| BASICS OF TWO-SIDED LAPLACE AND Z TRANSFORMS | |||
|---|---|---|---|
| DEFINITION | APPLICATIN | IMPORTANCE | CONVOLUTION |
| L[x(t)]=∫ x(t)e-stdt | Differential eqns. | L[dxN/dtN]=sNX(s) | L[x(t)*y(t)]=X(s)Y(s) |
| Z[x(n)]=∑ x(n)z−n | Difference 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 SCALE | REVERSAL | INITIAL VALUE | TIME 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 | |||
| IMPULSE | UNIT STEP | CAUSAL EXP. | ANTICAUSAL EXP. |
| L[δ(t)]=1 | L[u(t)]=1/s | L[eatu(t)]=1/(s-a) | L[−eatu(−t)]=1/(s-a) |
| Z[δ(n)]=1 | Z[u(n)]=z/(z-1) | Z[anu(n)]=z/(z-a) | Z[−anu(-n-1)]=z/(z-a) |
| SIGNAL TYPE | CAUSAL | ANTICAUSAL | TWO-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] |