# Output of marginally stable systems

In' b.p lathi's book of systems and signal it is written that

"marginally stable systems have one important application in the oscillator, which is a system that generates a signal on its own without the application of an external input. Consequently the oscillator output is a zero input response "

By this statement I have two problem:

1. if let's say oscillator system is a simple L-C circuit then how can it produce output without input and if it does produce then isn't it a violation of linearity?

2. and if initial conditions are zero then how can L-C circuits have zero input response? And then in that case how does it generates signal without initial conditions?

A second-order system without damping (such as an ideal LC circuit) will not produce any output without input and with zero initial conditions. It will produce output only if there is either a non-zero input signal or non-zero initial conditions.

One very common definition of linearity in system theory is that if $$y_1(t)$$ and $$y_2(t)$$ are the responses to input signals $$x_1(t)$$ and $$x_2(t)$$, respectively, then the response to $$x(t)=a_1x_1(t)+a_2x_2(t)$$ is given by $$y(t)=a_1y_1(t)+a_2y_2(t)$$. According to this definition, any system producing an output with zero input must be non-linear. Note that time-variance is not sufficient. In this sense, a system with non-zero initial conditions is non-linear because it produces an output with zero input.

1. An LC-circuit without input produces an output signal due to non-zero initial conditions, and - according to above definition of linearity - it is non-linear.

2. The circuit does not produce any output if the input as well as the initial conditions are zero. In practice, an oscillator is triggered by noise.

Note that another definition of linearity views the initial state not as part of the system but as another input. Defining linearity in this way, the LC circuit in your example with non-zero initial conditions is still linear, even without a signal at its input. It produces an output due to the initial conditions, which are now part of the system's input.

An ideal LC circuit has no resistance, and hence no resisitive loss mechanism. Therefore if it starts to oscillate (due to, for example, a non-zero initial voltage or current stored in the L or C), then it will oscillate indefinetely on its own, yielding a situation as described in your question.

Zero input response of a linear system is the response under no input excitation but due to non-zero initial conditions.

Note the distinction between the terms linear time-invariant (LTI) and the broader term linear alone. The discussion in the Lathi's book is about linear systems in the more general sense.

Note also that a linear (but not necessarily LTI) system can have non-zero initial conditions, and hence zero-input response; but a strictly LTI system cannot have non-zero initial conditions, because one property of LTI systems is the initial rest property. Hence zero-input response of LTI systems is zero by definition.

So by definition a system with zero-input response cannot be LTI, but it can be linear nevertheless...

• Time-variance doesn't actually play a role here. If the initial conditions are viewed as part of the system (Oppenheim), then a system with non-zero initial conditions is non-linear (instead of linear but time-varying). If the initial conditions are viewed as another input (Lathi) then a system that would be linear according to both definitions with zero initial conditions, is still linear with non-zero initial conditions. Jul 21, 2020 at 10:32