# Fourier Series Representation of Continuous Periodic Signals-1 Study notes For EC/EE

By BYJU'S Exam Prep

Updated on: September 25th, 2023

In the present article, you will find the study notes on Fourier Series & Representation of Continuous Periodic Signal which will cover the topics such as Fourier Theorem, Fourier Coefficients, Fourier Sine Series, Fourier Cosine Series, Orthogonality Relations of Fourier Series Exponential Fourier Series Representation, Even & Odd Symmetry

Table of content

## Periodic Functions

The Periodic functions are the functions which can be defined by the relation f(t + P) = f(t) for all t. For example, if f(t) is the amount of time between sunrise and sunset at a certain latitude, as a function of time t, and P is the length of the year, then f(t + P) = f(t) for all t, since the Earth and Sun are in the same position after one full revolution of the Earth around the Sun.

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## Fourier Series

Any arbitrary continuous-time signal x(t) which is periodic with a fundamental period To, can be expressed as a series of harmonically related sinusoids whose frequencies are multiples of fundamental frequency or first harmonic. In other words, any periodic function of (t) can be represented by an infinite series of sinusoids called as Fourier Series.

The periodic waveform is expressed in the form of Fourier series, while the non-periodic waveform may be expressed by the Fourier transform.

The different forms of the Fourier series are given as follows.

(i) Trigonometric Fourier series

(ii) Complex exponential Fourier series

(iii) Polar or harmonic form Fourier series.

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## Trigonometric Fourier Series

Any arbitrary periodic function x(t) with fundamental period T0 can be expressed as follows

This is referred to as the trigonometric Fourier series representation of signal x(t). Here, ω0 = 2π/T0 is the fundamental frequency of x(t) and coefficients a0, an, and bn are referred to as the trigonometric continuous-time Fourier series (CTFS) coefficients. The coefficients are calculated as follows.

Fourier Series Coefficient

From equation (ii), it is clear that coefficient a0 represents the average or mean value (also referred to as the dc component) of signal x(t).

In these formulas, the limits of integration are either (–T0/2 to +T0/2) or (0 to T0). In general, the limit of integration is any period of the signal and so the limits can be from (t1 to t1 + T0), where t1 is any time instant.

## Trigonometric Fourier Series Coefficients for Symmetrical Signals

If the periodic signal x(t) possesses some symmetry, then the continuous-time Fourier series (CTFS) coefficients become easy to obtain. The various types of symmetry and simplification of Fourier series coefficients are discussed below.

Consider the Fourier series representation of a periodic signal x(t) defined in the equation below as

Even Symmetry: x(t) = x(–t)

If x(t) is an even function, then product x(t) sinωot is odd and integration in equation (iv) above becomes zero. i.e bn = 0 for all n and the Fourier series representation is expressed as

For example, the signal x(t) shown below figure has even symmetry so bn = 0 and the Fourier series expansion of x(t) is given as

The trigonometric Fourier series representation of even signals contains cosine terms only. The constant a0 may or may not be zero.

Odd Symmetry: x(t) = –x(–t)

If x(t) is an odd function, then product x(t) cosωot is also odd and integration in equation (iii) above becomes zero i.e. an = 0 for all n. Also, a0 = 0 because an odd symmetric function has a zero-average value. The Fourier series representation is expressed as

For example, the signal x(t) shown below figure is odd symmetric so an = a0 = 0 and the Fourier series expansion of x(t) is given as

## Fourier Sine Series

The Fourier Sine series can be written as

——(2)

• Sum S(x) will inherit all three properties:
• (i): Periodic S(x +2π)=S(x); (ii): Odd S(−x)=−S(x); (iii): S(0) = S(π)=0
• Our ﬁrst step is to compute from S(x), the term bk that multiplies sin kx.

Suppose S(x)=∑ bn sin nx. Multiply both sides by sin kx. Integrate from 0 to π in Sine Series in equation (2)

• On the right side, all integrals are zero except for n = k. Here the property of “orthogonality” will dominate. The sines make 90o angles in function space when their inner products are integrals from 0 to π.
• Orthogonality for Sine Series

Condition for Orthogonality:

——(3)

• Zero comes, if we integrate the term cos mx from 0 to π. ⇒  = 0-0=0
• Integrating cos mx with m = n−k and m = n + k proves the orthogonality of the sines.
• The exception is when n = k. Then we are integrating (sin kx)2 = 1/2 − 1/2 cos2kx

——(4)

• Notice that S(x) sin kx is even (equal integrals from −π to 0 and from 0 to π).
• We will immediately consider the most important example of a Fourier sine series. S(x) is an odd square wave with SW(x) = 1 for 0

Example:

Find the Fourier sine coeﬃcients bk of the square wave SW(x) as given above.

Solution:

For k =1 ,2,…using the formula of sine coefficient with S(x)=1 between 0 and π:

• Then even-numbered coeﬃcients b2k are all zero because cos2kπ = cos 0 = 1.
• The odd-numbered coeﬃcients bk =4/πk decrease at the rate of 1/k.
• We will see that same 1/k decay rate for all functions formed from smooth pieces and jumps. Put those coeﬃcients 4/πk and zero into the Fourier sine series for SW(x).

## Fourier Cosine Series

The cosine series applies to even functions with C(−x)=C(x) as

—–(5)

Cosine has period 2π shown as above in figure two even functions, the repeating ramp RR(x) and the up-down train UD(x) of delta functions.

• That sawtooth ramp RR is the integral of the square wave. The delta functions in UD give the derivative of the square wave. RR and UD will be valuable examples, one smoother than SW, and one less smooth.
• First, we ﬁnd formulas for the cosine coeﬃcients a0 and ak. The constant term a0 is the average value of the function C(x):

—–(6)

• We will integrate the cosine series from 0 to π. On the right side, the integral of a0=a0π (divide both sides by π). All other integrals are zero.

• Again the integral over a full period from −π to π (also 0 to 2π) is just doubled.

## Orthogonality Relations of Fourier Series

Since from the Fourier Series Representation, we concluded that a periodic Signal, it can be written as

——-(7)

The condition of orthogonality is as follow:

Proof of the orthogonality relations:

This is just a straightforward calculation using the periodicity of sine and cosine and either (or both) of these two methods:

### Energy in Function = Energy in Coeﬃcients

There is also another important equation (the energy identity) that comes from integrating (F(x))2. When we square the Fourier series of F(x) and integrate from −π to π, all the “cross terms” drop out. The only nonzero integrals come from 12 and cos2 kx and sin2 kx, multiplied by a02,ak2 bk2.

• Energy in F(x) equals the energy in the coeﬃcients.
• The left-hand side is like the length squared of a vector, except the vector is a function.
• The right-hand side comes from an inﬁnitely long vector of a’s and b’s.
• If the lengths are equal, which says that the Fourier transforms from function to vector is like an orthogonal matrix.
• Normalized by constants √2π and √π, we have an orthonormal basis in function space.

## Complex Fourier Series

• In place of separate formulas for a0 and ak and bk, we may consider one formula for all the complex coeﬃcients ck.
• So that the function F(x) will be complex, The Discrete Fourier Transform will be much simpler when we use N complex exponentials for a vector.

The exponential form of the Fourier series of a periodic signal x(t) with period T0 is defined

where ω0 is the fundamental frequency given as ω0 = 2π /T0. The exponential Fourier series coefficients cn are calculated from the following expression

• Since c0 = a0 is still the average of F(x), because e0 = 1.
• The orthogonality of einx and eikx is to be checked by integrating.

Example:

Compute the Fourier series of f(t), where f(t) is the square wave with period 2π. which is defined over one period by.

The graph over several periods is shown below.

Solution:

Computing a Fourier series means computing its Fourier coef­ficients. We do this using the integral formulas for the coefficients given with Fourier’s theorem in the previous note. For convenience, we repeat the theorem here.

By applying these formulas to the above waveform we have to split the integrals into two pieces corresponding to where f(t) is +1 and where it is −1.

thus for n ≠ 0 ;

for n = 0

We have used the simplification cos nπ = (−1)n to get a nice formula for the coefficients bn.

This then gives the Fourier series for f(t)

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