# Why consider Linear Time-Invariant systems?

Okay this could be a very silly question but I am asking it anyway.

Why do we consider in most cases of signal processing that the system is Time-invariant?

Is it because most signals are linear and time-invariant or is there a more compelling reason to consider a system as LTI while looking at problems in this field?

• In a lot of cases you want your system to be time-invariant! Or do you want your audio equipment, car brake control, or heating system to behave one way from 12:00 .. 12:15 and another way from 12:15 .. 12:30? Commented Nov 4, 2013 at 6:39

is there a more compelling reason to consider a system as LTI while looking at problems in this field?

What makes the analysis of LTI systems attractive are the following:

Linearity:

• If $y_1(t)$ is the output due to the input $x_1(t)$ and
• $y_2(t)$ is the output due to the input $x_2(t)$ then
• $y = ay_1(t) + by_2(t)$ is the output due to the input $x = ax_1(t) + bx_2(t)$.

Time (or shift) invariance:

• If $h(t)$ is the output due to the input $\delta(t)$ then
• $h(t - \tau)$ is the output due to the input $\delta(t - \tau)$.

We then call $h(t)$ the impulse response of the system.

If and only if the above are true of a system do we have:

$y(t) = h(t) * x(t) = \int_{-\infty}^{+\infty}h(t-\tau)x(\tau)d\tau$

and

$Y(s) = H(s) X(s)$

Now, no real LTI system is truly LTI but are effectively so and thus we may use the above "tricks" to analyze them.

You mix two independent parameters of the system: linearity and time-invariance.

Linear systems are the ones which have linear relationship between outputs and inputs. In mathematical terms, the system is linear if for every input vector $\bar{x}$ the output vector $\bar{y}$ is given by:

$$\bar{y}=A\bar{x}$$

where A is some matrix representing a linear transformation.

Linear systems are the most interesting ones because the majority of systems are either linear, or they can be approximated by linear systems.

Furthermore, any non-linear system can be approximated by linear equation at any point. By numerically integrating these linear equations over consecutive points you may solve the initial non-linear equation (with some error though).

So, the importance of linear systems arise from the fact that we know how to treat them mathematically and computationally, and that any system may be analyzed in a framework of linear systems.

Time invariance just states that the parameters of the system itself do not change over time. The inputs and the outputs may change, but the system is the same over the time period of interest.

If the system is not time-invariant it may be either:

• Partitioned into non-overlapping time periods during which the system is time-invariant
• Approximated by time invariant systems over short periods of time. Integration of these approximations will provide an approximate solution of the initially time-variant system.

In summary:

LTI systems theory is the most fundamental in signal analysis and applies for much wider spectrum of problems you might've been initially guessing (even non-linear and time-varying).

Mainly because most systems are time invariant, and assuming time invariance generally makes solving problems much simpler. As soon as you throw out linearity and/or time invariance, things can get very complicated.

Signals are not linear time invariant. (A time invariant "linear" signal could be a constant , a particular case of useless signal which doesn't transmit any information).

We consider linear time invariant systems in signal processing, but also non-linear systems are present in a lot points of the signal path: mixers, samplers, limiters, compressors ...

By the way, it's true that it feels like linear is important. In my degree there are courses with "linear systems" in the title, but there are none called "non-linear systems". Maybe this is because each non-linear system must be studied apart, but you can study every linear system using the same set of tools.

They even tell you that non-linear systems are bad, because real systems are modelled as ideal linear systems with some non-linear (bad) effects. But they are essential (the systems, not the undesirable effects).