Periodic and Non-Periodic Signals

Non-Periodic (Aperiodic) Signals

• Non-periodic signals are also referred to as aperiodic signals.

Periodic Signals

• A signal is periodic if it repeats itself after a regular interval of time.
• Definition: A signal x(t) is periodic if x(t) = x(t + nT₀), where n is an integer and T₀ is the fundamental period.

Fundamental Period (T₀)

• Definition: The smallest positive value of time for which the signal is periodic.
• Example: For a signal with ΔT = 5 seconds, x(t) = x(t + 5).

Understanding with Example: Sine Wave

• Sine Wave: sin(θ) is our example for understanding periodicity.
• When θ = 0 or θ = π, sin(θ) = 0.
• The signal is not periodic with T₀ = π as values differ in sine function before and after π.
• Fundamental Period for Sine: 2π
  • sin(θ) = sin(θ + 2π), making the fundamental period T₀ = 2π.
  • Any multiple of the fundamental period (4π, 6π, etc.) will also show periodicity, but they are not the smallest period.

Fundamental Frequency (F₀)

• Definition: F₀ = 1/T₀
• Measured in cycles per second or Hertz (Hz).

Fundamental Angular Frequency (ω₀)

• Definition: ω₀ = 2πF₀
• Measured in radians per second.

Discrete-Time Signals

• Denoted by x[n].
• Periodic if x[n] = x[n ± mN], where m is an integer and N is the fundamental period.
• Capital N (fundamental period) must be an integer.
• Nominal Frequency: F = 1/N.

DC Value

• The statement "DC value is periodic" is true.
• Example: x(t) = 2, plot remains constant for all values of time T.
• Fundamental period is undefined for a DC value; hence, F₀ = 0.
• Relation F₀ = 1/T₀ is invalid for DC value.

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Remember to revise these key points before the exam.

Next Lecture

• Calculation of the fundamental time period.