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The system: $$ T(x[n]) = ax[n] + bx[n-3] $$

For me it seems that the system is linear:

$$ \begin{align} T(\alpha_1x_1[n]+\alpha_2x_2[n]) & = a(\alpha_1x_1[n]+\alpha_2x_2[n]) + b(\alpha_1x_1[n-3]+\alpha_2x_2[n-3]) \\ & = \alpha_1(ax_1[n] + bx_1[n-3]) + \alpha_2(ax_2[n] + bx_2[n-3]) \\ & = \alpha_1T(x_1[n])+\alpha_1T(x_2[n]) \end{align}$$

Thus it's linear, however in the presentation I got it says it's not linear. (without reasoning) Where I'm making the mistake (or maybe there is a mistake in the presentation)

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  • $\begingroup$ Could you please share a document saying it is not linear? Maybe from the context we could get it $\endgroup$ – Laurent Duval Nov 28 '18 at 18:31
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    $\begingroup$ It could be hard as I don't have the access to the presentations. But it was in no context, just list of a few systems with the properties like linearity, time in-variance, causality, memorylessness, stability. $\endgroup$ – sswwqqaa Nov 28 '18 at 19:35
  • $\begingroup$ Maybe you could talk to the source to double check there is no misunderstanding $\endgroup$ – Laurent Duval Nov 28 '18 at 20:53
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I believe there's either a mistake in the presentation or the presentation is using a different definition of linear.

For example, the system is linear in $x$ from a system perspective, but it's affine in $x[n]$ (and, therefore not linear) because of the $bx[n-3]$ offset.

On this site, we tend to go with the system definition rather than split hairs about linearity versus affine-ness.

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    $\begingroup$ The system is just a two-tap FIR filter. So I wouldn't say it's affine. It would be affine (if one was willing to make that distinction) if it were something like $y[n]=ax[n]+b$, i.e., just adding a constant (like a non-zero initial condition). That would make it non-linear in a signals&systems context. $\endgroup$ – Matt L. Nov 28 '18 at 17:26
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    $\begingroup$ Indeed, I have never heard about "being affine" in the ordinal variable. $\endgroup$ – Laurent Duval Nov 28 '18 at 18:29
  • $\begingroup$ @MattL. Understood! I've just seen some mathematicians get their knickers in a twist over systems like in the original post. Without knowing the source of the claim that it's not linear, it's hard to rebut it. $\endgroup$ – Peter K.♦ Nov 28 '18 at 18:43
  • $\begingroup$ I can't resist pedantry here :) maybe $a$ and $b$ are elements of a non-additive magma? $\endgroup$ – Laurent Duval Nov 28 '18 at 20:17
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    $\begingroup$ @LaurentDuval 😅😂🤣😜 $\endgroup$ – Peter K.♦ Nov 28 '18 at 23:43
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[Note: it may happen that a teacher makes a oral mistake, that puzzles the audience. So here is an alternative explanation on this system being non-something]

This system is, as far as Peter K., Matt L. and I know, nicely linear. You already did the computations. With a little more work, among classical properties, it is also time-invariant, causal, stable.

The only basic property it does not possess is "to be memoryless", unless one of the constant $a$ or $b$ (including both) is zero (see for instance an extended conception of a memoryless system from What is a memory less system?).

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    $\begingroup$ It's indeed linear and time-invariant, not affine. It would only be affine if in its output there were a constant term that is independent of the input signal, which is not the case here. It's just a plain FIR filter. $\endgroup$ – Matt L. Nov 28 '18 at 20:42
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    $\begingroup$ hmm let me put a (n almost) verbatim copy of your answer and wait to see if I get any upvotes :-)) and see if it is only you ;-) $\endgroup$ – Fat32 Nov 29 '18 at 9:12
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    $\begingroup$ Go ahead and I'd suggest you to add the non-additive magma notion $\endgroup$ – Laurent Duval Nov 29 '18 at 9:31
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A system $\mathcal{T}$ is linear if its response to a weighted sum of two signals equals the weighted sum of its individual responses to those two input signals:

$$\mathcal{T}\big\{\alpha x_1+\beta x_2\big\}=\alpha\mathcal{T}\big\{x_1\big\}+\beta\mathcal{T}\big\{x_2\big\}\tag{1}$$

with arbitrary constants $\alpha$ and $\beta$, and arbitrary input sequences $x_1[n]$ and $x_2[n]$. A system $\mathcal{T}$ satisfying $(1)$ is completely characterized by its impulse response $h[n]$, and its input-output relation can be formulated as a convolution of the input sequence with its impulse response:

$$\mathcal{T}\big\{x\big\}=\sum_{k=-\infty}^{\infty}h[k]x[n-k]\tag{2}$$

It is easily shown that the given system

$$\mathcal{T}\big\{x\big\}=y[n]=ax[n]+bx[n-3]\tag{3}$$

satisfies $(1)$, and that it is characterized by the impulse response

$$h[n]=a\delta[n]+b\delta[n-3]\tag{4}$$

Clearly, its input-output relation $(3)$ can be written as a convolution sum $(2)$. Equivalently, the $\mathcal{Z}$-transform of its response is given by the multiplication of the $\mathcal{Z}$-transform of its input sequence and its transfer function $H(z)=\mathcal{Z}\{h[n]\}$:

$$Y(z)=X(z)H(z)\tag{5}$$

By contrast, an affine system does not satisfy $(1)$, and it cannot be characterized by an impulse response or, equivalently, by a transfer function. The given linear system $(3)$ could be made affine by adding a constant $c$ ($c\neq 0$) to its output:

$$\mathcal{T}\big\{x\big\}=y[n]=ax[n]+bx[n-3]+c\tag{6}$$

This input-output relation cannot be formulated in terms of a convolution $(2)$, and it can easily be checked that the system $(6)$ doesn't satisfy $(1)$. Such a system is not linear, since part of the output (the constant $c$) does not depend on the input signal $x[n]$.

There is no reasonable definition of linearity according to which the given system $(3)$ could be classified as being non-linear.

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