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Several of the most important properties of the Fourier transform are derived.

Properties of the fourier transform

The Fourier Transform (FT) has several important properties which will be useful:

  1. Linearity:
    α x 1 ( t ) + β x 2 ( t ) α X 1 ( j Ω ) + β X 2 ( j Ω )
    where α and β are constants. This property is easy to verify by plugging the left side of [link] into the definition of the FT.
  2. Time shift:
    x ( t - τ ) e - j Ω τ X ( j Ω )
    To derive this property we simply take the FT of x ( t - τ )
    - x ( t - τ ) e - j Ω t d t
    using the variable substitution γ = t - τ leads to
    t = γ + τ
    and
    d γ = d t
    We also note that if t = ± then τ = ± . Substituting [link] , [link] , and the limits of integration into [link] gives
    - x ( γ ) e - j Ω ( γ + τ ) d γ = e - j Ω τ - x ( γ ) e - j Ω γ d γ = e - j Ω τ X ( j Ω )
    which is the desired result.
  3. Frequency shift:
    x ( t ) e j Ω 0 t X ( j ( Ω - Ω 0 ) )
    Deriving the frequency shift property is a bit easier than the time shift property. Again, using the definition of FT we get:
    - x ( t ) e j Ω 0 t e - j Ω t d t = - x ( t ) e - j ( Ω - Ω 0 ) t d t = X ( j ( Ω - Ω 0 ) )
  4. Time reversal :
    x ( - t ) X ( - j Ω )
    To derive this property, we again begin with the definition of FT:
    - x ( - t ) e - j Ω t d t
    and make the substitution γ = - t . We observe that d t = - d γ and that if the limits of integration for t are ± , then the limits of integration for γ are γ . Making these substitutions into [link] gives
    - - x ( γ ) e j Ω γ d γ = - x ( γ ) e j Ω γ d γ = X ( - j Ω )
    Note that if x ( t ) is real, then X ( - j Ω ) = X ( j Ω ) * .
  5. Time scaling: Suppose we have y ( t ) = x ( a t ) , a > 0 . We have
    Y ( j Ω ) = - x ( a t ) e - j Ω t d t
    Using the substitution γ = a t leads to
    Y ( j Ω ) = 1 a - x ( γ ) e - j Ω γ / a d γ = 1 a X Ω a
  6. Convolution: The convolution integral is given by
    y ( t ) = - x ( τ ) h ( t - τ ) d τ
    The convolution property is given by
    Y ( j Ω ) X ( j Ω ) H ( j Ω )
    To derive this important property, we again use the FT definition:
    Y ( j Ω ) = - y ( t ) e - j Ω t d t = - - x ( τ ) h ( t - τ ) e - j Ω t d τ d t = - x ( τ ) - h ( t - τ ) e - j Ω t d t d τ
    Using the time shift property, the quantity in the brackets is e - j Ω τ H ( j Ω ) , giving
    Y ( j Ω ) = - x ( τ ) e - j Ω τ H ( j Ω ) d τ = H ( j Ω ) - x ( τ ) e - j Ω τ d τ = H ( j Ω ) X ( j Ω )
    Therefore, convolution in the time domain corresponds to multiplication in the frequency domain.
  7. Multiplication (Modulation):
    w ( t ) = x ( t ) y ( t ) 1 2 π - X ( j ( Ω - Θ ) ) Y ( j Θ ) d Θ
    Notice that multiplication in the time domain corresponds to convolution in the frequency domain. This property can be understood by applying the inverse Fourier Transform [link] to the right side of [link]
    w ( t ) = 1 2 π - 1 2 π - X ( j ( Ω - Θ ) ) Y ( j Θ ) e j Ω t d Θ d Ω = 1 2 π - Y ( j Θ ) 1 2 π - X ( j ( Ω - Θ ) ) e j Ω t d Ω d Θ
    The quantity inside the brackets is the inverse Fourier Transform of a frequency shifted Fourier Transform,
    w ( t ) = 1 2 π - Y ( j Θ ) x ( t ) e j Θ t d Θ = x ( t ) 1 2 π - Y ( j Θ ) e j Θ t d Θ = x ( t ) y ( t )
  8. Duality: The duality property allows us to find the Fourier transform of time-domain signals whose functional forms correspond to known Fourier transforms, X ( j t ) . To derive the property, we start with the inverse Fourier transform:
    x ( t ) = 1 2 π - X ( j Ω ) e j Ω t d Ω
    Changing the sign of t and rearranging,
    2 π x ( - t ) = - X ( j Ω ) e - j Ω t d Ω
    Now if we swap the t and the Ω in [link] , we arrive at the desired result
    2 π x ( - Ω ) = - X ( j t ) e - j Ω t d t
    The right-hand side of [link] is recognized as the FT of X ( j t ) , so we have
    X ( j t ) 2 π x ( - Ω )

The properties associated with the Fourier Transform are summarized in [link] .

Fourier Transform properties.
Property y ( t ) Y ( j Ω )
Linearity α x 1 ( t ) + β x 2 ( t ) α X 1 ( j Ω ) + β X 2 ( j Ω )
Time Shift x ( t - τ ) X ( j Ω ) e - j Ω τ
Frequency Shift x ( t ) e j Ω 0 t X ( j ( Ω - Ω 0 ) )
Time Reversal x ( - t ) X ( - j Ω )
Time Scaling x ( a t ) 1 a X Ω a
Convolution x ( t ) * h ( t ) X ( j Ω ) H ( j Ω )
Modulation x ( t ) w ( t ) 1 2 π - X ( j ( Ω - Θ ) ) W ( j Θ ) d Θ
Duality X ( j t ) 2 π x ( - Ω )

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Source:  OpenStax, Signals, systems, and society. OpenStax CNX. Oct 07, 2012 Download for free at http://cnx.org/content/col10965/1.15
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