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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

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