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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?

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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.

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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).

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  • $\begingroup$ I ask if you could please provide reference for: Can not be realized. $\endgroup$ Commented Feb 19, 2023 at 8:52
  • $\begingroup$ @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 ?? $\endgroup$
    – Fat32
    Commented Feb 22, 2023 at 15:58
  • $\begingroup$ No just a question. Maybe it's possible to implement something other than LCCDE but, LCCDE implementation is guaranteed. $\endgroup$ Commented Feb 23, 2023 at 11:02
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    $\begingroup$ @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... $\endgroup$
    – Fat32
    Commented Feb 27, 2023 at 19:24

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