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I'm still learning DSP and referring to Oppenheim video lectures.

In that lectures, differential difference equation is obtained for IIR filter design, in Lecture 14.

$$\mathcal{L}[\frac{\mathrm d}{\mathrm dx}y_a(t)] = sY_a(S)$$ $$\mathcal{Z}[\frac{y[n+1] - y[n]}{T}] = \frac{z-1}{T}Y(z)$$

And using these two equations, difference equation $z-1=sT$ is obtained.

To calculate the z-transform of the differential, $y(n+1)$ is used. By using that term, are we making our system non-causal..? (As our equation depends on future values)

Thank you for your time..

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Replacing the derivative by forward differences is just a way to define a transformation from the $s$-plane to the $z$-plane, and it has nothing to do with the causality of the resulting discrete-time filter. As an exercise you could try to do the same with backward differences.

Note that this method for transforming a continuous-time filter to a discrete-time filter is useless in practice because stability is generally not preserved, and the transformed frequency response may be drastically different from the original response.

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  • $\begingroup$ I know that it is not using due to stability problems. But I had the problem that it won't be causal as it was deduced by a non causal equation.. $\endgroup$
    – Ramesh-X
    Feb 13, 2016 at 3:15
  • $\begingroup$ @Ramesh-X: You could add that example to your question, so we can discuss it. $\endgroup$
    – Matt L.
    Feb 13, 2016 at 20:25
  • $\begingroup$ I understood that it won't be a problem for the causality of the system.. Thanks for your help..! $\endgroup$
    – Ramesh-X
    Feb 14, 2016 at 0:03

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