Fourier Series Properties

Fourier series properties let you find coefficients of modified signals without re-integrating from scratch. These properties — linearity, time shifting, differentiation, integration, and Parseval theorem — are the main computational shortcuts on GATE and university examinations.

Core Concept

A Fourier series property tells you how an operation on x(t) changes its Fourier coefficients c_k. You do not need to recompute the integral. You apply the property algebraically. Linearity is the simplest: scaling or adding signals just scales or adds the corresponding coefficients.

Time shifting is more subtle. Delaying x(t) by t0 to get x(t - t0) does not change any coefficient magnitude. It only rotates each c_k by a phase of e^(-jkω0t0). The phase rotation is proportional to both the harmonic number k and the delay t0. This is why a delayed signal looks the same on an amplitude spectrum but different on a phase spectrum.

Differentiation multiplies each c_k by jkω0. This amplifies high harmonics and suppresses DC. Integration divides by jkω0 — the reverse operation. Parseval theorem says the average power of a signal equals the sum of squared magnitudes of all its Fourier coefficients. It converts time-domain energy calculations into spectral ones.

Key Formula

Linearity: F{ax(t)+by(t)} = a·c_k + b·d_k. Time shift: F{x(t-t0)} = c_k · e^(-jkω0t0). Frequency shift: F{x(t)e^(jmω0t)} = c_{k-m}. Differentiation: F{dx/dt} = jkω0 · c_k. Integration: F{∫x dt} = c_k/(jkω0) for k≠0. Time reversal: F{x(-t)} = c_{-k}. Parseval: (1/T)∫|x(t)|² dt = Σ|c_k|² summed over all k.

Problem: x(t) is a square wave with coefficients c_k known.
         Find the Fourier coefficients of y(t) = x(t - T/4).

Given: x(t) has c_k = 0 for even k,
        c_k = 1/(jkπ) · [1 - e^(-jkπ)] for general derivation,
        or simply c_k = A/(jkπ) · (1 - (-1)^k) / 2 (standard result).
        Let's use c_k of x(t) = (1-(-1)^k)/(jkπ) for |A|=1.

Step 1: Apply time-shift property.
         Delay t0 = T/4, so ω0·t0 = (2π/T)·(T/4) = π/2.

Step 2: New coefficients d_k = c_k · e^(-jk·π/2)
         = c_k · e^(-jkπ/2)

Step 3: For k = 1: d_1 = c_1 · e^(-jπ/2) = c_1 · (-j)
For k = 3: d_3 = c_3 · e^(-j3π/2) = c_3 · (j)
For even k: d_k = 0 (since c_k = 0)

Step 4: Magnitude |d_k| = |c_k| (time shift does not change spectrum magnitude).

Final Answer:
  |d_k| = |c_k| for all k.
  Phase of d_k = phase of c_k - kπ/2.
  Amplitude spectrum unchanged; only phase spectrum rotated.

Exam Tip: The time-shift property is the most tested Fourier series property in GATE. Remember that shifting x(t) by t0 multiplies c_k by e^(-jkω0t0), changing phase but not magnitude. For Parseval theorem, note that average power P = (1/T)∫|x|² dt = Σ|c_k|², summing over all k from -∞ to +∞. If using trigonometric form, P = a_0² + (1/2)Σ(a_k² + b_k²) for k ≥ 1.

Properties Summary

• Linearity: F{ax+by} = a·c_k + b·d_k; superposition applies directly to coefficients.
• Time shift: F{x(t-t0)} = c_k · e^(-jkω0t0); magnitude unchanged, phase rotated by kω0t0.
• Time reversal: F{x(-t)} = c_{-k}; the spectrum is frequency-reversed.
• Differentiation: F{dx/dt} = jkω0 · c_k; each coefficient scaled by jkω0.
• Integration: F{∫x dt} = c_k/(jkω0) for k≠0; DC must be handled separately.
• Parseval: average power = Σ|c_k|² summed over all integer k from -∞ to +∞.
• Conjugate symmetry: for real x(t), c_{-k} = c_k*; amplitude spectrum is even, phase spectrum is odd.

Quick Revision

• Time shift → phase change only; amplitude spectrum stays the same.
• Differentiation → multiply c_k by jkω0; integration → divide by jkω0.
• Parseval theorem: sum of |c_k|² equals average power of the signal.
• Linearity: coefficients of a sum are sums of the coefficients.
• Frequency shift (modulation): multiplying by e^(jmω0t) shifts the spectrum index by m.
• Time reversal reverses the spectrum: coefficient at index k becomes coefficient at index -k.
• For real signals, knowing c_k for k ≥ 0 is enough; c_{-k} = c_k* gives the rest.
• Exam trap: in Parseval for trigonometric form, the formula is a_0² + (1/2)Σ(a_k²+b_k²), not simply Σ(a_k²+b_k²) — the DC term is not halved.