Periodic Signals

A periodic signal repeats the same waveform pattern over and over with a fixed time duration called its period. Fourier series analysis decomposes any periodic signal into a sum of sinusoids, which is the mathematical foundation for frequency-domain analysis in communication systems and audio equalizers.

Core Concept

A signal x(t) is periodic with period T if x(t) = x(t + T) for every value of t. The smallest positive value of T satisfying this equation is the fundamental period. The corresponding fundamental frequency is f0 = 1/T Hz, or omega_0 = 2*pi/T radians per second.

Every periodic signal with finite power can be expressed as a Fourier series: a sum of complex exponentials at integer multiples of the fundamental frequency. These multiples are called harmonics. The n-th harmonic has frequency n*f0. The complex Fourier coefficients c_n describe how much of each harmonic is present.

Periodic signals have infinite energy but finite average power. Power is distributed among the harmonics. The sum of |c_n|^2 over all n equals the average power of the signal. This is Parseval's theorem for periodic signals.

Key Formula

Complex Fourier series of a periodic signal x(t) with period T:

x(t) = sum_{n=-inf}^{inf} c_n * exp(j*n*omega_0*t)

Fourier coefficients: c_n = (1/T) * integral from 0 to T of x(t) * exp(-j*n*omega_0*t) dt

Parseval for periodic signals: P = (1/T) * integral |x(t)|^2 dt = sum_{n=-inf}^{inf} |c_n|^2

Problem: Find the Fourier coefficients c_n for a square wave:
  x(t) = +1 for 0 <= t < T/2
  x(t) = -1 for T/2 <= t < T
  Period = T, omega_0 = 2*pi/T

Step 1: Apply formula for c_n
  c_n = (1/T) * [ integral_0^{T/2} e^{-jn*omega_0*t} dt
                - integral_{T/2}^{T}  e^{-jn*omega_0*t} dt ]

Step 2: Evaluate for n not equal to 0
  c_n = (1/T) * [ (-1/jn*omega_0)(e^{-jn*pi} - 1)
                - (-1/jn*omega_0)(e^{-j2n*pi} - e^{-jn*pi}) ]

Step 3: Use e^{-j2n*pi} = 1 and e^{-jn*pi} = (-1)^n
  c_n = (1 - (-1)^n) / (j*n*pi)

Step 4: For even n: c_n = 0
  For odd n:  c_n = 2 / (j*n*pi)

Step 5: DC component c_0
  c_0 = (1/T) * [ T/2 * (+1) + T/2 * (-1) ] = 0

Final Answer: c_n = 2/(j*n*pi) for odd n, 0 for even n, c_0 = 0.

Exam Tip: To check periodicity of a sum x(t) = x1(t) + x2(t), find the periods T1 and T2, then check if T1/T2 is rational. If yes, the sum is periodic with period LCM(T1, T2). If T1/T2 is irrational, the sum is not periodic. For example, cos(t) + cos(sqrt(2)*t) is not periodic because T1/T2 = 1/sqrt(2) is irrational.

Properties Summary

• Periodic condition: x(t) = x(t+T) for all t; T is the fundamental period.
• Fundamental frequency: omega_0 = 2*pi/T rad/s; harmonics at n*omega_0.
• Fourier coefficient: c_n = (1/T) integral_T x(t) e^{-jn*omega_0*t} dt.
• Power: P = sum |c_n|^2 (Parseval's theorem for periodic signals).
• Symmetry shortcuts: real even signal -> c_n real and even. Real odd signal -> c_n imaginary and odd.
• Half-wave symmetry: x(t+T/2) = -x(t) implies c_n = 0 for all even n.
• Sum of periodic signals: periodic only if ratio of periods is rational.

Quick Revision

• Periodic: same waveform every T seconds. Fundamental period is the smallest such T.
• Periodic signals are always power signals (finite P, infinite E).
• Square wave has only odd harmonics: c_n nonzero only for odd n.
• Half-wave symmetric signal: only odd harmonics present.
• Parseval: average power = sum of squared magnitudes of Fourier coefficients.
• LCM of periods gives the period of a sum of periodic signals (when rational).
• Exam trap: assuming the sum of two periodic signals is always periodic. It is not periodic when the ratio of their periods is irrational.