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Physical Audio Signal Processing
Delay-Line and Signal Interpolation
Lagrange Interpolation
Faust Code for Lagrange Interpolation
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Faust Code for Lagrange Interpolation
The Faust programming language for signal processing [453,450] includes support for Lagrange fractional-delay filtering, up to order five, in the library file filter.lib. For example, the fourth-order case is listed below:
// fourth-order (quartic) case, delay d in [1.5,2.5]
fdelay4(n,d,x) = delay(n,id,x) * fdm1*fdm2*fdm3*fdm4/24
+ delay(n,id+1,x) * (0-fd*fdm2*fdm3*fdm4)/6
+ delay(n,id+2,x) * fd*fdm1*fdm3*fdm4/4
+ delay(n,id+3,x) * (0-fd*fdm1*fdm2*fdm4)/6
+ delay(n,id+4,x) * fd*fdm1*fdm2*fdm3/24
with {
o = 1.49999;
dmo = d - o; // assumed nonnegative
id = int(dmo);
fd = o + frac(dmo);
fdm1 = fd-1;
fdm2 = fd-2;
fdm3 = fd-3;
fdm4 = fd-4;
};
An example calling program is shown in Fig.4.12.
// tlagrange.dsp - test Lagrange interpolation in Faust
import("filter.lib");
N = 16; % Allocated delay-line length
% Compare various orders:
D = 5.4;
process = 1-1' <: fdelay1(N,D),
fdelay2(N,D),
fdelay3(N,D),
fdelay4(N,D),
fdelay5(N,D);
// To see results:
// [in a shell]:
// faust2octave tlagrange.dsp
// [at the Octave command prompt]:
// plot(db(fft(faustout,1024)(1:512,:)));
// Alternate example for testing a range of 4th-order cases
// (change name to "process" and rename "process" above):
process2 = 1-1' <: fdelay4(N, 1.5),
fdelay4(N, 1.6),
fdelay4(N, 1.7),
fdelay4(N, 1.8),
fdelay4(N, 1.9),
fdelay4(N, 2.0),
fdelay4(N, 2.1),
fdelay4(N, 2.2),
fdelay4(N, 2.3),
fdelay4(N, 2.4),
fdelay4(N, 2.499),
fdelay4(N, 2.5);
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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.
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