# Cramer rao bound on estimating amplitude of the signal using complex demodulation

Started by July 14, 2003
```Hello,

I am finding the amplitude of the signal using complex demodulation
(quadrature demodulation). I wanted to know maximum accuracy to which
the amplitude of the one of the frequency contained in phase modulated
signal(which is the input) can be estimated.The accuracy to which the
amplitude obtained should be 10^-8 (between actual and estimated by
this method). So i wanted to know if there is any cramer rao bound
(accuracy to which amplitude of the signal can be estimated). The SNR
of the signal is about 55 to 60 dB.

with regards
praveen
```
```praveenkumar1979@rediffmail.com (praveen) wrote in message news:<ff8a3afb.0307140206.15505f8@posting.google.com>...
> Hello,
>
> I am finding the amplitude of the signal using complex demodulation
> (quadrature demodulation). I wanted to know maximum accuracy to which
> the amplitude of the one of the frequency contained in phase modulated
> signal(which is the input) can be estimated.The accuracy to which the
> amplitude obtained should be 10^-8 (between actual and estimated by
> this method). So i wanted to know if there is any cramer rao bound
> (accuracy to which amplitude of the signal can be estimated). The SNR
> of the signal is about 55 to 60 dB.
>
> with regards
> praveen

A good lullaby on "Crammer-Rao Bound"
by Peter Kootsookos is interesting - see
http://www.dspguru.com/comp.dsp/fun/crbound.htm!!

From my limited stock I can say, the CR bound provides a lower bound
on the estimator variance of an unbiased estimator. If we know your
signal's pdf we
can calculate Fisher information matrix and its inverse to get
variance estimate
what is known as CR lower bound. It looks like your SNR is high. I am
wondering
what accuracy you expect at low SNR- Is it the same? I guess CR bound
at low SNR and high SNR may not be same - please correct me if I am
wrong.

Regards,
Santosh
```
```praveenkumar1979@rediffmail.com (praveen) writes:

> I am finding the amplitude of the signal using complex demodulation
> (quadrature demodulation). I wanted to know maximum accuracy to
> which the amplitude of the one of the frequency contained in phase
> modulated signal(which is the input) can be estimated. The accuracy
> to which the amplitude obtained should be 10^-8 (between actual and
> estimated by this method). So i wanted to know if there is any
> cramer rao bound (accuracy to which amplitude of the signal can be
> estimated). The SNR of the signal is about 55 to 60 dB.

Kay's book (, page 542) says that the CRLBs for estimating A, f and phi in:

x[n] = A exp( j 2 pi f n ) + complex white Gaussian noise

are:

var(Ahat) >= sigma^2 / ( 2 N )

var(fhat) >= 6 sigma^2 / ( (2 pi)^2 A^2 N (N^2-1) )

var(phihat) >= sigma^2 ( 2 N - 1 ) / ( A^2 N (N + 1) )

where sigma^2 is the complex noise variance, N is the number of
samples of x[n] that you have.

 S. M. Kay, "Fundamentals of Statistical Signal Processing Vol 1:
Estimation Theory", Prentice-Hall, 1993.

Ciao,

Peter K.

--
Peter J. Kootsookos

"Na, na na na na na na, na na na na"
- 'Hey Jude', Lennon/McCartney
```
```"Peter J. Kootsookos" <p.kootsookos@remove.ieee.org> wrote

> Kay's book (, page 542) says that the CRLBs for estimating A, f and phi
in:
>
> x[n] = A exp( j 2 pi f n ) + complex white Gaussian noise

D'oh, I left out phi.... this should read something like:

x[n] = A exp( j ( 2 pi f n  + phi ) ) + complex white Gaussian noise

--
Peter J. Kootsookos

"Na, na na na na na na, na na na na"
- 'Hey Jude', Lennon/McCartney

```
```Hello santhosh,
You are right, CR bound varies for different SNR. The problem in my
case is that amplitude of the frequency is to be estimated with a
error of 10^-8. The main problem is because of the noise signal
present in it. If the ampliude of the noise is less than 10^-3(when
SNR is greater than 90 dB) and less the accuracy estimated is of the
order 10^-8. But when SNR is about 60 dB, noise ampltude is greater
than 10^-3 and hence estimated amplitude is of the order 10^-5.