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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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If 1. Linearity then 2. Time Shifting If then
Especially, 3. Time and Frequency Scaling If then
Note!---Afrequency shiftis equivalent to the product of the time function and the exponential factor 4. Frequency Shifting If then
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.
Example 1: determine the Fourier Transform of u'(t) because: so: Solution 2
Example 2: determine the Fourier Transform of u"(t) because: so:
Summary • Frequency Shifting • Differentiation in time-domain • Differentiation in frequency-domain
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