A periodic signal is a signal that repeats its sequence of values exactly after a fixed length of time, known as the period. Mathematically, a continuous-time signal x(t)x(t)x(t) is periodic if there exists a positive constant TTT such that for all ttt, x(t)=x(t+T)x(t)=x(t+T)x(t)=x(t+T). This TTT is called the fundamental period, representing the duration of one complete cycle of the signal
. For discrete-time signals x(n)x(n)x(n), periodicity means there exists a positive integer NNN such that x(n)=x(n+N)x(n)=x(n+N)x(n)=x(n+N) for all integers nnn, where NNN is the fundamental period in terms of samples
. The period TTT is constant for a given periodic signal, and the reciprocal of the period is the frequency f=1Tf=\frac{1}{T}f=T1, which indicates how many cycles occur per second
. Common examples of periodic signals include sinusoidal waves (sine and cosine functions), square waves, triangle waves, and sawtooth waves. These signals exhibit repeating patterns indefinitely over time
. In summary, periodic signals are characterized by their repeating patterns at regular intervals, which can be mathematically expressed and visualized as signals that replicate their shape over time or samples
