Fourier transform
f(u)=-∞∫∞g(v)exp(-iuv)dv
Duality___.
|X(w)|²dw/2π
Amount of energy in signal x(t) that lies in the frequency band between w and w+dw___.
π[δ(w-w₀)+δ(w+w₀)] π[δ(w-w₀)-δ(w+w₀)]/i
cos(w₀t) fourier transform___. sin(w₀t) fourier transform___.
iwX(w)
d(x(t))/dt=___.
ik2πa(k)/T₀
dx(t)/dt fourier series coefficients___.
1/a+iw 1/(a+iw)²
exp(-at)u(t) Re(a)>0 fourier transform___. texp(-at)u(t) Re(a)>0 fourier transform___.
a(k-M)
exp(iM2πt/T₀)x(t) fourier series coefficients___.
X(w)={1,|w|<W 0,|w|>W
sin(Wt)/πt fourier transform___.
1/(a+iw)ⁿ
tⁿ⁻¹exp(-at)u(t)/(n-1)! Re(a)>0 fourier transform___.
1/iw+πδ(w)
u(t) fourier transform___.
a-k
x(-t) fourier series coefficients___.
2πδ(w)
x(t)=1 fourier transform___.
∑(k=-∞to∞)2sin(kw₀T₁)δ(w-kw₀)/k
x(t)={1,|t|<T₁ 0,T₁<|t|≤T₀/2} x(t+T₀)=x(t)
2sin(wT₁)/w
x(t)={1,|t|<T₁ 0,|t|>T₁ fourier transform
∑(l=-∞to∞)a(l)b(k-l)
x(t)y(t) fourier series coefficients___.
a(k)exp(-ik2πt/T₀)
x(t-t₀) fourier series coefficients___.
a(k)
x(αt)
a*-k
x*(t) fourier series coefficients___.
Re{X(w)} iIm{X(w)}
xe(t)=Ev{x(t)} F(xe(t)) xo(t)=Od{x(t)} F(xo(t))
1
δ(t) fourier transform___.
exp(-iwt₀)
δ(t-t₀) fourier transform___.
2π∑(k=-∞to∞)a(k)δ(w-kw₀)
∑(k=-∞to∞)a(k)exp(ikw₀t) Fourier transform___.
2π∑(k=-∞to∞)δ(w-(2πk/T))/T
∑(n=-∞to∞)δ(t-nT) fourier transform
T₀a(k)b(k)
∫T₀ x(τ)y(t-τ)dτ fourier series coefficients___.
X(w)/iw+πX(0)δ(w)
F(-∞∫t x(t)dt)=___.
X(w-w₀)
F(exp(iw₀t)x(t))=___.
F(f(t))=2πg(-w)
F(g(t))=f(w)
id(X(w))/dw
F(tx(t))=___.
X(w/a)/|a|
F(x(at))=___.
X(w)Y(w)
F(x(t)*y(t))=___.
X(w)*Y(w)/2π
F(x(t)y(t))=___.
exp(-iwt₀)X(w)
F(x(t-t₀))=___.
X*(-w)
F(x*(t))=___.
∫T₀ |x(t)|²dt/t₀=∑(k=-∞to∞)|a(k)²|
Parseval relation for Periodic Signals___.
-∞∫∞|x(t)|²dt=-∞∫∞|X(w)|²dw/2π
Parseval's relation for aperiodic signals
a(k)(1/ik(2π/T₀))
-∞∫t x(t)dt fourier series coefficients___.
