# FFT scaling consistency

Hi,

Yesterday someone in the chat raised an interesing issue. The absolute
scale of the various FFT GUI sinks.

If you take:

[signal source ampl=1.0] -> [gui fft sink]

(use a source freq that fits exactly on 1 fft bin)

With complex + WXGUI it peaks at -3dB
With complex + QTSink it peaks at -9dB
With float + WXGUI it peaks at -9dB
With float + QTSink it peaks at -15dB

So basically:

• The WXGui + Complex should be 0dB IMHO.
A complex cosine of ampl 1.0 should be our ‘absolute’ reference for
0dB

• QTSink is always 6dB lower
-> That doesn’t sound logical, they should just match. GUI
selection should have no influence on the absolute reading when
working with purely synthetic data.

• Float is always 6dB lower than complex
-> I’m torn on this one … can’t decide
If you integrate the energy (x^2) over one cycle between a
complex cosine and a float cosine, you get half the energy. So this
should be -3 dB right ? And I guess the other 3dB comes from the fact
that half of this energy is in the positive spectrum and half of it in
the negative one (so if you take the total energy in the whole
spectral domain, it does indeed match the one in the time domain) ?
The question is then, when displaying a float fft, should we
just raise by 3dB ?

Anyway, I hope this make sense

Thoughts ?

Cheers,

Sylvain

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Hi Sylvain,

at first I hm’ed, but then I hm!ed.
So, first thing, yes, using a single tone as reference does sound
logical, especially since it’s so intuitive from an analysis point of
view, considering RX signals to be a sum of weighted complex sinusoids.

Then: no, QT GUI does that, see [1] (generated by [2]). The tone at
samp_rate / fft_length * 10 has a power of 0 dB; also, Gaussian noise
with an amplitude of 1 has an average (picture is heavily averaged)
power of -20 dB == 1/fft_length. Perfect!

To the complex vs. real discussion: Picture [3] tells me that I see
the noise at -23 dB; now, this will get extremely philosophical
whether when observing a real signal this should be -20 dB instead. I
like to think of it as “don’t add energy when you don’t specify where
that comes from, or you’ll end up building virtual perpetuum mobiles”:
The real noise can be considered complex noise with $\Im{x} \equiv 0$,
so it logically has less energy than the complex circular gaussian
noise of [1]. If we display it having as much energy as the latter,
we’re acting as if we knew that the imaginary part wasn’t zero. That
can’t end well, especially when looking at stuff like SNRs.

Cheers,
Marcus

[1] http://marcus.hostalia.de/test_fft_power.png
[2] http://marcus.hostalia.de/test_fft_power.grc
complex sinusoid label. It’s a plain-as-water real sine.)

On 11/26/2014 10:38 AM, Sylvain M. wrote:

(use a source freq that fits exactly on 1 fft bin)

total energy in the whole spectral domain, it does indeed match the

Sylvain

mailing list [email protected]

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Hi,

Then: no, QT GUI does that, see [1] (generated by [2]). The tone at
samp_rate / fft_length * 10 has a power of 0 dB; also, Gaussian noise
with an amplitude of 1 has an average (picture is heavily averaged)
power of -20 dB == 1/fft_length. Perfect!

Oh, that’s interesting.

I didn’t see that when I did the test here. Turns out it’s the window
selection, if you select Rectangular, you have 0dB and it’s consistent
between WX and Qt.

If you select something else, then it’s no longer 0dB and it becomes
inconsistent between the WX and Qt widgets.
Now obviously windowing will have some effect on the peak value, but:

• I didn’t expect it to be this large, I expected more like a few
tenth of dB, not precisely 3 or 6dB.
• I certainly don’t see why Wx or Qt would make a difference

To the complex vs. real discussion: Picture [3] tells me that I see
the noise at -23 dB; now, this will get extremely philosophical
whether when observing a real signal this should be -20 dB instead.

Yes, it’s definitely more of a philosophical question than anything
else.
Honestly I don’t really care either way, I just though I’d bring this
up at the same time and see if there was a “convention” that was
expected.

Cheers,

Sylvain

On 11/26/2014 12:07 PM, Sylvain M. wrote:

If you select something else, then it’s no longer 0dB and it becomes
inconsistent between the WX and Qt widgets.
Now obviously windowing will have some effect on the peak value, but:

• I didn’t expect it to be this large, I expected more like a few
tenth of dB, not precisely 3 or 6dB.

The peak loss is a function of the window length and its type. You can
calculate the value by:

loss = 1/(N * l2_norm(w)) * |\sum_{n=0}^{N-1} w(n)|^2

…hope my pseudo-latex makes sense

For a blackman-harris window, it’s very close to 3 dB for most window
lengths. For complex, you’re applying it twice, though, hence the
difference.

• I certainly don’t see why Wx or Qt would make a difference

That must be a bug or something. Looks like the WX results are closer to
the truth.

M

I normalized the wx complex display a long time ago. As you have
discovered, windowing affects this, but in two separate ways. First,
windowing reduces total signal power because most bins are less than 1.
We
normalize for that, so the total energy remains constant. You will see
that if you run with awgn.

affect
anything you expected to remain in one bin, like a sine wave. There
isn’t
any way to fix this as its not broken. To find the power in a sine
you’ll

Matt

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Hi,

at least Window scaling differing between QT and WX is in fact a bit
irritating…

Cheers

Marcus

On 11/26/2014 12:07 PM, Sylvain M. wrote:

window selection, if you select Rectangular, you have 0dB and it’s

To the complex vs. real discussion: Picture [3] tells me that I

Sylvain

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