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B.2 Fourier transformation

As discussed above, the Fourier transformation of a function h(t) is given by

∫ ∞ H (f ) = h(t)ei2πftdt. −∞
(102)

Once the Fourier transformation H(f) is known, the original function h(t) can be reconstructed via

∫ ∞ −i2πft h (t) = H (f)e df. −∞
(103)

[Note that the signs in the exponential of Eq. (100) and (101) are opposite. Which one should be minus or positive is actually a matter of convention because a trivial variable substitution f′ = −f can change the sign between minus and positive. Proof. In terms of f′, Eq.  (103) is written

∫ h(t) = −∞ H (− f ′)ei2πtf′d(− f′), ∞ ∫ ∞ ′ i2πtf′ ′ = − ∞ H(− f)e df, (104)
Define
H(f′) ≡ H(− f′) ∫ ∞ ′ = h(t)e−i2πtf dt. (105) −∞
Then Eq. (104) is written
∫ ∞ -- ′ i2πtf′ ′ h (t) = −∞ H (f )e df
(106)

The signs in the exponential of Eqs. (105) and (106) are opposite to Eqs. (102) and (103), respectively.]

[Some physicists prefer to use the angular frequency ω ≡ 2πf rather than the frequency f to represent the Fourier transformation. Using ω, equations. (102) and (103) are written, respectively, as

∫ H(ω) = ∞ h(t)eiωtdt, −∞
(107)

∫ h (t) =-1- ∞ H (ω)e−iωtdω, 2π −∞
(108)

where we see that the asymmetry between the Fourier transformation and its inverse is more severe in this representation: besides the opposite-sign in the exponents, there is also a 1∕2π factor difference between the Fourier transformation and its inverse. Whether the 1∕2π factor appears at the forward transformation or inverse one is actually a matter of convention. The only requirement is that the product of the two factors in the forward and inverse transformation is equal to 1∕2π. To obtain a more symmetric pair, one can adopt a factor 1∕√2-π- at both the forward and inverse transformation. The representation in Eqs. (102) and (103) is adopted in this note. But we should know how to change to the ω representation when needed.]

[

∫ ∫ ∫ ∫ ∞ dh(t)e−i2πtfdt = h(t)e−i2πtf|+ ∞− ∞ h(t)de−i2πtf dt = − ∞ h(t)de−i2πtfdt = i2πf ∞ h(t)e−i2πtfdt = i2πfH (f) −∞ dt − ∞ −∞ dt −∞ dt −∞
(109)

]

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