written 8.3 years ago by
teamques10
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modified 8.3 years ago

Autocorrelation
Definition: If the process {X(t)} is stationary either in the strict sense or in the wide sense, then E{X(t).X(tτ)} is a function of τ, denoted by $R_xx (τ)$or $R(τ)$ or $R_x (τ)$. This function $R(τ)$ is called the autocorrelation function of the process {X(t)}.
Properties:
 R(t) is a even function of $τ$
i.e. R($τ$)=R($τ$)
2. R($τ$)is maximum at $τ$=0
3. If the autocorrelation function R(t) of a real stationary process {X(t)} is continuous at τ=0, it is continuous at every other point
4. If R($τ$) is the autocorrelation function of a stationary process {X(t)} with no periodic component, then $lim_{τ→∞} R(τ)=μ_x^2 $, provided the limit exists
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 $S_x (f)=∫_{∞}^∞R(τ) e^{i2πfτ} dτ$
**Properties:**
1. The value of the spectral density at zero frequency is equal to the total area under the graph of the autocorrelation function
2. The mean square value of a wide sense stationary process is equal to the total area under the graph of the spectral density.
3. The spectral density function of a real random process is an even function
i.e. $S_x (ω)=S_x (ω)$
4. The Spectral density of a process {X(t)}, real or complex, is a real function of ω and non negative.
5. The spectral density and the autocorrelation function of a real WSS process form a Fourier Cosine transform pair
6. If $X_T$ (ω)is the Fourier transform of the truncated random process defined as
$X_T (t)=X(t) $ for t≤T
$=0$ $for t>T$
where {X(t)} is a real WSS process with power spectral density function S(ω) then
$$S(ω)=\lim{T→∞}\frac{1}{2T} {X_T (ω) ^2}$$