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Signals And Systems

Signals And Systems. Chapter 3 Fourier Transform. 3.7 Properties of Fourier Transform ( 3 ). Part I: Review. If. 1. Linearity. then. 2. Time Shifting. If. then. Especially,. 3. Time and Frequency Scaling. If. then.

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Signals And Systems

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  1. Signals And Systems

  2. Chapter 3Fourier Transform

  3. 3.7 Properties of Fourier Transform (3)

  4. PartI: Review

  5. If 1. Linearity then 2. Time Shifting If then

  6. Especially, 3. Time and Frequency Scaling If then

  7. Note!---Afrequency shiftis equivalent to the product of the time function and the exponential factor 4. Frequency Shifting If then

  8. PartII: Properties of the Fourier Transform

  9. 5. Differentiation in time-domain If then This is particularly important property, as it replaces the operation of differentiation in the time domain with that of multiplication by jw in the frequency domain. We will find the substitution to be extremely useful on the use of Fourier transform for the analysis of LTI systems described by differential equations.

  10. Example 1: determine the Fourier Transform of u'(t) because: so: Solution 2

  11. Example 2: determine the Fourier Transform of u"(t) because: so:

  12. 6. Differentiation in frequency-domain If then

  13. PartIII: Conclusion

  14. Summary • Frequency Shifting • Differentiation in time-domain • Differentiation in frequency-domain

  15. Exercise 1: Given that x(t) has the Fourier transform Using the properties of Fourier transform, but no integral operation to find: (1) (2) 4

  16. Exercise 2:3-28(1)3-29

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