Question

The following are properties of auto correlation except

• A. \(|R_{XX}(\tau)|\le R_{XX}(0)\)
• B. \(|R_{XX}(-\tau)|=R_{XX}(\tau)\)
• C. \(|R_{XX}(\tau)|=E[x(t)]\)
• D. If \(x(t)\) has a periodic component, then \(R_{XX}(\tau)\) will have a periodic component with the same period

Hint:

Autocorrelation is always an even function and attains its maximum magnitude at \(\tau=0\).

Answer 1

The Correct Option is C

Concept: The autocorrelation function is defined as \[ R_{XX}(\tau)=E[X(t)X(t+\tau)] \] It possesses several standard properties.

Step 1: Check property (A).
\[ |R_{XX}(\tau)|\le R_{XX}(0) \] This is a valid property.

Step 2: Check property (B).
Autocorrelation is an even function. \[ R_{XX}(\tau)=R_{XX}(-\tau) \] Hence valid.

Step 3: Check property (D).
If the signal contains a periodic component, the autocorrelation also exhibits the same periodicity. Hence valid.

Step 4: Identify the incorrect statement.
\[ |R_{XX}(\tau)|=E[X(t)] \] is not a standard autocorrelation property. Therefore this statement is incorrect. \[ \boxed{\text{Correct Option (C)}} \]