# Determining Causality From Discrete Impulse Response

Trying to wrap my mind around the concepts of this one...

Consider the following impulse response $h[n]$ for a linear, time-invariant system: $$h[n] =\left\{\underline{1} , -2, 2, -1\right\}$$ where n=0 starts at 1 for h[n]

• (a) Is the system causal? Why or why not?
• (b) Is the system stable? Why or why not?
• (c) Is the system FIR or IIR? Give a reason for your answer.
• (d) Does the system have memory? why or why not?

Based on my understanding I've concluded...

• (a): ?
• (b): Yes the system is stable because the summation of |h[n]| is less than infinity.
• (c): The system is an FIR (Finite Impulse System) because the duration of h[x] is not infinite. It has a finite number of values.
• (d): This system is not memoryless, therefore has memory. This is because in order to be memoryless (to my understanding) an impulse response should not have values at any times other than n=0. h[n] has multiple values

Updated the question to reflect how the values relate to time. The underlined 1 in h[n] denotes the origin n=0

• This is homework-style kind of question, the question won't be well-received unless you provide what you have tried or tell the community where you are stuck. – Gilles Feb 13 '18 at 6:36
• @OlliNiemitalo h(n) is the system's response to the input. where n is a unit in time. h is the impulse response of the system – crazyCoder Feb 13 '18 at 8:05
• All your conclusions are correct, but (a) can't be answered, as mentioned by Olli, because we don't know where in time to place those values given for $h[n]$. – Matt L. Feb 13 '18 at 8:44
• @OlliNiemitalo yes it is. I updated the question to reflect that – crazyCoder Feb 13 '18 at 17:13

As soon as you've figured out how the given values for $h[n]$ relate to the time index $n$ (which is unclear in the current problem formulation), it should be easy to decide whether the given system is causal or not by considering the input-output relation of a discrete-time LTI system:
$$y[n]=\sum_{k=-\infty}^{\infty}h[k]x[n-k]\tag{1}$$
where $x[n]$ is the input sequence, and $y[n]$ is the output sequence.
For the system to be causal, the output at time $n$ must only depend on the current value or on past values of the input $x[n]$. What does that imply for the impulse response $h[n]$?
• @crazyCoder: If you look at Eq. (1) in my answer you see that only the current and past values of $x[n]$ are used to compute $y[n]$ if and only if $h[n]=0$ for $n<0$. So that's simply the condition for causality. – Matt L. Feb 13 '18 at 22:09