FPGA Central

What is wrong with this thinking about computing a running integral over a
data sample:

y(n) = ?x(i) * cos(w*i) where the sum is over the previous N data points.
y(n+1) = ?x(i+1) * cos(w*(i+1)) again where the sum is over the previous N
data points.

What this means is that
y(n+1) = y(n) + x(i+1)*cos(w*(i+1)) - x(n-N)*cos(w*(n-N)).

Thus one can compute the running integral with 2 multiplies and 2 adds.

Seems way too simple, though nothing says to me that I should not be able to
move the cosine function one phase increment each sample increment thus
avoiding the sum over N multiplies and adds.

What I want to do is an integrate and dump scheme for demodulating an FSK
signal. Perhaps use a Goertzel filter for each frequency as well (if I can
figure out how to apply it to a data stream.) Any sources or help greatly
appreciated. I am sure someone has figured out how to do a great FSK
demodulator (much better than I would come up with) and I don't want to
re-invent the wheel!

Thanks!

Brian Reinhold

In article UATub.191555$275.664852@attbi_s53, Brian Reinhold at
[email protected] wrote on 11/19/2003 19:07:

> What is wrong with this thinking about computing a running integral over a
> data sample:
>
> y(n) = ?x(i) * cos(w*i) where the sum is over the previous N data points.
> y(n+1) = ?x(i+1) * cos(w*(i+1)) again where the sum is over the previous N
> data points.
>
> What this means is that
> y(n+1) = y(n) + x(i+1)*cos(w*(i+1)) - x(n-N)*cos(w*(n-N)).
>
> Thus one can compute the running integral with 2 multiplies and 2 adds.

i dunno what the 2 multiplies has to do with the "moving sum" (what is
commonly called this thing you're calling a "running integra").

x[i]*cos(w*i) is just a definition of the input to your moving summmer. the
multiplication is part of that signal definition. let's just call the input
"x[n]" and make our lives easier.

the moving sum operation:

N-1
y[n] = SUM{ x[n-i] }
i=0

is an FIR filter that can be computed with the recursion equation:

y[n] = y[n-1] + x[n] - x[n-N]

this is probably the simplest non-trivial example of what has been dubbed by
Julius Smith a "Truncated IIR" filter. it really is an FIR but it is
implemented with an IIR (in this case the IIR is an integrator), a delay
line and some subtraction.

you can pretty easily extend this concept to 1st order decaying exponential
impulse responses:

y[n] = alpha*y[n-1] + x[n] - (alpha^N)*x[n-N]

that's the same as

N-1
y[n] = SUM{ (alpha^i)*x[n-i] }
i=0

with a little more thought, you can extend it to 2nd order or any other IIR
filter where you can truncated the "infinite" impulse response at some
defined finite limit.

> Seems way too simple, though nothing says to me that I should not be able to
> move the cosine function one phase increment each sample increment thus
> avoiding the sum over N multiplies and adds.
>
> What I want to do is an integrate and dump scheme for demodulating an FSK
> signal. Perhaps use a Goertzel filter for each frequency as well (if I can
> figure out how to apply it to a data stream.) Any sources or help greatly
> appreciated. I am sure someone has figured out how to do a great FSK
> demodulator (much better than I would come up with) and I don't want to
> re-invent the wheel!

it would be doing that.

r b-j

Oops. I see that my "summation" sign (which looked fine when I typed it
using Alt-228) turned out to be a question mark. But you answered the
question anyways.

This integral operation is more like a Goertzel detector than a filter,
isn't it? I would suspect that the output of this running integral (if the
input x(n) were the frequency of the sinusoid) would approach a constant.

Brian

> > What is wrong with this thinking about computing a running integral over
a
> > data sample:
> >
> > y(n) = ?x(i) * cos(w*i) where the sum is over the previous N data
points.
> > y(n+1) = ?x(i+1) * cos(w*(i+1)) again where the sum is over the previous
N
> > data points.
> >
> > What this means is that
> > y(n+1) = y(n) + x(i+1)*cos(w*(i+1)) - x(n-N)*cos(w*(n-N)).
> >
> > Thus one can compute the running integral with 2 multiplies and 2 adds.
>
> i dunno what the 2 multiplies has to do with the "moving sum" (what is
> commonly called this thing you're calling a "running integra").
>
> x[i]*cos(w*i) is just a definition of the input to your moving summmer.
the
> multiplication is part of that signal definition. let's just call the
input
> "x[n]" and make our lives easier.
>
> the moving sum operation:
>
> N-1
> y[n] = SUM{ x[n-i] }
> i=0
>
> is an FIR filter that can be computed with the recursion equation:
>
> y[n] = y[n-1] + x[n] - x[n-N]
>
> this is probably the simplest non-trivial example of what has been dubbed
by
> Julius Smith a "Truncated IIR" filter. it really is an FIR but it is
> implemented with an IIR (in this case the IIR is an integrator), a delay
> line and some subtraction.
>
> you can pretty easily extend this concept to 1st order decaying
exponential
> impulse responses:
>
> y[n] = alpha*y[n-1] + x[n] - (alpha^N)*x[n-N]
>
> that's the same as
>
> N-1
> y[n] = SUM{ (alpha^i)*x[n-i] }
> i=0
>
> with a little more thought, you can extend it to 2nd order or any other
IIR
> filter where you can truncated the "infinite" impulse response at some
> defined finite limit.
>
> > Seems way too simple, though nothing says to me that I should not be
able to
> > move the cosine function one phase increment each sample increment thus
> > avoiding the sum over N multiplies and adds.
> >
> > What I want to do is an integrate and dump scheme for demodulating an
FSK
> > signal. Perhaps use a Goertzel filter for each frequency as well (if I
can
> > figure out how to apply it to a data stream.) Any sources or help
greatly
> > appreciated. I am sure someone has figured out how to do a great FSK
> > demodulator (much better than I would come up with) and I don't want to
> > re-invent the wheel!
>
> it would be doing that.
>
> r b-j
>
>