Why Average Power Equals Rₓ(0)? Derivation with Example

 

Why is the Average Power Equal to the Autocorrelation Function at Zero Lag?

One of the most important results in Random Process Theory is the relationship between the average power of a stationary random process and its autocorrelation function. Students often memorize the formula without understanding where it comes from.

In this article, we will derive the result step-by-step and show mathematically why:

Average Power = R_x(0)

where R_x(0) is the autocorrelation function evaluated at zero time delay.

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Step 1: Definition of Average Power

For a random process X(t), the average power is defined as the expected value of the square of the process:

P_s = E[X²(t)]

Here:

• E[ ] denotes the expectation operator.
• X(t) is the value of the random process at time t.
• The square operation ensures that positive and negative values contribute positively to power.

This definition is analogous to electrical power, where power is proportional to the square of voltage or current.

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Step 2: Definition of Autocorrelation Function

The autocorrelation function measures the similarity between a signal and a delayed version of itself. It is defined as:

R_x(τ) = E[X(t) · X(t + τ)]

where:

• τ (tau) represents the time delay (lag).
• R_x(τ) tells us how strongly the process at time t is related to its value at time t + τ.

If the value of R_x(τ) is large, the signal remains highly correlated after that delay. If it is small, the signal changes significantly over time.

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Step 3: Evaluate Autocorrelation at Zero Delay

To find the relationship with power, substitute:

τ = 0

into the autocorrelation formula:

R_x(0) = E[X(t) · X(t + 0)]

Since:

X(t + 0) = X(t)

we obtain:

R_x(0) = E[X(t) · X(t)] = E[X²(t)]

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Final Result

Comparing this expression with the definition of average power:

P_s = E[X²(t)]

and

R_x(0) = E[X²(t)]

we conclude that:

P_s = R_x(0)

This is a fundamental property of wide-sense stationary (WSS) random processes and is frequently used in communication systems, signal processing, and stochastic analysis.

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Intuitive Understanding

Think of autocorrelation as comparing a signal with itself.

• When τ ≠ 0, you compare the signal with a delayed version.
• When τ = 0, you compare every value with itself.
• A quantity compared with itself produces its square.
• The expected value of these squares is exactly the average power.

Therefore, the autocorrelation function reaches its maximum value at zero lag, and that value equals the signal power.

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Solved Example

Suppose the autocorrelation function of a random process is:

R_x(τ) = (2/3) cos(2πf₀τ)

Find the average power of the process.

Solution

Using the property:

P = R_x(0)

Substitute τ = 0:

P = (2/3) cos(2πf₀ × 0)

Since:

cos(0) = 1

Therefore:

P = (2/3) × 1

P = 2/3 Watts