MIMO with Zero Forcing Successive Interference Cancellation equalizer

The post on MIMO with Zero Forcing equalizer discussed a probable way of equalizing a 2×2 MIMO channel. The simulated results with the 2×2 MIMO system 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 try to improve the bit error rate performance by trying out Successive Interference Cancellation (SIC). We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

The background material on the MIMO channel has been described in the post on Zero Forcing equalizer. The text is repeated again for easy readability.

2×2 MIMO channel

In a 2×2 MIMO channel, probable usage of the available 2 transmit antennas can be as follows:

1. Consider that we have a transmission sequence, for example

2. In normal transmission, we will be sending in the first time slot, in the second time slot, and so on.

3. However, as we now have 2 transmit antennas, we may group the symbols into groups of two. In the first time slot, send and from the first and second antenna. In second time slot, send and from the first and second antenna, send and in the third time slot and so on.

4. Notice that as we are grouping two symbols and sending them in one time slot, we need only time slots to complete the transmission – data rate is doubled ! 🙂

5. This forms the simple explanation of a probable MIMO transmission scheme with 2 transmit antennas and 2 receive antennas.

Figure: 2 Transmit 2 Receive (2×2) MIMO channel

Other 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 experience by each transmit antenna is independent from the channel experienced by other transmit antennas.

3. For the transmit antenna to receive antenna, each transmitted symbol gets multiplied by a randomly varying complex number . As the channel under consideration is a Rayleigh channel, the real and imaginary parts of are Gaussian distributed having mean and variance .

4. The channel experienced between each transmit to the receive antenna is independent and randomly varying in time.

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

with and .

7. The channel is known at the receiver.

Zero forcing equalizer for 2×2 MIMO channel

Let us now try to understand the math for extracting the two symbols which interfered with each other. In the first time slot, the received signal on the first receive antenna is,


The received signal on the second receive antenna is,



, are the received symbol on the first and second antenna respectively,

is the channel from transmit antenna to receive antenna,

is the channel from transmit antenna to receive antenna,

is the channel from transmit antenna to receive antenna,

is the channel from transmit antenna to receive antenna,

, are the transmitted symbols and

is the noise on receive antennas.

For convenience, the above equation can be represented in matrix notation as follows:



To solve for , The Zero Forcing (ZF) linear detector for meeting this constraint . is given by,


To do the Successive Interference Cancellation (SIC), the receiver needs to perform the following:

Zero Forcing with Successive Interference Cancellation (ZF-SIC)

Using the Zero Forcing (ZF) equalization approach described above, the receiver can obtain an estimate of the two transmitted symbols , , i.e.


Take one of the estimated symbols (for example ) and subtract its effect from the received vector and , i.e.


Expressing in matrix notation,


The above equation is same as equation obtained for receive diversity case. Optimal way of combining the information from multiple copies of the received symbols in receive diversity case is to apply Maximal Ratio Combining (MRC).

The equalized symbol is,


This forms the simple explanation for Zero Forcing Equalizer with Successive Interference Cancellation (ZF-SIC) approach.

Simulation Model

The Matlab/Octave script performs the following

(a) Generate random binary sequence of +1’s and -1’s.

(b) Group them into pair of two symbols and send two symbols in one time slot

(c) Multiply the symbols with the channel and then add white Gaussian noise.

(d) Equalize the received symbols with Zero Forcing criterion

(e) Take the symbol from the second spatial dimension, subtract from the received symbol

(f) Perform Maximal Ratio Combining for equalizing the new received symbol

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

(h) Repeat for multiple values of and plot the simulation and theoretical results.

Click here to download Matlab/Octave script for simulating BER for BPSK modulation in 2×2 MIMO with Zero Forcing and Successive Interference Cancellation equalization (in Rayleigh channel)

Figure: BER plot for BPSK in 2×2 MIMO channel with Zero Forcing Successive Interference Cancellation equalization


Compared to Zero Forcing equalization alone case, addition of successive interference cancellation results in around 2.2dB of improvement for BER of .

The improvement is brought in because decoding of the information from the first spatial dimension () has a lower error probability that the symbol transmitted from the second dimension. However, the assumption is that is decoded correctly may not be true in general. We can discuss alternate approaches in future posts. 🙂


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

[WIRELESS-TSE, VISWANATH] Fundamentals of Wireless Communication, David Tse, Pramod Viswanath

64 thoughts on “MIMO with Zero Forcing Successive Interference Cancellation equalizer

  1. Dear Krishna,

    Can you guide me on the accuracy of the BER analysis based on the number of times the Monte Carlo simulations are run?

    I want to know after how many Monte Carlos can I be sure that my BER graph is accurate for a Rayleigh Fading Channel

    1. @Razi: Typically, one need to run the simulations sufficiently large number of times to ensure that the simulation results are accurate. When computing the bit error rate, one solution is to ensure that we have atleast 100 errors for each Eb/N0 point

  2. Hi krishna!

    Excellent work.

    I’m trying to implement physically a 2×2 MIMO system, do you think I would need to add something else (besides mixers, amplifiers and LNAs) that the code for the transmission and reception as used in matlab?

  3. No offence to the writer but i’ve already spotted like 2-3 typo’s and havent even finished reading lol — do they charge you for spellchecker in your time zone (just kidding) nice layout though I’ll give you that 🙂

  4. Hi krishna
    I want to build simple 2×2 MIMO-CDMA system using 16-QAM and zero forcing detector in MATLAB. So please help….

  5. Hi Krishna Sankar,

    Your article is great!
    I am reading your code to more understand, so please help me:
    Why to calculate the inverse matrix as in the source code rather than using “inv” and “transpose” code in Matlab?
    % Inverse of a [2×2] matrix [a b; c d] = 1/(ad-bc)[d -b;-c a]



  6. hello sir,
    is it possible that we use the ZF receiver with 2*1 ,and 1*2 antenna system.?
    please guide me how is that possilbe?

  7. i nazia parveen want matlab code for interference cancellation using mimo using parallel interference cancellation algorithm

  8. please Mr. Krishna Sankar Iwant to build simple mimo ofdm system to simulate snr vs. ber but I want it the mimo system to be flexable not only 2×2 so ican change the number of transmitter and recievers please help me if you have the matlab code please send it to me

    1. @eng_dina: In most of the articles which I have discussed I have used 2×2 MIMO case. And to increase the speed of the Matlab simulations, I have not used inv() operation in Matlab. You can try using inv() operation and increase the speed of the simulations

  9. hello,
    with match respect can you mr send me a explain shéma of MMSE,ZF,ZF_SICand MMSE_SIC.
    Thank you

  10. Hi,
    Thanks for your high quality contribution in this site.Could you please mention some refernce books for understanding the theory what you have used here.

  11. I have done simulation with MMSE-SIC but there is no diversity gain compared with MMSE equalizer. I have tried to find out the reason but I haven’t got it. Would you please give me an explain. Thank you in advance

      1. Sorry, I found my mistake, MMSE-MRC is still better than MMSE . By the way, thank you so much for your post. It helps me a lot!

  12. what exactly is the meaning of successive interfernce cancellation?

    when we subtract x2 from y1 to get x1,,, does this mean that we are cancelling noise common to both spatial stream?


  13. hi krishna………hi sir…………
    I am working multiuser detection successive interference cancellation with goldcode 31 but still error .can you help me?
    emailme in yutt.pangestu@gmail.com. thanks…..
    this the programs :


    clc,clear all;
    format long;

    P=[1 1 1 1 1];

    BER_sic=sim_sic(SNR_dB, P, Nb, squence, chan_type)

    semilogy(SNR_dB, BER_sic, ‘bv-‘)
    xlabel(‘SNR dalam dB’);
    ylabel(‘probabilitas bit error’);
    legend(‘sic awgn’);
    title(‘simulasi sic’);
    hold on;


    % BER_sic=sim_sic_fn(SNR_dB, P, Nb, code_matrix, chan_type)
    % return bit error rate of the sic receiver in AWGN or rayleigh fading
    % SNR_dB=signal-ti-nopise ratio in dB
    % P=power control vector, P(i)=transmitted power of i-th user
    % Nb=number of transmitted bit
    % code_matrix=matrix of the spreading code used
    % chan_type=channel type(1=AWGN, 2=rayleigh fading channel)
    % output:
    % BER_sic=bit error of the sic receiver
    function BER_sic=sim_sic(SNR_dB, P, Nb, squence, chan_type)

    K=length(P); %number of user
    N=size(squence,2); %spreading factor


    %generate the crosscorrelation matrix R

    for p=1:length(SNR_dB),



    %loop for Nb symbol bits
    for n=1:Nb,

    %generate a bernoulli symbol, i.e. b=1 or 0

    if chan_type==1
    elseif chan_type==2
    disp(‘invalid channel type’);

    %transmitted sinal

    %generate the observed data

    %assume that the codes are known and that they are
    %perfectly synchronized
    %matched filter output

    %short users according tho their received powers

    %detect users successively

    for u=K-1:-1:1,

    %if detected symbol is incerrect, increment the error_count
    if b_hat(user_index)~=b(1)


    %carculating Bit Error rate i.e. the percentage of erroneous
    %symbol estimates


    % r=awgn_chan(signal,SNR)
    % return the awgn channel output given the input signal and the SNR
    % Vsignal=normalized anti podal signal (-1,+1); can be a matrix
    % SNR=linear signal-to-noise ratio for the AWGN channel
    % OUTPUT:
    % r=AWGN channel output
    function r=awgn_chan(signal,SNR)



    % A=rayleigh(omega,K)
    % return a vector of rayleigh-distrimuted random variables
    % omega=variance of the rayleigh-distributed random variables
    % K=length of vector (i.e. number of users)
    % OUTPUT:
    % A=vector of rayleigh-distributed RVS
    function A=rayleigh(omega,K)


    % b=input_symbols(K,Nb)
    % Return the randomly generated antipodal symbol(+1,-1)
    % K=number of users
    % Nb=number of symbol for each user
    % OUTPUT:
    % b=the randomly generated antipodal symbols
    function b=input_symbols(K,Nb)


    function [gold_codes]=gold_codes
    Xa=[0 0 0 1 0];
    Xb=[0 0 0 1 0];
    for i=1:31

    Xa=[mod(Xa(3)+Xa(5),2) Xa(1:4)];
    Xb=[mod(Xb(2)+Xb(4)+Xb(5),2) Xb(1:4)];


  14. hi good work thank u
    i want to ask if we used alamouti code and we want to do the mmse-sic reciever
    what will be different than the above algorithm?
    thx in advance

  15. Hi,
    Iwould like knowing if we can use VBLAST with multiuser(CDMA).If yes you can send me a code matlab for that.Think you.

    1. @samira: The modeling of V-BLAST and multiuser communication is kind of similar (in V-BLAST the other spatial stream is the interferer and in multiuser the other user is the interferer). I did not quite understand your intention, when you said you want to put together V-BLAST with multi user.

  16. Hi Krishna
    This is very useful site,keep it up, do u have any MATLAB code for multicarrier delay diversity moduulation(MDDM)for MIMO

  17. Hello Krishna, can you start tutorials for multiuser detection for cdma and ofdma. Thanks

  18. Hi Sir,
    I am doing work on wireless communication.would like to send me a matlab code for frequency modulation (BW=200 kHz).
    2- how to plot the spectrum of it?
    kind regard

  19. hi krishna

    thanks, ur alamouti code helped me. But in that URL u give for v-blast, i cant find any specific v-blast codings. All are with ZF,SIC etc.. except v-blast. can u plz let me know…In v-blast processing steps there shld be a optimal ordering step, but here mimo with zf and optimal ordering is available. whether it is v-blast or any specific v-blast coding is available?

    1. @darshini: As I understand, in V-Blast we transmit the different symbol simultaneously from two tx antennas. Whether we do optimal ordering for decoding is receiver dependent. The simplest equalization scheme uses Zero Forcing, and then we can have decoding schemes like MMSE, MMSE-SIC, ML etc.

  20. hi
    I am working on BER performance of MIMO with V-BLAST spatial multiplexing and space time codes like golden code, almouti code, linear dispersion code etc. Can u provide me the simulation code, if u tried for any of the above(especially v-blast).

  21. hi

    I am working on BER performance of MIMO with V-BLAST spatial multiplexing and space time codes like golden codes, almouti code and linear dispersion codes
    can u provide me the simulation code if u tried for any of above.

  22. great work Krishna. congrats. your coding style is simple and easy to understand. hope you can post some work about MIMO-OFDM for example using alamouti or CDD. keep posting.

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