Re: DFT spectrum and coefficient insight

On Apr 15, 6:59 pm, "Blocher's spokesman" <n...@xxxxxxxx> wrote:
"dbd" <d...@xxxxxxxx> wrote in message
Another way to look at this might be that if you are going to take the
logarithm you don't need to take the square root (or square it again).
That's one of the reasons to use '10 log'.

Agreed, but it glosses over the fact that the magnitude of the DFT is the
square root of the square of the
Imag and real parts.



While the power spectrum can be -calculated- as the sum of the squares
of the real and imaginary components, the formalists usually -define-
it as the product of the complex coefficient and its complex
conjugate. It might be useful to understand why, and explain it at
this point.

I spent all afternoon thinking about this. Power *seems* so straight
forward, but it is not in my opinion.
I finally got it I think. I really had to look at Parsevals theorom for the
half frequency point. Parsevals theorom works at the half freq coefficient,
but that does not mean that the DFT coefficient properly reflects the
correct power of the signal there.

Any how, I started with an amplitude 1V cosine signal on an exact harmonic
of the windowing interval. After reviewing my basic EE theory (that took
longer than I care to admit to remember that this 1V-peak signal (across 1
ohm) gives 0.5 Watt). I took the DFT coefficients , which had a magnitude
of 0.5 when normalized. Then resquare the magnitudes to get power , and
wholla, 0.25 + 0.25 for the conjugate = 0.5 V.

And it works beautiful at DC, because the power of a 1vdc signal is 1 watt,
and the normalized DFT coefficient is 1 , instead of 0.5, so the power works
out correctly there too.

Any how your post got me thinking to pull the, so called, straight forward
power calculations together.

Brent

If it is your goal to understand DSP and spread knowledge of DSP you
would do better to dispense with the association of DFT coefficients
and concepts like voltage, impedance and any correspondence to an
analog concept of power until you are able appreciate and explain the
impact of deterministic, random and transient signals on the meaning
of the values calculated from DFT coefficients.

Dale B. Dalrymple
.

Relevant Pages

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