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Properties of the Fourier Transform

Module by: Carlos E. Davila. E-mail the author

Summary: 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:
    αx1(t)+βx2(t)αX1(jΩ)+βX2(jΩ)αx1(t)+βx2(t)αX1(jΩ)+βX2(jΩ)
    (1)
    where αα and ββ are constants. This property is easy to verify by plugging the left side of Equation 1 into the definition of the FT.
  2. Time shift:
    x(t-τ)e-jΩτX(jΩ)x(t-τ)e-jΩτX(jΩ)
    (2)
    To derive this property we simply take the FT of x(t-τ)x(t-τ)
    -x(t-τ)e-jΩtdt-x(t-τ)e-jΩtdt
    (3)
    using the variable substitution γ=t-τγ=t-τ leads to
    t=γ+τt=γ+τ
    (4)
    and
    dγ=dtdγ=dt
    (5)
    We also note that if t=±t=± then τ=±τ=±. Substituting Equation 4, Equation 5, and the limits of integration into Equation 3 gives
    -x(γ)e-jΩ(γ+τ)dγ=e-jΩτ-x(γ)e-jΩγdγ=e-jΩτX(jΩ)-x(γ)e-jΩ(γ+τ)dγ=e-jΩτ-x(γ)e-jΩγdγ=e-jΩτX(jΩ)
    (6)
    which is the desired result.
  3. Frequency shift:
    x(t)ejΩ0tX(j(Ω-Ω0))x(t)ejΩ0tX(j(Ω-Ω0))
    (7)
    Deriving the frequency shift property is a bit easier than the time shift property. Again, using the definition of FT we get:
    -x(t)ejΩ0te-jΩtdt=-x(t)e-j(Ω-Ω0)tdt=X(j(Ω-Ω0))-x(t)ejΩ0te-jΩtdt=-x(t)e-j(Ω-Ω0)tdt=X(j(Ω-Ω0))
    (8)
  4. Time reversal:
    x(-t)X(-jΩ)x(-t)X(-jΩ)
    (9)
    To derive this property, we again begin with the definition of FT:
    -x(-t)e-jΩtdt-x(-t)e-jΩtdt
    (10)
    and make the substitution γ=-tγ=-t. We observe that dt=-dγdt=-dγ and that if the limits of integration for tt are ±±, then the limits of integration for γγ are γγ. Making these substitutions into Equation 10 gives
    --x(γ)ejΩγdγ=-x(γ)ejΩγdγ=X(-jΩ)--x(γ)ejΩγdγ=-x(γ)ejΩγdγ=X(-jΩ)
    (11)
    Note that if x(t)x(t) is real, then X(-jΩ)=X(jΩ)*X(-jΩ)=X(jΩ)*.
  5. Time scaling: Suppose we have y(t)=x(at),a>0y(t)=x(at),a>0. We have
    Y(jΩ)=-x(at)e-jΩtdtY(jΩ)=-x(at)e-jΩtdt
    (12)
    Using the substitution γ=atγ=at leads to
    Y(jΩ)=1a-x(γ)e-jΩγ/adγ=1aXΩaY(jΩ)=1a-x(γ)e-jΩγ/adγ=1aXΩa
    (13)
  6. Convolution: The convolution integral is given by
    y(t)=-x(τ)h(t-τ)dτy(t)=-x(τ)h(t-τ)dτ
    (14)
    The convolution property is given by
    Y(jΩ)X(jΩ)H(jΩ)Y(jΩ)X(jΩ)H(jΩ)
    (15)
    To derive this important property, we again use the FT definition:
    Y(jΩ)=-y(t)e-jΩtdt=--x(τ)h(t-τ)e-jΩtdτdt=-x(τ)-h(t-τ)e-jΩtdtdτY(jΩ)=-y(t)e-jΩtdt=--x(τ)h(t-τ)e-jΩtdτdt=-x(τ)-h(t-τ)e-jΩtdtdτ
    (16)
    Using the time shift property, the quantity in the brackets is e-jΩτH(jΩ)e-jΩτH(jΩ), giving
    Y(jΩ)=-x(τ)e-jΩτH(jΩ)dτ=H(jΩ)-x(τ)e-jΩτdτ=H(jΩ)X(jΩ)Y(jΩ)=-x(τ)e-jΩτH(jΩ)dτ=H(jΩ)-x(τ)e-jΩτdτ=H(jΩ)X(jΩ)
    (17)
    Therefore, convolution in the time domain corresponds to multiplication in the frequency domain.
  7. Multiplication (Modulation):
    w(t)=x(t)y(t)12π-X(j(Ω-Θ))Y(jΘ)dΘw(t)=x(t)y(t)12π-X(j(Ω-Θ))Y(jΘ)dΘ
    (18)
    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 (Reference) to the right side of Equation 18
    w(t)=12π-12π-X(j(Ω-Θ))Y(jΘ)ejΩtdΘdΩ=12π-Y(jΘ)12π-X(j(Ω-Θ))ejΩtdΩdΘw(t)=12π-12π-X(j(Ω-Θ))Y(jΘ)ejΩtdΘdΩ=12π-Y(jΘ)12π-X(j(Ω-Θ))ejΩtdΩdΘ
    (19)
    The quantity inside the brackets is the inverse Fourier Transform of a frequency shifted Fourier Transform,
    w(t)=12π-Y(jΘ)x(t)ejΘtdΘ=x(t)12π-Y(jΘ)ejΘtdΘ=x(t)y(t)w(t)=12π-Y(jΘ)x(t)ejΘtdΘ=x(t)12π-Y(jΘ)ejΘtdΘ=x(t)y(t)
    (20)
  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(jt)X(jt). To derive the property, we start with the inverse Fourier transform:
    x(t)=12π-X(jΩ)ejΩtdΩx(t)=12π-X(jΩ)ejΩtdΩ
    (21)
    Changing the sign of tt and rearranging,
    2πx(-t)=-X(jΩ)e-jΩtdΩ2πx(-t)=-X(jΩ)e-jΩtdΩ
    (22)
    Now if we swap the tt and the ΩΩ in Equation 22, we arrive at the desired result
    2πx(-Ω)=-X(jt)e-jΩtdt2πx(-Ω)=-X(jt)e-jΩtdt
    (23)
    The right-hand side of Equation 23 is recognized as the FT of X(jt)X(jt), so we have
    X(jt)2πx(-Ω)X(jt)2πx(-Ω)
    (24)

The properties associated with the Fourier Transform are summarized in Table 1.

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

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