10.4 Properties of the Z-Transform

1. Linearity. If E( z) is the z-transform of the sequence { e(0), e(1), } and F( z) is the z-transform of the sequence { f(0), f(1), }, then aE( z) + bF(z) is the z-transform of the sequence { ae(0) + bf (0), ae(1) + bf(1), }. The proof is a simple application of the definition of the z-transform.

2. Final Value. If the sequence e(k) has a limit as k ? ?, then:

   
   

   The proof of this result is not trivial. Let us start with a sequence, a( k), that converges to zero. That is, for every ? there exists an N for which a( k) < ? for every k > N. Let us consider the z-transform of a( k). We find that:

   
   

   We find that:

   
   

   But ? can be made arbitrarily small and the limit must be non-negative, so we find that:

   
   

   From this it is clear that:

   
   

   if the sequence a( k) tends to zero.

   Suppose that e( k) ? c. Then write e( k) as:

   
   

   where a( k) ? 0 as k ? ? The z-transform of e( k) is then:

   
   

   We find that:

   
   

   just as it should.

3. Initial Value. It is easy to see...