Is the frequency response dependent on the frequency type? (I believe it is) If so, why is it depended on the shape of the signal, square signal vs sine vs triangle. what equations govern it.
Every kind of waveform (square, triangle, etc) can be expressed in terms of the superposition of a number of sinusoidal components. Taking the Fourier transform of any waveform will find those sinusoidal components, their amplitudes, and their phases.
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For example, this square wave is made from the fundamental frequency, plus all its odd harmonics. That is, if this is a 1000 Hz square wave, then it's a combination of sine waves at 1000 Hz, 3000 Hz, 5000 Hz, 7000 Hz, and so on. For an ideal square wave these harmonics go on forever, but as they get higher they decrease in amplitude.
The frequency response of something (like a filter) describes how it will affect each of these sinusoidal components. In a linear system (as most filters are), it's valid to consider what would happen to each component independently, given the frequency response, and then add all the results back together to see the net result.
Frequency response describes the response to a sine wave as its frequency is changed.
Luckily, other signal shapes, such as square wave or triangle wave, can be constructed by summing up different sine waves. Normally we use the frequency response when we are considering linear systems, so that the response to a sum of different inputs is equal to the sum of the responses of the components of the input. Then we can use the frequency response to determine the response to a sine wave, square wave, triangle wave, or any other signal shape we like.