2 Commonly used z-transforms

2.1 Unit impulse sequence (delta sequence)

This is a simple but important sequence denoted by δ n and defined as

δ n = 1 n = 0 0 n = ± 1 , ± 2 ,

The significance of the term ‘unit impulse’ is obvious from this definition.

By the definition (1) of the z-transform

{ δ n } = 1 + 0 z 1 + 0 z 2 + = 1

If the single non-zero value is other than at n = 0 the calculation of the z-transform is equally simple.

For example,

δ n 3 = 1 n = 3 0 otherwise

From (1) we obtain

{ δ n 3 } = 0 + 0 z 1 + 0 z 2 + z 3 + 0 z 4 + = z 3
Task!

Write down the definition of δ n m where m is any positive integer and obtain its z-transform.

δ n m = 1 n = m 0 otherwise { δ n m } = z m

Key Point 3

{ δ n m } = z m m = 0 , 1 , 2 ,

2.2 Unit step sequence

As we saw earlier in this Workbook the unit step sequence is

u n = 1 n = 0 , 1 , 2 , 0 n = 1 , 2 , 3 ,

Then, by the definition (1)

{ u n } = 1 + 1 z 1 + 1 z 2 +

The infinite series here is a geometric series (with a constant ratio z 1 between successive terms).

Hence the sum of the first N terms is

S N = 1 + z 1 + + z ( N 1 ) = 1 z N 1 z 1

As N S N 1 1 z 1 provided z 1 < 1

Hence, in what is called the closed form of this z-transform we have the result given in the following Key Point:

Key Point 4
{ u n } = 1 1 z 1 = z z 1 U ( z ) say , z 1 < 1

The restriction that this result is only valid if | z 1 | < 1 or, equivalently | z | > 1 means that the position of the complex quantity z must lie outside the circle centre origin and of unit radius in an Argand diagram. This restriction is not too significant in elementary applications of the z-transform.

2.3 The geometric sequence { a n }

Task!

For any arbitrary constant a obtain the z-transform of the causal sequence

f n = 0 n = 1 , 2 , 3 , a n n = 0 , 1 , 2 , 3 ,

We have, by the definition in Key Point 1,

F ( z ) = { f n } = 1 + a z 1 + a 2 z 2 +

which is a geometric series with common ratio a z 1 . Hence, provided a z 1 < 1 , the closed form of the z-transform is

F ( z ) = 1 1 a z 1 = z z a . The z -transform of this sequence { a n } , which is itself a geometric sequence is summarized in Key Point 5.

Key Point 5
{ a n } = 1 1 a z 1 = z z a | z | > | a | .

Notice that if a = 1 we recover the result for the z-transform of the unit step sequence.

Task!

Use Key Point 5 to write down the z-transform of the following causal sequences

  1. 2 n
  2. ( 1 ) n , the unit alternating sequence
  3. e n
  4. e α n   where α is a constant.
  1. Using a = 2 { 2 n } = 1 1 2 z 1 = z z 2 z > 2
  2. Using a = 1   { ( 1 ) n } = 1 1 + z 1 = z z + 1 z > 1
  3. Using a = e 1 { e n } = z z e 1 z > e 1
  4. Using a = e α { e α n } = z z e α z > e α

The basic z-transforms obtained have all been straightforwardly found from the definition in Key Point 1. To obtain further useful results we need a knowledge of some of the properties of z-transforms.