Is there a special order of applying Z-transform properties?

[YehiaMedhat]

I have this problem to get the [imath]\mathcal{Z}[/imath] transform of this expression: [imath]f(t) = t2^t u(t-2)[/imath]
My approach to solve the problem:
Let's apply the shift first:
[math]\mathcal{z}\{u(t-2)\} = \frac{z}{z-1} z^{-2} = \frac{1}{z(z-1)}[/math]Then apply the scaling:
[math]\mathcal{z}\{2^tu(t-2)\} = \frac{1}{\frac{z}{2}(\frac{z}{2}-1)} = \frac{4}{z(z-2)}[/math]Then apply the differentiation:
[math]\mathcal{z}\{t2^tu(t-2)\} = -z\frac{d}{dz}\frac{4}{z^2-2z)} = -z \frac{4(2z-2)}{(z^2-2z)^2}[/math][math]= \frac{8-8z}{z(z-2)^2}[/math]But, if I try some other order of applying the properties, I get some other solution. Let me try to deriviate first then scale then shift:
[math]\mathcal{z}\{t\} = -z\frac{d}{dz}\frac{z}{z-1} = \frac{z}{(z-1)^2}[/math][math]\mathcal{z}\{tu(t-2)\} = \frac{1}{z(z-1)^2}[/math][math]\mathcal{z}\{t2^tu(t-2)\} = \frac{8}{z(z-2)^2}[/math]This is not completely different solution, but it's significantly affecting my choice in an exam for instance. So, am I right that there's an order to follow when solving [imath]\mathcal{z}[/imath] transform problems, or I have already missed something in the above solutions?
If there's an order what is it?
And thank you all in advance.

 

[blamocur]

I am familiar with Z-transforms of sequences, but you seem to be applying something similiar to functions -- is there a definition somewhere?
Also, what is [imath]u(t-2)[/imath] ?

 

[mario99]

I have this problem to get the [imath]\mathcal{Z}[/imath] transform of this expression: [imath]f(t) = t2^t u(t-2)[/imath]
My approach to solve the problem:
Let's apply the shift first:
[math]\mathcal{z}\{u(t-2)\} = \frac{z}{z-1} z^{-2} = \frac{1}{z(z-1)}[/math]Then apply the scaling:
[math]\mathcal{z}\{2^tu(t-2)\} = \frac{1}{\frac{z}{2}(\frac{z}{2}-1)} = \frac{4}{z(z-2)}[/math]Then apply the differentiation:
[math]\mathcal{z}\{t2^tu(t-2)\} = -z\frac{d}{dz}\frac{4}{z^2-2z)} = -z \frac{4(2z-2)}{(z^2-2z)^2}[/math][math]= \frac{8-8z}{z(z-2)^2}[/math]But, if I try some other order of applying the properties, I get some other solution. Let me try to deriviate first then scale then shift:
[math]\mathcal{z}\{t\} = -z\frac{d}{dz}\frac{z}{z-1} = \frac{z}{(z-1)^2}[/math][math]\mathcal{z}\{tu(t-2)\} = \frac{1}{z(z-1)^2}[/math][math]\mathcal{z}\{t2^tu(t-2)\} = \frac{8}{z(z-2)^2}[/math]This is not completely different solution, but it's significantly affecting my choice in an exam for instance. So, am I right that there's an order to follow when solving [imath]\mathcal{z}[/imath] transform problems, or I have already missed something in the above solutions?
If there's an order what is it?
And thank you all in advance.

Multiplication property (which involves differentiation) depends on scaling and shifting properties, so it must be done after them!

 

[YehiaMedhat]

I am familiar with Z-transforms of sequences, but you seem to be applying something similiar to functions -- is there a definition somewhere?
Also, what is [imath]u(t-2)[/imath] ?

The definition we have is:
[math]\mathcal{Z}\{f(t)\} = \sum_{k=0}^{\infty} f(k) z^{-k}[/math]The [imath]u(t-2)[/imath] is the unit step function, which is equal to one whenever its input is positive, which is in this case [imath]t \geq 2[/imath]

 

[YehiaMedhat]

Multiplication property (which involves differentiation) depends on scaling and shifting properties, so it must be done after them!

Yes, exactly, so is there any textbook, any article that captures in simple words the order in which I should be applying the properties, or it's just the order in which the problem is written?

 

[mario99]

Yes, exactly, so is there any textbook, any article that captures in simple words the order in which I should be applying the properties, or it's just the order in which the problem is written?

The order is obvious. Scaling property depends on shifting property, so it must come after it. And Multiplication property depends on shifting and scaling properties, so it must come after them.

So, the order is:

1. Shifting.
2. Scaling.
3. Multiplication (involves differentiation).