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.

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**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 +{a_{1}cos(x) + a_{2}cos(2x) + a_{3}cos(3x)}+ ........+{b_{1}sin(x) + b_{2}sin(2x) + b_{3}sin(3x)}+ ...**Complete Fourier series**** -------(1)**

**where a**_{0}, a_{1},... and b_{1}, b_{2}. . . called Fourier coefficients & are computed as- 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 a
_{n}and b_{n}coefficients measure the strength of contribution from each "harmonic"

**Fourier Sine Series**

The Fourier Sine series can be written as

S(x) = b_{1} sinx + b_{2} sin 2x + b_{3} sin 3x + ........(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 b
_{k}that multiplies sinkx.

Suppose S(x)=∑ b_{n} sinnx. 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 90
^{o}angles in function space, when their inner products are integrals from 0 to π. - Orthogonality for sine Series
**Orthogonality**.........(3) - Zero comes quickly if we integrate the term comx from 0 to π. ⇒
_{0}∫^{π}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 - 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.

**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:

- Then even-numbered coeﬃcients b
_{2k}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).

**Fourier Cosine Series**

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

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):
- We will integrate the cosine series from 0 to π. On the right side, the integral of a
_{0}= a_{0}π (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 for a periodic Signal it can be written as**The condition of orthogonality 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:

- Method 2: use the trig identity and 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 1^{2} and cos2 kx and sin2 kx, multiplied by a_{0}^{2},a_{k}^{2} b_{k}^{2}.

- 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 a
_{0}and a_{k}and b_{k}, we may consider one formula for all the complex coeﬃcients C_{k}. - 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:

- If every c
_{n}= c_{−n}, we can combine e^{inx}with e^{−inx}into 2cosnx. Then last equation is the cosine series for an even function. - If every c
_{n}=−c_{−n}, we use e^{inx}−e^{−inx}=2i sinnx. Then last equation is sine series for an odd function and the c’s are pure imaginary. - To ﬁnd c
_{k}, multiply (8) by e^{−ikx}(not e^{ikx}) and integrate from −π to π. - Since the complex exponentials are orthogonal. Every integral on the right side is zero, except for the term (when n = k and (e
^{ikx})(e^{−ikx})= 1). The integral of 1 is 2π.which only term gives the formula for c_{k}. - Since c
_{0}= a_{0}is still the average of F(x), because e_{0}= 1. - The orthogonality of e
^{inx}and e^{ikx}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.

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

This then gives the Fourier series for f(t)

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39 videos|7 docs|19 tests

### Fourier Series (Part -1)

- Video | 19:06 min
### Fourier Series (Part -2)

- Video | 18:30 min
### Fourier Series (Part -3)

- Video | 17:11 min
### Fourier Series (Part -4)

- Video | 14:24 min
### Fourier Series (Part -5)

- Video | 15:55 min
### Fourier Series (Part -6)

- Video | 15:04 min

- Fourier Transform and its Properties
- Doc | 5 pages