What is exactly Power Spectrum Density?

Question

In papers or textbooks, I have found several different definitions of PSD. I think I understand the meaning of PSD and all these papers or textbooks agrees that PSD and Fourier Series share a proportional relationship. But still there are different formulas from P(f)=1/T(〖F(f)〗^2), P(f)=〖∆t〗^2/T(〖F(f)〗^2), P(f)=1/4∆f(〖F(f)〗^2), P(f)=(〖F(f)〗^2) or others. From what I learn, I think on the point of conception of power(or energy) and Parseval's theorem, discrete PSD should be described as simply P(f)=1/T(〖F(f)〗^2). So why is there so many different formulas? Are they wrong or do I misunderstand something?

Answer 1

Power Spectrum Density (PSD) represents the spectral distribution of total power of a given WSS random signal. Conventionally, it's defined as the Fourier transform of the Auto-Correlation function (sequence) of the WSS random process as:

$$ P_x(\omega) = \mathcal{ F } \{ r_{xx}[m] \} $$

where the auto-correlation sequence (ACS) $r_{xx}[m]$ of the WSS $x[n]$ is defined by:

$$ r_{xx}[m] = E\{ x[n]x^*[n-m] \} $$

These are theoretical entities, you may not compute them in practice where only a single or a few realizations of the random process are available. In such a case, the PSD is estimated and there are different estimation techniques, one of the most common and simplest is known as the periodogram estimate of PSD and given by

$$ \hat{P}_{x}(\omega) = \frac{1}{N} |X(\omega)|^2 $$

where $X(\omega)$ is the discrete-time Fourier transform of the observed signal $x[n]$ of length $N$.