From: jacobfenton on
I am trying to find the mathmatical magnitude response of the following FM
demodulation equation:

I[n-1]*(Q[n]-Q[n-2])-Q[n-1]*(I[n]-I[n-2])
-----------------------------------------
I[n-1]^2+Q[n-1]^2

How do I represent I and Q in terms of some x[n] to find the z transform of
the equation?
I know I[n]=x[n]*cos(phi) and Q[n]=x[n]*-sin(phi).
But phi is also a function of 'n'. Not sure what to do here.

Thanks.

-Jacob Fenton



From: Steve Pope on
jacobfenton <jacob.fenton(a)n_o_s_p_a_m.gmail.com> wrote:

>I am trying to find the mathmatical magnitude response of the following FM
>demodulation equation:
>
>I[n-1]*(Q[n]-Q[n-2])-Q[n-1]*(I[n]-I[n-2])
>-----------------------------------------
> I[n-1]^2+Q[n-1]^2

>How do I represent I and Q in terms of some x[n] to find the z transform of
>the equation?
>I know I[n]=x[n]*cos(phi) and Q[n]=x[n]*-sin(phi).
>But phi is also a function of 'n'. Not sure what to do here.

There is not enough information here. If you know the statistics
of phi(n), you can then compute the correlations between pairs
of signals such as Q[n] and Q[n-2], I[n] and Q[n], and you
can then come up with an analytic form for the magnitude of the
above ratio.

(Or if you happen to know all these signals are uncorrelated then
the answer is simple, but they are almost certainly not.)

Steve
From: Vladimir Vassilevsky on


jacobfenton wrote:

> I am trying to find the mathmatical magnitude response of the following FM
> demodulation equation:
>
> I[n-1]*(Q[n]-Q[n-2])-Q[n-1]*(I[n]-I[n-2])
> -----------------------------------------
> I[n-1]^2+Q[n-1]^2

~ 2WT

School math.


> Not sure what to do here.

Break your stupid head against the wall.

From: Clay on
On May 24, 3:44 pm, "jacobfenton" <jacob.fenton(a)n_o_s_p_a_m.gmail.com>
wrote:
> I am trying to find the mathmatical magnitude response of the following FM
> demodulation equation:
>
> I[n-1]*(Q[n]-Q[n-2])-Q[n-1]*(I[n]-I[n-2])
> -----------------------------------------
>             I[n-1]^2+Q[n-1]^2
>
> How do I represent I and Q in terms of some x[n] to find the z transform of
> the equation?
> I know I[n]=x[n]*cos(phi) and Q[n]=x[n]*-sin(phi).
> But phi is also a function of 'n'. Not sure what to do here.
>
> Thanks.
>
> -Jacob Fenton

First let's assume your analytic signal is truly analytic, then feed a
sinusoid into the system and see what you get:


Thus I(n) is A*cos(2*pi*n*f/fs) and Q(n)=A*sin(2*pi*n*f/fs)

plug it in and reduce (you only need a few trigonometric identities),
and you will get

sin(2*pi*f/fs) for your result

fs is the sample rate, f is the frequency and A is the arbitrary
amplitude.

IHTH,
Clay

From: Clay on
On May 25, 10:11 am, Clay <c...(a)claysturner.com> wrote:
> On May 24, 3:44 pm, "jacobfenton" <jacob.fenton(a)n_o_s_p_a_m.gmail.com>
> wrote:
>
> > I am trying to find the mathmatical magnitude response of the following FM
> > demodulation equation:
>
> > I[n-1]*(Q[n]-Q[n-2])-Q[n-1]*(I[n]-I[n-2])
> > -----------------------------------------
> >             I[n-1]^2+Q[n-1]^2
>
> > How do I represent I and Q in terms of some x[n] to find the z transform of
> > the equation?
> > I know I[n]=x[n]*cos(phi) and Q[n]=x[n]*-sin(phi).
> > But phi is also a function of 'n'. Not sure what to do here.
>
> > Thanks.
>
> > -Jacob Fenton
>
> First let's assume your analytic signal is truly analytic, then feed a
> sinusoid into the system and see what you get:
>
> Thus I(n) is A*cos(2*pi*n*f/fs) and Q(n)=A*sin(2*pi*n*f/fs)
>
> plug it in and reduce (you only need a few trigonometric identities),
> and you will get
>
> sin(2*pi*f/fs) for your result
>
> fs is the sample rate, f is the frequency and A is the arbitrary
> amplitude.
>
> IHTH,
> Clay

I left out a factor of two, the result is 2*sin(2*pi*f/fs)

Clay