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Selection Diversityby KRISHNA SANKAR on SEPTEMBER 6 , 2008
This is the first post in the series discussing receiver diversity in a wireless link. Receiverdiversity is a form of space diversity, where there are multiple antennas at the receiver. Thepresence of receiver diversity poses an interesting problem – how do we use ‘effectively‘ theinformation from all the antennas to demodulate the data. There are multiple ways to approachthe problem. The three typical approaches to be discussed are – selection diversity, equalgain combining and maximal ratio combining. In this post we will discuss selectiondiversity. For the discussion, we will assume that the channel is a flat fading Rayleighmultipath channel and the modulation is BPSK.
Background1. 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 onlyone tap. So, the convolution operation reduces to a simple multiplication. For a more rigorousdiscussion on flat fading and frequency selective fading, may I urge you to review Chapter 15.3Signal TimeSpreading from [DIGITAL COMMUNICATIONS: SKLAR]
3. The channel experienced by each receive antenna is randomly varying in time. For the receive antenna, each transmitted symbol gets multiplied by a randomly varying complexnumber . As the channel under consideration is a Rayleigh channel, the real and imaginaryparts of are Gaussian distributed having mean 0 and variance 1/2.
4. The channel experience by each receive antenna is independent from the channelexperienced by other receive antennas.
5. On each receive antenna, the noise has the Gaussian probability density function with
with and .
The noise on each receive antenna is independent from the noise on the other receiveantennas.
6. At each receive antenna, the channel is known at the receiver. For example, on the receive antenna, 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.
7. In the presence of channel , the instantaneous bit energy to noise ratio at receive
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6/23/2016 Selection Diversity
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antenna is . For notational convenience, let us define,
.
From the discussion on chisquare random variable, we know that, if is a Rayleighdistributed random variable, then is a chisquared random variable with two degrees of
freedom. The pdf of is
.
What is selection diversity?Consider a scenario where we have a single antenna for transmission and multiple antennasat the receiver (as shown in the figure below).
Figure: Receive diversity in a wireless link
At the receiver we have now N copies of the same transmitted symbol. Which then poses theproblem – how to effectively combine them to reliably recover the data.
Selection diversity approach is one way out – With selection diversity, the receiver selectsthe antenna with the highest received signal power and ignore observations from the otherantennas. The chosen receive antenna is one which gives .
Outage probability in Selection DiversityThe equations in the post refers the note on Receive diversity by Prof. RaviRaj Adve.
To analyze the bit error rate, let us first find the outage probability on the receive antenna.Outage probability is the probability that the bit energy to noise ratio falls below a threshold.The probability of outage on receive antenna is,
.
is the defined threshold for bit energy to noise ratio.
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In N reveive antenna case, the probability that all bit energy to noise ratio on all the receiveantenna are below the threshold is,
,
where
are the bit energy to noise ratio on the 1st, 2nd and so on till the Nth receiveantenna.
Since the channel on each antenna is assumed to independent, the joint probability is theproduct of individual probabilities.
.
Note that the equation above defines the probability that the effective bit energy to noise ratiowith N receive antennas (lets call ) is lower than the threshold . This is infact thecumulative distribuition function (CDF) of . The probability density function (PDF) is then thederiviate of the CDF.
.
Given that we know the PDF of , the average output bit energy to noise ratio is,
.
I do not know how to reduce the above integral to this simple sum. This means that,
with two receive antennas the effective bit energy to noise ratio is 1.5 times ,
with three receive antennas, the effective bit energy to noise ratio is 1.833 times ,
with four receive antennas, the effective bit energy to noise ratio is 2 times and so on.
If you recall the results from the AWGN with receive diversity case,
Effective bit energy to noise ratio in a N receive antenna case is N times the bit energy tonoise ratio for single antenna case.
With selection diversity we are seeing that the effective SNR improvement is not a linearimprovement with increasing the number of receive antennas. The returns diminish.
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Figure: SNR gain with selection diversity
Click here to download Matlab/Octave script for computing the effective SNR in Ralyeighchannel with selection diversity
Bit Error probability with selection diversity
If you recall, in the post on BER computation in AWGN, with bit energy to noise ratio of ,
the bit error rate for BPSK in AWGN is derived as
.
Given that the effective bit energy to noise ratio with selection diversity is , the total bit errorrate is the integral of the conditional BER integrated over all possible values of .
.
This equation reduces to
.
Refer Equation 11.24 in Section 11.3.2 Performance with Selection combining in [DIGCOMMBARRYLEEMESSERSCHMITT]. Again, I do not know the proof
BER Simulation ModelThe 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.
Peak to Average Power Ratio for OFDM 20 votes
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(c) At the receiver, find the receive path with maximum power
(d) Chose that receive path, 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 simulating BER for BPSK in Rayleigh channelwith selection diversity
Figure: BER plot for BPSK in Rayleigh channel with Selection Diversity
ObservationsAround 16dB improvement at BER point by with two receive antenna selection diversity
ReferencesReceive diversity – Notes by Prof. Raviraj Adve
[DIGCOMMBARRYLEEMESSERSCHMITT] Digital Communication: Third Edition, by JohnR. Barry, Edward A. Lee, David G. Messerschmitt
Happy learning.
Related posts:
Maximal RatioCombining (MRC)
Equal GainCombining (EGC)
Receive diversityin AWGN
Transmitbeamforming
Tagged as: BPSK, Diversity, PSK, Rayleigh
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entitles you to download the free eBook on BER of BPSK/QPSK/16QAM/16PSK in AWGN.
56 comments… read them below or add one
bhusan February 3, 2013 at 8:30 pm
Hi, sir . In selection diversity, You have calculated the effective snr as: EbN0EffSim(ii,jj) = mean(hSel.*conj(hSel)); EbN0EffThoery(ii,jj) = sum(1./[1:nRx(jj)]); But I think it should beEbN0EffSim(ii,jj) = mean(ySel.*conj(ySel))/nRx(jj); EbN0EffThoery(ii,jj) = (1 + (nRx(jj)1)*pi/4); If i am wrong please help me out and tell me the reason of using “hSel” instaed of “ySeal”.Thank You.
REPLY
Krishna Sankar February 5, 2013 at 5:51 am
@bhusan: The term ySel includes the effect of noise too. As I did not want tofactor in the effect of noise, used hSel. How is the pi/4 term coming?
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zeyad January 27, 2013 at 2:13 am
Hi sirare the receiving diversities (e.g MRC EGC) designed for SIMO system only? or it is used inMIMO system? I mean does the STBC decoder acts as receiving diversity?
REPLY
Krishna Sankar February 1, 2013 at 5:17 am
@zeyad: A simple STBC decoder with 2 transmit antenna and 1 receiveantenna. It is exploiting the diversity, but I reckon it would fall into the transmit diversitybucket.
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REPLY
yangkai November 13, 2012 at 11:36 am
Dear Sankar,
I have read some of your topics. It’s wonderful! It’s easy to understand for freshman.
There are two unsolved problems in this post. Maybe I can give you some advice.
1. the average output bit energy to noise ratio
The final result can be derived by using Mathematical induction [R1].
2. the total bit error rate
After expanding $[1exp(\gamma/(Eb/N0))]^N1$ and with the aid of [R2, eq. (5A.2)], thefinal equation can be solved after some manipulations.
Good luck.
yangkai
[R1] http://en.wikipedia.org/wiki/Mathematical_induction[R2] M. K. Simon and M.S. Alouini, Digital Communication over Fading Channels—A UnifiedApproach to Performance Analysis
REPLY
Krishna Sankar November 18, 2012 at 7:08 am
@yangakai: Thanks much.
REPLY
Soumya September 27, 2012 at 12:13 pm
Hi Sir,
Your posts have been so helpful to me. Thanks a lot. I have a question,if have to combine selection diversity and Equal gain combining i.e SC2,what modifications have to be done?I tried a lot, but my results seem to be wrong.
Please help.
REPLY
Krishna Sankar October 1, 2012 at 6:43 am
@Soumya: Sorry, did not understand your question. Did you mean having thetwo set of results in the same plot?
REPLY
Soumya October 1, 2012 at 8:37 am
I meant, if there are L branches, 2 branches with the best SNR(i.e secondorder selection combining) will be selected and combined using EGC…Please Help..
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Krishna Sankar October 5, 2012 at 5:15 am
@Soumya: Aah, am hoping that you will be able to adapt the existingmatlab code(s) on Selection Diversity and Equal Gain Combining to meet yourrequirements.http://www.dsplog.com/2008/09/06/receiverdiversityselectiondiversity/ http://www.dsplog.com/2008/09/19/equalgaincombining/
REPLY
Soumya October 6, 2012 at 7:59 pm
hi sir, i tried a lot…but i am not able to find the second highest SNRproperly..how do i do that?
Please guide me..
Navneet June 24, 2012 at 1:16 am
Hi!Will you please guide me on ‘How can we implement different diversity schemes likefrequency, time, spatial diversity on Awgn and Rayleigh channel in case of BPSK andQPSK??’Thanks..
REPLY
Krishna Sankar June 26, 2012 at 6:18 am
@Navneet: Please look athttp://www.dsplog.com/tag/diversity
REPLY
Suchita Chatterjee June 2, 2012 at 10:30 am
Hello Sir, your site is really helpful…. I am doing my research in Turbo coded OFDM over frequencyselective fading channels… I am using 1/2 and 1/3 rate turbo codes to reduce BER.. Couldyou help me on 1/3 rate turbo codes????
REPLY
Krishna Sankar July 2, 2012 at 5:12 am
@Suchita: Sorry, I have not yet posted anything on Turbo codes
REPLY
ali ghori December 12, 2011 at 7:05 am
just wana say 4 all of ur topics..”simply awsm” . GOD BLESS U
REPLY
paritosh February 19, 2011 at 12:12 am
Awesome Man that really helped me out, bookmarking you thank you so much
REPLY
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M. Ahmed September 4, 2010 at 2:36 pm
Dear krishna,How could DPSK be implemented in Selection diversity? and how to be simulated?thanks in advance
REPLY
Krishna Sankar September 6, 2010 at 5:13 am
@Ahmed: Should be possible. I have not tried though
REPLY
M. Ahmad September 4, 2010 at 7:08 am
Dear krishna,How could DPSK be implemented in Selection diversity?thanks in advance
REPLY
kimos_kh March 4, 2010 at 9:31 pm
Please,Can you give me a matlab program for MxN MIMO OFDM system thattransmitting independent data streams from each antenna (Spatial multiplexing)
REPLY
Krishna Sankar March 30, 2010 at 4:32 am
@kimos_kh: I have not discussed the MxN MIMO OFDM case. However, youcan find articles on 2×2 MIMO case at http://www.dsplog.com/tag/mimo
REPLY
vijayendra desai December 20, 2009 at 5:25 pm
i have doubt in this program/
1) why same equation for the AWGN noise and Raylaigth channel. 2) how can we find power using this equation: hPower = h.*conj(h);
please help me as soon as possible.
REPLY
Krishna Sankar December 23, 2009 at 5:45 am
@vijayendra: My replies1) Where did you see the same equation for AWGN and Rayleigh channel? 2) Assume h = a+jb. h*conj(h) = (a+jb)*(ajb)= a^2+b^2 ==> power of h
REPLY
Obinna O December 2, 2009 at 8:30 am
Krishna Pillai,
Hellos sir,
6/23/2016 Selection Diversity
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I am told to assume Rayleigh fading channel with BPSK modulation. Using MATLAB plot bit error probability (BEP) under coherent and noncoherent detectionwhen receiver is equipped with three antennas to exploit diversity. Your figures will include plots from simulation. Use average SNR (complex) from 5 to 20 dB.
Please help me resolve this, thank you so much
REPLY
Krishna Sankar December 7, 2009 at 5:06 am
@Obinna O: Hope you have finished the project by now.
REPLY
Obinna O December 8, 2009 at 7:58 am
No sir,
I have not, I was awaiting your reply and assistance. I am still in need of yourassistance.
REPLY
Krishna Sankar December 10, 2009 at 5:35 am
@Obinna: Please take a look ata) BER for BPSK in flat fading Rayleigh channelhttp://www.dsplog.com/2008/08/10/berbpskrayleighchannel/ b) BER for BPSK using selection diversity in flat fading Rayleigh channel (two rxantenna)http://www.dsplog.com/2008/09/06/receiverdiversityselectiondiversity/
REPLY
Obinna O December 11, 2009 at 7:33 am
Thank you so much sir. It was very helpful
Bhavesh Neekhra November 4, 2009 at 9:08 am
Hi Krishna,
Thanks for the well written post.
REPLY
Krishna Sankar November 8, 2009 at 8:49 am
@Bhavesh: Thanks
REPLY
Ayazkorea September 14, 2009 at 10:15 am
Antenna selection can be implemented at both transmit and receive ends.In Transmit antenna selection (TAS), Source has multiple antennas and receiver has singleanenna, with the feedback information from the receiver to source, the single antenna willbe selected for tranmission which has highet SNR.
The PDF and CDF of both schemes are same, so we can say that the perofrmance of bothshould be same.
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For exampleIn SC, the receiver selects the largest SNR branch, i.e,SNR_SC =max[h1, h2,... hk]In TAS, through the feed back information from receiver to source, the source will select oneof its antenna for transmission, which has highest SNR, i.e,SNR_TAS=max[h1, h2,... hk]
So, only single channel will be seen by receiver from tranmit antennas amog all.
About PDF this defination is valid for both:The probability density function of selecting the largest random variable amog kindependent and identically random variables can be writen as:
PDF=k[ pdf of SNR of single channel]^k1 *cdf of single channel
I think u can get my point now , further discussion will be welcomedReferences:[1] Shuping Chen, “Performance of AmplifyandForward MIMO Relay Channels withTransmit Antenna Selection and MaximalRatio Combining” 2009. WCNC 2009. IEEE [2] Zhuo Chen , “Analysis of transmit antenna selection/ maximalratio combining in Rayleighfading channels”, July 2005, IEEE
REPLY
Krishna Sankar September 18, 2009 at 5:30 am
@Ayazkorea: Thanks, I got you. Yes we select the best transmit antenna out ofh1, h2, h3. So the final channel model is same in both transmit selection and receiveselection. I agree with your comments.
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Ayazkorea September 13, 2009 at 10:44 pm
According to my study, there is no difference in performance of Transmit AntennaSelection (at source) and Selection Diversity (at destination), wether by implementing themultiple antennas at source or destination, (like transmit beamforming and receivebeamforming)
Are you agree with this??????
REPLY
Krishna Sankar September 14, 2009 at 5:37 am
@Ayazkorea: I did not quite understand, how you do the transmit selection?When we send from multiple transmit antennas, the new channel as seen by the singleantenna receiver isy = (h1+h2+h3)x + n, whereh1, h2, h3 are channel seen by the receiver from transmit antenna 1, 2, 3 respectively. Agree?In this case, how do we do the selection?
REPLY
Ayhem August 26, 2009 at 3:59 pm
could you please explain : how you have model white Gaussian noise as a tworandom variables multiblyed by Eb/No ??as you have given Rayliey noise the code ( h) and white Gaussian noise ( n) while they areboth the same .
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do you have a referenc you have based on in your model
REPLY
Krishna Sankar September 7, 2009 at 5:07 am
@Ayhem: This concept is discussed in most text books on digitalcommunications. You may refer Digital Communications by John Proakis.
Wiki references:http://en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noisehttp://en.wikipedia.org/wiki/Rayleigh_fading
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bluray August 26, 2009 at 9:05 am
May I know what type of real life applications uses simo? Is handphone (1 antenna)to base station (many antenna) 1 of the application?
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Krishna Sankar September 7, 2009 at 4:57 am
@bluray: Yes, uplink communication (handphonetobasestation) is a validapplication. However, in future, we might be seeing handphones with multiple antennas
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wayan June 1, 2009 at 7:28 am
what is different symbol eeror rate with bit error rate
REPLY
Krishna Sankar June 7, 2009 at 1:56 pm
@wayan: By symbol, we typically refer to the constellation symbol as defined bythe modulation scheme like BPSK, QPSK, 16QAM, 16PSK etc. By symbol error rate, wecount the number of errors where the received constellation symbol differs from thetransmitted constellation.
Each constellation symbol may correspond to a group of bits. So, each symbol error mayconstitute one or more bit errors.
The post on BER with 16QAM should be helpful to understand the concepthttp://www.dsplog.com/2008/06/05/16qambiterrorgraymapping/
Good luck.
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lila May 20, 2009 at 7:03 pm
hello khrisna how to diversity selection in riciab fading channel…pleasee
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Krishna Sankar May 22, 2009 at 5:29 am
@lila: I have not tried modeling in Rician fading channel. However, I would think
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that the concept remains the same – chose the path with higher signal to noise ratio
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mr.Bean May 3, 2009 at 11:06 pm
Thank you for all your posts. No other way is easier to start matlab in wirelesscommunication as effective as reading your posts. I wish you all the best, and hope that ultimate coding techniques will be discussed furthersuch as: LDPC, Turbo Coding,…
REPLY
Krishna Sankar May 12, 2009 at 5:02 am
@Mr.Bean: Thanks, kind words indeed. Sure, I do hope to move on to LDPC, Turbo decoding etc (as soon as I learn them)…;)
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Krishna Sankar May 12, 2009 at 5:02 am
@Mr.Bean: Thanks, kind words indeed. Sure, I do hope to move on to LDPC, Turbo decoding etc (as soon as I learn them)…;)
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debabandana March 19, 2009 at 2:24 pm
how to reliase selection combination with QAM16 modulation…please help me sir
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Krishna Sankar March 21, 2009 at 3:57 pm
@debabandana: I think it should be reasonably easy for you to adapt the code inthis post Selection diversity with BPSK to 16QAM case. For BER/SER with 16QAM, youmay look @(a) http://www.dsplog.com/2007/12/09/symbolerrorratefor16qam/ (b) http://www.dsplog.com/2008/06/05/16qambiterrorgraymapping/ Hope this helps
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well March 13, 2009 at 9:03 am
hi,,i want to know and i need help about BER for QPSK in rayleigh and rician fading channelwith selection diversity,,would you please explain to me..
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Krishna Sankar March 21, 2009 at 7:43 am
@well: This post discuss BER for BPSK in Rayleigh channel with selectiondiversity. I would think it would be reasonably easy to adapt to QPSK case.
I have not tried modeling Rician channel. Plan to do so in future.
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Stefan February 23, 2009 at 7:15 am
Can you help me about SIR based selection diversity script?How to implement these schemes with Nakagamim and alfamu fading distribution?
REPLY
Krishna Sankar February 24, 2009 at 5:31 am
@Stefan: I have not tried modeling the Nakagamim or alfamu fading model.You may use, http://en.wikipedia.org/wiki/Nakagami_distribution
I would think that the selection algorithm still remains the same. Its only the noise andchannel model which changes.
REPLY
Bhumika February 6, 2009 at 11:41 pm
Can you tell me how to implement these schemes with Rician fading distribution?
REPLY
Krishna Sankar February 10, 2009 at 8:01 pm
@Bhumika: I think the algorithm for doing the selection diversity still remains thesame. It is the channel model which changes from Rayleigh to Rician. I have not studiedthe modeling of Rician channel.
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Mohammad August 16, 2010 at 3:21 am
sorry I read, not wrote
REPLY
Krishna Sankar October 8, 2012 at 5:31 am
@Soumya: Find the received power on each antenna and sort them
REPLY
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