Why Signal Amplitude Reduces After Filtering

In real-world filters, the amplitude of a signal is often scaled due to unity energy normalization, which is applied to preserve the total signal power. This normalization ensures that the filtered signal maintains the same power as the original but results in a reduction in amplitude.

1. Signal Power Before Filtering

For a sinusoidal signal:

x(t) = A cos(2πf₍c₎t)

The power P_x of the signal is given by:

Pₓ = (1/T) ∫ |x(t)|² dt = A²/2

2. Bandpass Filter and Unity Energy Normalization

A bandpass filter with a constant gain H(f) over the passband ensures power normalization by scaling the gain such that:

∫₍f₁₎⁽f₂⁾ |H(f)|² df = 1

For a filter with bandwidth B = f₂ − f₁, the gain is:

|H(f)|² = 1 / B

The filter scales the signal by 1/√B to normalize the power.

3. Effect on Signal Amplitude

After filtering, the power of the filtered signal is the same as the original, but the amplitude is reduced. For sinusoidal signals:

Pᵧ = Pₓ = A² / 2

The amplitude of the filtered signal A_y is scaled as:

Aᵧ = A × √(1 / B)

4. Example: Amplitude Halving

Consider a sinusoidal signal:

x(t) = cos(2π × 1000t)

If the filter has a bandwidth B = 2, the amplitude of the filtered signal becomes:

Aᵧ = A × 1 / √2 ≈ 0.707A

Thus, the amplitude is reduced by approximately 29.3%.

If filter bandwidth B = 4, then the amplitude of the filtered signal reduces to 50%, and so on.

5. Why This Happens

Real-world filters are designed to prioritize power preservation rather than amplitude. This normalization ensures the filter does not artificially boost or reduce the signal's power.

Further Reading