Download to simulate an awgn channel with bpsk modulation

TO SIMULATE AN AWGN CHANNEL WITH BPSK MODULATION
We need to send an uncoded BPSK modulated binary signal over a noisy channel. The
noise is an additive white Gaussian noise. The simulation process can be divided into a
number of smaller tasks:
1. Generating AWGN Noise: In generating a Gaussian noise, we take the help of
central limit theorem. It states:
If the variable X has a non-normal distribution with mean µ and standard deviation σ,
then the limiting distribution of
_
z = ( x - µ )/(σ/√n) as n  ∞
is the standard normal distribution (i.e. with mean 0 and SD 1).
X must have finite mean and variance. n >= 25 is regarded as large.
Thus in order to generate Gaussian noise we generate a number of uniformly distributed
samples varying uniformly between a = -0.5 to b = +0.5.
Its mean is given by µ = (a + b)/2 = 0
and variance σ2 = (b-a) 2 /12 = 1/12
_
Now from the samples we can determine x and using the central limit theorem we can
generate a number of values z corresponding to each sample vector. The values of z thus
obtained are normally distributed with mean 0 and variance 1. We can obtain the normal
distribution of desired variance simply by multiplying the values in Z vector by the
corresponding standard deviation.
2. Generating the Message Signal: In order to generate a message signal where the
probability of occurrence of 0 and 1 is same, first we generate a sample uniformly
distributed within a range. Then we set the threshold exactly at the mid point such that if
the sample value is greater than the threshold, then the message value is taken as 1
otherwise 0.
3. BPSK Modulation: BPSK modulation of the binary message signal can be performed
simply by converting the unipolar signal (0,1) into the bipolar signal (-1,1).
4. Noise Addition: Now the Gaussian noise is added to the message signal. This addition
must be performed sample wise i.e. a noise sample must be added to the corresponding
message signal sample.
5. Hard Decision and decoding: At the receiver side, we need to demodulate and decode
the received signal. This can be done by comparing the received signal with a threshold
value. If the received signal amplitude is greater than the threshold, 1 is received
otherwise 0 is received. Generally the threshold value is set to be zero if the probabilities
of occurrence of 1 and 0 in a message bit stream are equal. Otherwise the optimum
threshold value (Dopt) is decided by the relation:
Dopt
where
=
(σ2/A)ln(P0/P1)
P0 = probability of occurrence of 0
P1 = probability of occurrence of 1
σ2 = noise variance
A = signal amplitude
6. Calculating Theoretical and Experimental Bit Error Rate: After simulating the
AWGN channel following the steps mentioned above, we need to verify whether our
experimental model is close to the theoretical one or not. Thus we need to calculate the
theoretical and experimental bit error rate and compare them.
Theoretical bit error rate is given by the relation:
BERtheo = 0.5*erfc(√(Eb/No))
Now if the amplitude of the message signal 1 is A, its power is given by A2. Power of the
message signal 0 is 0. If probability of occurrence of 1 and 0 is same then average signal
power is given by A2/2. And noise power is given by σ2. Hence signal to noise ratio is
given by A2/2σ2. Thus experimental BER is given by:
BERexp = 0.5*erfc(A/√2 σ)
7. Plotting the Results: Finally, to compare the experimental model with that of
theoretical model, we need to plot the corresponding BER on a single canvas. BER is
plotted against the signal to noise (SNR) ratio. SNR value is taken in dB. For a properly
designed model, error between the two plots must be very small.
.