Use the table of Fourier transforms (Table) and the table of properties (Table) to find th...

Time Domain Signal

Fourier Transform

f(t)

F(ω)

δ(t)

1

Aδ(t – t0)

u(t)

1

2πδ(ω)

K

2πKδ(ω)

sgn (t)

2πδ(ω – ω0)

cos ω0t

π[δ(ω – ω0) + δ(ω + ω0)]

sin ω0t

rect(t/T)

T sinc(ωT/2)

cos (ω0t)u(t)

sin (ω0t)u(t)

rect(t/T)cos (ω0t)

rect(t/2β)

tri(t/T)

T sinc2(Tω/2)

sinc2(Tt/2)

e–atu(t),Re{a} > 0

te–atu(t),Re{a} > 0

tn–1e–atu(t),Re{a} > 0

e–a|t|u(t),Re{a} > 0

δT(t)

Operation

Time Function

Fourier Transform

Linearity

af1(t) + bf2(t)

aF1(ω) + bF2(ω)

Time shift

f(t – t0)

Time reversal

f(–t)

F(–ω)

Time scaling

f(at)

Time transformation

f(at – t0)

Duality

F(t)

2πf(–ω)

Frequency shift

F(ω – ω0)

Convolution

f1(t)*f2(t)

F1(ω)F2(ω)

Modulation (Multiplication)

f1(t)f2(t)

Integration

Differentiation in time

(jω)n F(ω)

Differentiation in time

(–jt)nf(t)

Symmetry

f(t) real

F(–ω) = F*(ω)