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State and explain various properties of autocorrelation function and power spectral density function.

Mumbai University > Electronics and Telecommunication > Sem5 > Random Signal Analysis

Marks: 10M

Year: May 2015

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## Autocorrelation

Definition: If the process $\{X(t)\}$ is stationary either in the strict sense or in the wide sense, then $E\{X(t).X(t-τ)\}$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:

1. 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 $$\displaystyle\lim _{τ→∞}⁡R(τ)=μ_x^2$$ , provided the limit exists.

## Power Spectral Density

Definition:

If $\{X(t)\}$ is a stationary process (either in a strict sense or wide sense) with auto correlation 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 auto correlation 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 $X_τ (ω)$ is the Fourier transform of the truncated random process defined as

$X_τ (t)=X(t) \ \ \ \ \ \ for |t| ≤T$

$\ \ \ \ \ \ \ \ \ =0 \ \ \ \ \ \ \ \ \ \ \ for |t| \gt T$

where $\{X(t)\}$ is a real WSS process with power spectral density function $S(ω)$ then

$$S(ω)=\displaystyle\lim_{τ→∞}\frac{⁡1}{2τ} E{|X_τ (ω) |^2}$$