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Suppose we have these two functions and their z-transforms are

[tex]P(r,z)=\sum_{t=0}^{\infty}P(r,t)z^t[/tex]

and

[tex]F(r,z)=\sum_{t=0}^{\infty}F(r,t)z^t[/tex].

Now we are going to transform the following convolution of P and F:

[tex]\sum_{t'\le{t}}F(r,t')P(0,t-t')[/tex].

The result is said to be

[tex]F(r,z)P(0,z)[/tex].

But I don't know how to obtain the result. There are two difficulties:(1)the upper limit of t' is t, which is finite. (2)the two summations are coupled.

Anyone can help?

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# Z-transform of a discrete convolution

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