The bilinear transformation is a mathematical mapping of variables.
In digital filtering, it is a standard method of mapping the s or analog plane into the z or digital plane.
It transforms analog filters, designed using classical filter design techniques, into their discrete equivalents.
The bilinear transformation maps the s-plane into the z-plane by H(z)=H(s)|s=2fzz−1z+1
This transformation maps the jΩ axis (from Ω = –∞ to +∞) repeatedly around the unit circle (ejw, from ω = –π to π) by ω=2tan−1(Ω2fs)
bilinear can accept an optional parameter Fp that specifies prewarping. fp, in hertz, indicates a "match" frequency, that is, a frequency for which the frequency responses before and after mapping match exactly.
In prewarped mode, the bilinear transformation maps the s-plane into the z-plane with H(z)=H(s)|s=2πfptan(πfpfz)(z−1)(z+1)
With the prewarping option, bilinear maps the jΩ axis (from Ω = –∞ to +∞) repeatedly around the unit circle (ejw, from ω = –π to π) by ω=2tan−1(Ωtan(πfpfs)2πfp)
In prewarped mode, bilinear matches the frequency 2πfp (in radians per second) in the s-plane to the normalized frequency 2πfp/fs (in radians per second) in the z-plane.
The bilinear function works with three different linear system representations: zero-pole-gain, transfer function, and state-space form.
Zero-Pole-Gain [zd,pd,kd] = bilinear(z,p,k,fs) and [zd,pd,kd] = bilinear(z,p,k,fs,fp) convert the s-domain transfer function specified by z, p, and k to a discrete equivalent.
Inputs z and p are column vectors containing the zeros and poles, k is a scalar gain, and fs is the sampling frequency in hertz.
bilinear returns the discrete equivalent in column vectors zd and pd and scalar kd.
The optional match frequency, fp is in hertz and is used for prewarping.
Transfer Function [numd,dend] = bilinear(num,den,fs) and [numd,dend] = bilinear(num,den,fs,fp) convert an s-domain transfer function given by num and den to a discrete equivalent.
Row vectors num and den specify the coefficients of the numerator and denominator, respectively, in descending powers of s.
Let B(s) be the numerator polynomial and A(s) be the denominator polynomial.
The transfer function is: B(s)A(s)=B(1)sn+⋯+B(n)s+B(n+1)A(1)sm+⋯+A(m)s+A(m+1) fs is the sampling frequency in hertz.
bilinear returns the discrete equivalent in row vectors numd and dend in descending powers of z (ascending powers of z–1).
fp is the optional match frequency, in hertz, for prewarping.
State-Space[Ad,Bd,Cd,Dd] = bilinear(A,B,C,D,fs) and[Ad,Bd,Cd,Dd] = bilinear(A,B,C,D,fs,fp) convert the continuous-time state-space system in matrices A, B, C, Dx˙=Ax+Buy=Cx+Du to the discrete-time system:x[n+1]=Adx[n]+Bdu[n]y[n] =Cdx[n]+Ddu[n] fs is the sampling frequency in hertz.
bilinear returns the discrete equivalent in matrices Ad, Bd, Cd, Dd. The optional match frequency, fp is in hertz and is used for prewarping.
[zd,pd,kd] = bilinear(z,p,k,fs) [zd,pd,kd] = bilinear(z,p,k,fs,fp) [numd,dend] = bilinear(num,den,fs) [numd,dend] = bilinear(num,den,fs,fp) [Ad,Bd,Cd,Dd] = bilinear(A,B,C,D,fs) [Ad,Bd,Cd,Dd] = bilinear(A,B,C,D,fs,fp)
Numerator cannot be higher order than denominator.