z-transform of unit step function?

i have this n-domain function H[n]= (0.5^n)*u[n] and i need to find z transformation of this function. i've tried this one:

>> syms n z
>> y = ((0.5)^n)*heaviside(n);
>> yz = ztrans (y,n,z)

yz =

1/(2*z - 1) + 1/2

isn't supposed to be 1/(-0.5^z-1)+1 ? is there any solution to solve this z transform?

[Fardin Ahasan Maraz]

syms a n f 
n = 0;
f = a^n

f = 

1

ztrans(f)

ans = 

Answer 1

[Geoff Hayes]

I don't have the Symbolic Toolbox but I think that the provided answer is correct. If

 x[n] = 0.5^n * u[n]
 => x[n] = {…,0,0,0.5^0 * 0, 0.5^1 * 1, 0.5^2 * 1,…}

then the z-transform is

 sum(n=-Inf:+Inf)(0.5^n)*(z^-n)
    = sum(n=0:+Inf)(0.5^n)*(z^-n)  % ignore all n<0
    = 0.5 + sum(n=1:+Inf)(0.5^n)*(z^-n)
    = 0.5 + sum(n=1::+Inf)(0.5/z)^n
    = 0.5 + (0.5/z)/(1-(0.5/z))    % assuming abs(0.5/z)<1
    = 0.5 + 0.5/(z-0.5)
    = 1/2 + (1/2)/(z-1/2)
    = 1/(2*z - 1) + 1/2

[Diamond]

is it same with the 3rd row formula of this table ? or am i wrong to define u[n] by heaviside (n)?

[Geoff Hayes]

As Azzi mentions below, MATLAB's heaviside defines H(0) to be 0.5, whereas other definitions of the same function define H(0) to be 1. (See http://en.wikipedia.org/wiki/Heaviside_step_function#Zero_argument for details.)

The formula in the third row of your table assumes that H(0)=1 and can be derived similar to above.

I think that it is fine to define u[n] with heaviside(n) as long as your are aware (and state) that heaviside(0) = 0.5.

Answer 2

[Amine BENZOUBIR]

use this commande to change the origine to 1

v=1;
sympref('HeavisideAtOrigin', v);

[Nour]

Thanks!

Answer 3

[Azzi Abdelmalek]

The heaveaside function of Matlab is defined with heaviside(0) equal to 0.5

If you look at the table using another definition of heaviside (e(0)=1), you will find the z-transform of a^n is z/(z-a) . The heaviside defined in Matlab can be written as

heaviside(n)=e(n)-delta(n)  (delta is Kronecker function),  the z-transform is  z/(z-a)-0.5

In your case replace a by 0.5

[Diamond]

is it same with the 3rd row formula of this table? or am i wrong to define u[n] by heaviside (n)?