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

Example

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)

Let

Now Let Change of variable Since

Proof 3-Time-Shifting

Example Find the Fourier Transform of the pulse function Solution From previous Example

4-Time Transformation Proof

5-Duality ازدواجية

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

Method 2

7-Differentiation

Using integration by parts

Since x(t) is absolutely integrable

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

Proof

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

Method 3

Fourier Transform

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