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Maximal Ratio Combining (MRC)

by Krishna Sankar on September 28, 2008

This is the third post in the series discussing receiver diversity in a wireless link. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. The presence of receiver diversity poses an interesting problem – how do we use ‘effectively‘ the information from all the antennas to demodulate the data. In the previous posts, we discussed selection diversity and equal gain combining (EGC).

In this post, we will discuss Maximal Ratio Combining (MRC). For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

Background

We use the same constraints as defined in the Selection Diversity and Equal Gain Combining (EGC) post. Let me repeat the same.

1. We have N receive antennas and one transmit antenna.

2. 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]

3. The channel experienced by each receive antenna is randomly varying in time. For the $i^{th}$ receive antenna, each transmitted symbol gets multiplied by a randomly varying complex number $h_i$ . As the channel under consideration is a Rayleigh channel, the real and imaginary parts of $h_i$ are Gaussian distributed having mean $\mu_{h_i}=0$ and variance $\sigma^2_{h_i}=\frac{1}{2}$ .

4. The channel experience by each receive antenna is independent from the channel experienced by other receive antennas.

5. On each receive antenna, the noise $n$ has the Gaussian probability density function with

$p(n) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(n-\mu)^2}{2\sigma^2}$ with $\mu=0$ and $\sigma^2 = \frac{N_0}{2}$ .

The noise on each receive antenna is independent from the noise on the other receive antennas.

6. At each receive antenna, the channel $h_i$ is known at the receiver.

7. In the presence of channel $h_i$ , the instantaneous bit energy to noise ratio at $i^{th}$ receive antenna is $\frac{|h_i|^2E_b}{N_0}$ . For notational convenience, let us define,

$\gamma_i = \frac{|h_i|^2E_b}{N_0}$ .

Maximal Ratio Combining (MRC)

On the $i^{th}$ receive antenna, the received signal is,

$y_i = h_ix+n_i$ where

$y_i$ is the received symbol on the $i^{th}$ receive antenna,
$h_i$ is the channel on the $i^{th}$ receive antenna,
$x$ is the transmitted symbol and
$n_i$ is the noise on $i^{th}$ receive antenna.

Expressing it in matrix form, the received signal is,

$\mathbf{y} = \mathbf{h}x+\mathbf{n}$ , where

$\mathbf{y} = [y_1 y_2 \ldots y_N]^T$ is the received symbol from all the receive antenna

$\mathbf{h} = [h_1 h_2 \ldots h_N]^T$ is the channel on all the receive antenna

$x$ is the transmitted symbol and

$\mathbf{n} = [n_1 n_2 \ldots n_N]^T$ is the noise on all the receive antenna.

The equalized symbol is,

$\hat{\mathbf{x}} = \frac{\mathbf{h}^H\mathbf{y}}{\mathbf{h}^H\mathbf{h}} = \frac{\mathbf{h}^H\mathbf{h}x}{\mathbf{h}^H\mathbf{h}}+\frac{\mathbf{\mathbf{h}^Hn}}{\mathbf{h}^H\mathbf{h}} = x+\frac{\mathbf{\mathbf{h}^Hn}}{\mathbf{h}^H\mathbf{h}}$ .

It is intuitive to note that the term,

$\mathbf{h}^H\mathbf{h} = \sum_{i=1}^{N}|h_i|^2$ i.e sum of the channel powers across all the receive antennas.

Note: The equations in the post refers the note on Receive diversity by Prof. RaviRaj Adve.

Effective Eb/No with Maximal Ratio Combining (MRC)

Earlier, we noted that in the presence of channel $h_i$ , the instantaneous bit energy to noise ratio at $i^{th}$ receive antenna is

$\gamma_i = \frac{|h_i|^2E_b}{N_0}$ .

Given that we are equalizing the channel with $\mathbf{h}^H$ , with the $N$ receive antenna case, the effective bit energy to noise ratio is,

$\begin{eqnarray}\gamma &= &\sum_{i=1}^{N}\frac{|h_i|^2E_b}{N_0}\\ & & \\& = & N\gamma_i\end{eqnarray}$ .

Effective bit energy to noise ratio in a N receive antenna case is N times the bit energy to noise ratio for single antenna case. Recall, this gain is same as the improvement which we got in Receive diversity for AWGN case

Click here to download Matlab/Octave script for plotting effective SNR with Maximal Ratio Combining in Rayleigh channel

Figure: Effective SNR with Maximal Ratio Combining in Rayleigh fading channel

Error rate with Maximal Ratio Combining (MRC)

From the discussion on chi-square random variable, we know that, if $h_i$ is a Rayleigh distributed random variable, then $h_i^2$ is a chi-squared random variable with two degrees of freedom. The pdf of $\gamma_i$ is
$p(\gamma_i)=\frac{1}{(E_b/N_0)}e^{\frac{-\gamma_i}{(E_b/N_0)}$ .

Since the effective bit energy to noise ratio $\gamma$ is the sum of $N$ such random variables, the pdf of $\gamma$ is a chi-square random variable with $2N$ degrees of freedom. The pdf of $\gamma$ is,
$p(\gamma) = \frac{1}{(N-1)!(E_b/N_0)^N}\ \gamma^{N-1}e^{\frac{-\gamma}{(E_b/N_0)}},\ \ \ \ \gamma \ge 0$ .

If you recall, in the post on BER computation in AWGN, with bit energy to noise ratio of $\frac{E_b}{N_0}$ , the bit error rate for BPSK in AWGN is derived as

$P_b=\frac{1}{2}erfc\left({\sqrt{\frac{E_b}{N_0}}}\right)$ .

Given that the effective bit energy to noise ratio with maximal ratio combining is $\gamma$ , the total bit error rate is the integral of the conditional BER integrated over all possible values of $\gamma$ .

$\begin{eqnarray}P_{e}&=&\int_0^{\infty}\frac{1}{2}erfc\left(\sqrt{\gamma}\right)p(\gamma)d\gamma\\<br /> &=&\int_0^{\infty}\frac{1}{2}erfc\left(\sqrt{\gamma}\right) \frac{1}{(N-1)!(E_b/N_0)^N}\ \gamma^{N-1}e^{\frac{-\gamma}{(E_b/N_0)}}d\gamma\end{eqnarray}$ .

This equation reduces to

$\begin{eqnarray}P_{e}&=p^N\sum_{k=0}^{N-1}\left(\begin{eqnarray}N-1+k\\k\end{eqnarray}\right)\left(1-p\right)^k\end{eqnarray}$ , where

$p=\frac{1}{2}-\frac{1}{2}\left(1+\frac{1}{E_b/N_0}\right)^{-1/2}$ .

Refer Equation 11.12 and Equation 11.13 in Section 11.3.1 Performance with Maximal Ratio Combining in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]. Again, I do not know the proof

BER Simulation Model

The Matlab/Octave 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.

(d) Perform hard decision decoding and count the bit errors

(e) Repeat for multiple values of $\frac{E_b}{N_0}$ and plot the simulation and theoretical results.

Click here to download Matlab/Octave script for simulating BER for BPSK in Rayleigh channel with Maximal Ratio Combining

Figure: BER plot for BPSK in Rayleigh channel with Maximal Ratio Combining

Reference

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Receive diversity – Notes by Prof. Raviraj Adve

Tagged as: BPSK, Diversity, PSK, Rayleigh

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{ 139 comments… read them below or add one }

yangkai November 13, 2012 at 11:49 am

Dear Sankar,

The ber formula can be solved with the aid of [R1, eq. (5A.2)].

[R1] M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels—A Unified Approach to Performance Analysis

Krishna Sankar November 18, 2012 at 7:07 am: @yangakai: Thanks. Will take a look

Reply

Dineshbabu Muthukrishnan October 6, 2012 at 11:57 am

Sir,
yours website is very useful.Thank you so much.

Shruti September 21, 2012 at 8:09 pm

Hi, Thanks a lot for the code..i have a doubt..if you keep increasing the no. of antennas, the BER curve should tend towards the AWGN BER of BPSK right…but in this case…i observed..say we have 5 antennas,the curve is becoming less the AWGN BER of BPSK curve..please explain..

Krishna Sankar September 22, 2012 at 6:07 am

@Shruti: Yes, as you said more the number of receive antennas, the ber curve will tend towards the AWGN BPSK BER.
When you simulated, did you have a non-unity gain for the channel?

Shruti September 25, 2012 at 8:54 am

Thanks for your reply,sir,
i just tried modifying the existing code just by increasing the no. of receivers.
how do i model non-unity gain? please explain this concept.

Thanks a lot

Shruti September 25, 2012 at 9:12 am

is it okay if the curves(Rx>5) as we increase the antennas become smaller than the AWGN curve? What is its physical meaning?

Krishna Sankar September 26, 2012 at 5:28 am: @Shruti: Hmm… it should not be better than AWGN

Shruti September 25, 2012 at 9:14 am

is it okay BER curves for Rx>6 become smaller than the AWGN curve? What is its physical meaning?

Krishna Sankar September 26, 2012 at 5:27 am: @Shruti: Well, I do not think it should be better than AWGN

Krishna Sankar September 26, 2012 at 5:29 am

@Shruti: The gain is caused by the channel term h in u = x*h + n.
Make sure that over many realizations, the E(|h|^2} = 1 for unity gain

Shruti September 26, 2012 at 8:30 pm: I have the same code. but just added more receivers, and taken
h = 1/sqrt(2)*[randn(nRx(jj),N) + j*randn(nRx(jj),N)]
as you have taken.
How do I change this expression so as to get unity gain?

antonella August 11, 2012 at 11:54 am

Dear Krishna
your webpage is very useful for me
thank you

Humayun July 12, 2012 at 7:21 pm

How is this different from Equal Gain Combining (EGC) scheme ??

The output SNR of EGC is the sum of the SNRs on all brances. Also, as Prof. RaviRaj Adve mentioned in his note on Receive Diversity that for Maximal Ratio Combining, “The output SNR is, therefore, the sum of the SNR at each element.” (Just next to eq. 26, page 7)

I’d really appreciate if you can clear out my confusion.

Krishna Sankar July 13, 2012 at 5:22 am: @Humayun: In equal gain combining, all the received copies of the signal are used with the same weight. However, in maximal ratio combining, the received copy with lower strength is given lower weightage.
Helps?

Reply

AncientGlory June 3, 2012 at 12:17 pm

Hi Krishna,

Thanks for this article and the code. It is very useful. I have a minor question. When you generate the Rayleigh channel are you considering those samples to be frequency domain responses of the channel?

Krishna Sankar June 11, 2012 at 5:23 am: @AncientGlory: Am considering them as realizations of a flat fading channel.

Reply

Arinze Nwodogwu May 25, 2012 at 9:34 am

hello Krishna,
please help me.i need to know if this post on MRC is for GSM or CDMA system.thanks.

Krishna Sankar July 2, 2012 at 5:19 am: @Arinze: This post discuss the general concept – not specific to GSM or CDMA system

Reply

Arinze Nwodogwu May 18, 2012 at 8:29 am

Hello Krishna,
thanks for this wonderful post on MRC. please is this code for GSM or CDMA system?
thanks.

Taiyyab Husain Ansari May 13, 2012 at 9:13 pm

Sir , I want to know the algorithm that you have used for Matlab/Octave script for plotting effective SNR with Maximal Ratio Combining in Rayleigh channel AS THAT OF BER MODEL. PLEASE ! Tell me which code represents MRC and which for RAYLEIGH CHANNNEL ., variable jj & ii what represents . please help me to know
sir .Please!!!

Krishna Sankar May 15, 2012 at 5:48 am: @Taiyyab: The algorithm is to multiply by conjugate of the channel h. The term ii is for power levels Eb/N0 and jj is for the number of receive antennas

Reply

HugoWong May 4, 2012 at 7:21 pm

Hi krishna,

Your codes are really useful. But I donot understand why the “effective SNR ” is
EbN0EffSim(ii,jj) = mean(abs(yHat)) given in the “script_maximal_ratio_combining_effective_snr.m”
Thanks~

Krishna Sankar May 8, 2012 at 5:24 am: @HugoWong: yHat = conj(h)*y which captures the post equalization signal to noise ratio

Reply

Gopu S April 28, 2012 at 1:34 pm

Thank you. Can you help me with the codes of a relay using Amplify and forward (AAF) protocol.

Krishna Sankar May 2, 2012 at 4:54 am: @Gopu: Have not studied that topic

Reply

Gopu S April 26, 2012 at 4:10 pm

Hi krishna,
Can you please send me the codes of a Rayleigh fading channel

Krishna Sankar April 27, 2012 at 5:45 am: @Gopu: Please take a look at
http://www.dsplog.com/2008/08/10/ber-bpsk-rayleigh-channel/
http://www.dsplog.com/2008/08/26/ofdm-rayleigh-channel-ber-bpsk/

In general, you can take a look at http://www.dsplog.com/tag/rayleigh/

Reply

Gopu April 25, 2012 at 10:43 am

Hi krishna,
Have you done any work on cooperative diversity in wireless communication especially using a relay (Decode and forward,amplify and forward) and a mrc combiner ? If yes, can you please send me the codes.

Krishna Sankar April 26, 2012 at 5:49 am: @Gopu: Hmm… no. Sorry

Reply

Rabi March 10, 2012 at 9:58 pm

How can I change it for MRC for QPSK. Does the curve change for QPSK case? Please help me with the codes.

Krishna Sankar March 12, 2012 at 4:41 am: @Rabi: Yes, the curve should change

Reply

Sooraj March 2, 2012 at 8:30 pm

Hi krishna,

Your codes are really useful. Have you done any work MIMO for LTE ?? especially spatial multiplexing? If yes, can you send me the codes.

Thank you.

Krishna Sankar March 5, 2012 at 5:36 am: @Sooraj: No, have not studies LTE

Reply

Jayashree February 8, 2012 at 6:17 pm

Hi Krishna,

how should i simulate SINR for cooperative MIMO scheme(considering interference from other cells).The SINR equation is given below

SINR =

( |h11 |2 + |h12|2 ) E{x12} /
σ2 + ∑ k=312 |h1k|2 E {xi2}
my input is in terms of packets.
please help me.

Krishna Sankar February 10, 2012 at 6:01 am: @Jayashree: I have not tried simulating cooperative mimo schemes

Reply

Raj February 7, 2012 at 4:27 am

Hey,
Thank you so much for the wonderful work. It helped me a lot to start using Matlab for my work in wireless communications.
Do you have any article written on Coordinated Multipoint or Base Station Cooperation? Can you also give me a good reference to better understand MRC?
I have another question about your post. You have mentioned in the post that
“Given that we are equalizing the channel with h^H ”
Where h^H means the Hermitian transpose. But we are equlizing the channel with h+. Which is the pseusdo inverse of h. Aren’t we??

Krishna Sankar February 10, 2012 at 5:58 am

@Raj: Replies:
1/ Thanks.
2/ No, nothing on Coordinated Multipoint
3/ Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt is a good reference
4/ Well, guess both are same. Agree?

dedy July 5, 2012 at 9:50 am

how i read the book for free? i’m only student… and i don’t have money to buy it…. please helpme to read the book…^_^!

Krishna Sankar July 6, 2012 at 5:44 am: @dedy: emailed you the instructions

Reply

tu_winner February 1, 2012 at 9:20 pm

Hi Mr. Khrisna, can you show me why there is a negative sign in front of
-Eb_N0_dB(ii) in this code y = h.*sD + 10^(-Eb_N0_dB(ii)/20)*n;
Thanks so much.

Krishna Sankar February 4, 2012 at 11:18 am: @tu_winner: The -ve sign to scale the noise voltage for attaining the given Eb/N0

Reply

tuncay ince January 20, 2012 at 3:37 am

hi.. what can we do for more antennas ?? how can we simulate this
I change the code but not clearly true

nRx = [1 2 4];

theoryBer_nRx1 = 0.5.*(1-1*(1+1./EbN0Lin).^(-0.5));
p2 = 1/2 – 1/2*(1+1./EbN0Lin).^(-1/2);
theoryBer_nRx2 = p2.^2.*(1+2*(1-p2));
p3 = 1/2 – 1/2*(1+1./EbN0Lin).^(-1/2);
theoryBer_nRx3 = p3.^3.*(1+3*(1-p3));
close all
figure
semilogy(Eb_N0_dB,theoryBer_nRx1,’bp-’,'LineWidth’,2);
hold on
semilogy(Eb_N0_dB,simBer(1,:),’mo-’,'LineWidth’,2);
semilogy(Eb_N0_dB,theoryBer_nRx2,’rd-’,'LineWidth’,2);
semilogy(Eb_N0_dB,simBer(2,:),’ks-’,'LineWidth’,2);
semilogy(Eb_N0_dB,theoryBer_nRx3,’gd-’,'LineWidth’,2);
semilogy(Eb_N0_dB,simBer(3,:),’ks-’,'LineWidth’,2);
axis([0 35 10^-5 0.5])
grid on
legend(‘L=1 (theory)’, ‘L=1 (sim)’, ‘L=2 (theory)’, ‘L=2 (sim)’,'L=3 (theory)’, ‘L=3 (sim)’);
xlabel(‘Eb/No(dB)’);
ylabel(‘Bit Error Rate’);
title(‘BPSK modülasyonu Maximal Ratio Combining (Rayleigh channel) için BER’);

Krishna Sankar January 23, 2012 at 5:19 am

@tuncay: The code change looks okay. I did not verify the theoretical equation though. What’s the issue which you are observing?

tuncay January 26, 2012 at 5:53 pm

I think my theorical equation is not valid.do you know true formula ..if you debug this code you can understand me I want to triple curve

Krishna Sankar January 29, 2012 at 5:50 am

@tuncay: For the theoretical equation, did you refer Equation 11.12 and Equation 11.13 in Section 11.3.1 Performance with Maximal Ratio Combining in Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

tuncay ince January 31, 2012 at 6:10 pm: thank you for your intrest

tuncay ince January 11, 2012 at 5:30 am

hi.. what can we do more antennas ?? how can we simulate

Krishna Sankar January 14, 2012 at 6:56 am: @tuncay: Did you mean what can we do with more antennas?

Reply

Pranjal Gogoi December 21, 2011 at 8:54 pm

hi Krishna, I am Pranjal from Guwahati. I have a question for you’ in MRC how can i use DPLL(digital phase lock loop) before decision ckt

Krishna Sankar January 4, 2012 at 5:38 am: @Pranjal:Why do you wish to use a digital PLL. To handle frequency errors?

Reply

bilbo January 30, 2011 at 5:53 am

hi krishni!thanks so so so so … very much for leading codes!:D

Jim March 6, 2010 at 1:31 am

Hi Krishna, I am using 12 antennas with 30 degree spacing in a full circle. Each antenna has a 90 degree 3 db beam width , therefore I have 1080 degrees of coverage in a 360 degree circle (lots of overlap). I found by experimentation that I realized combining gain with this arrangement. I am using maximum ratio combining and the results are amazing. I did this on a hunch and really was surprised by the result. Each antenna had 15 db gain and most of the time I had 3 antennas with some signal all the time! I realized below threshold demodulation when this happened! How can I calculate “combining gain”!?
Thanks!

Krishna Sankar March 30, 2010 at 4:32 am: @Jim: Well, I did not quite understand the setup. Its 12 antennas for transmitter or receiver or split? How did you measure the gains.

Reply

Adem January 14, 2010 at 10:20 pm

Also, if the receiver received two signals from a cooperative path and a direct path, how to combine these two signals using MRC and how to find the overall bit bpsk error probability?
my best regards
Adem

Adem January 14, 2010 at 10:08 pm

Hi Krishna,
This is my first time to deal with a Rayleigh fading channel. could you clarify for me the meaning of hi and how to find it using a numerical example?
In a wireless cooperative scheme in which the node that’s in between the receiver and the transmitter recieves the transmitted BPSK signal and amlify it and forward it to the receiver. How to find the overall BPSK bit error probability (in the case of a Rayleigh channel?

Phill November 29, 2009 at 1:21 pm

Hello Mr. Khrisna, I’m very thankful for ur explanation ’bout this topic…
may I ask ’bout ur code:
y = h.*s + 10^(-Eb_N0_dB(ii)/20)*n;

-Why the rayleigh channel (h) multiplied with signal(s)…can I change the .* with ‘conv’?
-And is AWGN noise and multipath channel always complex value?can I just change them, so it just only hav the real part (without imajiner value)?
-and my last question: what should i do with AWGN & rayleigh ch codes and another codes if I change the data(BPSK) into 2D matrix (e.g. 3×10^6)?

sorry for my terrible english…thanks again…
my rgrds,
Phill

Krishna Sankar December 7, 2009 at 4:48 am: @Phill: My replies
a) Yes, you may change .* to conv. I use .* as the channel is a single tap (flat fading) hence .* is equivalent to conv.
b) Well, per the model both are complex values (i.e. it affects transmission on sine and cosine transmissions). For you trial, you may try to play with it
c) Well, you might want to make h and awgn from vectors to matrices.

Your english is good.

Reply

Talib October 25, 2009 at 1:20 am

Hello Krishna,

I wanted to ask you, is this the same as a ‘RAKE’ receiver in CDMA?
Thanks,

Krishna Sankar October 27, 2009 at 5:39 am: @Talib: I would think so. We have multiple copies of the transmitted symbol and we want to combine them optimally.

Reply

aydar October 23, 2009 at 5:57 pm

Hi,
I got a bit confused. Are ZF and MRC the same?
Thank you.

Krishna Sankar October 27, 2009 at 5:36 am

@aydar: Well,
a) ZF tries to address the problem where there is interfernce from an undesired waveform and we try to force that to zero.
b) MRC tries to optimally combine two copies of the same information so as to result in minimal BER.

aydar October 27, 2009 at 1:30 pm

Thank you for the reply,

MRC maximizes received SNR after antenna combining. I think, so does ZF in SIMO case too (when ithere is no inter-cell interference), in MIMO it tries to supress interferering MIMO user/stream => maximizes SINR. So, my question is can you write me a formula of MRC for MIMO case or is MRC defined only for SIMO?

Thank you in advance Krishna.

Krishna Sankar November 8, 2009 at 7:20 am: @aydar: I agree with your comment. Sorry, I do not have the precise equation for MRC in MIMO case where there is interference.

Reply

blwf October 15, 2009 at 10:44 am

Hi,
do you know any relation between the snr of the received signal and the BER.
In my case, using MREC i manage to get the same snr of MRC , however still the bit error rate of MREC is 0.5, therefore i think if there is a relation between BER and snr i could find the source of the problem..

regards,

Krishna Sankar October 17, 2009 at 4:02 am: @blwf: For different modulation types and based on the channel model, there exists a relation between BER and SNR. For eg,
for BPSK in AWGN channel, BER = 1/2erfc(sqrt(Eb/N0)) and so on…

For solving your 0.5 BER problem, I would suggest you to try with no noise, unity channel etc and work towards identifying the bug. Good luck.

Reply

eric October 7, 2009 at 1:23 pm

Hi krishna,

you mentioned in one of your earlier replies that the equalizer is xHat = (h’*h)^{-1}*h’y where ()’ is the conjugate transpose.

So does the above equation correspond to the following in your code?

% equalization maximal ratio combining
yHat = sum(conj(h).*y,1)./sum(h.*conj(h),1);

I assume u used xhat and yhat interchange? Thank you.

Krishna Sankar October 8, 2009 at 5:40 am: @eric: Yes, that code snippet which you pasted corresponds to the equalization part.

Reply

eric October 7, 2009 at 1:06 pm

hi krishna,

for the statement
y = h.*sD + 10^(-Eb_N0_dB(ii)/20)*n;

can i verify that the -Eb_N0_dB(ii)/20 is the weight factor?
why is there a negative sign in front of -Eb_N0_dB(ii). And why we divided it by 20, not 10? Thanks!

Krishna Sankar October 8, 2009 at 5:39 am: @eric: My replies
1/ We are scaling the noise voltage for simulating the effect of various Eb/N0 values.
2/ I defined Eb/N0 as the ratio between signal power and noise power. Given that the signal power is unity, I scaled the noise voltage to achieve the given Eb/N0 value. Division by 20 happens because we are dealing with voltage signals and not power.

Reply

blwf September 20, 2009 at 1:18 pm

thanks for the program.
1. but dont you think you should add average BER ?
because if the channel was not fast fading (changing with every samply-symbol-) you will not get the same performance(already tried it)
2. Do you know any way helps to apply the MREC using the channel auto correlation matrix. whenever i want to apply it i get inconvenient results.
please help/

regards

Krishna Sankar October 1, 2009 at 4:54 am

@blwf: My replies:
1/ Am assuming that the channel is fast fading
2/ What is MREC ?

blwf October 1, 2009 at 2:50 pm

It is maximum ration eigen combining. using the eignevectors of the received signal as weights?

Krishna Sankar October 8, 2009 at 5:15 am: @blwf: Hmmm, am not familiar with that approach.

Reply

Street hawk September 17, 2009 at 7:48 pm

Why we are taking normalized signal energy equal to 1 while doing simulation? Why not actual signal energy in some joule?

Krishna Sankar September 18, 2009 at 5:56 am: @Street hawk: In simulations our objective is to characterize the performance in comparison with other types of algorithms. Hence it makes sense to normalize the transmit power to 1 for all of our simulations to allow a fair comparison.

Reply

Neetu September 4, 2009 at 1:12 pm

Hi Krishna
I am working with Weibull channel.but the base to this is Rayleigh channel.
Weibull channel envelope can be obtain from Rayleigh envelope.
I was doubtful whether i can use these Diversity techniques for Weibull channel?
Regards,
Neetu

Krishna Sankar September 9, 2009 at 5:38 am: @Neetu: Though I have not tried modeling Weibull channel, I guess you can try these diversity techniques to see the performance gains, if any.

Reply

karan July 27, 2009 at 6:02 pm

hello krishna pillai
i am doing M.Tech project in cooperative communication in MIMO cellular networks, i don’t know how to start write matlab code for cooperative communication in mimo cellular networks. pls help me.

with regards
karan

Krishna Sankar July 28, 2009 at 4:36 am: @karan: Sorry, I have not tried modeling co-operative communication networks.

Reply

J_tce July 21, 2009 at 8:08 am

Hi Krishna,

Can you describe the velocity effect on the communication channel such as BER vs Ebn0 with varying the velocity in doppler frequency ?

I want to simulation it in matlab.

Thank you very much.

Krishna Sankar July 24, 2009 at 3:57 am: @J_tce: Till now, most of the simulation assume that the channel experienced by each symbol is independent. I have not yet discussed Doppler frequency till date. Will add to my to-do list.

Reply

Khattak July 5, 2009 at 3:33 pm

thanks your mrc code is exellent i learned a lot from it.

Krishna Sankar July 6, 2009 at 7:15 pm: @Khattak: Good, glad to hear that

Reply

Yazeed June 16, 2009 at 6:25 pm

hi
how can i write a code to plot BER with SNR for the theoritical result of MRC with L number of idepentent paths?
thank you

Krishna Sankar June 20, 2009 at 10:59 am: @Yazeed: You may use the equation for BER which is provided towards the end of the post. That is for N received antenna case.

Reply

wayan June 1, 2009 at 7:25 am

hellow Mr Krishna
can you hel me to creat BER for QPSK with rain attenuation in matlab

Krishna Sankar June 7, 2009 at 1:52 pm: @wayan: I am not familiar with modeling rain attenuation. However, I have written a post on symbol error rate for QPSK in AWGN
http://www.dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/

Hope this helps.

Reply

leth May 13, 2009 at 12:02 pm

hi,

i mean when you make sum. to all snr from branches we assume that there are no correlation can you help me if there are correlation ?

thanks

Krishna Sankar May 20, 2009 at 5:09 am: @leth: I have not tried modeling systems where there is correlation between the channel seen by each antenna.

Reply
himanshu dadheech June 23, 2009 at 1:33 am: @leth: just define the covariance matrix according to your problem using the correlation coefficient formula,then use mvrnd(MU,SIGMA) function to form columns of random variables and then separate them to use…

Reply

leth May 10, 2009 at 10:40 am

i am still wait

i hope you can help me

leth May 8, 2009 at 11:39 am

hi all,

in the simulation of MRC, it depend that the branches are i.i.d can you send me matlab code for MRC with correlated branch.

i need it ASAP.

thanks

Krishna Sankar May 12, 2009 at 5:22 am: @leth: Sorry, I have not tried modeling scenarios where there is correlation between the channel observed at both the antennas.

Reply

sahil May 6, 2009 at 8:31 am

respected sir,

I have seen your code for BER for BPSK in Rayleigh channel with Maximal Ratio Combining and i want to inplement this in ofdm system can you help me what should be my approach.

Thanks

Krishna Sankar May 12, 2009 at 5:18 am: @sahil: If I may pust simplistically,
a) in the transmitter group bits and allocate them to each OFDM symbol, take ifft add cyclic prefix and send.
b) in the receiver, as we have two receive paths, take fft of OFDM symbol from each path, and then apply maximal ratio combining on the signal at the output of fft.

Hope the post on BER for BPSK with OFDM modulation http://www.dsplog.com/2008/06/10/ofdm-bpsk-bit-error/ will be of help.

Reply

R@y April 14, 2009 at 5:45 am

I hope you can help me with this matlab code:
Consider QPSK transmission over flat fading Rayleigh channels and the following three systems: i) 1×1 system , ii) 1×2 system employing maximum ratio combining (MRC), and iii) 1×2 system (two transmit antenna and a receive antenna) employing Alamouti space-time coding. The channels in the last two systems are assumed to have the same variance and the noise samples are assumed to be uncorrelated and circularly symmetric Gaussian variables with the same variance.
Let p denote the correlation between the receive (transmit) antennas for the 1×2 (2×1) system. Estimates the average (over the channels) bit error rate (BER) for the three systems for different values of the average signal-to-noise ratio (SNR).

Krishna Sankar April 16, 2009 at 5:53 am: @ R@y: I have not tried simulating the BER in the case where channel is correlated. For posts on alamouti STBC, you may refer to http://www.dsplog.com/tag/alamouti.

Good luck.

Reply

Maha April 13, 2009 at 2:41 am

Me too i am interested in this question, is it possible to change MRC code by considering a constant nRx?

Krishna Sankar April 16, 2009 at 5:38 am: @Maha: Yes, indeed. Just change nRx variable accordingly.

Reply

Mathieu April 11, 2009 at 6:19 pm

Perfect! thank you that was really helpful, before i visit your website i was little bit confused in using MRC EGC SC but it’s more clear right now.
Concerning your MRC code, i was trying to consider a constant number of receiver channels ‘nRx=4′ by removing the variable ‘jj’, but the system displays this error message:
‘Warning: Input arguments must be scalar.’
i was wondering how could to avoid this problem, and is there any other variable that we have to change in order to have a constant nRx?

Krishna Sankar April 16, 2009 at 5:25 am: @Mathieu: I just checked. To simulate 4 receive chains, just change nRx = [1 2]; to nRx = [4]; Just make sure that the calls to simBer variable in the plots are modified.

Reply

Mathieu April 10, 2009 at 5:26 pm

Hi Krishna

I have a question about your MRC code

Could please give us more details about this section :

” % counting the errors
nErr(jj,ii) = size(find([ip- ipHat]),2); ”

i don’t see why did you use the function “find”?

Thank you in advance

Krishna Sankar April 11, 2009 at 7:46 am: @Mathieu: find() gives the index of non-zero elements in the error vector [ip - ipHat]. And then size() gives the number of non-zero elements -> which is equal to number of erraneous bits.

Reply

commengr April 8, 2009 at 8:42 pm

I trying to learn multiuser detection techniques for CDMA and OFDM. I’m a beginner so do you have any advice for me?

Are you planning to post tutorials regarding that subject?

I’m a real rookie so tell me about some resource thats starts off at the basic, not the book of Verdu!! please

Krishna Sankar April 11, 2009 at 7:21 am: @commengr: The multi user detection is a close cousin of problem of MIMO demodulation. In MIMO, the interference is caoused by transmission in spatial dimension (read other antenna in the same devide); in multi user scenario, the interference is caused by other users.
You may look at the posts on receiver structures for MIMO @ http://www.dsplog.com/tag/mimo
Hope this helps.

Reply

sara March 26, 2009 at 9:04 pm

thanx for ur reponse to my previous post i am trying to RAKE

sara March 26, 2009 at 8:24 pm

hi krishna

tell me in the case of transmit diversity
can we use MRC at the receiver???
and do we use combiner before equalizer or after that

regards
sara

Krishna Sankar April 4, 2009 at 8:06 am: @sara: Well, it depends on the transmit diversity scheme employed. Hopefully, the attached link on 2 transmit, 1 receive alamouti STBC should give your pointers
http://www.dsplog.com/2008/10/16/alamouti-stbc/

Reply

rahul March 20, 2009 at 12:58 pm

hello..
i have derieved average probilty of error of bpsk in nakagami fading by using moment generating function.now i need to get the formula in presence of mrc diversity.how can i do so ?

Krishna Sankar March 21, 2009 at 4:50 pm: @rahul: Sorry, I am not familiar with derivations. Good luck. Just in case: if you wish to publish your results from the derivations in dsplog.com, please do mail me.

Reply

sara March 15, 2009 at 8:40 pm

hi krishna!
nice job!!!!
well done!!!

i want to combine two signals at the receivers that are comming from different paths(i-e a multipath enviornment) where second signal is being received at the reciver with a delay of about 3 to 4 symbols.(currently i am using EGC)
now what i am getting at the receiver is very high BER, even though i am using rrc filter and DFE equalizer. can u plz suggest me how to address this problem.
also suggest me some good combining and equalization techniques for this problem!

waiting for ur response
regards
sara

Krishna Sankar March 21, 2009 at 8:00 am: @sara: Delay of 3 to 4 symbols! Quite high, aint it?
Anyhow, if you know the delays, you may add a rake receiver (to collect energy from all the delay elements and make them add coherently) to improve the BER. This may improve your receiver’s latency. But, should not matter in your simulatins where you want to prove your algorithm. Hope this helps.

Reply

maria March 13, 2009 at 3:37 pm

hi krishna….
i want simulation of ofdm without noise…
plz can u help me…
regards
maria

Krishna Sankar March 21, 2009 at 7:51 am: @maria: Please look at http://www.dsplog.com/2008/02/03/understanding-an-ofdm-transmission/

Reply

Sintat March 9, 2009 at 8:47 pm

Hi Krishna,

Nice derivation. For mathematical clarity, it is appropriate to replace Eb/No, in the last equaiton, by the average SNR (which can be different from Eb/No in the case where E[|h_i|^2] is different from 1.

Krishna Sankar March 21, 2009 at 6:58 am: @Sintat: Thanks for the comment. I agree. However, as this example is pertinent to flat fading Rayleigh channel with E[|h_i|^2] = 1, may I just continue using Eb/No. Thanks.

Reply

Martin March 3, 2009 at 12:06 pm

dear sir,
1) can you tell me what mean of nRx = [1, 2]?

2) n = 1/sqrt(2)*[randn(nRx(jj),N) + j*randn(nRx(jj),N)]
can you explain the element of randn(nRx(jj), N)?

3) yHat = sum(conj(h).*y,1)./sum(h.*conj(h),1)
is it possible to modify to become h*d^-2 and repeat it to cal. the error prob.?

Krishna Sankar March 5, 2009 at 5:24 am: @Martin:

(1) nRx = 1, showed results with flat fading with only one receive chain. nRx = 2 showed results with 2 receive chains using MRC. Both were provided to facilitate an easy comparison.

(2) randn() generates gaussian random variable with mean 0 and variance 1.

(3) In general,there can be multiple coding styles to reach the same result. What I chose need not be the way you chose.

Reply

Martin March 3, 2009 at 11:55 am

Can I use this program to produce a 1*m rayleigh fading? 1 is transmitter m is receivers
please help, thanks

Krishna Sankar November 9, 2008 at 7:27 pm

@davis: I have not tried modeling spatial correlation. However, from a quick googling, came up with the following
page:
Simulation of the Spatial Covariance Matrix
http://users.ece.utexas.edu/~rheath/papers/2003/multi_mode/index.htm

Further, may I suggest that you are mixing two things. The doppler frequency corresponds to correlation in time. The antenna spacing results in spatial correlation. I recall reading that 0.5 wavelength spacing should results in spatially uncorrelated channels.

davis November 7, 2008 at 5:04 pm

If I would want to simulate the MRC with corelation in indoor environment, how chould you suggest to make the two rx to be dependent on the distanse of the antennas?
i read that the 0.48wavelength they chould be uncorellated because of dopler frequency, but in indoor where the RX are static is there any dopler effect and can I get uncorelated chanels?

Krishna Sankar October 31, 2008 at 6:07 am

@davis: Yes, you are correct. In the article and simulations, no correlation between the channels was assumed.

Whether this is a valid assumption depends on the channel model and antenna spacing. For eg, if the antennas are far apart, then each antenna seeing an independent channel is highly probable. I would expect typical multi-antenna systems to provide enough spacing (more than half wavelength) between antennas to reap the benefits from diversity.

davis October 30, 2008 at 9:22 pm

Thank you for you replay
I see that in your code there is no corelation between the two 2 chanels ,Is it correct to do that?

Krishna Sankar October 29, 2008 at 10:19 am

@davis : Well, since I have not studied the topic, I cannot comment on absolute terms. However, given than BFSK has 3dB poorer results than BPSK in AWGN, I would think that you should obtain comparable results.

May I recommend to proceed as follows:
(a) Make 1×1 BFSK simulation in a single tap Rayleigh channel. I would think that you should obtain a curve which is 3dB poorer than the corresponding BPSK case.
(b) Once you have obtained that, then you may move to 1×2 case. With 1×2 case, am guessing that you should see the improvement as seen for BPSK case (around 15dB lower BER at 10^-4 point).

Hope this helps. Please do report your findings. All the best.

Krishna Sankar October 29, 2008 at 10:13 am

@nihmot: You can find posts discussing symbol error rate an BER for typical QAM modulation schemes in AWGN here.
URI: http://www.dsplog.com/tag/qam/
I hope it will be reasonably easy for you to adapt the concepts and write a simulation model to suit your requirements

Krishna Sankar October 29, 2008 at 10:08 am

@ila: Sorry for the delayed response, I was down the viral fever.
To answer all your queries
(Q1) can u pls kindly explain why do we use this?
sD = kron(ones(nRx(jj),1),s);
[Ans1] This is to repeat the same transmit symbol to be received by nRx receive antennas. For eg,
lets say s = [1 2 3 4 5];
kron(ones(3,1),s) = [1 2 3 4 5;
1 2 3 4 5;
1 2 3 4 5]
(Q2) Do you consider the weight in the code?
[Ans2]Yes, this is taken care. For eg, consider channel h = [1;2];
pinv(h) = 1/(5)[ 1 2].
For equalization, we multiply pinv(h) with received [y1;y2]. We can see that y2 is given more weightage than y1 during equalization.

Does this help?

himanshu dadheech June 24, 2009 at 4:51 pm

i’m using MRC in hoyt channel, so my code is slight different…i just want to ask that the weight that we multiply with the received signal is h’/h*h’ or h’/N
where h is channel and N is noise power…

i’m just confused about how to take the weights…
pls do reply soon…

Krishna Sankar June 25, 2009 at 5:52 am: @himanshu: I have not tried modeling hoyt fading channel. However, in the current post, the weighting factor was formed as follows:
Assume
y – [nrx x 1] received vector
h – [nrx x 1] channel vector
x – [1 x 1] transmit symbol
n – [nrx x 1] noise vector

y = hx + n

The equalizer is
xHat = (h’*h)^{-1}*h’y where
()’ is the conjugate transpose.

Reply

davis October 28, 2008 at 2:57 pm

hi Krishna
i am doing project on BER with bfsk on rayleigh channek also using MRC and i get better performans with 2 reicevers instead of one more than 3db in some case i got 8db better
is this posible? what is the maximum improvement should I expect?
THANKS

nihmot October 28, 2008 at 8:04 am

hi love this but am working on BER performance with 4-,8-,16-QAM over flat rayleigh fading channel using MRC tachnique. need a simulation code !!! can anyone help here….

ila October 26, 2008 at 1:37 pm

Also, in Wiki, it’s stated: http://en.wikipedia.org/wiki/Diversity_combining
“MRC: The received signals are weighted with respect to their SNR and then summed”.

do you consider the weight in the code?

ila October 26, 2008 at 1:35 pm

hi krishna,

can u pls kindly explain why do we use this?
sD = kron(ones(nRx(jj),1),s);

ila October 26, 2008 at 1:23 pm

hi Krishna,

thanks for the great posts u have put on. very helpful!

i dont really understand the part where:
sD = kron(ones(nRx(jj),1),s);
why do we do this?

i read in Wiki “The received signals are weighted with respect to their SNR and then summed”
http://en.wikipedia.org/wiki/Diversity_combining
i cant find where the calculation of weight in the code? is there assumptions made that this can be ignored?

thanks & look fwd to your kind reply

Krishna Sankar October 16, 2008 at 9:23 pm

@farie: I recently wrote a post on BER simulation of BPSK in Rayleigh channel with 2Tx, 1Rx antenna Alamouti STBC coding.
URI: http://www.dsplog.com/2008/10/16/alamouti-stbc/

Hope this helps.

farie October 14, 2008 at 6:06 pm

I was just browsing through your posts and wanted to know if you have a code for simulating BER for BPSK/QPSK/16QAM in Rayleigh channel with Alamouti code of 2×1 and 2×2 (two transmit one receiver and two transmit and two receivers).

Krishna Sankar October 12, 2008 at 5:37 pm

@Michael: Firstly, sorry for the delayed response. I was away from my blog desk for while.

Yes, you are right and captured a typo. It should have been (N-1+k) chose k rather than (m-1+k) chose k. I corrected the equation in the post. Thanks again.

Michael October 8, 2008 at 9:45 pm

First of all, very nice post!

I’ve got one question. In your bit error calculation for mrc you say you calculate the sum of (m-1+k) over k. What is m and where does it come from?

Krishna Sankar October 1, 2012 at 6:38 am

@Shruti: Hmm… i see your point. Try
h = 1/sqrt(2)*1/sqrt(nRx(jj))*[randn(nRx(jj),N) + j*randn(nRx(jj),N)]

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Maximal Ratio Combining (MRC)

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Maximal Ratio Combining (MRC)

Effective Eb/No with Maximal Ratio Combining (MRC)

Error rate with Maximal Ratio Combining (MRC)

BER Simulation Model

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Maximal Ratio Combining (MRC)

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Maximal Ratio Combining (MRC)

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