Transform lowpass analog filters to bandstop
[bt,at] = lp2bs(b,a,Wo,Bw)
[At,Bt,Ct,Dt] = lp2bs(A,B,C,D,Wo,Bw)
lp2bs transforms analog lowpass filter prototypes with a cutoff
angular frequency of 1 rad/s into bandstop filters with desired bandwidth and
center frequency. The transformation is one step in the digital filter design process
for the butter, cheby1, cheby2, and ellip functions.
lp2bs can perform the transformation on two different linear system
representations: transfer function form and state-space form. In both cases, the input
system must be an analog filter prototype.
[bt,at] = lp2bs(b,a,Wo,Bw)
transforms an analog lowpass filter prototype given by polynomial coefficients into
a bandstop filter with center frequency Wo and bandwidth
Bw. Row vectors b and a
specify the coefficients of the numerator and denominator of the prototype in
descending powers of s.
Scalars Wo and Bw specify the center
frequency and bandwidth in units of rad/s. For a filter with lower band edge
w1 and upper band edge w2, use
Wo = sqrt(w1*w2) and
Bw = w2-w1.
lp2bs returns the frequency transformed filter in row vectors
bt and at.
[At,Bt,Ct,Dt] = lp2bs(A,B,C,D,Wo,Bw)
converts the continuous-time state-space lowpass filter prototype in matrices
A, B, C,
D shown below
into a bandstop filter with center frequency Wo and bandwidth
Bw. For a filter with lower band edge w1
and upper band edge w2, use
Wo = sqrt(w1*w2) and
Bw = w2-w1.
The bandstop filter is returned in matrices At,
Bt, Ct, Dt.
lp2bs is a highly accurate state-space formulation of the classic
analog filter frequency transformation. If a bandstop filter is to have center frequency
ω0 and bandwidth
Bw, the standard s-domain
transformation is
where Q = ω0/Bw and p = s/ω0. The state-space version of this transformation is
Q = Wo/Bw; At = [Wo/Q*inv(A) Wo*eye(ma);-Wo*eye(ma) zeros(ma)]; Bt = -[Wo/Q*(A\B); zeros(ma,n)]; Ct = [C/A zeros(mc,ma)]; Dt = D - C/A*B;
See lp2bp for a derivation of the bandpass version of this
transformation.