In case you don't have the Matlab Signal Processing Toolbox

Without kaiserord, we would need to implement Kaiser's formula [102,59] for estimating the Kaiser-window required to achieve the given filter specs:

where is the desired stop-band attenuation in dB (typical values in audio work are to ). Note that this estimate for becomes too small when the filter passband width approaches zero. In the limit of a zero-width pass-band, the frequency response becomes that of the Kaiser window transform itself. A non-zero passband width acts as a ``moving average'' lowpass filter on the sidelobes of the window transform, which brings them down in level. The kaiserord estimate assumes some of this sidelobe smoothing is present.

A similar function from [179] for window design (as opposed to filter design) is

where now is the desired side-lobe attenuation in dB (as opposed to stop-band attenuation). A plot showing Kaiser window sidelobe level for various values of is given in Fig.3.19.

Similarly, the filter order is estimated from stop-band attenuation and desired transition width using the empirical formula

where is in radians between 0 and .

Without the function fir1, we would have to manually implement the window method of filter design by (1) constructing the impulse response of the ideal bandpass filter (a cosine modulated sinc function), (2) computing the Kaiser window using the estimated length and from above, then finally (3) windowing the ideal impulse response with the Kaiser window to obtain the FIR filter coefficients . A manual design of this nature will be illustrated in the Hilbert transform example of §B.5.

---

Order a Hardcopy of Spectral Audio Signal Processing

Previous: Results
Next: Comparison to the Optimal Chebyshev FIR Bandpass Filter

---

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.