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Note that in this case you can see that the system is causal only from the given implementation. It's important to understand that you can't see it from the difference equation (if no initial conditions are given), and in general you can't see it from the transfer function either (if no region of convergence is given). The only case for which the expression ...

4

This is three years later, but since I don't see the real answer posted here, I will post it. The correct answer is that if we are literally interpreting the original statement as a purely mathematical claim, taken at face value, then it is incorrect. There do exist causal filters, even minimum-phase ones with nice closed-form Fourier domain expressions, ...

4

Purely by inspection of the block diagram the system is causal, because the output is the sum of the current input sample and stuff that's delayed -- there's no $z$ blocks in there to predict the future, just $z^{-1}$ block to react to the past. Also by your method of finding the transfer function, the system is causal -- with a $3^{rd}$ order numerator and ...

3

but what does the stability mean when talking about a system? It means that all poles are INSIDE the unit circle. and if so how does it turn out that when a system is causal and stable its also min phase. Sorry, you got this wrong. Causal, stable and LTI does NOT imply minimum phase. A simple counter examples is a one-sample delay. It's causal, stable and ...

3

Is there a way to make this filter non-causal? Remember that non-causal filters aren't possible to interpret in any case, because "non-causal" literally means there's output caused by input that comes later. Just because you can buffer something if latency doesn't matter doesn't mean you've actually built a non-causal system - your buffering adds ...

2

Try writing your equation (2) as $Y(e^{j\omega})=W(e^{j(\omega + \pi)})=X(e^{j\omega})H_1(e^{j(\omega + \pi)})$, and now try solving for $H(e^{j\omega})$. Remember that phase shifting by $\pi$ and $-\pi$ gives the same result.

2

The system is causal, provided that the recursion is forward; i.e., it's recursed for increasing $k$. Seeing that you are confused about causality tests, let me elaborate on it. Let's put the definition of causality from Oppenheim's Signals & Systems book : A system is causal if the output at any time depends only on values of the input at the present ...

2

Clearly, for negative values of $t$, the system needs to know the future in order to determine its output. Hence, the system can't be causal. Since the system is also time-varying (show it!), its response to an impulse doesn't say much about its general behavior, unlike it would be the case for a linear time-invariant (LTI) system. So the given system's ...

1

It should be clear that a property of a system, such as causality, cannot be determined by looking at its input signals. For a linear time-invariant system, it is its impulse response $h(t)$ from which properties such as causality or stability can be determined. Only the second definition in the question is correct: a causal LTI system has an impulse ...

1

There's a tendency when doing "pure" DSP to call a filter "noncausal" because you started by designing it to be symmetric around t = 0, but then you're running it with a bunch of delay, or because you're running filtfilt on the data (where you scan the filter forward, and then backward), or some other simulated non-causal behavior. But ...

1

Hint Substitute $s = t-1$. That gets you the equations in a more standard form $y(s) = ...$ Then go through the same excercise.

1

As near as we can tell by experiment, causality is nature's way of doing its thing. Causality says that if you have a system $y(t) = h\left(x(t), t\right)$, and it is causal, then $y(t_0)$ is dependent only on values of $x(t)$ for $t < t_0$*. Causality doesn't say you can't combine systems into larger systems (as a rather pertinent example, you can ...

1

HINT: Just write down the values of the output signal $y[n]$ for $x[n]=\delta[n]$ for values of $n$ from $0$ to $N-1$ (you don't need any specific value for $N$, just use $n=0,1,2,\ldots$ and then you'll see what happens at $n=N-1$). Then figure out what happens at $n=N$, and what consequences this has on the output values for $n>N$. You may be surprised ...

1

Overlap-add or overlap-save/scrap with zero-padded data are the common methods of using block based convolution on streaming data. Pad the convolution (FFT/IFFT fast, or linear) by at least the length of the impulse response above your desired noise floor, minus 1. The basic idea is that these methods save the remainder of the impulse response that doesn't ...

1

Although this question is over 2 years old now, I think it's interesting to consider the solution, assuming that the original interpretation was incorrect, and that $u[n]$ represents the system input, not the unit step sequence. If so, then the first thing to recognize is that the system is just an accumulator. For a time-domain demonstration, simply make ...

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