A PERIODIC function can be represented as a sum of sine and cosine functions with some coefficients. Such a representation is known as Fourier series. The condition of periodicity is necessary for the Fourier series. There are few more conditions, known as Dirichlet’s conditions.
Consider such a function of period and defined over . It can be written as
The constants and are known as Fourier coefficients. They are calculated as:
This is the Fourier series.
Fourier Series : Complex Representation
Using and , we can write
Substituting this in the Fourier series for ,
Let , and .
Using this, we can write
We know that the definite integral is actually a sum. Further, in reality, the function may not be always periodic (e.g. an unrepeated pulse in a circuit). In such cases, the sum is replaced by an integral. Thus, we now allow the function to be aperiodic or . We introduce a new term , so .
Clearly, as , . Thus, can be written as
Taking limit, as , we can apply the fundamental theorem of integral calculus,
This is the Fourier integral representation of for .
Simplified Fourier Integral
I) Separating and ,
II) Using Euler’s identity,
III) We can now split the integrals of and . Sine being an odd function, is always , hence the 2nd integral is always . Hence,
The inner integral is known as the Fourier Transform of and is denoted by . Thus,
The inverse Fourier transform is given by
Fourier Cosine Transform (For even functions)
Fourier Sine Transform (For odd functions)