Fourier Transform FT EET 206 ELECTRIC CIRCUIT II

Fourier Transform (FT) EET 206 – ELECTRIC CIRCUIT II 1

CONTENTS • • • Introduction to Fourier transforms (FT) Comparison to Laplace T and Phasors Definition of Fourier Transform Fourier transforms of Basic Functions Properties of the Fourier Transforms Circuit applications using Fourier Transforms 2

Introduction To Fourier Transform Recall that: • Fourier Series enable us to represent a periodic function as a sum of sinusoids Now know that : • Fourier Transform (FT) is integral representation of a non periodic function Therefore: • In another words, FT is an integral transformation of f(t) from the time domain to the frequency domain 3

Comparison to Laplace Transform and Phasors • FT, Laplace T and phasors are all in frequency domain, where we can transform back to time domain form of the final solution by using inverse methods • Phasors is only applicable when input is sinusoidal periodic function, but FT can be used for non periodic and non sinusoidal function • Laplace only works for t > 0 with initial condition being stated, but FT can be used for t < 0 as well as for t > 0 4

Definition of Fourier Transforms 5

Inverse Fourier Transforms 6

FT of Basic Functions • We will show to obtain the FT of some basic functions using the (integral) definition of FT. • For more complex functions where the integral becomes tedious, we may temporarily replace jω by s and then replace back s with jω at the end. • The inverse transform may be obtained using partial fraction and the table as in Laplace T. 7

Example Obtain the Fourier transform of the “switchedon” exponential function as shown below 8

Solution: Given function is: 9

Its Fourier Transform is: 10

Example 11

Solution: 12

Example 13

Solution: 14

Example • Obtain the Fourier transform of the function in figure below: 15

Solution Euler’s formula 16

Example • Determine the Fourier transform of: g(t) = 4 u(t - 1) – 4 u(t - 2) • Solution: 17

Properties of Fourier Transforms 18

Properties of Fourier Transforms 19

Properties of Fourier Transforms • Multiplication by a constant 20

• Addition and subtraction 21

• Time Differentiation 22

Example • Find the Fourier transform of the function in Fig. below. 23

Solution • It is much easier to find the FT using the derivative property. • The first derivative: Now: f(t) = u(t+1) – u(t) – [u(t) – u(t-1)] = u(t+1) – u(t) + u(t-1)] = u(t+1) – 2 u(t) + u(t-1)] 24

Solution • The second derivative: f’’(t) = δ(t+1)-2 δ(t)+ δ(t-1) 25

Solution • Using the time differentiation property: (jω)2 F(ω) = ejω-2+e-jω = -2+2 cosω So, F(ω) = 2(1 -cosω) ω2 26

• Time Integration 27

• Time Scaling 28

• Time shifting Delay in the time domain corresponds to the phase shift in the frequency domain 29

• Frequency shifting Frequency shift in the frequency domain adds phase shift to the time domain 30

• Modulation 31

• Convolution in frequency domain: 32

• Convolution in time domain 33

Example on Finding Inverse FT: • Determine the inverse Fourier Transforms (FT) for the function below: 34

Solution To avoid complex algebra, we can replace jω with s for the moment. Using partial fraction 35

Solution 36

• A and B value: • Then replace back s = jω • Inverting the frequency domain to time domain, we get the inverse transform: 37