Fourier Series and Fourier Transform

Fourier analysis breaks a signal into sinusoids. Periodic signals have a Fourier series (discrete frequencies); aperiodic signals have a Fourier transform (continuous spectrum).

Fourier series

For a periodic signal with period T₀ (ω₀ = 2π/T₀):

x(t) = Σ c_k e^(jkω₀t), with c_k = (1/T₀) ∫ x(t) e^(−jkω₀t) dt.

Trigonometric form: a₀ + Σ (a_k cos kω₀t + b_k sin kω₀t).

Symmetry shortcuts:

• Even signal → only cosine terms (b_k = 0).
• Odd signal → only sine terms (a₀ = a_k = 0).
• Half-wave symmetry x(t ± T/2) = −x(t) → only odd harmonics.

Square wave (±A, even): x(t) = (4A/π)[cos ω₀t − (1/3) cos 3ω₀t + (1/5) cos 5ω₀t − …].

Parseval: average power = Σ |c_k|².

Fourier transform

X(ω) = ∫ x(t) e^(−jωt) dt; inverse x(t) = (1/2π) ∫ X(ω) e^(jωt) dω.

Properties: linearity; time shift → multiply by e^(−jωt₀); scaling x(at) → (1/|a|) X(ω/a); convolution in time ↔ multiplication in frequency; multiplication in time ↔ (1/2π) convolution in frequency; differentiation → jω X(ω); duality; Parseval: ∫|x|²dt = (1/2π)∫|X|²dω.

Frequency response

For an LTI system, H(jω) = Fourier transform of h(t). An input e^(jωt) gives output H(jω) e^(jωt): sinusoids pass through scaled by |H| and shifted by ∠H.

• Distortionless transmission: H(jω) = K e^(−jωt_d) (constant gain, linear phase).
• Ideal filters (low-pass, high-pass, band-pass) are non-causal.

Worked example

A rectangular pulse of height 2 and width 0.5 s.

• X(0) = area = 2 × 0.5 = 1. The first zero of the spectrum is at f = 1/τ = 2 Hz.

Practice