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Chapter 4 The Fourier Transform EE 207 Dr. Adil Balghonaim
Let x p (t) be a periodical wave, then expanding the periodical function Rewriting x p (t) and X n
Fourier Transform Pairs
Finding the Fourier Transform
Example Find the Fourier Transform for the following function
It was shown previously
The Fourier Transform for the following function
Example Find the Fourier Transform for the delta function x(t) = (t)
1-Linearity Proof Properties of the Fourier Transform
LetThen Proof Change of variable 2-Time-Scaling (compressing or expanding)
Now Let Change of variable Since
Example Find the Fourier Transform of the pulse function Solution From previous Example
4-Time Transformation Proof
Step 1 from Known transform from the F.T Table Step 2
Multiplication in FrequencyConvolution in Time Proof 6- The convolution Theorem
Now substitute x 2 (t- ) ( as the inverse Fourier Transform) in the convolution integral
Exchanging the order of integration, we have
Proof Similar to the convolution theorem, left as an exercise The multiplication Theorem Applying the multiplication Theorem
Find the Fourier Transform of following Solution Since
System Analysis with Fourier Transform
Proof 6- Frequency Shifting
Example Find the Fourier Transform for
Find the Fourier Transform of the function
Since and Therefore Method 1
Using integration by parts
Since x(t) is absolutely integrable
Example Find the Fourier Transform of the unit step function u(t) 7- Integration
Find the Transfer Function for the following RC circuit we can find h(t) by solving differential equation as follows Method 1
We will find h(t) using Fourier Transform Method rather than solving differential equation as follows Method 2
From Table 4-2
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
Find y(t) if the input x(t) is Method 1 ( convolution method) Using the time domain ( convolution method, Chapter 3) Example
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
Example Find y(t) Method 1 ( convolution method)
Method 2 Fourier Transform
Ch 3 Analysis and Transmission of Signals
Transformations of Continuous-Time Signals Continuous time signal: Time is a continuous variable The signal itself need not be continuous. Time Reversal.
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Laplace Transforms 1. Standard notation in dynamics and control (shorthand notation) 2. Converts mathematics to algebraic operations 3. Advantageous for.
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Digital Control Systems The z-Transform. The z Transform Definition of z-Transform The z transform method is an operational method that is very powerful.
Chapter 3 1 Laplace Transforms 1. Standard notation in dynamics and control (shorthand notation) 2. Converts mathematics to algebraic operations 3. Advantageous.
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Leo Lam © Signals and Systems EE235 Lecture 31.
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