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BER for BPSK in ISI channel with MMSE equalization

by Krishna Sankar on January 24, 2010

In the past, we had discussed BER for BPSK in flat fading Rayleigh channel and BER for BPSK in a frequency selective channel using Zero Forcing Equalization. In this post, lets discuss a frequency selective channel with the use of Minimum Mean Square Error (MMSE) equalization to compensate for the inter symbol interference (ISI). For simplifying the discussion, we will assume that there is no pulse shaping at the transmitter. The ISI channel is assumed to be a fixed 3 tap channel.

Transmit symbol

Let the transmit symbols be modeled as

$s(t) = \sum_{n=-\infty}^{\infty}a_ng(t-mT)$ , where

$T$ is the symbol period,

$a_n$ is the symbol to transmit,

$g(t)$ is the transmit filter,

$n$ is the symbol index and

$s(t)$ is the output waveform.

For simplicity, lets assume that the transmit pulse shaping filter is not present, i.e $g(t)=\delta(t)$ .

So the transmit symbols can be modeled by the discrete time equivalent

$s[k]=a_n$

Figure: Transmit symbols

Channel Model

Lets us assume the channel to be a 3 tap multipath channel with spacing $T$ i.e.

$h[k]=\left[\begin{array}h_1 & h_2 & h_3 \end{array}\right]$

Figure: Channel model (3 tap multipath)

In addition to the multipath channel, the received signal gets corrupted by noise $n$ , typically referred to as Additive White Gaussian Noise (AWGN). The values of the noise $n$ follows the Gaussian probability distribution function, $p(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}$ with

mean $\mu=0$ and

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

The received signal is

$y[k]=s[k] \otimes h[k] + n$ , where

$\otimes$ is the convolution operator.

MMSE Equalization

In Minimum Mean Square Error solution, for each sample time $k$ we would want to find a set of coefficients $c[k]$ which minimizes the error between the desired signal $s[k]$ and the equalized signal $c[k]\otimes y[k]$ , i.e.

$\begin{array}{lll}E\left(e[k]\right)^2 & = & E\left(s[k]-c[k]\otimes y[k] \right)^2\\ & = & E(s[k]-c^Ty)(s[k]-c^Ty)^T\\ & = &E(s[k])^2-E(c^Tys[k]) -E(s[k]y^Tc)+E(c^Tyy^Tc)\\ & = & E(s[k])^2-c^TR_{ys} -R_{sy}c+c^TR_{yy}c\end{array}$ ,

where,

$e[k]$ is the error at sample time $k$ ,

$c$ is column vector of dimension $[K\mbox{ x }1]$ storing the equalization coefficients,

$y$ is column vector of dimension $[K\mbox{ x }1]$ storing the received samples,

$K$ is the number of taps in the equalizer,

$R_{ys}=E(ys[k])$ is the cross correlation between received sequence and input sequence ,

$R_{sy}=E(s[k]y^T)$ is the cross correlation between received sequence and input sequence and

$R_{yy}=E(yy^T)$ is the auto-correlation of the received sequence.

For solving the Minimum Mean Square Error (MMSE) criterion, we need to find a set of coefficients $c$ which minimizes $E(e[k])^2$ .

Differentiation with respect to $c$ and equating to 0,

$\begin{array}{rrl}\frac{\partial}{\partial c} \left[E(s[k])^2-c^TR_{ys} -R_{sy}c+c^TR_{yy}c\right]&=& 0\\ -R_{sy}+R_{yy}c&= & 0\\c & = &R^{-1}_{yy} R_{sy}\end{array}$ .

Simplifying,

$\begin{array}{lll}R_{sy}&=&E(s[k]y^T)\\&=&E(s[k](hs[k]+n)^T)\\&=&h^TE(s^2[k])+E(s[k]n)\\ & = & h\end{array}$ ,

$\begin{array}{lll}R_{yy}&=&E(yy^T)\\&=&E((hs[k]+n)(hs[k]+n)^T)\\&=&E(hh^T)E(s^2[k])+hE(s[k]n)+E(n[k]s[k])h^T+E(n^2)\\ & = & E(hh^T) + E(n^2) \end{array}$

Note :

a) $E(s^2[k])=1$ is the variance of the input signal

b) $E(s[k]n[k])=0$ (as there is no correlation between input signal and noise)

Simulation Model

Click here to download: Matlab/Octave script for computing BER for BPSK with 3 tap ISI channel with MMSE Equalization

The attached Matlab/Octave simulation script performs the following:

(a) Generation of random binary sequence

(b) BPSK modulation i.e bit 0 represented as -1 and bit 1 represented as +1

(d) Adding White Gaussian Noise

(e) Computing the MMSE and ZF equalization filter at the receiver (with 7 taps in length)

(f) Demodulation and conversion to bits

(g) Counting the number of bit errors

(h) Repeating for multiple values of Eb/No

The simulation results are as shown in the plot below. BER plot for BPSK in a 3 tap ISI channel with MMSE equalizer

Figure: BER plot for BPSK in a 3 tap ISI channel with MMSE equalizer

Observations

1. Can see around 0.5dB gain with using MMSE equalizer

Reference

Complex to Real : Tutorial 26 – Filters, analog, digital and adaptive equalization

Tagged as: BPSK, ISI, MMSE

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{ 34 comments… read them below or add one }

1 Abrar January 25, 2010 at 2:04 am

Hi
I am not able to open the code file, is there any problem.
Please check:

Thanks

2 Krishna Sankar January 25, 2010 at 5:25 am: @Abrar: I fixed that. Thanks for pointing out.

Reply

3 Communications Engineer January 25, 2010 at 12:06 pm

Thanks for this post – really

4 mesange January 27, 2010 at 1:03 pm

Hi Krishna,
I hope you are doing well. I am simulating 16 QAM in a Rayleigh channel and am trying to compute the SER(& BER). I need to verify my results using the theoretical formula. Can you kindly tell me the formula of SER for 16 QAM in Rayleigh channel?
I shall be grateful
Thanking in advance,

5 Krishna Sankar January 28, 2010 at 5:29 am: @mesange: I have not discussed 16QAM in Rayleigh channel case. However, you can discussion on
a) Symbol error rate for 16QAM in AWGN
http://www.dsplog.com/2007/12/09/symbol-error-rate-for-16-qam/
b) Bit error rate for 16QAM in AWGN assuming Gray coded bit mapping
http://www.dsplog.com/2008/06/05/16qam-bit-error-gray-mapping/
c) BER for BPSK in Rayleigh channel
http://www.dsplog.com/2008/08/10/ber-bpsk-rayleigh-channel/

Hope you will be able to extend these results to Rayleigh channel 16QAM case. Good luck.

Reply

6 eng_dina January 27, 2010 at 4:37 pm

thanks for your your graet work
please I study for my master in frequency synchronization in mimo ofdm system but i have problem with the matlab code to simulate to find out if estimation of the CFO on one path is affected by the CFO values of the adjacent paths and examine the estimator accuracy in term of its mean and variance
please help me it’s urgent and necessary

f_max = f_Tofdm/(M_sub*T_s); % f_max =maximal Doppler frequency (Hz),M_sub*T_s = OFDM Symbol period (sec),
%f_ndopp = f_max*M_sub*T_s % Normalized Maximum Doppler Spread Freq.

% Area parameter
% ra Rural Area
% tu Typical Urban
% bu Bad Urban
% ht Hilly Terrain
% no no fading
AREA = ‘no’ ;

% start simulation loops
for ee = 1 : length(EcNo)
ee
[theo_fo_var, rfo_var, rfo] = FOE_mimo_fading(EcNo(ee), tfo_array, NTX, NRX, M_sub, N, f_max, AREA, T_s, model, L);
plot_theo_fo_var(:,:,ee)=theo_fo_var;
plot_rfo_var(:,:,ee)=rfo_var;
plot_theo_rfo(:,:,ee)=tfo_array;
plot_rfo(:,:,ee)=rfo;
end

% plot results
for ii_tx = 1 : NTX
for ii_rx = 1 : NRX

% rearrange array order for easier plotting
vect_plot_theo_rfo = permute(plot_theo_rfo(ii_tx,ii_rx, , [2 3 1]);
vect_plot_rfo= permute(plot_rfo(ii_tx,ii_rx, , [2 3 1] ) ;
vect_plot_theo_fo_var= permute(plot_theo_fo_var(ii_tx,ii_rx,:),[2 3 1]);
vect_plot_rfo_var= permute(plot_rfo_var(ii_tx,ii_rx, , [2 3 1]);

%plot mean
figure;
plot(EcNo, vect_plot_theo_rfo, ‘:’ , ‘Linewidth’,2) ;
hold;
plot(EcNo, vect_plot_rfo,’o-’,’MarkerSize’ , 6,’Linewidth’, 1) ;
grid on;
t_str=sprintf(’New_ Theo and Est Freq Offset %s M=%d N=%d %dx%d path:tx%d-rx%d fo=%1.3f’,AREA,M_sub,N,NTX,NRX,ii_tx,ii_rx,tfo_array(ii_tx,ii_rx));
title(t_str);
xlabel(’EcNo SNR range and step (in dB)’);
ylabel(’Norm Freq Offset’) ;
legend(’Theo’,’Est’,’Location’,’Northeast’);
% save graph
fstr=sprintf(’new_m%d_%dx%dp%d%d_%s’,M_sub,NTX,NRX,ii_tx,ii_rx,AREA);
hgsave(fstr);

% plot variance
figure;
semilogy(EcNo, vect_plot_theo_fo_var,’:’,’Linewidth’,2);
hold;
semilogy(EcNo, vect_plot_rfo_var,’o-’,’MarkerSize’,6,’Linewidth’,1);
grid on;
grid minor;
t_str=sprintf(’New_ Est Var and CRLB %s M=%d N=%d %dx%d path:tx%d-rx%d fo=%1.3f’,AREA,M_sub,N,NTX,NRX,ii_tx,ii_rx,tfo_array(ii_tx,ii_rx));
title(t_str);
xlabel(’EcNo (in dB)’);
ylabel(’Variance’);
legend(’CRLB’,’Est’,’Location’,’Northeast’);
% save graph
fstr=sprintf(’new_y%d_%dx%dp%d%d_%s’,M_sub,NTX,NRX,ii_tx,ii_rx, AREA) ;
hgsave(fstr);
end
end
% rename variables for export
new_plot_theo_fo_var=plot_theo_fo_var;
new_plot_rfo_var=plot_rfo_var ;
new_plot_theo_rfo=plot_theo_rfo;
new_plot_rfo=plot_rfo;

% save all variables
fstr=sprintf(’new_f%d_%dx%d_%s.mat’,M_sub,NTX,NRX,AREA);
save(fstr);
% save variables for plotting
fstr=sprintf(’new_p%d_%dx%d_%s.mat’,M_sub,NTX,NRX,AREA);
save(fstr,’new_plot_theo_fo_var’,’new_plot_rfo_var’,’new_plot_theo_rfo’,’new_plot_theo_rfo’);

% end of file

7 Krishna Sankar January 28, 2010 at 5:24 am: @eng_dina: How are you modeling the CFO for MIMO systems?

If all the chains have a common RF clock, then all the chains will have similar CFO and the estimate from all the chains can be combined to improve the accuracy of the CFO estimation.
If the chains have independent RF clock, then we need to estimate CFO on each chain independently.

Reply

8 bhasker January 30, 2010 at 12:32 pm

your script is really appreciable sir. it is really helpful to me
1) i want to implement MMSE equalization in OFDM. can u help me on this sir
2) actually i want to know where should i aplly the equalizer in ofdm. either just before removing cyclic prefix or after takeing FFT.
plz help sir

9 Krishna Sankar April 4, 2010 at 4:38 am: @bhasker: Remove cyclic prefix, take FFT. You can see an example BER for BPSK in OFDM in AWGN @ http://www.dsplog.com/2008/06/10/ofdm-bpsk-bit-error/

Reply

10 balakrishna February 21, 2010 at 2:54 pm

please help me on DESIGN OF “RFMEMS SHUNT SWITCHS ” IN MAT LAB

11 Krishna Sankar March 31, 2010 at 5:33 am: @balakrishna: Sorry, am not aware of RFMEMS shunt switches

Reply

12 lim February 24, 2010 at 7:37 am

Hi,
Thanks for your code. It has been a great help.
I have since modified your code slightly to a 3 tap rayleigh fading channel (all taps follow Rayleigh fading, i.e. no longer a fixed 3 tap fading).
But I am not sure if it is correct. Any suggestions to go about verifying it?

13 Krishna Sankar March 31, 2010 at 5:14 am: @lim: Try simulating the BER vs Eb/N0 curve. Am not aware of any theoretical results to compare

Reply

14 sajib February 25, 2010 at 12:30 pm

hello
i needthe matlab code for MMSE equalization in ofdm sysetm just to take idea how it works in OFDM ssytem.
or can u give u ur personal email address plz so tht i would contact wid u if i feel some problem during doing the code.
Regards

15 Krishna Sankar March 31, 2010 at 5:07 am: @sajib: You can find my details on http://www.dsplog.com/contact-us/

Reply

16 Gregg March 30, 2010 at 3:00 pm

i am new to the mobile communication field and am just getting familiar with tools like matlab. i have a project at hand and i need to put up some thing before the month ends. pls can some help me out with the matlab source code? the topic of the research is the study of “the effect of ISI and equalisation on a mobile communication channel”. from my understanding so far, i need to plot some waveform with ISI and another with equalisation, compare them and hence state their effects from the two plots.the idea is there but doing this with matlab is a hell of a job especially for some new like me.i need help. thanks

17 Krishna Sankar March 31, 2010 at 5:04 am: @Gregg: Due to time constraints, I wont be able to help in the coding part. However, you can ask queries. I will try answer to the best of my knowledge

Reply

18 Aristide April 12, 2010 at 8:05 pm

Hi ,Krishna i am impressed with the way you have simulated the BER of the MMSE and zero forcing equalkizers in presence of ISI.Would you please tell me what do you mean by taps .Is it the processing gain N?
Could you also tell me the System of transmission you have considered .Asynchronous symbol transmissions or synchronous?
Thank in advance

19 Krishna Sankar April 14, 2010 at 5:16 am: @Aristide:
a) In wireless transmissions, we get multiple copies of the transmitted signal at the receiver (due to reflections). The taps model that aspect.
b) Synchronous.

Reply

20 siddarth April 27, 2010 at 3:37 pm

please i need references and matlab code for channel estimation in ofdm system

would be grateful if u provide the needy

21 Krishna Sankar April 28, 2010 at 5:46 am: @siddarth: The post on OFDM BER in Rayleigh channel discuss that
http://www.dsplog.com/2008/08/26/ofdm-rayleigh-channel-ber-bpsk/

Reply

22 rajendrasingh kushwah April 27, 2010 at 5:46 pm

sir pls help me in matlab implemantation we should requir to implemant the ifdma based paper so i need some help

23 Krishna Sankar April 28, 2010 at 5:48 am: @rajendrasingh: ifdma or ofdma? i have not discussed ofdma, but you can see some posts on OFDM @ http://www.dsplog.com/tag/ofdm/

Reply

24 kaka May 12, 2010 at 7:59 am

glad to see it

25 Jovan S. May 12, 2010 at 5:43 pm

Drear Mr. Sankar,

I’m very thankful for your work. With your posts you have helped me several times. This post is excellent. I’ve lost several weeks trying to solve the problem by myself and haven’t succeeded. Thanks to your post it become more clear to me and I succeeded to make useful simulation. You are genius!

Thanks,
Jovan

26 Krishna Sankar June 24, 2010 at 4:14 am: @Jovan: Glad to be of help

Reply

27 behzad June 8, 2010 at 6:33 pm

hi
i want to simulate doppler effect in ofdm simulation but i have some problem:
1-how i can with ifft command in matlab define separation sub-carrier frequency
2-how i can effecting doppler in my simulation
please if have any source or m-file insert your website
best regards
mhmdbehzad

28 Krishna Sankar June 14, 2010 at 6:26 am: @behzad: My replies:
1/ subcarrier spacing is controlled by the sampling clock and the number of points in the FFT. For examples, if the sampling clock is 20MHz and the FFT size is 64, the subcarrier spacing is 20MHz/64 = 312.5kHz
2/ I have not tried simulating doppler cases.

Reply

29 fernando June 20, 2010 at 3:40 pm

i want to ask???
i want create mmse equalizer but channel (ht) not ht=[0.3 0.9 0.2]
i want the channel is reylegh channel but not in the mimo system just create mmse equalizer with BPSK in reylegh channel (not mimo)

can you help me???
thank’s
fernando

30 Krishna Sankar June 21, 2010 at 5:44 am: @fernando: You can try changing ht to have Rayleigh components. Make sure that the receiver is also adjusted accordingly.

Reply

31 joel July 20, 2010 at 2:11 pm

Hello,
can you please explain why are you using the toeplitz function ? You evaluate the E(hh^T) expression with following Matlab code :
hAutoCorr = conv(ht,fliplr(ht));
hM = toeplitz([hAutoCorr([3:end]) zeros(1,2*K+1-L)], [ hAutoCorr([3:end]) zeros(1,2*K+1-L) ]);
Can you please explain this ? Thank you !
Best Regards

32 Krishna Sankar July 21, 2010 at 6:21 am: @joel: Toeplitz matrices are used to represent convolution in a matrix multiplication form. Hope the brief post on toeplitz matrix @
http://www.dsplog.com/2007/04/21/using-toeplitz-matrices-in-matlab/ helps.

Reply

33 joel July 21, 2010 at 7:38 pm

ok, I see. Two more questions

1. Why is actually c=R_yy^-1R_sy implemented as a convolution ?(the expression c_mmse = [inv(hM)*d.'].’; is a convolution due to toeplitz matrix)
I don’t quite understand why do you not multiply the matrices but convolve them?

2. Can the derivation of E(e(k))^2 be found in literature ? I’m looking for detailed mathematical background to this formula.

Best Regards and Thank you !

34 Krishna Sankar July 22, 2010 at 6:01 am: @joel:
1) Well, [inv(hM)*d.'].’ is not convolution. Its just the matrix multiplication as we derived.
2) Try Digital Communication by Proakis.

Further, I liked the material presented in http://www.complextoreal.com/chapters/filters.pdf (pages 20-30)

Reply

BER for BPSK in ISI channel with MMSE equalization

Transmit symbol

Channel Model

MMSE Equalization

Simulation Model

Reference

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