03-09-2012, 01:40 PM
DESIGN OF FIR FILTERS BY WINDOWING
[attachment=33201]
As discussed in Section 7.1 commonly used techniques for the design of IIR filters are
based on transformations of continuous-time IIR systems into discrete-time IIR systems.
This is partly because continuous-time filter design was a highly advanced art before
discrete-time filters were of interest and partly because of the difficulty of implementing
a noniterative direct design method for IIR filters.
In contrast, FIR filters are almost entirely restricted to discrete-time
implementations. Consequently, the design techniques for FIR filters are based on
directly approximating the desired frequency response of the discrete-time systems.
Furthermore, most techniques for approximating the magnitude response of an FIR
system assume a linear phase constraint, thereby avoiding the problem of spectrum
factorization that complicates the direct design of IIR filters.
Many idealized systems are defined by piecewise-constant or piecewise-functional
frequency response with discontinuities at the boundaries between bands. As a result,
these systems have impulse responses that are noncausal and infinitely long. The most
straightforward approach to obtaining a causal FIR approximation to such systems is to
truncate the ideal response. Equation (7.40) can be thought of as a Fourier series
representation of the periodic frequency response Hd(ej), with the sequence hd[n] playing
the role of the Fourier coefficients. Thus, the approximation of an ideal filter by
truncation of the ideal impulse response is identical to the issue of the convergence of
Fourier series, a subject that has received a great deal of study. A particularly important
concept from this theory is the Gibbs phenomenon, which was discussed in Example
2.22. In the following discussion, we will see how this nonuniform convergence
phenomenon manifests itself in the design of FIR filters.
The fourth column of Table 7.1 shows the peak approximation error ( in dB ) for the windows of Eqs. (7.47). Clearly, the windows with the smaller side lobes yield better approximations of the ideal response at a discontinuity. Also, the third column, which shows the width of the main lobe, suggests the narrower transition regions can be achieved by increasing M. Thus, through the choice of the shape and duration of the window, we can control the properties of the resulting FIR filter. However, trying different windows and adjusting lengths by trial and error is not a very satisfactory way to design filters. Fortunately, a simple formalization of the window method has been developed by Kaiser (1974).
[attachment=33201]
As discussed in Section 7.1 commonly used techniques for the design of IIR filters are
based on transformations of continuous-time IIR systems into discrete-time IIR systems.
This is partly because continuous-time filter design was a highly advanced art before
discrete-time filters were of interest and partly because of the difficulty of implementing
a noniterative direct design method for IIR filters.
In contrast, FIR filters are almost entirely restricted to discrete-time
implementations. Consequently, the design techniques for FIR filters are based on
directly approximating the desired frequency response of the discrete-time systems.
Furthermore, most techniques for approximating the magnitude response of an FIR
system assume a linear phase constraint, thereby avoiding the problem of spectrum
factorization that complicates the direct design of IIR filters.
Many idealized systems are defined by piecewise-constant or piecewise-functional
frequency response with discontinuities at the boundaries between bands. As a result,
these systems have impulse responses that are noncausal and infinitely long. The most
straightforward approach to obtaining a causal FIR approximation to such systems is to
truncate the ideal response. Equation (7.40) can be thought of as a Fourier series
representation of the periodic frequency response Hd(ej), with the sequence hd[n] playing
the role of the Fourier coefficients. Thus, the approximation of an ideal filter by
truncation of the ideal impulse response is identical to the issue of the convergence of
Fourier series, a subject that has received a great deal of study. A particularly important
concept from this theory is the Gibbs phenomenon, which was discussed in Example
2.22. In the following discussion, we will see how this nonuniform convergence
phenomenon manifests itself in the design of FIR filters.
The fourth column of Table 7.1 shows the peak approximation error ( in dB ) for the windows of Eqs. (7.47). Clearly, the windows with the smaller side lobes yield better approximations of the ideal response at a discontinuity. Also, the third column, which shows the width of the main lobe, suggests the narrower transition regions can be achieved by increasing M. Thus, through the choice of the shape and duration of the window, we can control the properties of the resulting FIR filter. However, trying different windows and adjusting lengths by trial and error is not a very satisfactory way to design filters. Fortunately, a simple formalization of the window method has been developed by Kaiser (1974).