Symmetry Conditions in Fourier Series - GATE Study Material in PDF

In the previous GATE study material learnt the basics and properties of Fourier Series. In these GATE 2018 Notes, we will learn about the Symmetry Conditions in Fourier Series‼ These study material covers everything that is necessary for GATE EC, GATE EE, GATE ME, GATE CE as well as other exams like ISRO, IES, BARC, BSNL, DRDO etc. These notes can also be downloaded in PDF so that your exam preparation is made easy and you ace your exam.

You should probably go through the basics covered in previous articles, before starting off with this module.

Recommended Reading –

Laplace Transforms Limits, Continuity & Differentiability Mean Value Theorems Differentiation Partial Differentiation Maxima and Minima Methods of Integration & Standard Integrals Vector Calculus Vector Integration Time Signals & Signal Transformation Standard Time Signals Signal Classification Types of Time Systems 1 | P a g e

Introduction to Linear Time Invariant Systems Properties of LTI Systems Introduction to Fourier Series Properties of Fourier Series

Applying of symmetry conditions reduces the complexity in finding Fourier series or Fourier series coefficients. If we have the knowledge about the symmetric condition of the given signal, then we can directly calculate some coefficients.

Types of Symmetries in Continuous Time Signals: 1. Even Symmetry 2. Odd Symmetry 3. Half wave symmetry / Rotational symmetry

We can now see how it is applicable to Trigonometric and Exponential Fourier Series.

Symmetry Conditions in Trigonometric Fourier Series Even Symmetry If x(t) is an even periodic signal, then 1 2 a = ∫ x(t)dt = ∫ x(t)dt 0 T T T T ⁄2 4 2 a = ∫ x(t) cos(nω t)dt 푂푅 a = ∫ x(t) cos(nω t)dt n T 0 n T 0 T T ⁄2 & bn = 0 This is because cos(nω0t) is an even signal while sin(mω0t) is odd signal. We know that- Even × Odd = Odd and (odd signal)dt = 0 ∫T Therefore, for every even signal bn = 0. Hence, Fourier Series of an even signal contains DC term and cosine terms only.

Odd Symmetry

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If x(t) is an odd function then 1 1 a = ∫ x(t)dt = ∫(odd signal) dt = 0 0 T T T T 2 a = ∫ x(t) cos(nω t)dt = 0 (∵ ∫ odd = 0) n T 0 T T

2 b = ∫ x(t). sin(nω t)dt n T 0 T We know that, odd × odd = even 4 Hence, b = ∫ x(t) sin(nω t)dt n T 0 T ⁄2 as ∫ even = 2 ∫T even T ⁄2 Therefore, for every odd signal a0 = 0 and an = 0. Hence, Fourier series contains only sine terms.

Half-Wave Symmetry Now, a signal x(t) is half wave symmetric if T x(t) = – x (t ± ) 2 Where T is the fundamental period of the signal.

Half wave symmetry is defined only for periodic signals. For instance,

A half symmetric periodic signal waveform

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Previous signal after half of time period

Example 1: x(t) = A sin(ωot). Check the symmetry of this signal.

Waveform with half wave symmetry Solution:

Half symmetric waveform after half time period

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T π x(t ± ⁄ ) = A sin (ω0 (t ± )) 2 ω0

= A sin(ω0t ± π) = −A sin ω0t So this signal is half wave symmetric.

If x(t) is half wave symmetric then its Fourier series contains odd harmonics only. Odd harmonics are – ω0, 3ω0, 5ω0, ……

Example 2: Find TFS for the following signal

Solution: A ; − T⁄ < t < T⁄ x(t) = { 4 4 T T T T −A ; − ⁄2 < t < − ⁄4 and ⁄4 < t < ⁄2

∞ x(t) = a0 + ∑n=1(an cos(nω0t) + bn sin(nω0t)) Since, x(t) is even symmetric. Therefore, bn = 0 Since, x(t) is symmetric about horizontal axis. Therefore, average of x(t) is zero. Hence, DC term a0 = 0 Therefore, ∞ x(t) = ∑n=1 an cos(nω0t) 2 4 an = ∫ x(t)cos (nω0t)dt = ∫T x(t)cos (nω0t)dt T T T ⁄2 T 4 T⁄ T⁄ 4A sin(nω t) ⁄4 = { 4 A cos(nω t)dt + 2 −A cos nω t dt} = {+ 0 | + ∫0 0 ∫T 0 T ⁄4 T nω0t 0 T sin(nω t) ⁄2 (− 0 | )} nω0t T ⁄4 T T 4A ⁄4 ⁄2 4A nω0T T an = {sin (nω0t|0 − sin(nω0t)|T⁄ )} = {sin ( ) − sin(nω0 ⁄ ) + nω0T 4 nω0T 4 2 T sin(nω0 ⁄4)} 5 | P a g e

2π Since, T = ω0 ⇒ ω0T = 2π 4A 2πn 2πn 2πn a = {sin ( ) − sin ( ) + sin ( )} n 2πn 4 2 4 2A πn a = [2 sin ( ) − sin(πn)] n nπ 2 sin(πn) = 0 2A πn a = .2 sin ( ) n πn 2 4A πn a = sin ( ) n πn 2 ∞ x(t) = ∑n=1 an cos(nω0t) 4A nπ ⇒ x(t) = ∑∞ sin ( ) cos(nω t) n=1 nπ 2 0 4A π 4A 4A 3π x(t) = sin ( ) cos(ω t) + sin(π) cos(2ω t) + sin ( ) cos(3ω t) π 2 0 2π 0 3π 2 0 4A 1 1 x(t) = {cos ω t − cos 3ω t + cos 5ω t −}……… π 0 3 0 5 0

Hidden Symmetry There is also a phenomenon called hidden symmetry. The symmetry which is hidden by the DC component is known as hidden symmetry. This symmetry can be obtained by removing the offset (adding or subtracting the DC).

If x(t) has some hidden symmetry, then its Fourier series contains DC and sine or DC and cosine terms depending upon the symmetry. i.e. for hidden odd symmetry the Fourier Series will contain DC and sine terms. For hidden even symmetry the series will be having DC and cosine terms.

For instance,

Hidden DC signal in a periodic square pulse

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Original periodic square pulse

Symmetry Conditions in Exponential Fourier Series Even Symmetry

If x(t) is an even function, then bn = 0 then a a c = a , c = n and c = n 0 0 n 2 −n 2

i.e. cn = c-n

⇒ cn are real If x(t) is even signal, then EFS coefficients are real and even. An even periodic signal contains only DC (co ≠ 0) and cosine terms, so phase angle will be either zero degrees or ±180∘

Odd Symmetry

If x(t) is an odd signal, then a0 = 0 and an = 0 ⇒ c0 = 0

−jb jb c = n , c = n n 2 −n 2

i.e. cn = -c-n

⇒ cn are imaginary

If x(t) is odd, then its EFS coefficients are imaginary and odd. In terms of frequency spectrum co is always zero and phase angle is only ±π/2.

Half-Wave Symmetry

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In terms of Fourier spectrum, for half wave symmetric signal, cn = 0 ∀ n even. In other word, the spectrum of a half wave symmetric signal contains only odd harmonics. Half wave symmetry is defined only for periodic signals. Even and Half Wave Symmetry For even and half wave symmetric signals, T x(t) = x (t ± ). 2 For an even and half wave symmetric signal, the spectrum satisfies the following conditions:

cn = 0 ∀ n even

and ∠cn = 0∘ or 180∘ Odd and Half Wave Symmetry For odd and half wave symmetric signals, T x(t) = − x (t ± ). 2 For an odd and half wave symmetric signal, the spectrum satisfies the following conditions:

cn = 0 ∀ n even, ∘ and ∠cn = ±90 .

We will continue with the Fourier Transforms in the next article. Did you like this article on Symmetry Conditions in Fourier Series? Let us know in the comments. You may also enjoy –

Fourier Transform

Properties of Fourier Transform

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