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 $\{x_1, x_2, x_3, \ldots, x_n \}$

2. In normal transmission, we will be sending $x_1$ in the first time slot, $x_2$ in the second time slot, $x_3$ 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 $x_1$ and $x_2$ from the first and second antenna. In second time slot, send $x_3$ and $x_4$ from the first and second antenna, send $x_5$ and $x_6$ 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 $\frac{n}{2}$ 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 $i^{th}$ transmit antenna to $j^{th}$ receive antenna, each transmitted symbol gets multiplied by a randomly varying complex number $h_{j,i}$ . As the channel under consideration is a Rayleigh channel, the real and imaginary parts of $h_{j,i}$ are Gaussian distributed having mean $\mu_{h_{j,i}=0$ and variance $\sigma^2_{h_{j,i}}=\frac{1}{2}$ .

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 $n$ 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 $\sigma^2 = \frac{N_0}{2}$ .

7. The channel $h_{j,i}$ 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,

$y_1 =h_{1,1}x_1 + h_{1,2}x_2 + n_1 = [h_{1,1}\ h_{1,2}] \left[\begin{eqnarray}x_1 \\ x_2 \end{eqnarray}\right]+n_1$ .

The received signal on the second receive antenna is,

$y_2 = h_{2,1}x_1 + h_{2,2}x_2 + n_2 = [h_{2,1}\ h_{2,2}] \left[\begin{eqnarray}x_1 \\ x_2\end{eqnarray}\right]+n_2$ .

where

$y_1$ , $y_2$ are the received symbol on the first and second antenna respectively,

$h_{1,1}$ is the channel from $1^{st}$ transmit antenna to $1^{st}$ receive antenna,

$h_{1,2}$ is the channel from $2^{nd}$ transmit antenna to $1^{st}$ receive antenna,

$h_{2,1}$ is the channel from $1^{st}$ transmit antenna to $2^{nd}$ receive antenna,

$h_{2,2}$ is the channel from $2^{nd}$ transmit antenna to $2^{nd}$ receive antenna,

$x_1$ , $x_2$ are the transmitted symbols and

$n_1,\ n_2$ is the noise on $1^{st}, 2^{nd}$ receive antennas.

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

$\begin{eqnarray}\left[\begin{eqnarray}y_1 \\ y_2\end{eqnarray}\right] & = & {\left[\begin{array}{cc}h_{1,1}& h_{1,2} \\h_{2,1}&h_{2,2}\end{array}\right]}\left[\begin{eqnarray}x_1 \\ x_2 \end{eqnarray}\right]+\left[\begin{eqnarray}n_1\\n_2 \end{eqnarray}\right]\end{eqnarray}$ .

Equivalently,

$\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$

To solve for $\mathbf{x}$ , The Zero Forcing (ZF) linear detector for meeting this constraint $\mathbf{WH=I}$ . is given by,

$\mathbf{W}=\mathbf{(H^HH)^{-1}H^H}$ .

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 $x_1$ , $x_2$ , i.e.

$\left[\begin{array}\hat{x}_1\\\hat{x}_2\end{array}\right] = \mathbf{(H^HH)^{-1}H^H}\left[\begin{array}y_1\\y_2\end{array}\right]$ .

Take one of the estimated symbols (for example $\hat{x}_2$ ) and subtract its effect from the received vector $y_1$ and $y_2$ , i.e.

$\left[\begin{array}r_1\\r_2\end{array}\right] = \left[\begin{array}{ccl}y_1 &- & h_{1,2}&\hat{x}_2\\y_2 & - & h_{2,2}&\hat{x}_2\end{array}\right] = \left[\begin{array}{clc}h_{1,1}&x_1 &+ &n_1 \\ h_{2,1}&x_1 &+ &n_2\end{array}\right]$ .

Expressing in matrix notation,

$\left[\begin{array}r_1\\r_2\end{array}\right] = \left[\begin{array} h_{1,1}&amp\\h_{2,1}\end{array}\right]x_1 + \left[\begin{array}n_1\\n_2\end{array}\right]$ ,

$\mathbf{r} = \mathbf{h}x_1+\mathbf{n}$

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,

$\hat{\mathbf{x}}_1 = \frac{\mathbf{h}^H\mathbf{r}}{\mathbf{h}^H\mathbf{h}}$ .

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

(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 $\frac{E_b}{N_0}$ 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

Observations

Compared to Zero Forcing equalization alone case, addition of successive interference cancellation results in around 2.2dB of improvement for BER of $10^{-3}$ .

The improvement is brought in because decoding of the information from the first spatial dimension ( $x_1$ ) has a lower error probability that the symbol transmitted from the second dimension. However, the assumption is that $x_2$ is decoded correctly may not be true in general. We can discuss alternate approaches in future posts.

References

[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

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Tagged as: BPSK, MIMO, Rayleigh, SIC, ZF

{ 25 comments… read them below or add one }

1 karl November 22, 2008 at 11:19 pm

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.

2 Krishna Pillai November 25, 2008 at 6:10 am

@karl: Thanks Sure, will add posts on MIMO-OFDM.

3 darshini January 27, 2009 at 10:54 am

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.

4 Krishna Pillai January 28, 2009 at 6:03 am

@darshini: I have some posts on MIMO V-BLAST at
URI: http://www.dsplog.com/tag/mimo/

There is also a post on Alamouti STBC code @
http://www.dsplog.com/2008/10/16/alamouti-stbc/

Hope this helps.

5 moh March 26, 2009 at 7:37 pm

can find me code with mimi vblast

6 Krishna Pillai April 4, 2009 at 7:57 am: @moh: You meant, MIMO with V-BLAST. This post indeed talks about V-BLAST scheme. For MIMO V-BLAST with zero forcing equalizer, please refer to
http://www.dsplog.com/2008/10/24/mimo-zero-forcing/

Hope this helps.

7 darshini January 27, 2009 at 11:40 am

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).

8 darshini February 11, 2009 at 2:57 pm

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?

9 Krishna Pillai February 12, 2009 at 6:42 am: @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.

10 darshini February 11, 2009 at 3:01 pm

hi krishna
what about golden code? whether there r any posts about MIMO using golden codes?

11 Krishna Pillai February 12, 2009 at 6:40 am: @darshini: No, nothing yet on golden codes with MIMO

12 John Titor February 18, 2009 at 8:28 pm

hi, the ZF-SIC should be a non-linear approach, but why the result you got is so linear?

13 Krishna Pillai February 21, 2009 at 7:38 am: @John: Hmm… I did not follow your question. As the SNR increases, its reasonable to expect lower BER too….

14 Ronaldo April 7, 2009 at 6:15 pm

Can u give me some explain on the H^H operation in your Matlab script?

15 Krishna Pillai April 11, 2009 at 6:39 am: @Ronaldo: ()^H is the Hermitian operator. Hermitian operator means – conjgutate transpose of a matrix

16 tidjani April 8, 2009 at 10:19 am

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?
Thanks
kind regard

17 Krishna Pillai April 11, 2009 at 6:59 am: @tidjani: Sorry, I have not written posts on FM modulation.

18 Communications_engineer April 21, 2009 at 8:27 pm

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

19 Krishna Pillai April 30, 2009 at 5:02 am: @Communications_engineer: Sure. Further, the articles of receiver structures for V-BLAST can be directly applied to Multi user detection case too. Agree?

20 shah May 2, 2009 at 5:54 pm

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

21 Krishna Pillai May 12, 2009 at 4:58 am: @shah: Thanks.
No, I have not tried modeling multi carrier delay diversity modulation.

22 samira June 25, 2009 at 1:55 pm

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.

23 Krishna Pillai June 26, 2009 at 5:14 am: @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.

24 mohamed July 3, 2009 at 4:58 pm

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

25 Krishna Pillai July 6, 2009 at 5:47 pm: @mohamed: Well, with Alamouti coding we do not need MMSE-SIC receiver. A Zero Forcing equalization is optimal.

MIMO with Zero Forcing Successive Interference Cancellation equalizer

2×2 MIMO channel

Other Assumptions

Zero forcing equalizer for 2×2 MIMO channel

Zero Forcing with Successive Interference Cancellation (ZF-SIC)

Simulation Model

Observations

References

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