BER for BPSK in Rayleigh channel
Long back in time we discussed the BER (bit error rate) for BPSK modulation in a simple AWGN channel (time stamps states August 2007). Almost an year back! It high time we discuss the BER for BPSK in a Rayleigh multipath channel.
In a brief discussion on Rayleigh channel, wherein we stated that a circularly symmetric complex Gaussian random variable is of the form,
,
where real and imaginary parts are zero mean independent and identically distributed (iid) Gaussian random variables with mean 0 and variance .
The magnitude which has a probability density,
is called a Rayleigh random variable. This model, called Rayleigh fading channel model, is reasonable for an environment where there are large number of reflectors.
System model
The received signal in Rayleigh fading channel is of the form,
, where
is the received symbol,
is complex scaling factor corresponding to Rayleigh multipath channel
is the transmitted symbol (taking values +1’s and -1’s) and
is the Additive White Gaussian Noise (AWGN)
Assumptions
1. The channel is flat fading - In simple terms, it means that the multipath channel has only one tap. So, the convolution operation reduces to a simple multiplication. For a more rigorous discussion on flat fading and frequency selective fading, may I urge you to review Chapter 15.3 Signal Time-Spreading from [DIGITAL COMMUNICATIONS: SKLAR]
2. The channel is randomly varying in time - meaning each transmitted symbol gets multiplied by a randomly varying complex number . Since
is modeling a Rayleigh channel, the real and imaginary parts are Gaussian distributed having mean 0 and variance 1/2.
3. The noise has the Gaussian probability density function with
with
and
.
4. The channel is known at the receiver. Equalization is performed at the receiver by dividing the received symbol
by the apriori known
i.e.
where
is the additive noise scaled by the channel coefficient.
Bit Error Rate
The equations listed below refers Chapter 14.3 in [DIGITAL COMMUNICATION: PROAKIS]
If you recall, in the post on BER computation in AWGN, the probability of error for transmission of either +1 or -1 is computed by integrating the tail of the Gaussian probability density function for a given value of bit energy to noise ratio . The bit error rate is,
.
However in the presence of channel , the effective bit energy to noise ratio is
. So the bit error probability for a given value of
is,
,
where .
To find the error probability over all random values of , one must evaluate the conditional probability density function
over the probability density function of
.
Probability density function of
From our discussion on chi-square random variable, we know that if is a Rayleigh distributed random variable, then
is chi-square distributed with two degrees of freedom. since
is chi square distributed,
is also chi square distributed. The probability density function of
is,
.
Error probability
So the error probability is,
.
Somehow, this equation reduces to
.
Note:
1. I have not yet figured out the math to reduce the above integral to the answer. If some one knows, kindly drop in a comment. 2. Another way for finding the bit error rate might be to find the pdf of
Simulation Model
It will be useful to provide a simple Matlab/Octave example simulating a BPSK transmission and reception in Rayleigh channel. The script performs the following
(a) Generate random binary sequence of +1’s and -1’s.
(b) Multiply the symbols with the channel and then add white Gaussian noise.
(c) At the receiver, equalize (divide) the received symbols with the known channel
(d) Perform hard decision decoding and count the bit errors
(e) Repeat for multiple values of and plot the simulation and theoretical results.
Click here to download Matlab/Octave script for BER computation of BPSK in Rayleigh fading channel
Figure: BER plot of BPSK in Rayleigh fading channel
When compared to the AWGN case, around 25dB degradation due to the multipath channel (at the point). This is both good and bad: bad because we need to spend so much energy to get a reliable wireless link up (in this era of global warming), and good because we signal processing engineers are trying to figure out ways for improving the performance.
Reference
[DIGITAL COMMUNICATION: PROAKIS]
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Comments
Thank you! Kirishna, hope fully ur script is very very help full for my work but still i have doubt on how to write a matlab script to simulate a frequency selective fading channel.
for example a channel of 120 Hz Doppler frequency.
[...] Lealem Tamirat, in a comment on BER for BPSK in Rayleigh channel, wondered about the performance of an OFDM modulated system in a frequency selective Rayeligh [...]
hello!
i go through your posts of matlab code, these are very helpful. please help me to solve the following writing the matlab code.
Assume BPSK modulation with equal user transmit
powers. The channels between users (interuser) and to the
base station (uplink) are mutually independent with flat,
Rayleigh fading that remains constant over each source block
(quasi-static fading). Each receiver has perfect CSI and employs coherent detection, and we quantify the quality of each channel by its average received SNR. We use the RCPC code family with memory M = 4 and puncturing period P = 8
and choose overall code rate R = 1/4 with source block size K = 128 bits.
if more detail is needed please inform me…
it is very urgent for me. if u could do it for me i would be very thankful to you.
thanks
khurshid
thanks krishna
actually i have to show the performance comparison of different strategies of cooperative communication…..
different combining techniques at receiver may also be considered but not compulsory…..
it is quit similar to relaying…..
i have sent some matlab codes on krishna at dsplog
it shows some error which i fail to removed.
please help me in removing these errors……
Hi,
In order to find probability of error in a rayleigh fading channel we will average the conditional probability of error of the particular modulation scheme in AWGN channel over the rayleigh probability density function. Why we do this ? Please give explanation.
Thanks and Regards,
V Varun.
@Krishna Pillai
Thanks a lot for giving me a reply. So this is similar to finding the mean of a random variable right !!! Here the random variable is instantaneous SNR right !!! Comment on this statement Sir !!!
@ Pillai
Which equation ? Is it the probability of error in a Rayleigh fading channel or the probability of error in a AWGN channel ? Let me know so that i can come up with a solution if i know
You will find that the missing steps in the derivation above can be completed by writing open the final (double) integral, and changing the order of integration. That is, integrate first with respect to gamma from 0 to x^2 (where x is the argument of the complementary error function), and then with respect to x from 0 to infinity. The integration problem is finally reduced to single integrals from 0 to infinity over zero-mean Gaussian distributions.
Sir
How to derive the probability error of Non-coherent Non-Orhogonal modulation. TRhanking you sir!!
it can be obtained if the orthogonal condition is not satisfied. But the derivations become more complicated. the advantage is that it provide high spectral efficiency. ok sir thank u for ur fast response
Hi Kirshna:
hoping u r fine.
I just want to ask about equlization which u have done here just by dividing by channel. is it another method of equlization.
can u give some explation about equlization or refer some material about it, bec, I want to understand it.
Tahnks and take care.
abrar
Hello Kirshna
I doing project on bfsk with rayleigh multipath chanel
i wish to ask you if I use not an envelop but modulated fsk
0->cos(f1*t) and 1->cos(f2*t)
how can I simulate rayleigh multipath chanel for bfsk in this case?
thank you
Davis
Hi Krishna,
Jussi Poikonen is correct, you can determine the double integral in this fashion to obtain the desired result.
Hi,
Could you please explain on how you get
h = 1/sqrt(2)*[randn(1,N) + j*randn(1,N)]; for Rayleigh?
Coz I think, it should be just randn(1,N) + j*randn(1,N)
Thanks
[...] shift keying (BPSK) in Rayleigh channel. As usual, mr. google was my best friend, then I found a nice explanation about BER for BPSK in Rayleigh channel. Unfortunately, the author also hasn’t figured out how to derive the equation yet. But thank [...]
[...] with zero forcing equalizer showed matching results as obtained in for a 1×1 system for BPSK modulation in Rayleigh channel. In this post, we will discuss a different equalization approach called Minimum Mean Square Error [...]
Hi Kirshna
thanks for ur reply and sorry for the t I am saying thanks very late bec i am busy in my exam.Hoping U r fine, may GOD bless u. I am doing Postgraduate diploma in Massey University New Zealand.
I am doing simulation on MC-CDMA. So I need ur help in modeling frequency selective channel. if u have any code about frequency selective chanel estimation, it will helpful for me. I am try to do so I will send u code if I have have succeded to some extent.
waiting for ur reply.
Thanks nad Take care.
abrar
I am Muhammad from Pakistan,
I was in need to compute the Bit error rate of BPSK and QPSK in AWGN and Rayleigh fading channel.
After finding this document, i was very happy as it improve my knowledge and releive my burden.
Thanks for all
Hi Krishna,
I am trying to follow this tutorial and extend it further to simulations for 4-QAM over Rayleigh fading channels. Could you please tell me how the code to apply the fading channel would change from the BPSK case. thanks
@Muhammad: Thanks. Glad that you are finding http://www.dsplog.com useful for your studies. Happy learning!
Dear Krishna Pillai,
Thanks for the very interesting blog ..
I am wondering whether it makes any sense to apply MMSE and ML equalization on SISO communication systems in order to get over the ZF equalization major problem of noise enhancement (when channel is in deep fade).
In such a case, how is the theoretical BER performance of BPSK (for instance) over flat rayleigh (given by proakis, Digital communications) affected ? can u reccommend any papers that calculate the theoretical BER performance of ML and MMSE equalization over flat fading .. Most papers i got assume ZF equalization when it comes to SISO systems !
I have seen the simulation code which u used to simulate a Rayleigh fading channel. But i would like u to ask a hit how to simulate a frequency selective Rayleigh fading channel for OFDM application. I think the above simulation out put is a flat Rayleigh fading.
Thank you!!!