1. Properties of Fourier Transforms

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

3. Proof: Substitute

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

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

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

7. 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:

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

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

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

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

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

13. 5. Multiplication by t Proof:

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

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

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

17. Example 8 Use tables to find the FT of Hence find the FT of Tables: So ifthen

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

19. Now look at Tutorial 2