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Published byAndrea Bishop Modified over 9 years ago
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EE 207 Dr. Adil Balghonaim Chapter 4 The Fourier Transform
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Let xp(t) be a periodical wave, then expanding the periodical function
Rewriting xp(t) and Xn
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Fourier Transform Pairs
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Fourier Transform Pairs
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Fourier Transform Pairs
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Finding the Fourier Transform
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Example Find the Fourier Transform for the following function
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Example
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It was shown previously
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The Fourier Transform for the following function
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Example Find the Fourier Transform for the delta function x(t) = d(t)
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Properties of the Fourier Transform
1-Linearity Proof
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2-Time-Scaling (compressing or expanding)
Let Then Proof Change of variable
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Let
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Now Let Change of variable Since
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3-Time-Shifting Proof
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Example Find the Fourier Transform of the pulse function
Solution From previous Example
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4-Time Transformation Proof
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5-Duality ازدواجية
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Step 1 from Known transform from the F.T Table
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6- The convolution Theorem
Multiplication in Frequency Convolution in Time Proof
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Now substitute x2(t-l) ( as the inverse Fourier Transform) in the convolution integral
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Exchanging the order of integration , we have
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The multiplication Theorem
Proof Similar to the convolution theorem , left as an exercise Applying the multiplication Theorem
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Find the Fourier Transform of following
Solution Since
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System Analysis with Fourier Transform
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6- Frequency Shifting Proof
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Example Find the Fourier Transform for
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Find the Fourier Transform of the function
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Method 1 Since and Therefore
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Method 2
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7-Differentiation
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Using integration by parts
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Since x(t) is absolutely integrable
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7- Integration Example Find the Fourier Transform of the unit step function u(t)
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Proof
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Find the Transfer Function for the following RC circuit
Method 1 we can find h(t) by solving differential equation as follows
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Method 2 We will find h(t) using Fourier Transform Method rather than solving differential equation as follows
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From Table 4-2
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Method 3 In this method we are going to transform the circuit to the Fourier domain . However we first see the FT on Basic elements
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Method 3
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Method 3 Fourier Transform
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Fourier Transform
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Example Find y(t) if the input x(t) is Method 1 ( convolution method) Using the time domain ( convolution method , Chapter 3)
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Method 2 Fourier Transform
Sine Y(w) is not on the Fourier Transform Table 5-2 Using partial fraction expansion (will be shown later) From Table 5-2
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Example
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Method Phasor method Voltage Division
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Method 2 Fourier Transform method
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Let x(t) be a periodical signal
were Fourier Transform
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Constant with respect to Fourier Transform
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Example Find y(t) Method 1 ( convolution method)
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Method 2 Fourier Transform
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