A pop-quiz featured a system analysis of the following system: $y[n]=\sum_{n=0} ^{-\infty} x[n]$

The professor argued that this system exhibits the following properties:

  • Linearity
  • Causality
  • Instability
  • Memory
  • Time-invariance

In general the system seems nonsensical, as it sums all samples from the beginning of time up until now, even though these are not accessible to $x[n]$. Colleagues argued that the index $n$ might be "local" to the sum, translating to a more sensical description $y[n]=\sum_{k=0} ^{-\infty} x[n+k]$ with a distinguished index $k$.

As provided by the answer below, the most likely meant system is $y[n]=\sum_{k=-\infty} ^{n} x[k]$

We do not intend to use dsp.stackexchange as a lazy solution but as a source of discussion, as this topic is of personal curiosity.

  • 1
    $\begingroup$ If you're doing theoretical analysis, it's fair to claim knowledge of $x[n]\ \forall \ n \in (-\infty, 0]$. That's not the problem. However, if you look at the expression given, you'll see that it sums from $k = -\infty$ up to $k = 0$ (variable change mine) regardless of the current time. So it's just a constant. $\endgroup$
    – TimWescott
    Oct 31, 2022 at 20:55
  • $\begingroup$ For that matter, for really theoretical analysis, you can claim knowledge of $x[n]$ for all time -- it really depends on the nature of the analysis you're doing. E.g., Fourier transforms assume that a signal is known for all time, even though that's physically impossible. $\endgroup$
    – TimWescott
    Nov 1, 2022 at 15:08

1 Answer 1


The given equation relating the output to the input doesn't make sense. Note that on the left-hand-side the variable $n$ is used as time (or anything else) index, but on the right-hand-side it is used as the summation index. What is probably meant is the following system:


which is just an accumulator, and which has all the properties listed in the question.

Note that $(1)$ can be implemented recursively:


The system's impulse response is the unit step sequence: $h[n]=u[n]$. It is straightforward to show that $(1)$ is equivalent to

$$y[n]=(x\star h)[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]=\sum_{k=-\infty}^{\infty}x[k]u[n-k]\tag{3}$$


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