Q. Explain Periodic Signals and Fourier Series?

In the study of analog systems, predicting the response of circuits to a general time-varying voltage or currentwaveformx(t) is a difficult task.However, if x(t) can be expressed as a sumof sinusoids, then the principle of superposition can be invoked on linear systems and the frequency response of the circuit can be utilized to expedite calculations. Expressing a signal in terms of sinusoidal components is known as spectral analysis.

A periodic signal has the property that it repeats itself in time, and hence, it is sufficient to specify the signal in the basic time interval called the period. A periodic signal x(t) satisfies the property

x(t + kT ) = x(t)

for all t, all integers k, and some positive real number T, called the period of the signal. For discrete-time periodic signals, it follows that

x(n + kN) = x(n)

for all integers n, all integers k, and a positive integer N, called the period. A signal that does not satisfy the condition of periodicity is known as nonperiodic.