# LCCDE in simple words?

What is LCCDE?I only know its abbreviation/full form :linear constant-coefficient difference equation

I know that in s domain we have differential equations and in z domain we have difference equation but what is LCCDE?What is its need?What is difference between a normal difference equation and a LCCDE?

An $$N$$th-order linear constant-coefficient difference equation (LCCDE) is of the form

$$y[n]=\sum_{k=0}^{M}b_kx[n-k]-\sum_{k=1}^{N}a_ky[n-k]\tag{1}$$

It is linear because the sequences $$x[n]$$ and $$y[n]$$ appear linearly in $$(1)$$, and it has constant coefficients because the coefficients $$a_k$$ and $$b_k$$ do not depend on the index $$n$$.

LCCDEs are important because they can be used to describe many practically useful discrete-time (or discrete-space) systems, such as linear time-invariant (LTI) filters (which are also called linear shift-invariant (LSI) filters, if $$n$$ does not represent a time index).

Given appropriate initial conditions, Eq. $$(1)$$ can be used to recursively compute the sequence $$y[n]$$ given its past values and the current and past values of the sequence $$x[n]$$. For causal LTI filters, $$y[n]$$ is interpreted as the filtered output, and $$x[n]$$ is the input. In that case, Eq. $$(1)$$ describes an infinite impulse response (IIR) filter.

If $$a_k=0$$, $$k=1,\ldots,N$$, Eq. $$(1)$$ reduces to

$$y[n]=\sum_{k=0}^{M}b_kx[n-k]\tag{2}$$

which is a convolution sum, and the coefficients $$b_k$$ represent the system's impulse response. Note that the impulse response has finite length $$M+1$$, i.e., the system described by $$(2)$$ is a finite impulse response (FIR) filter.

A difference equation simultaneously characterises a system and enables the practical computation of its output $$y[n]$$ for a given input $$x[n]$$ and stated initial conditions.

An LCCDE (MattL Eq(1)) is a special form of a difference equation with constant coefficients; they define LTI systems (under initial rest condition) by providing their impulse response $$h[n]$$ as a solution to $$x[n] = \delta[n]$$.

One of the most important aspects of the difference equations is that if a system cannot be described by a (finite order) difference equation then it can not be exactly realized (implemented with a computer).

• I ask if you could please provide reference for: Can not be realized. Commented Feb 19, 2023 at 8:52
• @mohammadsdtmnd I don't have a specific reference entry point for that exact phrase, afaik it's virtually everwhere, in every DSP book etc. May be the statement is different though. What causes a problem in that statement to you? Do you have an example of a realizible system without a difference equation ?? Commented Feb 22, 2023 at 15:58
• No just a question. Maybe it's possible to implement something other than LCCDE but, LCCDE implementation is guaranteed. Commented Feb 23, 2023 at 11:02
• @mohammadsdtmnd ok then. To reinforce what I've said: if a discrete-time system is implementable, then by practice, a finite number of multiply-accumulate operations should be capable of representing its I/O relation ( a difference equation), otherwise you cannot compute the system output in finite time. A finite number of multiply-accumulate implies a finite order difference equation, etc... Furthermore, a frequency domain implementation also has similar constraints of having finite number of DFT samples to repsenent the system etc... Commented Feb 27, 2023 at 19:24