Fourier Transform Properties and Applications: Delay, Modulation, Scaling, Differentiation, Multiplication, Symmetry

Properties of Fourier Transforms

For any function f(t), obtain the graph of by translating b units to the right: 1. The Delay Property b Fourier Transform:

Proof: Substitute

Example 1 Use tables to find the Fourier Transform of and hence find the Fourier Transform of From tables So Using gives

2. Modulation Proof: an exercise for you to do!

Example 2 Use tables to find the FT of and hence find the FT of From tables: So Now use

3. The Scaling Property The graph of is half the width of the graph of The graph of is the width of the graph of The height of the graph? Unchanged Fourier Transform:

Example 3 Use tables to find the Fourier Transform of and hence find the Fourier Transform of From tables: Using

Note: These properties can sometimes be combined, for example… Delay Scaling

Example 4 If determine Hence find the FT of Eg 2 with Delay Scaling

4. Time Differentiation Proof: Another exercise for you to do! This property can be extended for higher derivatives:

Example 5 Use tables to find the FT of Hence find the Fourier Transform of Tables: If then Use Hence and so

5. Multiplication by t Proof:

Example 6 Use tables to find the FT of Hence find the FT of Tables: Use:

Example 7 Use tables to find the FT of Hence find the FT of Eg. 4 

6. The Symmetry Property Proof: another exercise for you to do!

Example 8 Use tables to find the FT of Hence find the FT of Tables: So if then

Example 9 Use tables to find the FT of Hence find the FT of Eg 4  So if then

Now look at Tutorial 2

Fourier Transform Properties and Applications: Delay, Modulation, Scaling, Differentiation, Multiplication, Symmetry

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