Mumbai University > Electronics and Telecommunication > Sem7 > Random Signal Analysis

Marks: 10M

Year: Dec 2014

Power Spectral Density

Definition:

If {X (t)} is a stationary process (either in a strict sense or wide sense) with autocorrelation function $R(τ)$, then the Fourier transform of $R(τ)$ is called the power spectral density function of {X(t)} and denoted as $S_xx (ω)$ or $S_x (ω)$.

Thus $S_x (ω)=∫_{-∞}^∞R(τ) e^{-iωτ} dτ$

Or $S_x (f)=∫_{-∞}^∞R(τ) e^{-i2πfτ} dτ$

Significance:

Power Spectral Density (PSD) is the frequency response of a random or periodic signal. It tells us where the average power is distributed as a function of frequency.

The PSD is deterministic, and for certain types of random signals is independent of time (The signal has to be stationary, which means that the statistics do not change as a function of time) .This is useful because the Fourier transform of a random time signal is itself random, and therefore of little use calculating transfer relationships (i.e., finding the output of a filter when the input is random.

To find power spectral density of X(t) = acos(bt+Y) we need to find the autocorrelation, assuming a and b are constants

$R_X (τ)=E(X(t)×X(t+τ)) \\ \ \ \ =E(a cos⁡(bt+Y).acos⁡(bt+bτ+Y))\\ \ \ \ =\frac{a^2}{2} E[2 cos⁡(bt+Y).cos⁡(bt+Y+bτ)]\\ \ \ \ =\frac{a^2}{2} E[cos⁡(2bt+bτ+Y)+cos⁡(bτ))]\\ \ \ \ =\frac{a^2}{2} [∫_{0}^2πcos⁡(2bt+bτ+Y).\frac{1}{2π} dY+ ∫_0^{2π}cos⁡(bτ).\frac{1}{2π} dY]$

$R(τ)=\frac{a^2}{2} cos⁡(bτ)$

Now power spectral density $S(ω)=F{R(τ)}=F \{{\frac{a^2}{2} cos⁡(bτ)}\}$

Fourier Transform of cosbt is

$$cosbt=π[δ(ω+b)+δ(ω-b)]$$

$$∴S(ω)=\frac{πa^2}{2}[δ(ω+b)+δ(ω-b)]$$