Sheet with useful_formulas
- 1. useful formulas name formula Euler’s formula ej θ = cos(θ) + j sin(θ) . . . for cosine cos(θ) = ejθ + e−jθ 2 . . . for sine sin(θ) = ejθ − e−jθ 2 j sinc function sinc(θ) := sin(π θ) π θ formulas for continuous-time LTI signals and systems name formula area under impulse Z δ(t) dt = 1 multiplication by impulse f(t) δ(t) = f(0) δ(t) . . . by shifted impulse f(t) δ(t − to) = f(to) δ(t − to) convolution f(t) ∗ g(t) = Z f(τ) g(t − τ) dτ . . . with an impulse f(t) ∗ δ(t) = f(t) . . . with a shifted impulse f(t) ∗ δ(t − to) = f(t − to) transfer function H(s) = Z h(t) e−st dt frequency response Hf (ω) = Z h(t) e−jωt dt . . . their connection Hf (ω) = H(jω) provided jω-axis ⊂ ROC selected Laplace transform pairs x(t) X(s) ROC x(t) Z x(t) e−st dt (def.) δ(t) 1 all s u(t) 1 s Re(s) > 0 e−a t u(t) 1 s + a Re(s) > −a cos(ωot) u(t) s s2 + ω2 o Re(s) > 0 sin(ωot) u(t) ωo s2 + ω2 o Re(s) > 0 e−a t cos(ωot) u(t) s + a (s + a)2 + ω2 o Re(s) > −a e−a t sin(ωot) u(t) ωo (s + a)2 + ω2 o Re(s) > −a Note: a is assumed real. Laplace transform properties x(t) X(s) a x(t) + b g(t) a X(s) + b G(s) x(t) ∗ g(t) X(s) G(s) dx(t) dt s X(s) x(t − to) e−s to X(s)
- 2. selected Fourier transform pairs x(t) Xf (ω) x(t) Z x(t) e−jωt dt (def.) 1 2π Z Xf (ω) ejωt dω Xf (ω) δ(t) 1 1 2 π δ(ω) u(t) π δ(ω) + 1 jω ejωot 2 π δ(ω − ωo) cos(ωo t) π δ(ω + ωo) + π δ(ω − ωo) sin(ωo t) j π δ(ω + ωo) − j π δ(ω − ωo) ωo π sinc “ωo π t ” ideal LPF cut-off frequency ωo symmetric pulse 2 ω sin „ T 2 ω « width T, height 1 impulse train impulse train period T, height 1 period, height ωo = 2π T Fourier transform properties x(t) Xf (ω) a x(t) + b g(t) a Xf (ω) + b Gf (ω) x(a t) 1 |a| X “ω a ” x(t) ∗ g(t) Xf (ω) Gf (ω) x(t) g(t) 1 2π Xf (ω) ∗ Gf (ω) x(t − to) e−jtoω X(ω) x(t) ejωot X(ω − ωo) x(t) cos(ωot) 0.5 X(ω + ωo) + 0.5 X(ω − ωo) x(t) sin(ωot) j 0.5 X(ω + ωo) − j 0.5 X(ω − ωo) dx(t) dt j ω Xf (ω)