MIMO with ZF SIC and optimal ordering

In previous posts, we had discussed equalization of a 2×2 MIMO channel with Zero Forcing (ZF) equalization and later, Zero Forcing equalization with successive interference cancellation (ZF-SIC). In this post, we will explore a variant of ZF-SIC called Zero Forcing Successive Interference Cancellation with optimal ordering. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

Brief description of 2×2 MIMO transmission, assumptions on channel model and the noise are detailed in the post on Zero Forcing equalization with successive interference cancellation

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,

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

Using the Zero Forcing (ZF) equalization, the receiver can obtain an estimate of the two transmitted symbols $x_1$ , $x_2$ , i.e.

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

Successive Interference Cancellation with optimal ordering

In classical Successive Interference Cancellation, the receiver arbitrarily takes one of the estimated symbols, and subtract its effect from the received symbol $y_1$ and $y_2$ . However, we can have more intelligence in choosing whether we should subtract the effect of $\hat{x}_1$ first or $\hat{x}_2$ first. To make that decision, let us find out the transmit symbol (after multiplication with the channel) which came at higher power at the receiver. The received power at the both the antennas corresponding to the transmitted symbol $x_1$ is,

$P_{x_1}=|h_{1,1}|^2 + |h_{2,1}|^2$ .

The received power at the both the antennas corresponding to the transmitted symbol $x_2$ is,

$P_{x_2}=|h_{1,2}|^2 + |h_{2,2}|^2$ .

If $P_{x_1}>P_{x_2}$ then the receiver decides to remove the effect of $\hat{x}_1$ from the received vector $y_1$ and $y_2$ and then re-estimate $\hat{x}_2$ .

$\left[\begin{array}r_1\\r_2\end{array}\right] = \left[\begin{array}{ccl}y_1 &- & h_{1,1}&\hat{x}_1\\y_2 & - & h_{1,2}&\hat{x}_1\end{array}\right] = \left[\begin{array}{clc}h_{1,2}&x_2 &+ &n_1 \\ h_{2,2}&x_2 &+ &n_2\end{array}\right]$ .

Expressing in matrix notation,

$\left[\begin{array}r_1\\r_2\end{array}\right] = \left[\begin{array} h_{1,2}&amp\\h_{2,2}\end{array}\right]x_2 + \left[\begin{array}n_1\\n_2\end{array}\right]$ ,

$\mathbf{r} = \mathbf{h}x_2+\mathbf{n}$

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,

$\hat{\hat{x}}_2 = \frac{\mathbf{h}^H\mathbf{r}}{\mathbf{h}^H\mathbf{h}}$ .

Else if $P_{x_1} \le P_{x_2}$ the receiver decides to subtract effect of $\hat{x}_2$ from the received vector $y_1$ and $y_2$ , and then re-estimate $\hat{x}_1$

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

Expressing in matrix notation,

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

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

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,

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

Doing successive interference cancellation with optimal ordering ensures that the reliability of the symbol which is decoded first is guaranteed to have a lower error probability than the other symbol. This results in lowering the chances of incorrect decisions resulting in erroneous interference cancellation. Hence gives lower error rate than simple successive interference cancellation.

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) Find the power of received symbol from both the spatial dimensions.

(f) Take the symbol having higher power, 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 computing BER for 2×2 MIMO channel equalized by ZF-SIC with optimal ordering

Figure: BER plot for BPSK in 2×2 MIMO equalized by ZF-SIC with optimal ordering

Observations

Compared to Zero Forcing equalization with successive interference cancellation case, addition of optimal ordering results in around 2.0dB of improvement for BER of $10^{-3}$ .

References

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

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

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{ 9 comments… read them below or add one }

1 lealem December 1, 2008 at 10:44 am

Hello! Kirishna how are u today? Kirishna.. i am doing a simulation for adaptive OFDM based on current SNR but i have got a problem in between. Can you please help me how can i relate the channel state information with SNR and how can i adjust the modulation scheme based on the channel impulse response. if you have a matlab script that can be used to simulate the performance of adaptive OFDM send me please!
Lealem

2 Krishna Pillai December 2, 2008 at 8:20 am

@lealem : From your comment, it seems like you want to have an adaptive modulation scheme where you chose between BPSK/QPSK/16QAM/64QAM based on the channel power and SNR. Sorry, I have not yet done such studies till date. Will add this to my to-study list

3 apri May 28, 2009 at 7:29 am

Krishna, which one it easy to adaptive modulation scheme on the channel power and SNR
many thanks
apri

4 Krishna Pillai May 31, 2009 at 8:34 pm: @apri: I just mailed you the download instructions. Plz check.

5 Bintang April 20, 2009 at 10:09 am

Hi Krishna, is this similar to V-BLAST MIMO (Wolniansky, 1998)? Thank you in advance..and many thanks too for putting up this great site.

6 Krishna Pillai April 25, 2009 at 7:28 am: @Bintang: Yes, this post refers to the landmark 1998 paper V-BLAST: An architeture for realizing very high data rates over the rich scattering wireless channel – P. W. Wolniansky, G. J. Foschini, G. D. Golden, R. A. Valenzuela for defining the transmission.

7 Gabriel September 30, 2009 at 7:52 pm

The sorting of the equalization matrix based on the channel power is not clear for me. What is hCof? Can you explain it a little bit more?

8 Krishna Sankar October 1, 2009 at 5:31 am: @Gabriel: The variable hCof corresponds to the co-factor matrix (used when computing the inverse). Then based on the channel power H, we sort the cofactor matrix accordingly.

9 skumar January 16, 2010 at 8:21 pm

yha i tested it, i am happy krishna that i coild mathch therotical results.

MIMO with ZF SIC and optimal ordering

Zero forcing equalizer for 2×2 MIMO channel

Successive Interference Cancellation with optimal ordering

Simulation Model

Observations

References

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