shift Keying
 A cos(c1t ) ,for symbol 1
s (t )  
 A cos(c2t ) ,for symbol 0
0t T
0t T
With T is an integer time1/f c . The probability of error Pe is
 A2T
Pe  Q 
 2N0


 Eb 
  Q 


N
0 


Where
Eb  A2T 2 is the average signal energy per bit
N 0 Noise power spectral density
C) Phase Shift Keying
,for symbol 1
 A cos(c1t )
s (t )  
 A cos(ct   ) ,for symbol 0
0t T
0t T
With T is an integer time 1/f c . The probability of error Pe is
 A2T
Pe  Q 
 N0


 2 Eb
  Q 

 N0




Where
Eb  A2T 2 is the average signal energy per bit
N 0 Noise power spectral density
Probability of Error for Binary Modulation
Example 1
If the average BER is 1 in 105, what is the probability of having
a) Single bit error
b) Single bit correct
c) At least one error in the 8-bit byte?
Solution
a) probability of having single bit error 1/105 = 0.00001
b) probability of having single bit correct 1- 0.00001= 0.99999
c) probability of having one or more errors in the 8-bit byte is
1-(0.99999)8=0.00008
Example 2
Find the bit error probability for BPSK system with a bit rate of 1 Mbit/S. the
received waveforms S1 (t )  A cos(0t ) and S2 (t )   A cos(0t ) . The value of A is
10 mV. Assume that the single-sided noise power spectral density is N 0  1011
W/Hz.
Solution
A
T=
2Eb
 102 V
T
1
 106
6
1  10
A2T
Eb 
2
2 Eb
 3.16
N0
 2 Eb 
4
Pe  Q 
  Q  3.16   8  10
 N0 
Example 3
Binary data is transmitted over an RF band-pass channel with usable bandwidth of
10MHz at a rate of (4.8) (106) bit/sec using an ASK signaling method. The carrier
amplitude at the receiver antenna is 1 mv and the noise power spectral density at the
receiver input is 2 x 10-15 watt/Hz. Find the error of probability at receiver.
Solution
 A2Tb
Pe  Q 
 4 N0





6

3  10
1

10



4.8 
A2Tb


 5.103
4 N0
4  2  1015
Q(5.103)  1.69  103
Example 4
An on-off binary system uses the pulse waveforms

 t 
 S1 (t )  A sin   , 0  t  T
S (t )  
T 
0
, 0 t T
 S2 (t ) 

Let A  0.2 mV and T=2  s . Additive white noise with a power spectral density
1015 Watt/Hz is added to the signal. Determine the probability of error when
P(s1 )  P(s2 )  1 / 2
Solution
 A2Tb
Pe  Q 
 4 N0





 (0.2  103 )2 (2  10 6 ) 
-6
Q
  Q(4.472) = 4.29  10

15


4  10

