Effect of an LTI Filter on Signal Mean and Variance

Why Do Mean and Variance Change After Passing Through an LTI Filter?

One of the most common questions in Digital Signal Processing (DSP) is: "Why does the mean or variance of a random signal change after passing through a Linear Time-Invariant (LTI) filter?" Understanding this concept is essential for topics such as random processes, communication systems, estimation theory, adaptive filtering, and noise analysis. This article explains the mathematics, intuition, and physical interpretation behind the change in signal statistics after LTI filtering.

What is an LTI Filter?

A Linear Time-Invariant (LTI) filter is completely characterized by its impulse response h[n]. If the input signal is

x[n]

then the output is obtained using convolution:

y[n] = h[n] * x[n]

Every output sample is a weighted combination of several input samples. Therefore, the statistical properties of the output generally differ from those of the input.

How Does the Mean Change?

Suppose the input has mean

μ_x = E{x[n]}

Using linearity of expectation,

E\{y[n]\} = E\left\{ \sum_{k=-\infty}^{\infty} h[k]x[n-k] \right\}

Since expectation is linear,

μ_y = μ_x \sum_{k=-\infty}^{\infty} h[k]

Important Result

Mean of Output = Mean of Input × Sum of Filter Coefficients

Interpretation

• If the input mean is zero, the output mean is also zero.
• If the sum of filter coefficients equals 1, the mean remains unchanged.
• If the sum is greater than 1, the mean increases.
• If the sum is less than 1, the mean decreases.
• If the filter has zero DC gain (for example, a differentiator), the output mean becomes zero.

How Does the Variance Change?

Variance measures the average power of fluctuations around the mean. Unlike the mean, variance depends on correlations between samples and therefore changes according to the filter characteristics.

Variance = Average Power Around the Mean

Physical Meaning

An LTI filter amplifies some frequencies while attenuating others. Since signal power is distributed across frequencies, changing these frequency components changes the total output power. Because variance represents power for a zero-mean signal, the variance also changes.

Case 1: White Noise Input

Assume

• Input is zero-mean white noise.
• Variance is σ²_x.
• Samples are mutually uncorrelated.

The output is

y[n] = Σ h[k]x[n-k]

The output variance becomes

σ²_y = σ²_x Σ h²[k]

Interpretation

• If Σh²[k] = 2 → variance doubles.
• If Σh²[k] = 0.5 → variance becomes half.
• If Σh²[k] = 1 → variance remains unchanged.

The quantity

Σ h²[k]

is called the energy of the filter.

Case 2: General Wide-Sense Stationary Random Process

When the input samples are correlated, the output variance depends on the input Power Spectral Density (PSD).

σ²_y = (1 / 2π) ∫ |H(e^jω)|² S_x(ω) dω

where

• H(e^jω) = frequency response of the filter
• S_x(ω) = input power spectral density

Interpretation

Each frequency component of the input is multiplied by

|H(e^jω)|²

Therefore, the filter redistributes signal power across frequencies. Integrating the filtered spectrum gives the total output variance.

Physical Intuition

Filter Type	Effect on Mean	Effect on Variance
Low-pass Filter	Usually unchanged if DC gain = 1	Reduces variance by removing high-frequency energy
High-pass Filter	Removes DC, mean becomes zero	Usually decreases variance
Amplifier	Increases mean	Increases variance
Attenuator	Reduces mean	Reduces variance
All-pass Filter	Mean unchanged	Variance unchanged because |H(ejω)| = 1

Key Differences Between Mean and Variance

Statistic	Depends On	Output Formula
Mean	DC Gain	μy = μx Σh[k]
Variance (White Noise)	Filter Energy	σ²y = σ²x Σh²[k]
Variance (General Signal)	Frequency Response & PSD	σ²y = (1/2π)∫|H(ejω)|²Sx(ω)dω

Summary

• An LTI filter performs a weighted sum of input samples.
• The output mean equals the input mean multiplied by the sum of filter coefficients.
• For a zero-mean input, the output also has zero mean.
• The output variance changes because the filter modifies the signal's energy distribution.
• For white noise, σ²_y = σ²_x Σh²[k].
• For a general stationary process, the output variance is obtained by integrating the filtered power spectral density.
• An all-pass filter preserves variance because it changes only phase, not magnitude.

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