04-01-2013, 09:18 AM
FIR Filter Design Using Windows
[attachment=46394]
Optimality
Theorem
For a given ideal frequency response and for a
given order, the filter obtained by impulse
response truncation has a minimum error
among all causal FIR filters of the given order.
The Gibbs Phenomenon
Peaks near the transition band:
Both within and outside the pass band
Peaks height ~0.09, regardless of the filter’s order
Become narrower as the order gets larger
Understanding
GibbsMain lobe width determine Distance of peaks from transition point
Side lobe level:
The ratio between main lobe and next lobe’s height
Roughly Determines peak height
2/(3p), or about -13.5db in Dirichlet kernel
FIR filter Design Using Windows
Define the ideal frequency response Hd (q)
For each pair {qp,qs} take the midpoint 0.5(qp+qs)
Obtain the ideal impulse response hd [n] as in the IRT method
Compute coefficients:
Criteria for Choosing a Window
Given the desired order of the FIR filter,
we would like the kernel function to have:
A main lobe that is as narrow at possible
Side lobes that are as low as possible
The Window Design Challenge
Choosing a window always involves a trade-off between
the width of the main lobe and the level of the side lobes
The rectangular window
Narrowest possible main lobe of all windows of the same length
But its side lobes are the highest
We are ready to increase the main-lobe width in order to reduce the side lobe level
As a consequence – we increase transition band width in order to reduce the pass band ripple
Our aim – choose a window with a good trade-off
[attachment=46394]
Optimality
Theorem
For a given ideal frequency response and for a
given order, the filter obtained by impulse
response truncation has a minimum error
among all causal FIR filters of the given order.
The Gibbs Phenomenon
Peaks near the transition band:
Both within and outside the pass band
Peaks height ~0.09, regardless of the filter’s order
Become narrower as the order gets larger
Understanding
GibbsMain lobe width determine Distance of peaks from transition point
Side lobe level:
The ratio between main lobe and next lobe’s height
Roughly Determines peak height
2/(3p), or about -13.5db in Dirichlet kernel
FIR filter Design Using Windows
Define the ideal frequency response Hd (q)
For each pair {qp,qs} take the midpoint 0.5(qp+qs)
Obtain the ideal impulse response hd [n] as in the IRT method
Compute coefficients:
Criteria for Choosing a Window
Given the desired order of the FIR filter,
we would like the kernel function to have:
A main lobe that is as narrow at possible
Side lobes that are as low as possible
The Window Design Challenge
Choosing a window always involves a trade-off between
the width of the main lobe and the level of the side lobes
The rectangular window
Narrowest possible main lobe of all windows of the same length
But its side lobes are the highest
We are ready to increase the main-lobe width in order to reduce the side lobe level
As a consequence – we increase transition band width in order to reduce the pass band ripple
Our aim – choose a window with a good trade-off