Signals and systems : why do we study causal signals?

Till now I have read that causal signals are right sided and anti-causal, left sided.

• Why did we need to classify a signal with respect to its position?
• What is it's physical interpretation?
• Similarly, why are LTI systems classified into causal and non-causal systems ?
• please note that the fundamental notion of causality stems from philosophy of physics and is borrowed from there by signal processing (which is fundamentally a mathematical subject and therefore do not pay any attention to the physical meaning of time or space, other than being mathematical objects). Therefore you shall better consult into some philosophy of phyiscs literature for cause and effect. Nov 3 '16 at 9:37
• to pick up on @Fat32, certainly when simulating or emulating a system, you need not simulate causality. when processing a sound file, i can arbitrarily set my "present time", $t$ to wherever i want in the sound file and look ahead to "future" samples and include them in my convolution summation (or whatever process). i like doing that so i can precisely simulate "zero-phase" filters with symmetrical (and acausal) impulse response. but real-time processing must be causal. Nov 3 '16 at 18:13

An LTI system (or even a system that isn't L or TI) that is "causal" has a prayer of being realized in real time whereas an "acausal" system cannot ever be realized in real time because an acausal system responds to input from the future. A causal system responds only to input of the present and/or input from the past.

For LTI systems, it can be shown that the output $y(t)$ depends strictly on the input $x(t)$ and the system impulse response $h(t)$. This dependence is expressed as the operation we call "convolution":

$$y(t) = h(t) \circledast x(t) \triangleq \int\limits_{-\infty}^{+\infty} h(\tau) \, x(t-\tau) \, d\tau$$

or

$$y(t) = \int\limits_{-\infty}^{+\infty} x(\tau) \, h(t-\tau) \, d\tau$$

If and only if $h(t) = 0$ for all $t<0$, then the output $y(t)$ has no dependence on future values of the input, $x(t)$.

$$y(t) = \int\limits_{0}^{+\infty} h(\tau) \, x(t-\tau) \, d\tau$$

or

$$y(t) = \int\limits_{-\infty}^{t} x(\tau) \, h(t-\tau) \, d\tau$$

That is the salient difference between causal and acausal systems.

Now since the impulse response $h(t)$ is called "causal" if it is zero for all time before $t=0$ (meaning the impulse response does not respond before the impulse hits the system input), that definition was extended to all signals, even signals that are not an impulse response.

The reason that it is significant is that it turns out that the real and imaginary part of the frequency response of a causal system must be Hilbert transform pairs; they must be related to each other as Hilbert transforms of each other.

• For me your last point is easy to understand by an analogy. If negative frequencies of a signal are filtered away by multiplying them by zero in the frequency domain, then the real and imaginary parts of the signal become Hilbert transform pairs: all frequencies rotate to the same direction with the phase of the imaginary part lagging the real part by 90 degrees. Conversely, an impulse response can be made causal by multiplying it by zero at negative times. The two actions just have the time and frequency domain swapped and the Fourier transform is its own inverse, up to a sign. Nov 3 '16 at 9:07

I think the "causal signal" is simply borrowed from the "causal system". For a system, the "causal" constrain is meaningful and fundamental, i.e., if the input does not occur, the system should not produce any output with respect to the input values. Then, for a LTI system, the "causal" nature means $h(t) =0, \text{ for } t<0$. Then, the concept is extended to the signal. However, since it is meaningful for a "non-causal" signal, this concept is not important.

In short:

(1) For a practical system, "causal" is a basic requirement, especially for analog system, a "non-causal" analog system does not exist indeed. "causal" comes from the time, i.e., what you do now should not be affected by the unknown future.

(2) For a LTI system, the "causal" is equivalent to $h(t)=0, \text{ for } t<0$.

(3) Then, the "causal" concept is extended to signals, i.e., if a signal $x(t)=0, \text{ for }t<0$, then $x(t)$ is called a "causal" signal. It is just because they are similar in their forms.

(4) But, unlike "non-causal" system, "non-causal" signal is meaningful and common in the world.

• can you explain it more, still I didn't get the answer. Nov 3 '16 at 6:50

I think a mathematical convenience is taken too literal here. The reason we are interested in causal systems is because we cannot differentiate robustly. In other words, theoretically there is no problem in a system where we depend on the fifth derivative of the input. But in reality measuring the fifth derivative is impossible except a set of physical systems of measure zero. That's why we never implement a PID controller as a pure derivative but with a lead/lag filter. I still don't know why we insist on teaching that way.

Since causal systems only depend on the causal part of the input it is mathematically equivalent to writing the current state as the initial condition of the state vector and running it along. Hence we can safely assume that at $t=0$, $x(0)=x_0$ and $x(t)=0$ for all $t<0$. Notice that having all values of the state zero for negative $t$ values is a convention. You can take arbitrary values as long as the initial condition is admissible. That's why we deal with left sided signals. Because we don't care.

The key assumption here is that our choice of causality is completely artificial! There is no causality dictated by the physics, say, in $F=ma$. Nothing forces us to choose force is due to position or vice versa. This is all because of engineering conveniences.

This point of view is the central observation of the so-called behavioral approach to control systems. Though you can't do much with it yet, it is a great way to understand control systems properly. The late great Jan Willems was the leading actor of that view. You can read about the weird anticausality of a door closing mechanism here because of our strange choice of input output mapping. Moreover, the so-called descriptor systems are such systems where the system is not restricted to be causal.

• Causality is implied by the ontological existance called the physical Nature. Behaviour of Nature is described by (mostly inexact) laws of physics in terms of mathematical equations which create a theoretical (mathematical) representation of Nature. Causality is not a mathematical concern but a Natural property. Hence mathemaical equations that quantify physical laws do not have restrictions of causailty. It's the metaphysics of Natura that empirically disables anti-causal working. Aug 4 '17 at 12:42
• @Fat32 Read the paper I've linked to. Causality is a construct in mathematical modeling. Aug 10 '17 at 9:42
• That is just a paper on control theory. Anyway; control theory uses mathematical models of reality. For example accelaration of a mass is a physical reality, and its mathematical model is $F=ma$. So is for causality. Causality is a physical reality and its mathematical model is $h(t)=0$ for $t<0$ for LTI systems, for example. For non LTI systems basic I/O relation is used. Mathematics alone do not care for real, imaginary, causal, noncausal things. It's the physics which care. Aug 10 '17 at 9:54
• @Fat32 $F=ma$ is not causal. That's the whole point of the paper. Just read it. Aug 10 '17 at 9:56
• $F=ma$ is just a mathematical equation no more important than $y=mx$ . Of course it won't be causal. causality is a phsyical law which states that future physically does not exist (yet); i.e., no cause from (non-existant) future can result in a current effect (response). But as a mathematician you can talk about tomorrow, next year, one bilion year etc. Draw a graphics from $t=-\infty$ to $t=\infty$ now you are plotting future values of a signal ? Hence mathematics does not care about past or future. Aug 10 '17 at 10:03