EE 207 Dr Adil Balghonaim Chapter 4 The

EE 207 Dr. Adil Balghonaim Chapter 4 The Fourier Transform

Let xp(t) be a periodical wave, then expanding the periodical function Rewriting xp(t) and Xn

Fourier Transform Pairs

Finding the Fourier Transform

Example Find the Fourier Transform for the following function

Example

It was shown previously

The Fourier Transform for the following function

Example Find the Fourier Transform for the delta function x(t) = d(t)

Properties of the Fourier Transform 1 -Linearity Proof

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

Let

Now Let Change of variable Since

3 -Time-Shifting Proof

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

6 - The convolution Theorem Convolution in Time Proof Multiplication in Frequency

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

Exchanging the order of integration , we have

The multiplication Theorem Proof Similar to the convolution theorem , left as an exercise Applying the multiplication Theorem

Find the Fourier Transform of following Solution Since

System Analysis with Fourier Transform

6 - Frequency Shifting Proof

Example Find the Fourier Transform for

Find the Fourier Transform of the function

Method 1 Since Therefore and

Method 2

7 -Differentiation

Using integration by parts

Since x(t) is absolutely integrable

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

Proof

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

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

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

Method 3

Method 3 Fourier Transform

Fourier Transform

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

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

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

Method 2 Fourier Transform