Electronic – Why consider Linear Time-Invariant systems

linearsignalsignal processing

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?

Best Answer

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.

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