Signal processing and properties question

Question

We have the relationship between input and output: $Y(k)=|x(k+1)| + x(k)+ kx(k-5)$.

Find the output of the system when $x(k)=d(k)$.

What does this output represent?

Show if the system is linear, invariant, causal, stable, etc

Edit : I am familiar with sistem properties ,but what I mostly dont understand in the question is how to find the output when when $x(k)=d(k)$ and what it represents.To be more clear,what does it mean when when $x(k)=d(k)$.

Is $d(k)$ the discrete impulse function, i.e. $d(0) = 1, d(k) = 0$ otherwise? — Daniel R, Jun 13 '14 at 11:43

score 1 · Accepted Answer · answered Jun 13 '14 at 11:55

1

If $d(k)=\delta(k)$, i.e. the discrete unit impulse, then the response of the system to the input $x(k)=\delta(k)$ is its impulse response. However, since the system is not linear and time-invariant, it is not fully characterized by its impulse response. These are the system's properties:

it is non-linear, because the absolute value is used to compute the output signal
it is time-varying, because the time index $k$ appears as a multiplicative constant in the system equation.
it is non-causal, because in order to compute $y(k)$ the input at time $k+1$ must be known.
it is not BIBO-stable, because even for a bounded input, the output is not bounded due to the multiplicative constant $k$ which grows without bounds.

answered Jun 13 '14 at 11:55

Matt L.

10,636

How do I find the impulse response? – fsdd Jun 13 '14 at 12:12
@fsdd: Use $x(t)=\delta(t)$ and compute $y(t)$. – Matt L. Jun 14 '14 at 16:44
$Y(k)=|d(k+1)| + d(k)+ kd(k-5)$= { 0 for x ≠ 0 and 2+ k for x=0.} Is this correct? – fsdd Jun 15 '14 at 12:39

Signal processing and properties question

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