DC Component of a Periodic Signal

DC Component (Zero-Frequency) – Complete Hand Calculation Guide

The DC component (zero-frequency component) represents the average value or total area of a signal, depending on the transform used.

1. DC Component of a Periodic Signal

The DC component of a periodic signal is its average value over one full period.

Definition

For a periodic signal x(t) with period T:

DC component = (1/T) ∫₀ᵀ x(t) dt

• Mean value
• Average value
• Zero-frequency component (in Fourier analysis)

Continuous-Time

X_DC = (1/T) ∫_{t₀}^{t₀ + T} x(t) dt

You can integrate over any full period.

Discrete-Time

X_DC = (1/N) Σ x[n],  n = 0 to N-1

2. Examples (Periodic Signals)

Example 1: Sine Wave

x(t) = A sin(ωt)
X_DC = 0

A pure sine (or cosine) has zero DC component.

Example 2: Offset Sine Wave

x(t) = A sin(ωt) + 3
X_DC = 3

The DC component equals the constant offset.

Example 3: Rectangular Wave (Unipolar)

If a square wave switches between 0 and 5 V with 50% duty cycle:

X_DC = (0 + 5)/2 = 2.5 V

Example 4: General Square Wave

• Amplitude = A
• Duty cycle = D (fraction of time signal is high)

X_DC = A · D

3. DC Component in Fourier Series

x(t) = a₀ + Σ ...
a₀ = (1/T) ∫₀ᵀ x(t) dt

The DC component is the zero-frequency term.

4. Zero-Frequency Component in Fourier Transform

Continuous-Time Fourier Transform (CTFT)

X(ω) = ∫_{-∞}^{∞} x(t)e^{-jωt} dt
X(0) = ∫_{-∞}^{∞} x(t) dt

DC component equals the total area under the signal.

Example: Rectangular Pulse

x(t) = A, for -T/2 ≤ t ≤ T/2
X(0) = A × T

Example: Exponential Signal

x(t) = e^{-at} u(t)
X(0) = 1/a

Discrete-Time Fourier Transform (DTFT)

X(e^{j0}) = Σ x[n]

DC component = Sum of all samples

5. Quick Comparison

Transform Type	DC Component
Fourier Series	Average value
Fourier Transform	Total area
DTFT	Sum of samples