Transform Pairs

Mathematics / transform-pairs.yaml
13 records
2 sections
fourier_transform_pairs 8 fourier_transform_properties 5

fourier transform pairs

NameTime DomainFrequency Domain
DC level 1 2*pi*delta(omega)
Unit step u(t) pi*delta(omega) + 1/(j*omega)
Complex exponential exp(j*omega_0*t) 2*pi*delta(omega - omega_0)
Cosine cos(omega_0*t) pi*[delta(omega - omega_0) + delta(omega + omega_0)]
Sine sin(omega_0*t) pi/j*[delta(omega - omega_0) - delta(omega + omega_0)]
Impulse train sum_n delta(t - n*T) (2*pi/T)*sum_m delta(omega - 2*pi*m/T)
Rectangular pulse rect(t/2b) 2b*sinc(omega*b/2)
Triangular pulse tri(t/2b) 2b*sinc^2(omega*b/2)

fourier transform properties

NameTime DomainFrequency Domain
Time shift g(t - t_0) exp(-j*omega*t_0)*G(omega)
Frequency shift exp(j*omega_0*t)*g(t) G(omega - omega_0)
Differentiation d^n g/dt^n (j*omega)^n * G(omega)
Convolution g1(t) * g2(t) G1(omega) * G2(omega)
Multiplication g1(t) * g2(t) (1/2*pi) * (G1 * G2)(omega)