# Rejection of particular frequency in a signal which contains many frequencies in its spectrum (fourier transform)

How does a filter know which frequency is to reject?

Detail:

A signal may contain a lot of frequencies. How could we surely say that the result of linear combination of the result of applying these frequencies individually will produce the same result as applying the whole signal?

At any point of time there will be a voltage level which will be applied to a circuit (linear combination of amplitude of contained frequencies.) Circuit only "sees" voltages, so how will it know that which frequency to reject? I mean the resultant signal amplitude values will be the value, after subtracting the value of rejected signal frequency amplitude from original input signal value at every time instant.

• What kind of filter are you asking about? An analog filter using capacitors, resistors and inductors? Or a digital filter which uses an algorithm? Note that capacitors and inductors have an impedance that depends on frequency. Dec 3 '19 at 12:33
• RLC filter .... I do understand that it depends on the impedance but the impedance for different frequencies are the result of applying them properly. But in the case of a signal we are applying voltages and signal is not periodic. Dec 3 '19 at 12:36
• Do es my question make sense ? Dec 3 '19 at 12:39
• Do you understand the concept of superposition? That's what allows you to consider the frequency components of the input signal individually, and then combine the results at the end. Dec 3 '19 at 12:39
• Scientist try to describe how nature works by defining laws, etc and hope nature will fit into it. But nature will not obey the laws dictated by humanity. So, how does a filter know? It doesn't know, it just does. I think a better question is: How do humans know which frequency components will be filtered by a filter? Then, DaveTweed remarks gives a start by noting nature's signals can be described by superposition of frequency components (Fourier formulated laws about it).. etc Dec 3 '19 at 12:46

I find the way you are phrasing your question a little confused but I think I know what you are looking for. I'll try to answer.

A circuit only sees (instantaneously) voltages - well yes, but there are elements on the circuit that are time-dependent. If we are talking about an analogue circuit, this is L's and C's. These elements store energy in one form or other, and release it with a rate which is variable depending on component value.

So now we have a circuit that has some "memory" characteristics. It's instantaneous state is dependent on the sum of all previous states.

There are a number of ways of analysing this mathematically, but eventually we find that with certain circuit configurations we can arrange that the circuit's response varies according to the frequency content (Fourier spectrum) of the input. (This is one form of analysis that works well for periodic signals - signals which are assumed to behave in the same way, over and over, through all of time. In practice of course, nothing does this, but, say, a 1MHz waveform running for a second or two is close enough for the assumption to hold.)

Likewise a digital filter works by computing the instantaneous output based on some conmbination of previous (sampled) inputs. Now the memory element is really memory, used to store some number of samples in a rolling buffer.

In the end the important fact is that circuit behaviour has a time (and therefor frequency) dependent element.

Circuits see more than voltages. They can well remember something of the past voltage values of a signal and form current output with some math rule from current and past input voltage values. That can attenuate different simultaneous sinewaves differently.

An example: A filter contains a 1 millisecond delay line and a summing circuit which creates the output by adding together the filter input and the 1ms delayed input signal.

The result: 500 Hz sinewave or 500 Hz component of more complex signal dies totally, but DC and 1 kHz components get doubled.

A filter applies the same math rule to every input signal.That rule is called "transfer function". But we have taken a habit to present signals as a sum of simultaneous sinewaves. As well we build filter circuits which do not smash signals in a too complex way, our filters are built to retain the frequencies of the simultaneously inputted sinewaves. We say our filtering circuits are linear. They cause only amplitude and phase changes to the sinewaves.

The filter itself knows nothing of how we describe its behaviour. As said, it applies the same math rule for every input.

Both filtering and frequency are concepts that to be useful require time. (A single point is an impulse that contains infinite frequencies, you can only remove them all by zeroing the point, or increase them all by boosting that point. Not very useful.)

A realistic LTI filter has an impulse response. It looks over a period of time to try and match the "shape" of a certain frequency or combination of spectral frequency bands and then afterwards produces some behavior. The longer the filter looks at the data, the better it can tell two spectral bands apart and perhaps treat them separately (e.g. block one and allow the other). The filter's result will modify a waveform over a period of time, as modifying just a single point will only produce a "glitch", or a startup transient. And that result will look different to another filter that is long enough to detect the specific frequencies that were modified.

So, although a digital filter will modify signals point by point, you can only describe what they are doing after letting them run for a bunch of points, sometimes a very very large bunch of points for very narrow frequency band changes. (assuming non-zero noise or any finite quantization)

But in the case of a signal we are applying voltages and signal is not periodic.

Two things.

First, periodic and continuous are not the same thing. By definition, a frequency is periodic. Also, the frequency components of a transient are periodic. The characteristics of an RLC circuit vary as a function of the frequency of the signal going through it. If the signal is a combination of two frequencies, the circuit appears to the signal have two different impedances or response curves, one for each frequency.

Second, for a long time, transient events were considered to be aperiodic - not having a frequency or frequencies. The genius of Fourier's work was that he showed that transients are composed of an infinite series of frequencies, and it is the combinations of those frequencies that makes the signal die out or stay continuous. This means that something that appears on a scope to be "not periodic", actually is from the point of view of the filter circuit. Fourier's initial work was a mathematical model of how heat spreads out into a cold mass when it comes in contact with a hot point. Doesn't get more aperiodic than that.