Derive closed form solution for a receiver

Started by January 5, 2010
```Hi All

I have simulated a performance of the receiver for time varying frequency
selective channel. I am using adaptive equalizer to combat multipath.

What I would like to do is to derive closed form solution such as
lower/upper bound or exact solution for BER.

However, there are many variables: time varying channel, no channel state

By time varying I mean, channel changes at every symbol.

How can I approach such problem?

Many thanks.

Chintan Shah

```
```On Jan 5, 12:38 pm, "cpshah99" <cpsha...@rediffmail.com> wrote:
> Hi All
>
> I have simulated a performance of the receiver for time varying frequency
> selective channel. I am using adaptive equalizer to combat multipath.
>
> What I would like to do is to derive closed form solution such as
> lower/upper bound or exact solution for BER.
>
> However, there are many variables: time varying channel, no channel state
>
> By time varying I mean, channel changes at every symbol.
>
> How can I approach such problem?
>
> Many thanks.
>
> Chintan Shah

Chintan,
First, think about whether you really need a closed-form expression.
In most cases, you will get better understanding by doing a simulation
and estimating the bit error rate.

If you indeed want to analytically find the closed-form expression,
you should probably consult a textbook like Digital Communications by
John Proakis. Upper and lower bounds are calculated for many cases
there.

Dilip.
```
```
cpshah99 wrote:

> Hi All
>
> I have simulated a performance of the receiver for time varying frequency
> selective channel. I am using adaptive equalizer to combat multipath.
>
> What I would like to do is to derive closed form solution such as
> lower/upper bound or exact solution for BER.

> However, there are many variables: time varying channel, no channel state
>
> By time varying I mean, channel changes at every symbol.
>
> How can I approach such problem?

Let the multipath energy be W, speading function S.
Equalizer reduces W by the factor E.
Assume equalizer is optimal Kalman filter, then E is a function of SNR
and S. The W times E adds to the signal; that is Raleigh process. Find
error rate from there.

> Many thanks.
> Chintan Shah

STUPIDENT MATLABI

```
```Hi Dilip

Thanks.

Although simulation is good to understand, somebody can ask how to derive
the same results thru analysis. Proakis does not have such treatment.
Please tell me the page number if you have come across.

>Let the multipath energy be W, speading function S.
>Equalizer reduces W by the factor E.
>Assume equalizer is optimal Kalman filter, then E is a function of SNR
>and S. The W times E adds to the signal; that is Raleigh process. Find
>error rate from there.
>

I am sorry but I did not understand it. It will be great if you can
expalin a bit further.

>
>STUPIDENT MATLABI
>
Yeah I think so.

```
```On Jan 6, 5:21&#2013266080;am, "cpshah99" <cpsha...@rediffmail.com> wrote:
> Hi Dilip
>
> Thanks.
>
> Although simulation is good to understand, somebody can ask how to derive
> the same results thru analysis. Proakis does not have such treatment.
> Please tell me the page number if you have come across.
>

In my 3rd edition, there is a section on "Performance Characteristics
of the MSE Equalizer" in a chapter titled "Communication through Band-
limited channels".
```
```>On Jan 6, 5:21=A0am, "cpshah99" <cpsha...@rediffmail.com> wrote:
>> Hi Dilip
>>
>> Thanks.
>>
>> Although simulation is good to understand, somebody can ask how to
derive
>> the same results thru analysis. Proakis does not have such treatment.
>> Please tell me the page number if you have come across.
>>
>
>In my 3rd edition, there is a section on "Performance Characteristics
>of the MSE Equalizer" in a chapter titled "Communication through Band-
>limited channels".
>

Hi Dilip

Thanks again.

I found the section in my 4th edition in Chapter 10.

I have seen this treatment but it considers fixed channel not 'Time
varying'.

I think the problem that I am trying to solve needs some assumptions.
Because it matters a lot how we model the time varying frequency selective
channel. Also if the receiver has the channel knowledge or not, which type
of adaptive algorithm we use etc...

Thanks again.

Chintan Shah

```
```Performance Analysis over fading channel can be quite complicated.

As general approach, assuming that the received signal is corrupted by
Gaussian noise, you need to:

1. Derive the SNR expression conditioned to the channel (i.e., for a given
channel realization). Then the instantaneous Bit Error Probability will in
the form of the Q() function.

2. Next, you need to average the instantaneous BEP over the statistic of
the channel fading (such as Rayleigh, Nakagami, etc. ). This is the
difficult part, and coming up with a close form solution is not trivial.

On my opinion, the most complete treatment on this subject is the book

"Digital Communication over Fading Channels, 2nd Edition"
Marvin K. Simon, Mohamed-Slim Alouini

If you are interested in performance analysis of wireless communication
systems over fading channels, this is absolutely the book you must look
at.

```
```>Performance Analysis over fading channel can be quite complicated.
>
>As general approach, assuming that the received signal is corrupted by
>Gaussian noise, you need to:
>
>1. Derive the SNR expression conditioned to the channel (i.e., for a
given
>channel realization). Then the instantaneous Bit Error Probability will
in
>the form of the Q() function.
>
>2. Next, you need to average the instantaneous BEP over the statistic of
>the channel fading (such as Rayleigh, Nakagami, etc. ). This is the
>difficult part, and coming up with a close form solution is not trivial.
>
>On my opinion, the most complete treatment on this subject is the book
>
>"Digital Communication over Fading Channels, 2nd Edition"
>Marvin K. Simon, Mohamed-Slim Alouini
>
>If you are interested in performance analysis of wireless communication
>systems over fading channels, this is absolutely the book you must look
>at.
>
>

%%%%

Hi Marco

Thanks.

I have seen this book vaguely. Most of the books deal with frequency flat