Classification of a system

Question

I am preparing for an examination and have a study guide that I feel has a couple of errors. The questions concern the classification of discrete time dynamical system. Here are the problems that I am questioning.

Questions

1. Is the system $ h[n] = \delta[n+2] + \delta[n+1] + \delta[n] $ a causal system?

   Given answer: Yes

2. Is the system $ y[n] = x[n] + 1 $ a linear system?

   Given answer: Yes

Here is why I think that both of these are incorrect.

Contesting the Answers

Is the system $ h[n] = \delta[n+2] + \delta[n+1] + \delta[n] $ a causal system?

The output of a causal system depends only on past and present inputs. If $ h[n] $ is the output of the system, then I believe that this system is causal (is this true?). However, $ h[n] $ is what we have used to denote the impulse response of a system. Is it not true that a necessary condition for a causal system is that the impulse response $ h[n] = 0 $ for all $ n < 0 $ ? here, we can see that $ h[n] = 1 $ for $n \in \{ -1,-2 \} $. If $ h[n] $ is indeed the impulse response, then wouldn't the output be $ y[n] = x[n+2] + x[n+1] + x[n] $. Clearly, such an output depends on future values of the input. Thus, I believe that the system is anti-causal.

Is the system $ y[n] = x[n] + 1 $ a linear system?

A linear system must obey the principle of superposition. Let's consider additivity,

Let $$ y_1[n] = x_1[n] + 1, ~~~~~ y_2[n] = x_2[n] + 1 $$ then: $$ y_1[n] + y_2[n] = x_1[n] + x_1[n] + 2 $$ It would appear to me that this test fails because of the constant. Is this correct? This is the part where I am the unsure.

If we check homogeneity then we have the following; $$ \alpha y[n] = \alpha (x[n] + 1) $$ where $\alpha$ is a constant.

It would appear to me that this test shows that the system has the homogeneity property since it can be written in the same form as the original system, only scaled by the constant $\alpha$.

Because this system fails the additivity test, I would think that we can classify it as a non-linear system.

Going further, I believe it is called an affine system (i.e. a linear system shifted). Is it not true that a linear system must pass through the origin?

Answer 1

1. You’re correct, that’s not a causal system.

2. You’re correct, that’s not a linear system. The additivity property doesn’t apply, but neither does the homogeneity. To see this, consider $$\alpha y[n] = \alpha (x[n] +1) = \alpha x[n] + \alpha$$ Clearly, that’s a different result than you would get from applying $\alpha$ to the input: $$\alpha x[n] +1$$