Properties of the Fourier transform Linearity x1()<=>X1(f,x2()<=>X2(=>ax1(t+βx2()<=>∞X1(6+BX2(1 Duality X(<=>x(t)=>x(<=>X(-t)andx(1f)<=>X(f) Time-shifting: x(t-)<=> X(fe-12mtt Scaling: FI(X(at)]=1/a x(t/a Convolution: x(t<=> X(f), y(t<=> Y(f then FIx(t*y(t)]= X(Y( Convolution in time corresponds to multiplication in frequency and vIsa versa x(0)*y()=|xt-)rrα β α β τ π τ Properties of the Fourier transform • Linearity – x1(t) <=> X1(f), x2(t) <=>X2(f) => αx1(t) + βx2(t) <=> αX1(f) + βX2(f) • Duality – X(f) <=> x(t) => x(f) <=> X(-t) and x(-f)<=> X(t) • Time-shifting: x(t-τ) <=> X(f)e-j2πfτ • Scaling: F[(x(at)] = 1/|a| X(f/a) • Convolution: x(t) <=> X(f), y(t) <=> Y(f) then, – F[x(t)*y(t)] = X(f)Y(f) – Convolution in time corresponds to multiplication in frequency and visa versa ∞ x t( ) * y(t) = x t − τ )y(τ )dτ ∫−∞( Eytan Modiano Slide 6