Commonly used z-transforms

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

$δ_{n} = \{\begin{matrix} 1 & n = 0 \\ 0 & n = \pm 1, \pm 2, \dots \end{matrix}$

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

By the definition (1) of the z-transform

\begin{array}{rcl} ℤ {δ_{n}} & = & 1 + 0 z^{- 1} + 0 z^{- 2} + \dots \\ = & 1 \end{array}

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

For example,

$δ_{n - 3} = \{\begin{matrix} 1 & n = 3 \\ 0 & otherwise \end{matrix}$

From (1) we obtain

\begin{array}{rcl} ℤ {δ_{n - 3}} & = & 0 + 0 z^{- 1} + 0 z^{- 2} + z^{- 3} + 0 z^{- 4} + \dots \\ = & z^{- 3} \end{array}

Task!

Write down the definition of $δ_{n - m}$ where $m$ is any positive integer and obtain its z-transform.

$δ_{n - m} = \{\begin{matrix} 1 & n = m \\ 0 & otherwise \end{matrix}$ $ℤ {δ_{n - m}} = z^{- m}$

Key Point 3

$ℤ {δ_{n - m}} = z^{- m} m = 0, 1, 2, \dots$

2.2 Unit step sequence

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

$u_{n} = \{\begin{matrix} 1 & n = 0, 1, 2, \dots \\ 0 & n = - 1, - 2, - 3, \dots \end{matrix}$

Then, by the definition (1)

$ℤ {u_{n}} = 1 + 1 z^{- 1} + 1 z^{- 2} + \dots$

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

\begin{array}{rcl} S_{N} & = & 1 + z^{- 1} + \dots + z^{- (N - 1)} \\ = & \frac{1 - z^{- N}}{1 - z^{- 1}} \end{array}

As $N \to \infty$ $S_{N} \to \frac{1}{1 - z^{- 1}}$ 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

ℤ {u_{n}} = \frac{1}{1 - z^{- 1}} = \frac{z}{z - 1} \equiv 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

$f_{n} = \{\begin{matrix} 0 & n = - 1, - 2, - 3, \dots \\ a^{n} & n = 0, 1, 2, 3, \dots \end{matrix}$

We have, by the definition in Key Point 1,

$F (z) = ℤ {f_{n}} = 1 + a z^{- 1} + a^{2} z^{- 2} + \dots$

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

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

Key Point 5

ℤ {a^{n}} = \frac{1}{1 - a z^{- 1}} = \frac{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

$2^{n}$
${(- 1)}^{n}$ , the unit alternating sequence
$e^{- n}$
$e^{- α n}$ where $α$ is a constant.

Using $a = 2$ $ℤ {2^{n}} = \frac{1}{1 - 2 z^{- 1}} = \frac{z}{z - 2} |z| > 2$
Using $a = - 1$ $ℤ {{(- 1)}^{n}} = \frac{1}{1 + z^{- 1}} = \frac{z}{z + 1} |z| > 1$
Using $a = e^{- 1}$ $ℤ {e^{- n}} = \frac{z}{z - e^{- 1}} |z| > e^{- 1}$
Using $a = e^{- α}$ $ℤ {e^{- α n}} = \frac{z}{z - e^{- α}} |z| > e^{- α}$

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.

2 Commonly used z-transforms

2.1 Unit impulse sequence (delta sequence)

Task!

Answer

Key Point 3

2.2 Unit step sequence

Key Point 4

2.3 The geometric sequence { a n }

Task!

Answer

Key Point 5

Task!

Answer

2.3 The geometric sequence ${a^{n}}$