Hilbert Transformers - Digital Filter Design Solutions - page 419

function [Fmod] = HilbertFmod(s, hin, hout,d,g);
%
%HilbertFmod Extracts the instantaneous modulation frequency Fmod

% from a signal s using the digital Hilbert transform.

%
%
% Usage: [Fmod] = HilbertFmod(s,hin,hout,d,g);
% Input: s, a narrowband signal of the form A*sin(pi*Fmod*t)
% g, vector of Hilbert transform coefficients of order L
% hin, vector of low-pass premixing filter coefficients of order L
% hout, vector of low-pass postmixing filter coefficients of order L2
% d, vector of differentiating filter coefficients of order L
%
% Output: Fmod, Instantaneous modulation frequency
%
% Notes. (i) Fmod is allowed small variations that satisfy narrowband conditions.
%
Lh1 = length(hin);
Lh2 = length(hout);
Lg = length(g);
Ld = length(d);
if ((Lh1 == Ld)&(Ld == Lg))
sF = filter(hin,1,s);
sH = filter(g,1,s);
sD = filter(d,1,s);
A = sH.^2 + sF.^2;
S = (2/pi)*(sD.*sH)./A;
Fmod = filter(hout,1,S);
end
if (Ld ~= Lg)|(Ld ~= Lh1)
disp(' HilbertFmod: Warning - filter lengths are different.');
end
return

Figure 7.25 A Matlab® function that determines the modulation frequency F m , using the Hilbert
transform.

2 2

behind premixing filters LPF1 and LPF2 are , respectively.

After mixing, the noise powers could be added together, neglecting higher-order
frequency-dependent terms and on the assumption that the noise processes in both
arms are largely uncorrelated. The sum of the noise power of the two premixing
filters becomes the input to the postmixing filter LPF3. The phase noise power,
2
ϕ

F σ and F σ

1 ϕ s 2 ϕ c

σ of the postmixing filter, and therefore the whole system, is given by

2 2 2

σ ≈ F ( F σ + F σ ) (7.33)

ϕ 3 1 ϕ s 2 ϕ c

Further, it is assumed that F 3 is less than both F 1 and F 2 . Note that if other unity
gain filters (e.g., band-pass) are used, the normalized cut-off F in all cases should
be replaced by the overall bandwidth of the new filter types. The output noise
power for m th order differentiators in the mixing process will be discussed in the