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Properties of Fourier Transforms

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.

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Properties of Fourier Transforms

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  1. Properties of Fourier Transforms

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

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

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

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

  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 if then

  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

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