The Frequency Response of a linear system ?


I watched the ( regarding this topic from the YouTube channel Physics videos by Eugene Khutoryansky. It helped me visualize how lateral numbers are useful to understand this phenomenon.

But it got me also confused about linear systems in general. Can somebody by starting of explaining what is a linear system in the first place and what are some real world engineering applications?

Thanks in advance.

In: Physics

Well linear system is a system where output if proportional to input. So basically output is linear function of input. To make it easier to imagine, system is e.g. electric motor, input is applied voltage, response is your motor revolution per second. So if your electric motor was a linear system, two times bigger voltage would mean two times higher revs. This is just an example to give something to imagine, but in reality its not an linear system. And you can describe basically anything in such way. So system is a plane, input is how much did pilot push the yoke, response is flight angle.

Linear systems have wonderful feature, that they satisfy the property of superposition. For understanding this, let as assume that we have some complicated input, and its possible to represent this input as sum of some simple inputs (we have function Fc – complicated, which is equal to F1+F2, which are simple). Then response to the complicated Fc input, is equal to response to F1 + response to F2. This is true for linear system, id does not have to be true for non-linear systems. Knowing this you can combine it with Fourier Theorem (that you can represent any function with sum of series of sin), and you can easily calculate response to different inputs.

So simplified flow will be like this: you test your system with e.g. impulse input (impulse contains of very big number of sin functions), and you can tell what is your system response to each frequency. Then you can take this information, and you can calculate output for arbitrary input. You first disassemble your input into sin functions, then you know response for each of those from previous experiment, and your output is just sum of those responses.

And what is neat is that you do not do it step by step like I described, those calculations can be done very neatly using complex numbers, many programming languages will have necessary functions available as ready libraries. And one last thing, real system are rarely strictly linear. But we can make some tricks and assumptions, so that we treat our system as liner. This simplifies calculations greatly. In relation to examples from first paragraph we can assume, that while our motor is not linear, in range from 100 to 120 revs it can be approximated as linear. So if what you need is to know motor behavior in this range, then you can go with this approximation.

*Linearity* in general essentially refers to the property that something (a function, system, relation, etc.) acts on the *whole* in a way determined solely by how it acts on the *parts*. For example, a function is linear when it obeys `f(a+b) = f(a) + f(b)`.

In the context of Fourier analysis, the central theme is that *any* function can be represented as a sum of *sine waves*. For example, let’s say we have some function `a(t) = A₁·sin(w₁·t) + A₂.sin(w₂·t) + …`. If some *linear* operator `F` acts on this function, that means the action of the operator on the function as a whole must be determined *solely* by how the operator acts on the individual sine waves of frequency ω₁, ω₂, and so on.

In other words, knowing how an operator acts on a sine wave of every frequency is sufficient to know how an operator acts on *any* function. This knowledge is called the *frequency response* of an operator, and it can be a useful tool to characterize or study the properties of linear operators acting on funbctions, which are otherwise somewhat hard to visualize intuitively.

(They can also be useful to simplify math in a lot of practical applications, for example signal theory – because it turns out that a convolution in the time domain is equal to a multiplication in the frequency domain, and multiplications are much easier to compute than convolutions for large signals)