# Causality, Linearity, and Time Invariance for Systems Described by Linear Constant Coefficient Differential Equations

I am currently using Signals and Systems by Alan Oppenheim as a reference to learn about LTI systems.

Before introducing systems represented by linear constant coefficient differential equations, the author first considers general systems of the form $x(t) \rightarrow y(t)$. When discussing linearity of systems, he introduces the "zero in-zero out" property of linear systems (not necessarily time invariant or causal), that is, given a linear system, if the input $x(t)$ is zero for all $t$, then the output $y(t)$ must be zero for all $t$. In addition, a linear system is causal if and only if $y(t) = 0$ for all $t < t_0$ if $x(t) = 0$ for all $t < t_0$, a condition known as initial rest, from which I concluded that initial rest for linear systems implies causality but not time invariance.

Now given a linear constant coefficient differential equation $$Ly(t) = x(t),$$ where $L$ is a linear differential operator with constant coefficients, the author emphasizes that although the differential equation is linear, the system that it represents may or may not be linear depending on the initial conditions. I find this confusing because given two input output pairs $x_1(t) \rightarrow y_1(t)$ and $x_2(t) \rightarrow y_2(t)$, we have $L(ay_1(t) + by_2(t)) = ax_1(t) + bx_2(t)$ for any constants $a, b \in \mathbb C$ by virtue of the linearity of the operator $L$. It seems that $ax_1(t) + bx_2(t) \rightarrow ay_1(t) + by_2(t)$ regardless of the initial conditions, why is the system not necessarily linear in that case? What am I missing here?

In addition, the author states that for the system to be linear, the initial conditions have to be zero, and for it to be causal and time invariant in addition to being linear, its initial conditions must obey initial rest. But isn't initial rest in general a condition required for causality, not time invariance? Besides, doesn't this imply that a differential equation can only represent a causal system if the input is zero before some time? Given an input that is nonzero for all time, how do we impose causality via initial conditions that correspond to initial rest? I hope my question makes sense. Thank you in advance.

• I'm not sure if this is a duplicate of this question. Does that one clear out your doubts? Commented Jan 29, 2018 at 19:13
• Not really, I'm still having a hard time understanding the issues I raised in the question.
– user33568
Commented Jan 29, 2018 at 19:29
• admittedly, the concepts of linearity, cuasality, time-invariance and their implications on the LCCDE representations of systems are the most confusing and the least explained parts of that book. Commented Jan 29, 2018 at 22:28
• Yeah, I even tried watching his MIT lectures and it is still not clear to me. It's really bugging me. Can you recommend any other references? I'm not a big fan of Haykin either, I think Oppenheim's book is better overall.
– user33568
Commented Jan 29, 2018 at 23:10
• Interestingly the first edition of Opp. Signals and Systems gives a better and more indepth discussion of the LCCDE issue... I recommend its 1st edition. Haykin , Proakis, Papoulis, Lathi are among the possibilities but afaik this discussion is always obscure. Commented Jan 30, 2018 at 0:18

Note that the system is not uniquely characterized by the linear differential operator. You need auxiliary conditions, and dependent on these conditions, the system may be linear, causal and time-invariant or not. So if you just use the operator $L$ to show that the system is linear then you're actually not showing anything because $L$ is a linear operator by definition.

The output of a system described by a linear constant coefficient differential equation can be split up into two contributions: the zero-state response (ZSR) and the zero-input response (ZIR). The ZSR is the response of the system with zero initial conditions, and, consequently, the ZSR is fully determined by the input signal. The ZIR response is the response of the system caused by non-zero initial conditions, and it is independent of the input signal. Since non-zero initial conditions cause an output term that does not depend on the input signal, scaling or shifting the input signal will not affect that term. Consequently, a system with non-zero initial conditions can neither be linear nor time-invariant, because if it were, the output should scale and shift according to the scaling and shifting of the input signal.

Causality simply means that the system's response at a given time does not depend on the future of the input signal. Clearly, causality is a property of the system, so it cannot depend on the input signal. A causal linear time-invariant (LTI) system is characterized by an impulse response $h(t)$ that satisfies $h(t)=0$ for $t<0$, and it is irrelevant whether the input signal is zero before a certain time or not.

Also take a look at this related answer.

• When solving differential equations in physics and engineering the input is sometimes “complexified” as in this example: math.stackexchange.com/questions/48099/… and the real/imaginary part of the output is computed as appropriate. Is this valid even if the auxiliary conditions are nonzero?
– user33568
Commented Feb 13, 2018 at 21:01
• @0MW: You can take the real part of the ZSR to get the ZSR corresponding to the real part of the input. Commented Feb 13, 2018 at 21:59
• And when we talk about the natural response of a system, are we referring to the ZIR or the homogeneous solution with initial conditions applied to the homogeneous solution + particular solution?
– user33568
Commented Feb 14, 2018 at 7:16
• @0MW: That term is used differently by different people (and even by the same people ...). Read this. Commented Feb 14, 2018 at 8:56