Properties of Fourier Series - GATE Study Material in PDF

Transcription

1 Properties of Fourier Series - GAE Study Material in PDF In the previous article, we learnt the Basics of Fourier Series, the different types and all about the different Fourier Series spectrums. Now, let us take the discussion further and learn about the Properties of Fourier Series. Using these Properties of Fourier Series, we can learn to manipulate Fourier Series, which is what we will learn in these GAE 018 notes. hese GAE Notes are useful for GAE EE, GAE EC, GAE ME, and GAE CE. hey are also useful for other exams like BARC, BSNL, IES, DRDO, ISRO etc. You can get these GAE study material downloaded in PDF so that your exam preparation is made easy and you ace your paper. Before you get started though, you should ensure that you are caught up with the basics of Engineering Mathematics. Laplace ransforms Limits, Continuity & Differentiability Mean Value heorems Differentiation Partial Differentiation Maxima and Minima Methods of Integration & Standard Integrals Vector Calculus 1 P a g e

2 Vector Integration ime Signals & Signal ransformation Standard ime Signals Signal Classification ypes of ime Systems Introduction to Linear ime Invariant Systems Properties of LI Systems Introduction to Fourier Series Properties of Continuous ime Fourier Series (CFS) he various properties of Fourier series have been listed explained below. Before going into them, let us get familiar with the representation convention. Let x(t) = and y(t) = c n n= D n n= e jnω 0t e jnω 0t hen x(t) CFS and y(t) CFS D n Here CFS stands for Continuous time Fourier series and cn and Dn are Fourier series coefficients. Now we will look into the properties of Fourier transform. he properties we will discuss are: 1. Linearity P a g e

3 . ime Shifting 3. Frequency Shifting 4. ime Scaling 5. ime Inversion 6. Differentiation in ime 7. Integration in ime 8. Convolution 9. Multiplication 10. Symmetry property Let us look at these properties in detail now. 1) Linearity If x(t) CFS c n and y(t) CFS D n hen, ax(t) + by(t) CFS ac n + bd n i.e. Fourier Series is a linear operation. ) ime Shifting If x(t) CFS hen according to time shifting property, x(t t 0 ) CFS e jnω 0t 0 c n x(t + t 0 ) CFS e jnω 0t 0 c n i.e. Magnitude of Fourier Series coefficients remains unchanged when the signal is shifted in time. 3) Frequency Shifting If x(t) CFS hen according to frequency shifting property, 3 P a g e

4 e jn 0ω 0 t x(t) CFS c (n n0 ) e jn 0ω 0 t x(t) CFS c (n+n0 ) 4) ime Scaling If x(t) is periodic with period then x(at) will be periodic with period /a ; a>0 If x(t) CFS hen x(at) CFS hus, after time scaling FS coefficients are the same. But, the spacing between the frequency components changes from ω 0 to aω 0 or from 1 to a 5) ime Inversion ime inversion property states that If x(t) CFS hen x( t) CFS c n 6) Differentiation in ime According to this property, if x(t) CFS hen d dt CFS x(t) (jnω 0 )c n = ( jπn ) c n 7) Integration If x(t) CFS hen x(t)dt CFS 1 c jnω n + c P a g e

5 8) Convolution If x(t) CFS hen c n and y(t) CFS D n x(t) y(t) CFS c n D n Hence, the convolution in time domain leads to multiplication of Fourier series coefficients in Fourier series domain. 9) Multiplication in ime Domain If signals are multiplied in the time domain, then the following phenomenon happens For x(t) CFS We have x(t)y(t) CFS. c n D n c n and y(t) CFS D n Multiplication in time domain leads to convolution in Fourier series domain. 10) Symmetry Symmetry properties state that If x(t) is real then c n = c n If x(t) is imaginary then c n = c n Example 1: Find the Fourier series of following signal - 5 P a g e

6 Solution: x 1 (t) = A ejnω 0t 0 A = 5, = 4, and ω 0 = π 4 = π x 1 (t) = 5 n= e jπ nt 4 Example : Find the Fourier series of following signal - Solution: x = x 1 (t 1) x 1 (t) CFS 5 4 x (t) = x 1 (t 1) CFS 5 4 e jnπ 1 = 5 4 e jnπ x (t) 5 = Σ n= 4 (e jnπ )e jnπt Example 3: Find the Fourier series of following signal - 6 P a g e

7 Solution: Recall, EFS of periodic train of pulses Aτ Here, A = 5, τ = 4, = 6 x 3 (t) = Aτ n= sinc ( nτ ) ejnω0t = x 3 (t) = 10 sinc 3 (n) 3 ejnπ 3 n= t sinc (nτ ) 5 4 n= sinc (n 4 ) e jnπ 6 t 6 6 Example 4: Solution: x 4 (t) = x 3 (t + 1) x 3 (t) CFS 10 3 sinc ( 3 n) x 4 (t) = x 3 (t + 1) CFS 10 sinc 3 ( n). ejnπ 6 1 [FSC] 3 = 10 3 sinc ( 3 n). ejnπ 3 x 4 (t) = 10 sinc 3 n= ( 3 7 P a g e n) ejnπ 3 t. e jnπ 3

8 x 4 (t) = 10 sinc 3 n= ( 3 n) ejnπ(t+1) 3 Note: Amplitudes of Fourier coefficients depend on the derivatives of the signal. 1. If the 1 st derivative of the signal tends to infinite (i.e. 1 st derivative of the signal is an impulse) then Fourier coefficients are proportional to 1/n.. If the nd derivative of the signal tends to infinite (i.e. nd derivative is an impulse) then Fourier coefficients are proportional to 1 n 3. If mth derivative is a impulse then Fourier coefficient are proportional to 1 n m Example 5: Find FS of the given periodic signals. Solution: A x(t) = {. t ; 0 < t < A (1 t ) ; < t < Since, x(t) is an even function herefore, bn = 0 8 P a g e

9 d x(t) dt i. e d x(t) dt = f(δ(t)) { Fourier coefficients 1 n a 0 = 1 x(t)d(t) = 1 A 0 = A [t] A t + [t ] 0 t dt + 1 a 0 = A 8 + A [( ) 1 ( 4 )] = A 4 + A [ ] a 0 = A 4 + A 8 = A a n = x(t) cos(nω 0t)dt = 4A A (1 t ) dt t cos(nω 0 t)dt + 4A cos(nω 0t)dt 4A 0 = A t cos(nω 0 0t)dt + A(1 t ) cos(nω 0 t)dt t cos(nω 0 t)dt = 4A [ 1 (nω 0 ) [cos(nω 0t) + nω 0 t sin(nω 0 t)]] 4A [ 1 (cos(nω (nω 0 ) 0t) + nω 0 t. sin(nω 0 t))] 0 + 4A sin(nω 0 t) nω 0 a n = 4A [ 1 (nω 0 ) {cos(nω 0 9 P a g e ) cos 0 + nω 0 4A [ 1 (nω 0 ) {cos(nω 0) cos(nω 0 ω 0 = π ) + nω 0 a n = 4A [ 1 {cos(nπ) 1 + nπ sin(nπ)} (nω 0 ) sin(nω 0 )}] 4A 1 {cos(nπ) (nω 0 ) cos(nπ) + nπ sin nπ nπ sin nπ} = 4A [ 1 4A (cosnπ 1)] [ 1 (1 cos nπ)] (nω 0 ) (nω 0 ) a n = ω 0 = π = 4A 4π n 4A (nω 0 ) ( cos nπ ) (cos nπ 1) sin(nω 0 ) nω 0 sin(nω 0 )}]

10 a n = A π n (cos(nπ) 1) = { ( πn ) A ; for odd value of n 0 ; for even value of n A x(t) = 1 + n=1 (cos(nπ) 1)(cos nω 0 t) π n π We will continue with the Symmetry Conditions ins Fourier series in the next article. Did you like this article on Properties of Fourier Series? Let us know in the comments. You may also enjoy Symmetry Conditions in Fourier Series Fourier ransform Properties of Fourier ransform 10 P a g e