4 Further z -transform properties

We showed earlier that the results

ℤ { v n + w n } = V ( z ) + W ( z ) and similarly ℤ { v n w n } = V ( z ) W ( z )

follow from the linearity property.

You should be clear that there is no comparable result for the product of two sequences.

ℤ { v n w n } is not equal to V ( z ) W ( z )

For two specific products of sequences however we can derive useful results.

4.1 Multiplication of a sequence by a n

Suppose f n is an arbitrary sequence with z-transform F ( z ) .

Consider the sequence { v n } where

v n = a n f n i.e. { v 0 , v 1 , v 2 , } = { f 0 , a f 1 , a 2 f 2 , }

By the z-transform definition

ℤ { v n } = v 0 + v 1 z 1 + v 2 z 2 + = f 0 + a f 1 z 1 + a 2 f 2 z 2 + = n = 0 a n f n z n = n = 0 f n z a n But F ( z ) = n = 0 f n z n

Thus we have shown that ℤ { a n f n } = F z a

Key Point 8

ℤ { a n f n } = F z a

That is, multiplying a sequence { f n } by the sequence { a n } does not change the form of the z-transform F ( z ) . We merely replace z by z a in that transform.

ℤ { cos n } = z 2 z cos 1 z 2 2 z cos 1 + 1

So, replacing z by z 1 2 = 2 z ,

ℤ 1 2 n cos n = ( 2 z ) 2 ( 2 z ) cos 1 ( 2 z ) 2 4 z cos 1 + 1

Task!

Using Key Point 8, write down the z-transform of the sequence { v n } where

v n = e 2 n sin 3 n

We have, ℤ { sin 3 n } = z sin 3 z 2 2 z cos 3 + 1

so with a = e 2 we replace z by z e + 2 to obtain

ℤ { v n } = ℤ { e 2 n sin 3 n } = z e 2 sin 3 ( z e 2 ) 2 2 z e 2 cos 3 + 1 = z e 2 sin 3 z 2 2 z e 2 cos 3 + e 4
Task!

Using the property just discussed write down the z-transform of the sequence { w n } where

w n = e α n cos ω n

We have, ℤ { cos ω n } = z 2 z cos ω z 2 2 z cos ω + 1

So replacing z by z e α we obtain

ℤ { w n } = ℤ { e α n cos ω n } = ( z e α ) 2 z e α cos ω ( z e α ) 2 2 z e α cos ω + 1 = z 2 z e α cos ω z 2 2 z e α cos ω + e 2 α
Key Point 9

ℤ { e α n cos ω n } = z 2 z e α cos ω z 2 2 z e α cos ω + e 2 α ℤ { e α n sin ω n } = z e α sin ω z 2 2 z e α cos ω + e 2 α

Note the same denominator in each case.

4.2 Multiplication of a sequence by n

An important sequence whose z-transform we have not yet obtained is the unit ramp sequence { r n } :

r n = 0 n = 1 , 2 , 3 , n n = 0 , 1 , 2 ,

Figure 5

No alt text was set. Please request alt text from the person who provided you with this resource.

Figure 5 clearly suggests the nomenclature ‘ramp’.

We shall attempt to use the z-transform of { r n } from the definition:

ℤ { r n } = 0 + 1 z 1 + 2 z 2 + 3 z 3 +

This is not a geometric series but we can write

z 1 + 2 z 2 + 3 z 3 = z 1 ( 1 + 2 z 1 + 3 z 2 + ) = z 1 ( 1 z 1 ) 2 z 1 < 1

where we have used the binomial theorem ( HELM booklet  16.3) .

Hence

&integers; { r n } = &integers; { n } = 1 z 1 1 z 2 = z ( z 1 ) 2 z > 1
Key Point 10

The z-transform of the unit ramp sequence is

&integers; { r n } = z ( z 1 ) 2 = R ( z ) (say)

Recall now that the unit step sequence has z-transform &integers; { u n } = z ( z 1 ) = U ( z )  (say) which is the subject of the next Task.

Task!

Obtain the derivative of U ( z ) = z ( z 1 ) with respect to z .

We have, using the quotient rule of differentiation:

d U d z = d d z z z 1 = ( z 1 ) 1 ( z ) ( 1 ) ( z 1 ) 2 = 1 ( z 1 ) 2

We also know that

R ( z ) = z ( z 1 ) 2 = ( z ) 1 ( z 1 ) 2 = z d U d z (3)

Also, if we compare the sequences

u n = { 0 , 0 , 1 , 1 , 1 , 1 , }

  

r n = { 0 , 0 , 0 , 1 , 2 , 3 , }

  

we see that r n = n u n , (4)

so from (3) and (4) we conclude that   &integers; { n u n } = z d U d z

Now let us consider the problem more generally.

Let f n be an arbitrary sequence with z-transform F ( z ) :

F ( z ) = f 0 + f 1 z 1 + f 2 z 2 + f 3 z 3 + = n = 0 f n z n

We differentiate both sides with respect to the variable z , doing this term-by-term on the right-hand side. Thus

d F d z = f 1 z 2 2 f 2 z 3 3 f 3 z 4 = n = 1 ( n ) f n z n 1 = z 1 ( f 1 z 1 + 2 f 2 z 2 + 3 f 3 z 3 + ) = z 1 n = 1 n f n z n

But the bracketed term is the z-transform of the sequence

{ n f n } = { 0 , f 1 , 2 f 2 , 3 f 3 , }

Thus if F ( z ) = &integers; { f n } we have shown that

d F d z = z 1 &integers; { n f n } or &integers; { n f n } = z d F d z

We have already (equations (3) and (4) above) demonstrated this result for the case f n = u n .

Key Point 11

If &integers; { f n } = F ( z ) then &integers; { n f n } = z d F d z

Task!

By differentiating the z-transform R ( z ) of the unit ramp sequence obtain the z-transform of the causal sequence { n 2 } .

We have

&integers; { n } = z ( z 1 ) 2

so

&integers; { n 2 } = &integers; { n . n } = z d d z z ( z 1 ) 2

By the quotient rule

d d z z ( z 1 ) 2 = ( z 1 ) 2 ( z ) ( 2 ) ( z 1 ) ( z 1 ) 4 = z 1 2 z ( z 1 ) 3 = 1 z ( z 1 ) 3

Multiplying by z we obtain

&integers; { n 2 } = z + z 2 ( z 1 ) 3 = z ( 1 + z ) ( z 1 ) 3

Clearly this process can be continued to obtain the transforms of { n 3 } , { n 4 } , etc.