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

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

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Fourier Analysis:Fourier Transform 3 Motivation for Fourier Transform n The Fourier Series representation is only valid for periodic signals. n We need a method of representing aperiodic signals in the frequency domain. 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.

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

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Fourier Analysis:Fourier Transform 5 Fourier Transform Representation n 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.

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

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Fourier Analysis:Fourier Transform 7 complex function of

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

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

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

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

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Fourier Analysis:Fourier Transform 12 Phase Response

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

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Fourier Analysis:Fourier Transform 14 Fourier Transform Tables n 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. FT

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Fourier Analysis:Fourier Transform 15 Example: Pulse n Find the Fourier Transform of: x(t) t -T 1 T1T1 0 1

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Fourier Analysis:Fourier Transform 16 Example: Pulse

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

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

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

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Fourier Analysis:Fourier Transform 20 Inverse Fourier Transform n 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).

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

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

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

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

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Fourier Analysis:Fourier Transform 25 Duality Property n Let: n Then:

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

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Fourier Analysis:Fourier Transform 27 Frequency Shifting Property n Let: n Then:

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Fourier Analysis:Fourier Transform 28 Time Domain Differentiation Property n Let: n Then:

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

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

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Fourier Analysis:Fourier Transform 31 Convolution Property Convolution n Let: n Then:

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

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

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

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

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

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Fourier Analysis:Fourier Transform 37 Amplitude Modulation X x(t) cos( o t) y(t) t x(t) t y(t)=x(t)cos( o t) FT FT -1 Y(j ) -o-o oo X(j )

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

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Fourier Analysis:Fourier Transform 39 Relation between X(s) and X(j ) 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

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Fourier Analysis:Fourier Transform 40 Example: Ramp Function n 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. t

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

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