# Digital Signal Processing/Z Transform

The **Z Transform** has a strong relationship to the DTFT, and is incredibly useful in transforming, analyzing, and manipulating discrete calculus equations. The Z transform is named such because the letter 'z' (a lower-case Z) is used as the transformation variable.

## z Transform Definition[edit | edit source]

For a given sequence x[n], we can define the z-transform X(z) as such:

[Z Transform]

- it is important to note that z is a continuous complex variable defined as such:

- where is the imaginary unit.

There can be several sequences which will generate the same z-transform with the different functions being differentiated by the convergence region of for which the summation in the z-transform will converge. These convergence regions are annuli centered at the orgin. In a given convergence region, only one will converge to a given .

### Example 1[edit | edit source]

- for

### Example 2[edit | edit source]

- for

Note that both examples have the same function as their z-transforms but with different convergence regions needed for the respective infinite summations in their z-transforms to converge. Many textbooks on z-transforms are only concerned with so-called *right-handed* functions which is to say functions for which for all less than some initial start point ; that is, for all . So long as the function grows at most exponentially after the start point, the z-transform of these so-called *right-handed* functions will converge in an open annulus going to infinity, for some positive real .

It is important to note that the z-transform rarely needs to be computed manually, because many common results have already been tabulated extensively in tables, and control system software includes it (MatLab,Octave,SciLab).

The z-transform is actually a special case of the so-called Laurent series, which is a special case of the commonly used Taylor series.

## The Inverse Z-Transform[edit | edit source]

The inverse z-transform can be defined as such:

[Inverse Z Transform]

Where C is a closed-contour that lies inside the unit circle on the z-plane, and encircles the point z = {0, 0}.

The inverse z-transform is mathematically very complicated, but luckily—like the z-transform itself—the results are extensively tabulated in tables.

## Equivalence to DTFT[edit | edit source]

If we substitute , where is the frequency in radians per second, into the Z-transform, we get

which is equivalent to the definition of the Discrete-Time Fourier Transform. In other words, to convert from the Z-transform to the DTFT, we need to evaluate the Z-transform around the unit circle.

## Properties[edit | edit source]

Since the z-transform is equivalent to the DTFT, the z-transform has many of the same properties. Specifically, the z-transform has the property of duality, and it also has a version of the convolution theorem (discussed later).

The z-transform is a linear operator.

## Convolution Theorem[edit | edit source]

Since the Z-transform is equivalent to the DTFT, it also has a convolution theorem that is worth stating explicitly:

- Convolution Theorem
- Multiplication in the discrete-time domain becomes convolution in the z-domain. Multiplication in the z-domain becomes convolution in the discrete-time domain.

Y(s)=X(s).H(s)

## Z-Plane[edit | edit source]

Since the variable z is a continuous, complex variable, we can map the z variable to a complex plane as such:

## Transfer Function[edit | edit source]

Let's say we have a system with an input/output relationship defined as such:

Y(z) = H(z)X(z)

We can define the transfer function of the system as being the term H(z). If we have a basic transfer function, we can break it down into parts:

Where H(z) is the transfer function, N(z) is the numerator of H(z) and D(z) is the denominator of H(z). If we set N(z)=0, the solutions to that equation are called the **zeros** of the transfer function. If we set D(z)=0, the solutions to that equation are called the **poles** of the transfer function.

The poles of the transfer function amplify the frequency response while the zero's attenuate it. This is important because when you design a filter you can place poles and zero's on the unit circle and quickly evaluate your filters frequency response.

### Example[edit | edit source]

Here is an example:

So by dividing through by X(z), we can show that the transfer function is defined as such:

We can also find the D(z) and N(z) equations as such:

And from those equations, we can find the poles and zeros:

- Zeros
- z → 0
- Poles
- z → -1/2

## Stability[edit | edit source]

It can be shown that for any causal system with a transfer function H(z), all the poles of H(z) must lie within the unit-circle on the z-plane for the system to be stable. Zeros of the transfer function may lie inside or outside the circle.

## Gain[edit | edit source]

Gain is the factor by which the output magnitude is different from the input magnitude. If the input magnitude is the same as the output magnitude at a given frequency, the filter is said to have "unity gain".

## Properties[edit | edit source]

Here is a listing of the most common properties of the Z transform.

Time domain | Z-domain | ROC | |
---|---|---|---|

Notation | ROC: | ||

Linearity | At least the intersection of ROC_{1} and ROC_{2}
| ||

Time shifting | ROC, except if and if | ||

Scaling in the z-domain | |||

Time reversal | |||

Conjugation | ROC | ||

Real part | ROC | ||

Imaginary part | ROC | ||

Differentiation | ROC | ||

Convolution | At least the intersection of ROC_{1} and ROC_{2}
| ||

Correlation | At least the intersection of ROC of X_{1}(z) and X_{2}()
| ||

Multiplication | At least | ||

Parseval's relation |

**Initial value theorem**

- , If causal

**Final value theorem**

- , Only if poles of are inside unit circle