# 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 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 realized (implemented with a computer).