# Power Spectral Density of Complex Signals.

Started by September 24, 2008
```I have N number of time-domain complex signals. And I want to find the
corresponding power spectral density of these signals. The usual way I
think is to find the autocorrelation of the signals. Then, the fourier
transform of it. My question is, how do I find the power spectral density
of complex signals? Autocorrelation of complex signals? Autocorrelation of
the magnitude of the complex signals? Am I have to focused just on
magnitude or the complex numbers?

```
```On 24 Sep, 15:08, "kbkien" <kbk...@hotmail.com> wrote:
> I have N number of time-domain complex signals. And I want to find the
> corresponding power spectral density of these signals. The usual way I
> think is to find the autocorrelation of the signals. Then, the fourier
> transform of it. My question is, how do I find the power spectral density
> of complex signals? Autocorrelation of complex signals?

The general approach that you know from real-valued signals still
applies, but you need to pay attention to some extra details.

First, you need to conjugate one of th eterms in the estimator
for the autocorrelation. I *think* the convention is

rxx[k] = E[ x[n] x'[n+k] ]

where x'[n] is the complex conjugate of x[n], but you might
want to confirm that.

The other detail to look out for is that the PSD is no longer
symmetrical around DC (or Fs/2), so you will want to plot
it on the full interval [DC, Fs].

Rune
```
```>On 24 Sep, 15:08, "kbkien" <kbk...@hotmail.com> wrote:
>> I have N number of time-domain complex signals. And I want to find the
>> corresponding power spectral density of these signals. The usual way I
>> think is to find the autocorrelation of the signals. Then, the fourier
>> transform of it. My question is, how do I find the power spectral
density
>> of complex signals? Autocorrelation of complex signals?
>
>The general approach that you know from real-valued signals still
>applies, but you need to pay attention to some extra details.
>
>First, you need to conjugate one of th eterms in the estimator
>for the autocorrelation. I *think* the convention is
>
>rxx[k] = E[ x[n] x'[n+k] ]
>
>where x'[n] is the complex conjugate of x[n], but you might
>want to confirm that.
>
>The other detail to look out for is that the PSD is no longer
>symmetrical around DC (or Fs/2), so you will want to plot
>it on the full interval [DC, Fs].
>
>Rune
>

So, what I have to do is basically to find the autocorrelation of the
complex signal as below:

rxx[k] = E[ x[n]*x'[n+k] ]

where x'[n] is the conjugate of the complex signal x[n].

Then, the fourier transform of rxx[k] as below:

PSD = FFT[Rxx]

I believe the PSD is therefore a sequence of real value. Am I right?

```
```On 25 Sep, 11:03, "kbkien" <kbk...@hotmail.com> wrote:
> >On 24 Sep, 15:08, "kbkien" <kbk...@hotmail.com> wrote:
> >> I have N number of time-domain complex signals. And I want to find the
> >> corresponding power spectral density of these signals. The usual way I
> >> think is to find the autocorrelation of the signals. Then, the fourier
> >> transform of it. My question is, how do I find the power spectral
> density
> >> of complex signals? Autocorrelation of complex signals?
>
> >The general approach that you know from real-valued signals still
> >applies, but you need to pay attention to some extra details.
>
> >First, you need to conjugate one of th eterms in the estimator
> >for the autocorrelation. I *think* the convention is
>
> >rxx[k] = E[ x[n] x'[n+k] ]
>
> >where x'[n] is the complex conjugate of x[n], but you might
> >want to confirm that.
>
> >The other detail to look out for is that the PSD is no longer
> >symmetrical around DC (or Fs/2), so you will want to plot
> >it on the full interval [DC, Fs].
>
> >Rune
>
> So, what I have to do is basically to find the autocorrelation of the
> complex signal as below:
>
> rxx[k] = E[ x[n]*x'[n+k] ]
>
> where x'[n] is the conjugate of the complex signal x[n].
>
> Then, the fourier transform of rxx[k] as below:
>
> PSD = FFT[Rxx]

You might want to use some window function.

> I believe the PSD is therefore a sequence of real value. Am I right?

Yes. The autocorrelation is complex-valued and conjugate
symmetric, so the PSD is real-valued. However, the PSD
is not symmetric.

Rune
```
```On Sep 25, 5:03&#2013266080;am, "kbkien" <kbk...@hotmail.com> wrote:
> >On 24 Sep, 15:08, "kbkien" <kbk...@hotmail.com> wrote:
> >> I have N number of time-domain complex signals. And I want to find the
> >> corresponding power spectral density of these signals. The usual way I
> >> think is to find the autocorrelation of the signals. Then, the fourier
> >> transform of it. My question is, how do I find the power spectral
> density
> >> of complex signals? Autocorrelation of complex signals?
>
> >The general approach that you know from real-valued signals still
> >applies, but you need to pay attention to some extra details.
>
> >First, you need to conjugate one of th eterms in the estimator
> >for the autocorrelation. I *think* the convention is
>
> >rxx[k] = E[ x[n] x'[n+k] ]
>
> >where x'[n] is the complex conjugate of x[n], but you might
> >want to confirm that.
>
> >The other detail to look out for is that the PSD is no longer
> >symmetrical around DC (or Fs/2), so you will want to plot
> >it on the full interval [DC, Fs].
>
> >Rune
>
> So, what I have to do is basically to find the autocorrelation of the
> complex signal as below:
>
> rxx[k] = E[ x[n]*x'[n+k] ]
>
> where x'[n] is the conjugate of the complex signal x[n].
>
> Then, the fourier transform of rxx[k] as below:
>
> PSD = FFT[Rxx]
>
> I believe the PSD is therefore a sequence of real value. Am I right?

The trick in getting a good PSD estimate is how you determine rxx[k].
Since the PSD is the Fourier transform of the process's correlation
function, there's an expected value in there that you have to account
for, so you have to do some kind of "averaging" or something that
could equate to an expectation. There are several well-known methods
for estimating a signal's PSD, like Welch's method (which is
nonparametric), or other more "modern" parametric methods. The exact
method you choose can affect nontrivially the PSD estimate that you
get out.

Jason
```