DC Component

The DC component (from “direct current”) is the zero-frequency component of a signal — its average value or constant offset.

Definition

For a signal $g(x)$, the DC component is:

$$ g_{DC} = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} g(x) \, dx $$

In terms of the Fourier transform, the DC component is the value at zero frequency:

$$ G(0) = \int_{-\infty}^{\infty} g(x) \, dx $$

Why “DC”?

The term comes from electrical engineering:

• DC (Direct Current): Constant, non-oscillating current
• AC (Alternating Current): Oscillating current with non-zero frequency

By analogy, the “DC component” of any signal is the part that doesn’t oscillate — the constant offset.

Properties

Property	Description
Frequency	$f = 0$
Period	Infinite (no oscillation)
Value	Average of the signal
In spectrum	Located at the origin

Examples

1. $g(x) = 3 + \cos(2\pi x)$: DC component is 3 (the constant offset)
2. $g(x) = \sin(x)$: DC component is 0 (averages to zero)
3. $g(x) = \text{rect}(x)$: DC component equals the area of the rectangle

In 2D

For a 2D function $g(x, y)$:

$$ G(0, 0) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x, y) \, dx \, dy $$

The DC component is the total “volume” under the surface.

Applications

• Image processing: The DC component is the average brightness of an image
• Signal processing: Removing the DC component centers a signal around zero
• Power systems: DC offset in AC signals can indicate faults

See Also

• Fourier transform
• Sampling