# Power difference in the power spectrum density after DFT/FFT

Started by November 18, 2009
```Hi,
I am a novice at DSP techniques. I did not find an answer regarding my
question in this board or DG.

I want to compare the difference in total power of two signal source. I
know that area under the power spectral density represents total power. I
also know that PSD is symmetric about the 0.5sampling frequency (Fs). So,
if I show PSD from 0-0.5Fs, its area only contains half the total power,
correct? And do you just multiply power from 0-0.5Fs range by 2, before you
compare the power of the two sources?

Take one step back, if I want to compare the magnitude at a given
frequency after I did a DFT on each of the two sources, do I multiply the
magnitude by 2 also?

I appreciate all you pro's input.

```
```On 18 Nov, 12:27, "shinchan75034" <shinchan75...@gmail.com> wrote:
> Hi,
> I am a novice at DSP techniques. I did not find an answer regarding my
> question in this board or DG.
>
> I want to compare the difference in total power of two signal source. I
> know that area under the power spectral density represents total power. I
> also know that PSD is symmetric about the 0.5sampling frequency (Fs). So,
> if I show PSD from 0-0.5Fs, its area only contains half the total power,
> correct?

Correct.

>  And do you just multiply power from 0-0.5Fs range by 2, before you
> compare the power of the two sources?

For *realtive* power measurements then yes, this should suffice.
Be aware, though, that the usual convensions is that the DFT/IDFT
pair contains some scaling factors that mess up the simple arguments.
The key is a scaling factor 1 in the DFT and 1/N in the IDFT, so
that the power in spectrum domain is a factor sqrt(N) larger than
one might expect it to be.

Finding the factor is easy: The sum of squared magnitudes in
time and spectrum domain should be equal when the DFT is scaled
correctly.

There is also the question about PSD estimators. There are several
ways of estimating the power spectrum density. Some of these
estimators use extensive weighting and averaging of the data, which
might mess with the accuracy of the numbers.

Rune
```
```On Nov 18, 6:27&#4294967295;am, "shinchan75034" <shinchan75...@gmail.com> wrote:
> Hi,
> I am a novice at DSP techniques. I did not find an answer regarding my
> question in this board or DG.
>
> I want to compare the difference in total power of two signal source. I
> know that area under the power spectral density represents total power. I
> also know that PSD is symmetric about the 0.5sampling frequency (Fs). So,
> if I show PSD from 0-0.5Fs, its area only contains half the total power,
> correct?

No. It contains more. The omitted "negative frequencies" don't
include
mirrors of DC and Nyquist (Fs/2)

> And do you just multiply power from 0-0.5Fs range by 2, before you
> compare the power of the two sources?

No the first and last components are only multiplied by 1.

> Take one step back, if I want to compare the magnitude at a given
> frequency after I did a DFT on each of the two sources, do I multiply the
> magnitude by 2 also?

Except for DC and Nyquist..

> I appreciate all you pro's input.

Warning: If you multiply by 1 and 2 to get the single-sided amplitude
spectrum, you cannot square those amplitudes to get the SS power
spectrum..

Hope this helps.

Greg

```