Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

Signal and System

Electrical Engineering (EE) : Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

The document Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Signal and System.
All you need of Electrical Engineering (EE) at this link: Electrical Engineering (EE)

Fourier Series Representation
The Periodic functions are the functions which can define by 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.

Fourier Theorem 

  • We will first state Fourier’s theorem for periodic functions with period P = 2π. In words, the theorem says that a function with period 2π can be written as a sum of cosines and sines which all have period 2π.

Suppose f(t) has period 2π then we have
f(x) ≅  a0/2 +{a1cos(x) + a2cos(2x) + a3cos(3x)}+ ........+{b1sin(x) + b2sin(2x) + b3sin(3x)}+ ...
Complete Fourier series
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev -------(1)

  • where a0, a1,... and b1, b2 . . . called Fourier coefficients & are computed as 
    Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
  • The above is called the Fourier series for the function.That is, sines and cosines, the simplest periodic functions, are the “building blocks" for more general periodic functions.
  • There are some terminology coming from acoustics and music: the n = 1 called the fundamental, and the frequencies n ≥ 2 are called the higher harmonics. 
  • cos(nt) and sin(nt) are periodic on the interval 2π for any integer n. The an and bn coefficients measure the strength of contribution from each "harmonic"

Fourier Sine Series
The Fourier Sine series can be written as
S(x) = b1 sinx + b2 sin 2x + b3 sin 3x + Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev  ........(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 number bk that multiplies sinkx.

Suppose S(x)=∑ bn sinnx. Multiply both sides by sin kx. Integrate from 0 to π in Sine Series in equation (2)
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

  • 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
    Orthogonality  Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev .........(3)
  • Zero comes quickly if we integrate the term comx  from 0 to π. ⇒ 0π comx dx = 0-0=0
  • Integrating cosmx with m = n−k and m = n + k proves orthogonality of the sines.
  • The exception is when n = k. Then we are integrating (sinkx)2 = 1/2 − 1/2 cos2kx
    Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
    Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
  • Notice that S(x)sinkx 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<x<π. It is an odd function with period 2 π, that vanishes at x=0 and x= π.
    Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

Example: Finding 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:
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

  • Then even-numbered coefficients b2k are all zero because cos2kπ = cos0 = 1.
  • The odd-numbered coefficients bk = 4/πk decrease at the rate 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).

Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

Fourier Cosine Series
The cosine series applies to even functions with C(−x)=C(x) as
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

Cosine has period 2π shown as above in the 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, 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):
    Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
  • 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.
    Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRevStudy Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
  • 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 for a periodic Signal it can be written as
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
The condition of orthogonality as follow:
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
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:

  • Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
  • Method 2: use the trig identity Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRevand the similar trig identies for cos(α) sin(β) and sin(α) sin(β).

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.

Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

  • Energy  in F(x) equals the energy in the coefficients.
  • Left hand side is like the length squared of a vector, except the vector is a function.
  • 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 transform 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.

Here we practice in advance with the complex infinite series for a 2π-periodic function:
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

  • If every cn = c−n, we can combine einx with e−inx into 2cosnx. Then last equation is the cosine series for an even function.
  • If every cn =−c−n, we use einx−e−inx =2i sinnx. Then last equation is sine series for an odd function and the c’s are pure imaginary.
  • To find ck, multiply (8) by e−ikx (not eikx) and integrate from −π to π.
    Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
  • Since the complex exponentials are orthogonal. Every integral on the right side is zero, except for the term (when n = k and (eikx)(e−ikx)= 1). The integral of 1 is 2π.which only term gives the formula for ck.
    Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
  • 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.
    Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

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.

Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev The graph over several periods is shown below.
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
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.
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
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.
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

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)
Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!
39 videos|7 docs|19 tests

Up next >

Dynamic Test

Content Category

Related Searches

Semester Notes

,

Extra Questions

,

past year papers

,

Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

,

Sample Paper

,

Objective type Questions

,

mock tests for examination

,

Viva Questions

,

Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

,

Important questions

,

MCQs

,

Study Notes for Fourier Series Representation of Continuous Periodic Signals -1 Electrical Engineering (EE) Notes | EduRev

,

Summary

,

pdf

,

Free

,

Exam

,

Previous Year Questions with Solutions

,

study material

,

ppt

,

shortcuts and tricks

,

video lectures

,

practice quizzes

;