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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 $lim_{τ→∞}⁡ R(τ)=μ_x^2$, 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 $$S(ω)=\lim{T→∞}⁡\frac{1}{2T} {|X_T (ω) |^2}$$