# How does an LTI system produce multiple values from a unit impulse

I understand that in a discreet system, an impulse is 1 at the origin point and 0 everywhere else. I've seen many examples showing the impulse response of an LTI system to be many points, have many values. How can this be when the impulse it's given is really just 1?

Does the system react to the zeroes as well?

Does the system react differently to zeroes after it encounters a one? If so then I've seen systems who have non zero points in the impulse response where supposedly the non zero points is a reaction to a zero BEFORE it encountered a one. Does the system have memory and can alter it's previous calculation after hitting the one?

How many values will an impulse response have? Is there a rule? Does the system produce values independent of the unit impulse?

I'm trying to understand this in the only way I can given my background. Which is programming. I imagine the system as a loop or process which when given a single input produces a single output. Ex. Given 1 it will multiply it by some constant.

• Consider an example. I loan you \$1 and you repay me 5% interest compounded monthly. The principle will be (.95)^m. where m is the number of months – Stanley Pawlukiewicz Dec 25 '18 at 18:59

You are probably confused by the memoryless systems whose output due to an impulse is just another impulse; a single value just as you expect.

However, there are systems with memory too, and some of them have infinite memories; their current output is affected by an impulse applied infinite samples ago. Incidentally they have infinite length impulse responses and called as IIR systems.

These infinite memory IIR systems are somewhat similar to physical systems described by second order differential equations; those exhibit simple harmonics motion with (damped) sinusoidal response which continue to oscillate indefinetely once an initial impulse is applied.

So there is nothig wrong or surprising with outputing values, even though there is no current input value, which happens for systems with memory.

Linear and Time-Invariant (LTI) systems have the properties of linearity AND time-invariance. Both combined entail that such systems can be uniquely described (or defined) by a response to a single impulse, called THE impulse response. The reasons for this are the following:

• An LTI system output $$y$$ boils down to an invariant linear combination of inputs, through convolution
• each discrete signal can be split into a linear sum of weights and delayed "impluses".

So we now know that, given it is LTI, a system can be probed through the use of a single impulse. let us come back to what happens when the system, or its properties, is unknown.

With your computer background, you may understand that the series of 'numbers' 00001000 and 00000100 can produce different outputs for a given program $$\mathcal{P}$$. A simple one is parity. If you sum bits modulo 2, both yield the same result. But if the program does the conversion to an integer, if interpreted as bits $$b_7b_6b_5b_4b_3b_2b_1b_0$$: they yield 8 or 4. But they could be interpreted as an impulse sequence. Hence, the conversion of bits into an integer is not a shift invariant system (it is not linear either).

But if you don't know what the program does, this is not very informative. If you have an unknown program that accepts 8-bit inputs, you can try to understand its behavior by trying all possible inputs, hoping it is repeatable, i.e. $$\mathcal{P}$$ does not change in behavior over time (or is time invariant). With 8-bit inputs, there are 2^8 possible inputs, and you can derive a function linking inputs to outputs.

For an unknown signal processing system $$\mathcal{P}$$, inputs are infinite sequences of real values, and it is in general impossible to probe the system with an infinity of possibilities.

But, under some generic conditions, some systems can be described partially or fully with a some well-chosen input sequences. Linear and Time-Invariant (LTI) systems are very interesting in this respect. Their structure is such that they can be fully described by a single response to an impulse (the "impulse response"). It uses zeroes after and before, similarly to a program that would sum all digits at once. Such a sum is a special case of a convolution, the operation behind LTI systems.

For your questions : Most of such systems have memory. The memory-less ones only amplify (or delay causally) the inputs. The impulse response can be very short as well of infinite length. The rule is the rule of convolution.

And "Does the system produce values independent of the unit impulse?" Any input can be written as a linear combination of delayed impulses, hence their output is the linear combination of delayed "impulse responses", so, no.

Now some visual demos: