Average Power Spectrum Density

August 7, 2009 at 8:01 pm · Filed under Signal Processing

While going over some topics in digital communication, I came across the
Average-PSD (Average Power Spectrum Density) term; the A-PSD is a statistical
character of WSCS (Wide Sense Cyclo-Stationary) random process much
like the PSD is a statistical character of WSS (Wide Sense Stationary) random
process. Following, I will talk about the PSD, how it is calculated,
and how it represents the average power density of a WSS process. Later,
I will do the same with A-PSD and WSCS process.

1. PSD (of a WSS process)

When a random process is said to be Wide-Sense Stationary (WSS) – it has a time-invariant mean and a 1D auto-correlation function; that is, for a WSS process x(t) we have:

\mu_{x}(t)=E[x(t)]=\mu_{x}

\tilde{R}_{xx}(t+\tau,t)\equiv E[x(t+\tau)x^{*}(t)]=R_{xx}(\tau)

And the PSD is defined as the Fourier transform of R_{xx}(\tau):

S_{xx}(w)=F\{R_{xx}(\tau)\}=\underset{-\infty}{\overset{\infty}{\int}}R_{xx}(\tau)\cdot e^{-jw\tau}d\tau

Let us do some math:

S_{xx}(w)=F\{R_{xx}(\tau)\}=\underset{-\infty}{\overset{\infty}{\int}}R_{xx}(\tau)\cdot e^{-jw\tau}d\tau=

=\underset{-\infty}{\overset{\infty}{\int}}E[x(t+\tau)x^{*}(t)]e^{-jw\tau}d\tau=\underset{-\infty}{\overset{\infty}{\int}}E[x(t+\tau)x^{*}(t)e^{-jw\tau}]d\tau=

=E[\underset{-\infty}{\overset{\infty}{\int}}x(t+\tau)x^{*}(t)e^{-jw\tau}d\tau]=E[x^{*}(t)\cdot\underset{-\infty}{\overset{\infty}{\int}}x(t+\tau)e^{-jw\tau}d\tau]=

=E[x^{*}(t)\cdot X(w)\cdot e^{jwt}]=E[X(w)\cdot(x(t)e^{-jwt})^{*}]

And this is true for every w and every t !

Note, that even though t appears on the right hand side of the equality, the result will be independent of t (since we assume that the process is WSS). Also note that we consider X(w) to be the Fourier transform of x(t) in the sense that it is a new “random process” of variable w. We also assume that all realizations of x(t) are bounded in time [-0.5T,0.5T], in order to avoid convergence issues. However, the boundaries shall meet the requirement of T being “very large” . Thus, from now on we will refer to the latter as X_{T}(w), as to point out that the Fourier transform is taken over a window with a size T.

Let us now integrate both sides of the equation with respect to t:

\underset{-0.5T}{\frac{1}{T}\overset{0.5T}{\int}}S_{xx}(w)dt=\underset{-0.5T}{\frac{1}{T}\overset{0.5T}{\int}}E[X_{T}(w)\cdot(x(t)e^{-jwt})^{*}]dt

S_{xx}(w)=E[X_{T}(w)\cdot\underset{-0.5T}{\frac{1}{T}\overset{0.5T}{\int}}(x(t)e^{-jwt})^{*}]

Now since T\rightarrow\infty, we get:

\mathbf{S_{xx}(w)=lim_{T\rightarrow\infty}\{E[\mid\frac{X_{T}(w)}{\sqrt{T}}\mid^{2}]\}}

To summaries, if we look at the Fourier transform of a “very long” (T\rightarrow\infty) window of x(t), we get X_{T}(w) which is a random variable for every w. The result shows that the PSD is essentially the mean of |X_{T}(w)|^{2}, thus being the “mean power” of the random process for every frequency w.

2. (Time) Averaged PSD (of a WSCS process)

When a random process is said to be Wide-Sense Cyclo-Stationary (WSCS) – it has a periodic mean and a 2D auto-correlation function which is periodic as well. That is, for a WSCS process x(t) we have:

\mu_{x}(t)=E[x(t)]=\mu_{x}(t+k\cdot T)

R_{xx}(t+\tau,t)\equiv E[x(t+\tau)x^{*}(t)]=R_{xx}(t+\tau+k\cdot T,t+k\cdot T)

where T is the “period”‘ of the random process.

Note that WSCS processes are very common in the field of digital communication, e.g. QAM signal.

Let us now define:

S_{xx}(w;t)\equiv F\{R_{xx}(t+\tau,t)\}\equiv\underset{-\infty}{\overset{\infty}{\int}}R_{xx}(t+\tau,t)\cdot e^{-jw\tau}d\tau

From here we go on pretty much like in the case of a WSS processes as shows in section 1, only this time we get:

\alpha\equiv\frac{1}{\bar{T}}\overset{0.5\bar{T}}{\underset{-0.5\bar{T}}{\int}}S_{xx}(w;t)dt=E[X(w)\cdot\frac{1}{\bar{T}}\overset{0.5\bar{T}}{\underset{-0.5\bar{T}}{\int}}(x(t)e^{-jwt})^{*}]=

=lim_{\bar{T}\rightarrow\infty}\{E[\mid\frac{X_{\bar{T}}(w)}{\sqrt{\bar{T}}}\mid^{2}]\}

Note that while the right han side remain un-changed, the spectrum is now a function of t as well – making the integral not-so-trivial as in the case of WSS processes.

However, by definition we can show that the spectrum is periodic T:

S_{xx}(w;t+k\cdot T)\equiv\underset{-\infty}{\overset{\infty}{\int}}R_{xx}(t+k\cdot T+\tau,t+k\cdot T)\cdot e^{-jw\tau}d\tau\underset{WSCS}{=}

=\underset{-\infty}{\overset{\infty}{\int}}R_{xx}(t+\tau,t)\cdot e^{-jw\tau}d\tau=S_{xx}(w;t)

Without the loss of generality, we assume \bar{T}=T\cdot M; thus, we can re-write the right hand side:

\alpha=\frac{1}{\bar{T}}\overset{0.5\bar{T}}{\underset{-0.5\bar{T}}{\int}}S_{xx}(w;t)dt=\frac{1}{TM}\cdot\underset{i=0}{\overset{M-1}{\sum}}\overset{(i+1)T}{\underset{iT}{\int}}S_{xx}(w;t)dt=

=\frac{1}{TM}\cdot\underset{i=0}{\overset{M-1}{\sum}}T\cdot S_{xx}^{AV}(w)=\alpha

where we define S_{xx}^{AV}(w)\equiv\frac{1}{T}\cdot\underset{<T>}{\int}S_{xx}(w;t)dt, the (time-) “average” PSD.

\alpha=\frac{1}{M}\cdot M\cdot S_{xx}^{AV}(w)=S_{xx}^{AV}(w)

So we get:

\mathbf{S_{xx}^{AV}(w)=lim_{\bar{T}\rightarrow\infty}\{E[\mid\frac{X_{\bar{T}}(w)}{\sqrt{\bar{T}}}\mid^{2}]\}\equiv\bar{S}_{xx}(w)}

Beautiful. The latter equation shows that the mean of |X_{\bar{T}}(w)|^{2} is essentially the “average” PSD for a WSCS process.

3. Conclusions

The result is interesting since:

  1. One way to compute the average PSD is to first compute the time-average autocorrelation function (averaged over a single period), and then take the Fourier transform. However, if one is not interested in the autocorrelation function,but only in the average PSD, he may want to use the method previously shown.
  2. The results also show that if we have a lab equipment such as a “spectrum analyzer” which finds the PSD of a WSS process by averaging |X_{\bar{T}}(w)|^{2}, it will find the average PSD of a WSCS process as well.

OK, it has been a long post already. Next time I will talk a little bit about WSCS processes in digital communication, and show how to use the results of the current post on QAM signals.

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