Fourier Series Representation of Periodic Signals - Electrical Engineering (EE) PDF Download

Fourier Series Representation of Periodic Signals

Consider a periodic signal x(t) with fundamental period T, i.e.     Then the fundamental frequency of this signal is defined as the reciprocal of the fundamental period, so that    Under certain conditions, a periodic signal x(t) with period T can be expressed as a linear combination of sinusoidal signals of discrete frequencies, which are multiples of the fundamental frequency of x(t). Further, sinusoidal signals are conveniently represented in terms of complex exponential signals. Hence, we can express the periodic signal in terms of complex exponentials, i.e.

Such a representation of a periodic signal as a combination of complex exponentials of discrete frequencies, which are multiples of the fundamental frequency of the signal, is known as the Fourier Series Representation of the signal
Inner product 

The set of periodic signals with period T form a vector space.

We define the following inner product: 

And the norm or magnitude of the signal is defined as:- 

Now we consider the set of vectors ,   that belong to this vector space.              (note)

 We shall first show these vectors are mutually orthogonal. In other words we show that;-

Further, you may verify :

Thus, we have shown that this set of complex exponentials forms an orthogonal set in the vector space of all periodic signals with period T. Indeed, if we restrict ourselves to a certain class of signals in this vector space (those that satisfy the Dirichlet Conditions, which will be discussed in the next lecture), one can show that the above set of complex exponentials forms a basis for this class. i.e.: signals in this class can be expressed as a linear combination of these complex exponentials. In other words, such signals permit a Fourier Series representation. 

Assuming the Fourier Series representation of a signal x(t), with period T exists, it is easy to find the Fourier Series coefficients, using the orthogonality of the basis set of complex exponentials.

Taking inner product with   on both sides

Frequency Domain Representation

From the above discussion, we can say that a periodic signal whose Fourier Series Expansion exists, can be represented uniquely in terms of it's Fourier co-efficients. These co-efficients correspond to a particular multiples of the fundamental frequency of the signal.
Thus, the signal may be equivalently represented as a discrete signal on the frequency axis:

This is called the Frequency domain representation of the signal.
We next discuss the conditions under which the Fourier Expansion is valid.

Conclusion:

In this lecture you have learnt:

• A representation of a periodic signal as a combination of complex exponentials of discrete frequencies, which are multiples of the fundamental frequency of the signal, is known as the Fourier Series Representation of the signal.
• Set of periodic signals, with period T are a vector space
• Orthogonality of
• Calculating the Fourier series coefficients for a periodic signal.
• Frequency Domain Representation of a periodic signal