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Introduction

One of the most important concepts of DSP is to be able to properly represent the input/output relationship to a given LTIsystem. A linear constant-coefficient difference equation (LCCDE) serves as a way to express just this relationship in a discrete-time system. Writing the sequenceof inputs and outputs, which represent the characteristics of the LTI system, as a difference equation help in understandingand manipulating a system.

difference equation
An equation that shows the relationship between consecutive values of a sequence and the differences amongthem. They are often rearranged as a recursive formula so that a systems output can be computed from the inputsignal and past outputs.

General formulas for the difference equation

As stated briefly in the definition above, a difference equation is a very useful tool in describing and calculatingthe output of the system described by the formula for a given sample n . The key property of the difference equation is its ability to help easily find thetransform, H z , of a system. In the following two subsections, we will look atthe general form of the difference equation and the general conversion to a z-transform directly from the differenceequation.

Difference equation

The general form of a linear, constant-coefficient difference equation (LCCDE), is shown below:

k 0 N a k y n k k 0 M b k x n k
We can also write the general form to easily express a recursive output, which looks like this:
y n k 1 N a k y n k k 0 M b k x n k
From this equation, note that y n k represents the outputs and x n k represents the inputs. The value of N represents the order of the difference equation and corresponds to the memory of the system being represented.Because this equation relies on past values of the output, in order to compute a numerical solution, certain past outputs,referred to as the initial conditions , must be known.

Conversion to z-transform

Using the above formula, [link] , we can easily generalize the transfer function , H z , for any difference equation. Below are the steps taken to convert any difference equation into its transferfunction, i.e. z-transform. The first step involves taking the Fourier Transform of all the terms in [link] . Then we use the linearity property to pull the transform inside thesummation and the time-shifting property of the z-transform to change the time-shifting terms to exponentials. Oncethis is done, we arrive at the following equation: a 0 1 .

Y z k 1 N a k Y z z k k 0 M b k X z z k
H z Y z X z k 0 M b k z k 1 k 1 N a k z k

Conversion to frequency response

Once the z-transform has been calculated from the difference equation, we can go one step further to define the frequencyresponse of the system, or filter, that is being represented by the difference equation.

Remember that the reason we are dealing with these formulas is to be able to aid us in filter design. A LCCDEis one of the easiest ways to represent FIR filters. By being able to find the frequency response, we will be able tolook at the basic properties of any filter represented by a simple LCCDE.
Below is the general formula for the frequency response of a z-transform. The conversion is simple a matter of takingthe z-transform formula, H z , and replacing every instance of z with w .
H w z w H z k 0 M b k w k k 0 N a k w k
Once you understand the derivation of this formula, look atthe module concerning Filter Design from the Z-Transform for a look into how all of these ideas of the Z-transform , Difference Equation, and Pole/Zero Plots play a role in filter design.

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Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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