20 Find the Z transform of sequence $x[n]$

20.1 Example 1

Find the Z transform for the unit step discrete function

Given the unit step function $x[n]=u[n]$ defined as $x=\{1,1,1,\cdots\}\,\$ for $n\geq0\,$ , find its Z transform.

Mathematica	Matlab
Remove["Global`*"]; ZTransform[UnitStep[n],n,z] Out[] = z/(-1+z)	syms n pretty(ztrans(heaviside(n))) 1 ----- + 1/2 z - 1

20.2 Example 2

Find the Z transform for $x[n]=\left( \frac{1}% {3}\right) ^{n}u\left( n\right) +\left( 0.9\right) ^{n-3}u\left( n\right)$

Mathematica	Matlab
f[n_]:=((1/3)^n+(9/10)^(n-3))UnitStep[n]; ZTransform[f[n],n,z] Out[18]= z (3/(-1+3 z)+10000/(729 (-9+10 z)))	syms n pretty(ztrans( ( (1/3)^n + (0.9)^(n-3) ) * heaviside(n) ) ) 100 1 ------------- + ------- + 1729/1458 81 (z - 9/10) 3 z - 1