FIR Digital Filter Design

FIR filters are basic in spectral audio signal processing. In fact,
the fastest way to implement long FIR filters in conventional
CPUs^{5.1} is by means of
*FFT convolution*.
The
*convolution theorem* for Fourier transforms (§2.3.5)
states that the convolution of an input signal
with a filter
impulse-response
is given by the inverse DTFT of the product of
the signal's spectrum
times the filter's frequency
response
, *i.e.*,

(5.1) |

where

and the DTFT is defined as (§2.1)

(5.2) |

As usual with the DTFT, the sampling rate is assumed to be . In practice, the DTFT is used in

This chapter provides a starting point in the area of FIR digital
filter design. The so-called ``window method'' for FIR filter design,
also based on the convolution theorem for Fourier transforms, is
discussed in some detail, and compared with an optimal
Chebyshev method. Other methods, such as least-squares, are discussed
briefly to provide further perspective. Tools for FIR filter design in
both Octave and the Matlab Signal Processing Toolbox are listed where
applicable. For more information on digital filter design, see,
*e.g.*, the documentation for the Matlab Signal Processing Toolbox and/or
[#!JOSFP!#,#!Treichler09!#,#!Burrus08!#,#!ParksAndBurrus!#,#!SteiglitzText96!#,#!RabinerAndGold!#,#!OppenheimAndSchafer!#,#!JOST!#].

- The Ideal Lowpass Filter
- Lowpass Filter Design Specifications

- Least-Squares FIR Filter Design

- Frequency-Sampling FIR Filter Design
- Window Method for FIR Filter Design

- Hilbert Transform Design Example
- Primer on Hilbert Transform Theory
- Preparing the Desired Impulse Response
- More General FIR Filter Design
- Comparison to Optimal Chebyshev FIR Filter
- Conclusions

- Generalized Window Method
- Minimum-Phase Filter Design
- Minimum-Phase and Causal Cepstra
- Optimal FIR Digital Filter Design

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University