## LAPLACE AND Z TRANSFORM COMPARISON

BASICS OF TWO-SIDED LAPLACE AND Z TRANSFORMS
DEFINITIONAPPLICATINIMPORTANCECONVOLUTION
L[x(t)]=∫ x(t)e-stdtDifferential eqns.L[dxN/dtN]=sNX(s)L[x(t)*y(t)]=X(s)Y(s)
Z[x(n)]=∑ x(n)z−nDifference eqns.Z[x(n-N)]=z−NX(z)Z[x(n)*y(n)]=X(z)Y(z)
PROPERTIES OF TWO-SIDED LAPLACE AND Z TRANSFORMS
TIME SCALEREVERSALINITIAL VALUETIME MULTIPLY
L[eatx(t)]=X(s-a)L[x(-t)]=X(−s)x(0)=lims→∞sX(s)L[tx(t)]=−dX(s)/ds
Z[anx(n)]=X(z/a)Z[x(-n)]=X(1/z)x(0)=limz→∞ X(z)Z[nx(n)]=−zdX(z)/dz
EXAMPLES OF TWO-SIDED LAPLACE AND Z TRANSFORMS
IMPULSEUNIT STEPCAUSAL EXP.ANTICAUSAL EXP.
L[δ(t)]=1L[u(t)]=1/sL[eatu(t)]=1/(s-a)L[−eatu(−t)]=1/(s-a)
Z[δ(n)]=1Z[u(n)]=z/(z-1)Z[anu(n)]=z/(z-a)Z[−anu(-n-1)]=z/(z-a)

## REGIONS OF CONVERGENCE

SIGNAL TYPECAUSALANTICAUSALTWO-SIDED
FINITE LENGTH{x(0)...x(N)}{X(-N)...x(0)}{x(-M)...x(N)}
{z: 0<|z|≤∞}{z: 0≤|z|<∞}{z: 0<|z|<∞}
ONE SINGLE
EXPONENTIAL
anu(n)-anu(-n-1)anu(n)+bnu(-n-1)
{z: |z|>|a|}{z: |z|<|a|}{z: |a|<|z|<|b|}
SUM SEVERAL
EXPONENTIALS
∑cipinu(n) ∑diqinu(-n-1)SUM OF THESE 2
|z|>max[|pi|]|z|<min[|qi|]max[|pi|]<|z|<min[qi]