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Chapter 4 The Fourier Transform EE 207 Dr. Adil Balghonaim

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Let x p (t) be a periodical wave, then expanding the periodical function Rewriting x p (t) and X n

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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) = (t)

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1-Linearity Proof Properties of the Fourier Transform

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LetThen Proof Change of variable 2-Time-Scaling (compressing or expanding)

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Let

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Now Let Change of variable Since

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Proof 3-Time-Shifting

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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 Step 2

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Multiplication in FrequencyConvolution in Time Proof 6- The convolution Theorem

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Now substitute x 2 (t- ) ( as the inverse Fourier Transform) in the convolution integral

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Exchanging the order of integration, we have

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Proof Similar to the convolution theorem, left as an exercise The multiplication Theorem 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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Proof 6- Frequency Shifting

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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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Since and Therefore Method 1

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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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Example Find the Fourier Transform of the unit step function u(t) 7- Integration

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Proof

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Find the Transfer Function for the following RC circuit we can find h(t) by solving differential equation as follows Method 1

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We will find h(t) using Fourier Transform Method rather than solving differential equation as follows Method 2

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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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Fourier Transform

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Find y(t) if the input x(t) is Method 1 ( convolution method) Using the time domain ( convolution method, Chapter 3) Example

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Using partial fraction expansion (will be shown later) From Table 5-2 Method 2 Fourier Transform Sine Y( ) is not on the Fourier Transform Table 5-2

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Example Find y(t) Method 1 ( convolution method)

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Method 2 Fourier Transform

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