# Can linear systems always be represented as differential or difference equations?

On my note, it was written that linear systems can always be represented as either differential equations or difference equations. I forgot the source of the quote. But I am not sure if it is correct.

For example, for a linear time-invariant system, its output is the convolution of the input and the system's impulse response, which I don't know how to put into differential or difference equations.

Thanks!

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For example, for a linear time-invariant system, its output is the convolution of the input and the system's impulse response, which I don't know how to put into differential or difference equations.

A (discrete) LTI system is described by $y[n] = h[n] \star x[n]$. If $h[n]$ has finite support (FIR) then the output is a linear combination of the input, which can be written as a difference equation. But, even when $h[n]$ hasn't finite support, the transfer can often be expressed in the form $$y[n] + b_1 y[n-1] + \cdots b_{M-1} y[n-M+1]=a_0 x[n] + x_1 x[n-1] + \cdots a_{N-1}x[n-N+1]$$

which corresponds to a filter with "$N$ zeroes and $M$ poles" (using the Z-transform, $H(z)$ is a rational function), and that's the general expresion of a difference equation (see eg. here). Often, but not always: most LTI filters that appear in signal processing have $H(z)$ rational, but that is not necessary. So, the assertion is only partially correct. It's correct if we either restrict to filters with $H(z)$ rational, or if we allow infinite-order difference equations, or if we interpret it as an approximation (a LTI filter can be expressed with arbritrary precision by a rational $H(z)$, etc).

For continuous-time systems, it's analogous, using differential equations instead of difference equations.

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Thanks! (1) For continuous LTI system, how is convolution be represented as differential equations? (2) What about linear systems instead of LTI? –  Tim Aug 25 '12 at 23:37
@Tim: 1) for continous LTI, its conceptually similar but less straightforward. The convolution is an integral equation, which can sometimes be transformed to a differential equation. For a simple example, try $h(t)=\alpha e^{-\alpha t }$ (exponential average, analogous to a one pole filter). In general, the usual way is writing the Laplace transform; again, if it's a rational funcion in $s$, then the differential equation is straightforward. –  leonbloy Aug 26 '12 at 0:36
@Tim: 2) The property is only for LTI. Think of the recyprocal: any difference equation can be expressed as the equation above, and that corresponds to a LTI filter. –  leonbloy Aug 26 '12 at 0:38
... that is, as long as we restrict to linear difference equations with constant coefficients –  leonbloy Aug 26 '12 at 12:33
Thanks, leonbloy! I still don't understand how the case for continuous LTI is like? Can you give a detail example for going from convolution to differential equation? –  Tim Sep 7 '12 at 15:40