17-03-2014, 06:10 PM
Abstract: A new method for obtaining series of window
functions is introduced. A novel form of frequency response
expression of digital filters is deduced which is sum of sinc
sum functions. One of the remarkable characteristics of the
form is that weights(coefficients of terms) of window
functions can be directly calculated according to the
expression. An example for how to find a window function is
illustrated in detail. An instance of series of window functions
is tabled with both window weights and filter performances.
Stopband attenuation of filters is from 32db to 63db with the
gap about 3db. With same performances of stopband
attenuation and transition width both the order and passband
ripple of filters using the new method is little better than that
using Kaiser window if passband attenuation is smaller than
50db and little worse if bigger than 52db. The obtained
window functions are as simple as Blackman window function.
The outstanding feature of the new approach is that it can
provide a very efficient way to find series of window
functions with both good performances and easy calculation.
Key words: Filter, FIR, Sinc sum, Window
1. Introduction
About the design of FIR digital filters there are some
methods available. Some of them are window method,
frequency sampling method, optimal method[1], fractional
delay filters[2], evolution algorithm[3], neural network
method[4], frequency response masking technique[5]. The
most common methods are window method and optimal
method, which is because of their remarkable advantages. The
fixed window method is simplest for calculation. The
adjustable window method, mainly Kaiser window and
Chebyshev window, takes account of small calculation and
narrow transition width with necessary passband ripple and
stopband attenuation. The optimal method can give equal
passband and stopband ripples with narrow transition width
and hence with best performances. Of course each method has
its disadvantages more or less. The absence of fixed window
method is that the cut-off frequencies of a filter is not easily
controlled and the order of it is some big. The lack of
adjustable window method is that the calculation is much
more complex compared to fixed window method and the
order of filters is bigger in comparison with optimal method.
The problem of the optimal method is that the calculation is
much more complicated compared with adjustable window
method. Because of the advantages and disadvantages of these
methods they are selected in different situations. Further about
window method, all existed windows that have been tried to
find are all based on the approach of window frequency
responses[6-11], of which the edge of the mainlobe or the
height of sidelobes in frequency domain must be small. Now a
new method for searching window functions was proposed.
Next we give a discussion about it.
functions is introduced. A novel form of frequency response
expression of digital filters is deduced which is sum of sinc
sum functions. One of the remarkable characteristics of the
form is that weights(coefficients of terms) of window
functions can be directly calculated according to the
expression. An example for how to find a window function is
illustrated in detail. An instance of series of window functions
is tabled with both window weights and filter performances.
Stopband attenuation of filters is from 32db to 63db with the
gap about 3db. With same performances of stopband
attenuation and transition width both the order and passband
ripple of filters using the new method is little better than that
using Kaiser window if passband attenuation is smaller than
50db and little worse if bigger than 52db. The obtained
window functions are as simple as Blackman window function.
The outstanding feature of the new approach is that it can
provide a very efficient way to find series of window
functions with both good performances and easy calculation.
Key words: Filter, FIR, Sinc sum, Window
1. Introduction
About the design of FIR digital filters there are some
methods available. Some of them are window method,
frequency sampling method, optimal method[1], fractional
delay filters[2], evolution algorithm[3], neural network
method[4], frequency response masking technique[5]. The
most common methods are window method and optimal
method, which is because of their remarkable advantages. The
fixed window method is simplest for calculation. The
adjustable window method, mainly Kaiser window and
Chebyshev window, takes account of small calculation and
narrow transition width with necessary passband ripple and
stopband attenuation. The optimal method can give equal
passband and stopband ripples with narrow transition width
and hence with best performances. Of course each method has
its disadvantages more or less. The absence of fixed window
method is that the cut-off frequencies of a filter is not easily
controlled and the order of it is some big. The lack of
adjustable window method is that the calculation is much
more complex compared to fixed window method and the
order of filters is bigger in comparison with optimal method.
The problem of the optimal method is that the calculation is
much more complicated compared with adjustable window
method. Because of the advantages and disadvantages of these
methods they are selected in different situations. Further about
window method, all existed windows that have been tried to
find are all based on the approach of window frequency
responses[6-11], of which the edge of the mainlobe or the
height of sidelobes in frequency domain must be small. Now a
new method for searching window functions was proposed.
Next we give a discussion about it.