Natural, Forced and Total System Response - Time domain Analysis of DT

Question

I am a self-learner of DSP by reading related books.

While reading Time domain Analysis of DT (Discrete Time) system, I suddenly got introduced to difference equation representing a system. It then further stated -

The response of any DT system can be decomposed as :

Total Response = Zero State Response + Zero Input response

Where Zero state response is system response due to input alone and zero input response is when input is zero.

It goes further on to explain Natural, Forced and Total System Response - but unfortunately I am unable to map it to a concept.

For me its like response of a car engine (example per say or anything - a system) is like response when engine is not started + when engine is started - but this does not makes any sense for me. The engine system why should it consider zero input response at all. Its just a change of its state.

Similarly I am trying to understand Natural / Forced response but do not get the concept quiet clear. Also What is the role of studying them and where can we apply these concepts?

What is meant by recursive and non-recursive system - though my understanding is when system output is fed again into system as its input, it would then make the system call as recursive system - but why does a system need to do that?

Answer 1

First of all, many textbooks use the following definition (but see this answer for a different one):

zero-state response (ZSR) = forced response

zero-input response (ZIR) = natural response

The zero-state response is the response of a system to an input signal given that the initial state of the system is zero. The zero-input response is the output of the system due to a non-zero initial state. For a discrete-time system, the initial state is given by the values stored in memory (i.e. in the delay elements). Since we're dealing with linear systems, the superposition principle can be applied, and the total response of the system is given by ZSR+ZIR. So if the initial state is zero, the total response equals the ZSR, and if there's no input signal, the output is given by the ZIR, which always decays if the system is stable.

In analog circuit theory you have exactly the same. There the initial state is the energy stored in capacitors and inductors. Your car engine analogy is less fortunate because it doesn't really work. You could come up with a mechanical analogy though as long as the system is linear.

It might be a good idea to formulate a new question concerning recursive systems (after having searched the internet and this site). But you're right that a system is recursive if not only the input but also past output values determine the current output. The reason why this is done is the fact that in this way certain system properties can be implemented much more efficiently than by only using the current and past input values. The problem with recursive systems is that they can be unstable, and that quantization effects (in signals and filter coefficients) can more easily result in numerical problems than for non-recursive systems.

PS: If you teach yourself I strongly recommend MIT's open courseware. You should probably start with 6.003 Signals and Systems.