# Estimating FIR Coefficient Bounds for Adaptive Filter

Started by June 23, 2008
```In order to provide adequate word size in an integer-arithmetic
adaptive FIR filter, I want to know the bounds on the filter
coefficients. I could probably place some bounds on parameters of the
desired response, but the desired response is generally unknown a
priori and should be considered arbitrary.

Since the desired response is produced by a pysical system, I imagine
that some properties of that system will help form the basis of
bounding its response parametrically.

The physical system could be the multipath propagation of a a wave.
One assumption, for example, may be that the impulse response has an
upper bound on its duration.

The number of coefficents may be large (>40) and complex. Real-time
processing rates and hardware limitations rule out anything but
integer arithmetic, for the time being.

I'm not sure how to begin, and would appreciate any references where
this has been addressed in either a general way or for a specific
case.

Thanks,
Jim
```
```>In order to provide adequate word size in an integer-arithmetic
>adaptive FIR filter, I want to know the bounds on the filter
>coefficients. I could probably place some bounds on parameters of the
>desired response, but the desired response is generally unknown a
>priori and should be considered arbitrary.
>
>Since the desired response is produced by a pysical system, I imagine
>that some properties of that system will help form the basis of
>bounding its response parametrically.
>
>The physical system could be the multipath propagation of a a wave.
>One assumption, for example, may be that the impulse response has an
>upper bound on its duration.
>
>The number of coefficents may be large (>40) and complex. Real-time
>processing rates and hardware limitations rule out anything but
>integer arithmetic, for the time being.
>
>I'm not sure how to begin, and would appreciate any references where
>this has been addressed in either a general way or for a specific
>case.
>
>Thanks,
>Jim
>

Hi Jim,

the behaviour of an adaptive filter's coefficients depends mainly on two
factors:

1.The statistical properties of the data that you feed to the adaptive
filter, namely the desired response d[i] and the input signal u[i].
2.The adaptive algorithm that you employ.

Finding a priori the bounds on the coefficients may be a formidable if not
impossible task. In applications where the coefficients must remain bounded
a technique called "leakage" is used.

So check as a start the "leaky LMS algorithm". The more leakage you
introduce the more bounded the coefficients become but the price for that
is a degradation in the performance of the adaptive filter. As usually, a