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Fourier Transform Table UBC M267 Resources for 2005 F(t) ?F(?) Notes (0) f(t) ? ? ?? f(t)e ?i?t dt Definition
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[PDF] EE2 Mathematics Solutions to Example Sheet 4: Fourier Transforms
EE2 Mathematics Solutions to Example Sheet 4: Fourier Transforms 1) Because f(t) = e?t = { e?t t > 0 et t < 0 } the Fourier transform of f(t) is
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F {f(t)}(s) = F(s) = / ? ?? f(t)e ?j2?st dt The inverse Fourier transform transforms a func- tion of frequency F(s) into a function of time f(t):
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e?i 2?nx L (3) Then using the mathematical identity The Fourier transform of a function of t gives a function of ? where ? is the angular frequency
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?? sinc ( ? 2?) 5 2 Some Fourier transform pairs The signal x(t) = e?btu(t) is absolutely integrable as
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EE2 Mathematics
Solutions to Example Sheet 4: Fourier Transforms
1)Becausef(t) =e-|t|=?e-t, t >0
e t, t <0? the Fourier transform off(t) is f(ω) =? e-iωt-|t|dt=? 0 et(1-iω)dt+? 0 e-t(1+iω)dt=21 +ω22)(i) DesignateF{f(t)}=
f(ω) withaa real constant of either sign. ThenF{f(at)}=?∞-∞e-iωtf(at)dt. Defineτ=atsodτ=adt. Whena >0 the limits (-∞,∞) forτcorrespond
to those fort, but whena <0 the direction reverses. ThusF{f(at)}=|a|-1?∞
e-i(ω a)τf(τ)dτ=|a|-1f?ωa? (ii) The 'shift property" (in the formula sheets)F{f(t-a)}=e-iωa f(ω) can simply be proved by definingτ=t-athenF{f(t-a)}=?
e-iωtf(t-a)dt=e-iωa?∞ e-iωτf(τ)dτ=e-iωa f(ω)3)To find the Fourier transform of the non-normalized Gaussianf(t) =e-t2we first complete
the square in the exponential f(ω) =? e-iωt-t2dt=e-14ω2?∞ e-(t+12iω)2dt=⎷πe-14ω2The normalized auto-correlation function ofe-t2is
γ(t) =?
-∞e-u2e-(t-u)2du -∞e-2u2du=e-12t2?∞
-∞e-2(u-12t)2du?∞ -∞e-2u2du=e-12t2 The integrals in the numerator & denominator cancel becausethey are equal; the origin of the former is shifted w.r.t. to the latter on the infiniteu-axis but its value is not affected.4)Withf(t) =e-at2andg(t) =e-bt2, a minor re-scaling of the results of Q3 shows that
f(ω) =?π ae-ω24ag(ω) =?π
be-ω2 4bThe convolution theorem says thatF??∞
-∞f(t?)g(t-t?)dt?? f(ω)g(ω). Therefore, with h(t) =? exp?-at?2?exp?-b(t-t?)2?dt?=f?g we have h(ω) =π⎷abe-ω2(a+b) 4ab. 15)Withf(t) =e-t2for whichf(ω) =⎷πe-14ω2andg(t) = cosatfor which
g(ω) =? e-iωtcosatdt=12? e-iωt(eiat+e-iat)dt 1 2? e-it(ω-a)+e-it(ω+a)? dt=π{δ(ω-a) +δ(ω+a)} Thus f(t)g(t)dt=π⎷2π?
e-14ω2{δ(ω-a) +δ(ω+a)}dω=⎷πe-14a2.
6)Writef(t) = (1 +t2)-1so that
-∞dt (1 +t2)2=? |f(t)|2dt=12π? |f(ω)|2dωHence we want to evaluate
f(ω)|2dω. To do this we must first findf(ω) =?∞ -∞e-iωtdt1+t2. To apply Jordan"s Lemma it is necessary to consider the two separate casesω <0 andω >0. (i)ω <0: Consider the complex integral? C Ue -iωzdz1+z2withCUa semi-circle in theupper12-plane
in which there is a simple pole atz=i. The residue at this pole iseω/2iand the integral over HR→0 asR→ ∞by Jordan"s Lemma.
R -Rω <0
?H+ R -→i•For the contourCUin the upper12-plane (ω <0):
πe C Ue -iωzdz1 +z2=?
R -Re -iωxdx1 +x2+? H Re -iωzdz1 +z2 (ii)ω >0: Consider the complex integral? C Le -iωzdz1+z2withCLa semi-circle in thelower12-plane
in which there is a simple pole atz=-i. The residue at this pole is-e-ω/2i& the integral overH-R→0 asR→ ∞by Jordan"s Lemma.
2R-R←-
?H- R-iω >0•For the contourCLin the lower1
2-plane (ω >0):
-πe-ω=? C Le -iωzdz1 +z2=?
-R Re -iωxdx1 +x2+? H Re -iωzdz1 +z2 Note the reverse order of the limits in the real integral.Thus, in the limitR→ ∞, we have
f(ω) =?πe-ω, ω >0 πeω, ω <0?
. Finally we can now calculate 12π?
|f(ω)|2dω=π22π? ?0 e2ωdω+? 0 e-2ωdω?12π.
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