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The book states:

Causality implies that, if x1[n] = x2[n] for n <= n0, then y1[n] = y2[n] for n <= n0.

But isn't it always the case, that for same inputs we're getting same outputs except for the case of time-variance, which this sentence is not about?

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3 Answers 3

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Consider, for example, a moving average window. At any time $n$ the output $y[n]$ will equal the average $\frac{x[n-1]+x[n]+x[n+1]}{3}$. Now let us choose:

$$x_1[n]=\begin{bmatrix}1 \\ 1 \\ 2 \end{bmatrix}, x_2[n]=\begin{bmatrix}1 \\ 1 \\ 3 \end{bmatrix}$$

Hence for $n_0=2$ we have:

$$x_1[n]=x_2[n]\forall n\leq n_0$$

$$y_1[2]=\frac{1+1+2}{3}=\frac{4}{3}\neq \frac{5}{3}=\frac{1+1+3}{3}=y_2[2]$$ $$y_1[2]\neq y_2[2]\forall n\leq n_0$$

Basically, as all answered where stated, causality wishes to show independence of the output from any future input. The complicated form of definition comes from continuous-time where a system is causal iff: $$x_1(t)=x_2(t)\forall t< t_0 \Rightarrow y_1(t)=y_2(t)\forall t< t_0$$ An example of a non-causal system here would be a differentiation system as a differential is defined around the area of the point and for any point, $t_0$ will require an $\epsilon$ to the past and an $\epsilon$ to the future. This is why we require a soft inequality with respect to $t_0$. Discrete-time does not present such a problem.

In general, most physical systems are causal (they can't see into the future). More to the subject of signal processing, any real-time signal processing method is, roughly, causal. I did heard of systems (radio\TV) that delay the signal for a buffer of a second in order to do some processing and transfer it. I am not sure if this is called real-time, though it will probably be non-causal (otherwise, why delaying??). Any off-line method may also be causal or non-causal.

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This is a formal way of saying that the output of the system at any time $n_0$ only depends on past (and present) samples. The fact that $x_1$ and $x_2$ can differ in the future (times $n > n_0$) doesn't affect the output of the system until, and including, $n_0$.

This definition works regardless of time invariance.

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Causality implies that only current and past values of the signal impact the output. Now suppose that the system is not causal (output depends on values of $n>=n_o$) and then $x_1[n]$ and $x_2[n]$ are completely different after $n>=n_o$, then clearly y[n] will be different at time $n=n_o$ even though both sequences are equal before $n<n_o$. Therefore that property holds only for causal systems.

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