I have a function of $j\Omega$ and $v$ (velocity), i.e.:

$$A(j\Omega,v)=\frac{(j\Omega)^2}{p + \frac{j\Omega}{v}}$$

where $p$ is the pole.

I want know how $A(j\Omega,v)$ modify its behavior, when I insert another cascade filter $F(j\Omega)$.

In this case I would like study the power of this system via Parseval's theorem:

$$P(v) = \frac{1}{2 \pi} \int_{-\infty}^{+\infty} \big|A(j \Omega,v) F(j\Omega)\big|^2 \text{d} \Omega$$

Until yesterday, I have used always the same $F(j\Omega)$, so the Matlab code was long, but simple:

acc = @(w,v) ...
    abs( ...
    ((1i.*w).^2).* ...
    (1./(.1+1i.*w./v)) .*...
          ((0.0612.*(1i.*w)+1)./ ...
          (3.375e-06.*(1i.*w).^4+0.0001057.*(1i.*w).^3+0.0167.* ...
          (1i.*w).^2+0.0612.*(1i.*w)+1)) ...
    ).^2;


q = zeros(v_min, v_max);

for v = v_min:v_max;
    q(v) = integral(@(w)acc(w,v), 0, Inf)./pi;
end

Where the second part of acc is the filter.

This code works and everything is OK.

Now, I do not want put that filter, but several kind of IIR Butterworth low-pass filters.

1. A series, with $F_c = constant$, but with order N = 1:10.
2. A second series, with $N = constant$, and Fc = [1 2 4 8 16 32 64 128]Hz.

I know that the code for generate these filters is:

h  = fdesign.lowpass('N,F3dB', N, Fc, Fs);
Hd = design(h, 'butter');

but my questions is: how concatenate this filter with a Matlab function, i.e.:

A = @(w,v)(((1i.*w).^2).*(1./(.1+1i.*w./v)));

for evaluate the integral?