Table of Fourier Transform Pairs of Energy Signals Function Time Domain x(t) Frequency Domain X(ω) name ∞ FT x(t) X (ω)==∫ xt()e− jtω dt F {}xt() −∞ 1 ∞ IFT xt()==∫ X()ω e jtω dωωF −1 {}X() X ()ω 2π −∞ T Rectangle tt1 t ≤ T rect =∏ ≡ 2 T sinc ω Pulse TT0 elsewhen 2π t Triangle t 1−≤tW 2 W Λ≡ W W sinc ω Pulse W 2π 0 elsewhen Sinc sin(π ⋅Wt) 1 ω sinc()Wt ≡ rect Pulse π ⋅Wt WW2π Exponen- 2a ea−at > 0 tial Pulse a22+ω Gaussian t 2 σ 22ω exp(− ) σπ2 exp(− ) Pulse 2σ 2 () 2 Decaying 1 Exponen- exp(−>at)u(t) Re{a} 0 aj+ ω tial 2 Sinc 2 1 ω sinc (Bt) Λ Pulse B 2π B
Rect Pulse Gaussian Pulse
1.5 1.5
T=1 σ2=1
1 1 ) a ( ect r 0.5 0.5
0 0 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 a a
Sinc Pulse Triangle Pulse
1.5 1.5 W=1 W=1 1 1 ) a (
c 0.5 n
Si 0.5
0 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 0 -0.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 a a
BAF Fall 2002 Page 1 of 4 Table of Fourier Transform Pairs of Power Signals Function Time Domain x(t) Frequency Domain X(ω) name
∞ FT x(t) X (ω)==∫ xt()e− jtω dt F {}xt() −∞ 1 ∞ IFT xt()==∫ X()ω e jtω dωωF −1 {}X() X ()ω 2π −∞
Impulse δ (t) 1
DC 1 2(πδω)
jjθθ− Cosine cos()ω0t +θ πδee()ω−+ω00δ(ω+ω)
jjθθ− Sine sin ()ω0t +θ −−jeπδ ()ωω00−eδ(ω+ω)
Complex exp()jω t 2(πδω−ω) Exponential 0 0 10t ≥ 1 Unit step ut()≡ πδ ()ω + 00t < jω 10t ≥ 2 Signum sgn(t) ≡ −10t < jω
Linear 1 − jπ sgn(ω) Decay t ∞ 22π ∞ π Impulse δ ()tn− T δω− k Train ∑ s ∑ n=−∞ TTssk =−∞ ∞ aejkω0t , where ∑ k ∞ Fourier k=−∞ 2π akδω()− ω Series 1 ∑ k 0 ax= ()te− jkω0t dt k=−∞ k T ∫ 0 T0
BAF Fall 2002 Page 2 of 4 Table of Fourier Transforms of Operations
FT Property Operation Given gt( ) ⇔ G(ω)
Linearity af (t) +⇔bg (t) aF (ω)(+bG ω)
− jtω 0 Time Shifting gt( −⇔t0 ) e G(ω)
1 ω Time Scaling ga()t ⇔ G aa 1 Modulation (1) gt()cos(ωωt)⇔−G()ω+G(ω+ω) 002 0
jtω0 Modulation (2) gt( )e ⇔−G(ω ω0 )
dg(t) Differentiation If ft()= , then Fj()ω =⋅ωωG() dt t 1 Integration If f ()tg= ∫ (α )dα , then FG()ω =+()ωπG(0)δ()ω −∞ jω gt( )∗⇔f(t) G(ω)⋅F(ω) , where Convolution ∞ gt()∗≡f()t ∫ g(α )f(t−αα)d −∞ 1 Multiplication ft()⋅⇔g(t) F()ω ∗G(ω) 2π
Duality If gt( ) ⇔ z(ω), then zt( )(⇔ 2π g−ω)
If g(t) is real valued then G-( ω)(= G* ω) Hermitian Symmetry ( G-( ω) = G (ω) and ∠G-( ω)(=−∠G ω))
Conjugation gt**( ) ⇔ G(−ω)
∞∞ 221 Pg==tdt Gω dω Parseval’s Theorem avg ∫∫() () −∞ 2π −∞
BAF Fall 2002 Page 3 of 4 Some Notes:
1. There are two similar functions used to describe the functional form sin(x)/x. One is the sinc() function, and the other is the Sa() function. We will only use the sinc() notation in class. Note the role of π in the sinc() definition: sin (π x) sin(x) sinc()x ≡≡;(Sa x) π x x
2. The impulse function, aka delta function, is defined by the following three relationships:
a. Singularity: δ (t − t0 )= 0 for all t ≠ t0 ∞ b. Unity area: ∫δ (t)dt =1 −∞
tb c. Sifting property: f (t)δ (t − t )dt = f (t ) for t < t < t . ∫ 0 0 a 0 b ta
3. Many basic functions do not change under a reversal operation. Other change signs. Use this to help simplify your results. 1 a. δδ()t=(−t) (in general, δδ()at ⇔ (t)) a b. rect (t)(=−rect t) c. Λ=()ttΛ(−) d. sinc(tt)(=−sinc ) e. sgn ()tt=−sgn (− )
4. The duality property is quite useful but sometimes a bit hard to understand. Suppose a known FT pair gt( ) ⇔ z(ω) is available in a table. Suppose a new time function z(t) is formed with the same shape as the spectrum z(ω) (i.e. the function z(t) in the time domain is the same as z(ω) in the frequency domain). Then the FT of z(t) will be found to be zt()⇔2π g(−ω) , which says that the F.T. of z(t) is the same shape as g(t), with a multiplier of 2π and with –ω substituted for t.
An example is helpful. Given the F.T. pair sgn(t) ⇔ 2 jω , what is the Fourier transform of x(t)=1/t? First, modify the given pair to jt2sgn( ) ⇔ 1 ω by multiplying both sides by j/2. Then, use the duality function to show that 1 tj⇔−2π 2sgn ( ωπ) =jsgn (−ω)(=−jπsgn ω).
BAF Fall 2002 Page 4 of 4