obtaining continuous time signal bin energy

>> +/- 1% frequency offset relative to fs.
correction: to the inverse of the measurement length.
-mn

On Sep 17, 4:13 pm, "mnentwig" <mnent...@elisanet.fi> wrote:
> One comment to the original poster:
>
> >> For the power, should it be 2*((a3/N)^2 + (a6/N)^2 + (a9/N)^2), when
>
> FFT
> that looks OK, but verify your scaling factors against known examples.
> Since the bins are complex, it might actually be better to use c * conj(c)
> for squaring.
> It might have been me who brought up the "a"s, but never mind...
>
> But:
>
> I get the impression you are processing measured data?
>
> If so, forget about the Nyquist limit!!
>
> Chances are 99.9% that you need to leave some frequency space unused to
> account for nonideal equipment.
> I'd try to do so in any case, just to avoid the "stepping-on-Nyquist's
> feet" issue discussed above. In typical measurement problems it's just
> plain common sense.

> And something else:
>
> If your measurement equipment is synchronized with the signal (reference
> clock), then you can sample the data using a rectangular window and
> immediately read from a single FFT bins. If not, you need to account for
> frequency error, consider adjacent bins, and use a non-rectangular
> window.
>
> Then, to make things a bit more complicated:
> So far the window sets your frequency resolution, but the actual frequency
> error may be higher (and: the longer your measurement time, the narrower
> the "aperture" from windowing on the frequency axis).
>
> If so, I could, for example:
>
> - sum up the power from adjacent bins, for example 990-1010, if 1000 is
> the center frequency => allows roughly +/- 1% frequency offset relative to
> fs.
>
> or:
> - Convolve the power spectrum with a window function, for example [1 1 1 1
> 1 1 1 1 1 1]/10 and read off the power from the center bin (this example
> window is effectively only half as wide as the previous)
>
> It's inconvenient from an analytical point-of-view, but that's also what a
> real (hardware) spectrum analyzer does: it sweeps a filter with a given
> filter response.
>
> -mn

Markus,

Yes we are processing measured data.  I will double check with the
hardware
provider whether there are space left.

Our measurement equipment is synchronized with the signal, so the
rectangular
is good enough.  Then do we still need the last two ways you
mentioned?  The
first way is clear.  I don't get the second one.  So the convolution
is done
in the frequency domain?  How will it affect a spectrum?  Could you
recommend
any reference for this?

Thanks,
Brian

Hello,

>Yes we are processing measured data.  I will double check with the
>hardware
>provider whether there are space left.

Probably you sample at a higher rate, and there are FFT bins that contain
only noise and can be discarded. If not, beware. What happens to signal
energy just above the Nyquist limit? 

>Our measurement equipment is synchronized with the signal, so the
>rectangular is good enough. 

As long as you capture a full number of periods it should be OK.
I hate to make simple things complicated.
BUT if it's not good enough, the problems may be easier to fix before you
buy the hardware.

>> So the convolution is done in the frequency domain?
Consider it a sneaky way to sum up the bins as in the first example. 
This convolution is not done on the signal, but on the power spectrum.
Look at it long enough and you'll see that it can give the same result on
the "nominal" bin as summing up the bins.

If the signal energy is clearly confined to one bin, then reading from one
bin is enough, and possibly even the most accurate. 
Just look at the power spectrum, and you'll see how much energy 'leaks'
into adjacent bins by whatever mechanism.

-mn