Math Review with Matlab Fourier Analysis Fourier Transform

Math Review with Matlab: Fourier Analysis Fourier Transform S. Awad, Ph. D. D. Cinpinski E. C. E. Department University of Michigan-Dearborn

Fourier Analysis: Fourier Transform n Motivation For Fourier Transform Energy Signal Definition Fourier Transform Representation Example: FT Calculation Example: Pulse Inverse Fourier Transform Properties Example: Convolution Parseval’s Theorem Relation between X(s) and X(j ) n Example: Ramp Function n n n n 2

Fourier Analysis: Fourier Transform Motivation for Fourier Transform We need a method of representing aperiodic signals in the frequency domain. n The Fourier Series representation is only valid for periodic signals. n The Fourier Transform will accomplish this task for us. However, it is important to note that the Fourier Transform is only valid for Energy Signals. n 3

Fourier Analysis: Fourier Transform What is an Energy Signal ? A signal g(t) is called an Energy Signal if and only if it satisfies the following condition. n 4

Fourier Analysis: Fourier Transform Representation The Fourier Transform of an Energy Signal x(t) is found by using the following formula. n There is a one to one correspondence between a signal x(t) and its Fourier Transform. For this reason, we can denote the following relationship. n 5

Fourier Analysis: Fourier Transform Example: FT Calculation x(t) = e-atu(t) 1 a>0 t n Note: If a<0, then x(t) does not have a Fourier transform because: 6

Fourier Analysis: Fourier Transform complex function of 7

Fourier Analysis: Fourier Transform Magnitude Response Let us now find the Magnitude Response. The expression for the magnitude response of a fraction is calculated as follows. n 8

Fourier Analysis: Fourier Transform Magnitude Response Now calculate the Magnitude Response of X(j ) n 9

Fourier Analysis: Fourier Transform Magnitude Response n We can now plot the Magnitude Response. w (rad/sec) Even function of 10

Fourier Analysis: Fourier Transform Phase Response The expression for the phase response of a fraction is calculated as follows. n 11

Fourier Analysis: Fourier Transform Phase Response 12

Fourier Analysis: Fourier Transform Phase Response n We can now plot the Phase Response. w(rad/sec) Odd function of 13

Fourier Analysis: Fourier Transform Tables We could go ahead and find the Fourier Transform for any Energy Signal using the previous formula. However, Signals & Systems textbooks usually provide a table in which these have already been computed. Some are listed here. n FT FT 14

Fourier Analysis: Fourier Transform Example: Pulse n Find the Fourier Transform of: x(t) 1 -T 1 0 T 1 t 15

Fourier Analysis: Fourier Transform Example: Pulse 16

Fourier Analysis: Fourier Transform Magnitude Response n Note: X(j ) = 0, when So: 17

Fourier Analysis: Fourier Transform Magnitude Response n We can now plot the Magnitude Response. 18

Fourier Analysis: Fourier Transform Phase Response n We can now plot the Phase Response. 19

Fourier Analysis: Fourier Transform Inverse Fourier Transform Recall that there is a one to one correspondence between a signal x(t) and its Fourier Transform X(j ). n If we have the Fourier Transform X(j ) of a signal x(t), we would also like be able to find the original signal x(t). n 20

Fourier Analysis: Fourier Transform Inverse Fourier Transform Let X(j ) = FT{x(t)} = n n x(t) = FT-1{X(j )} = n FT-1 is the inverse Fourier Transform of X(j ) 21

Fourier Analysis: Fourier Transform Properties There are several useful properties associated with the Fourier Transform: n Linearity Property Time Scaling Property Duality Property Time Shifting Property Frequency Shifting Property n n Time Domain Differentiation Property n n Time Domain Integration Property n n n Symmetry Property Convolution Property n Multiplication by a Complex 22

Fourier Analysis: Fourier Transform Linearity Property n n Let: Then: 23

Fourier Analysis: Fourier Transform Time Scaling Property n Let: n Then: where a is a real constant 24

Fourier Analysis: Fourier Transform Duality Property n Le t: n Then : 25

Fourier Analysis: Fourier Transform Time Shifting Property n Le t: n Then : n Note: “a” can be positive or negative 26

Fourier Analysis: Fourier Transform Frequency Shifting Property n Le t: n Then : 27

Fourier Analysis: Fourier Transform Time Domain Differentiation Property n Le t: n Then : 28

Fourier Analysis: Fourier Transform Time Domain Integration Property where: 29

Fourier Analysis: Fourier Transform Symmetry Property n n If x(t) is a real-valued time function then conjugate symmetry exists: Example: 30

Fourier Analysis: Fourier Transform Convolution Property n Le t: n Then : Convolution 31

Fourier Analysis: Fourier Transform Example: Convolution Filter x(t) through the filter h(t) x(t) LTI System y(t) where h(t) is the impulse response Convolution 32

Fourier Analysis: Fourier Transform Example: Convolution n Knowin g n n We can write Note: 33

Fourier Analysis: Fourier Transform Multiplication by a Complex Exponential n Le t: n Then : 34

Fourier Analysis: Fourier Transform Sinusoid Examples (i) Amplitude Modulation 35

Fourier Analysis: Fourier Transform Sinusoid Examples (ii) Amplitude Modulation 36

Fourier Analysis: Fourier Transform Amplitude Modulation y(t)=x(t)cos(wot) x(t) X y(t) t cos(wot) FT 1 t FT-1 Y(jw) X(jw) w -wo wo w 37

Fourier Analysis: Fourier Transform Parseval’s Theorem n n Let x(t) be an energy signal which has a Fourier transform X(j ). The energy of this signal can be calculated in either the time or frequency domain: Time Domain Frequency Domain 38

Fourier Analysis: Fourier Transform Relation between X(s) and X(j ) n If X(j ) exists for x(t): assuming x(t) = 0 for all t < 0 n Example: Frequency Response n H(s) is known as the transfer function 39

Fourier Analysis: Fourier Transform Example: Ramp Function The Fourier Transform exists only if the region of convergence includes the j axis. n To prove this point, let us look at the Unit Ramp Function. The Ramp Function has a Laplace Transform, but not a Fourier Transform. n t 40

Fourier Analysis: Fourier Transform Example: Ramp Function n If we define the Step Function as: n The Unit Ramp Function can now be rewritten as t*u(t) n The Laplace Transform is and the corresponding Region of Convergence (ROC) is Re(s) > 0 Since the ROC does not include the j axis, this means that the Ramp Function does not have a Fourier Transform n 41