Periodic Function

A ‘Periodic Function’ is a function that repeats itself in regular intervals or periods. Let’s take an example of a lyrical song or the heartbeat; it redoes the same action on an equal beat. If we see the graphical representation of a periodic function, it looks like a single pattern is being repeated over uniform intervals of time.

The repetition of action or values at uniform intervals is the function of a “periodic function”. The major characteristic of periodic functions is the “period of a function”. The “period of a function” helps to determine a periodic function. The equation of a periodic function having a period ‘T’ can be illustrated as  f(x + T) = f(x), where T represents a non-zero constant such that  f(x + T) = f(x), where all the values of x belong to real numbers. Some common examples of periodic function are seen when time is variable, such as the hands of a clock, the movements of planets around the Sun, phases of the moon etc, all of the above-mentioned actions represent periodic behaviour.

Periodic Function

A “Periodic Function” is a function that repeats itself in regular intervals or periods. The “period of a function” helps to determine a periodic function. The equation of a periodic function having a period ‘T ’ can be illustrated as  f(x + T) = f(x), where T represents a non-zero constant such that  f(x + T) = f(x), where all the values of x belong to real numbers. The periodic function has some important “periods”.

1. In terms of sin x and cos x, it is 2π
2. In terms of tan x and cot x, it is π
3. In terms of sec x and cosec x, it is 2π

Properties of Periodic Functions

• The graphical representation of a periodic function looks like a single pattern is being repeated over uniform intervals of time.
• The range of the periodic function is for a fixed interval. 
• The domain of the periodic function includes the values of the real number.
• Let’s consider f(y) as a periodic function with a period of P, then 1/f(y) will also be a periodic function with the same fundamental period.

Fourier Series

The Fourier series is named after the name of Jean Baptiste Fourier, who contributed to the study of trigonometric series. The “Fourier Series” is defined as the method of representing a periodic function as a sum of sine and cosine functions. The Fourier Series is known to be very helpful in solving various problems, which involve partial differential equations. The “Fourier Series” is an infinite series. The Fourier Series in terms of trigonometric functions represents an expansion or estimation of a periodic function.

Properties of Fourier Series

• Time reversal property

                        Fourier series coefficient

              x(-t)   <——————————-> fXn. 

• Linearity property 

                          Fourier series coefficient

Ax(t)+ By(t)   < ————————————> Af Xn    +  Bf Yn

•  Time scaling property 

          Fourier series coefficient

X(at) < —————————–> fXn 

• Convolution property

                 Fourier series coefficient

X(t) . Y(t) < —————————> T fXn . f Yn.    

Application of Fourier Series

Fourier Series are used in applied mathematics. The Fourier Series is especially used in subjects, such as physics and electronics, to express ‘periodic functions’, which comprise communications signal waveforms. The “Fourier Series” is an infinite series; it helps in the expansion of a function in terms of sine and cosine function. In physics and engineering, the expansion of functions in terms of sine and cosine is helpful, as it helps to represent difficult functions analytically. Fourier Series are used in the fields of electronics, quantum mechanics, electrodynamics, digital signal processing, and spectral analysis.

Conclusion

A function that repeats itself at regular intervals or periods is called a “Periodic function”. Let’s take an example of periodic functions. The trigonometric function repeats itself at intervals of the 2π period. ‘Periodic functions’ are used to study oscillation, movements of planets around the Sun, phases of the Moon, waves and phenomena related to periodicity. The function, which is not periodic, is called a ‘non-periodic function’. The periodic function can be described with the help of the “Fourier Series”. The “Fourier Series” is defined as the method of representing a periodic function as a sum of sine and cosine functions. Some properties of the Fourier Series are time-reversal property, linearity property, time scaling property, multiplication property, frequency shifting property, differentiation and integration property, and convolution property. Fourier Series are used in the fields of electronics, quantum mechanics, electrodynamics, digital signal processing, and spectral analysis.