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I am confused about the construction of filters for non-periodic signals which attempt to limit the output to a certain frequency range(by frequency I mean with respect to the Fourier Transform).

How are they constructed? How do you even find the non-periodic frequency response of a filter?

Lets say I want to construct an along filter which limits the output of an aperiodic signal to a certain frequency range(approximately) which has an impulse response approximately like the Sinc function.

What does it look like? How is it done?

Thank you for the information.

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A filter has its own frequency response, defined by the magnitude and the phase plot.

A filter works for both periodic and non-periodic signals: the reason is that while periodic signals can be represented as a superposition of infinite oscillations at frequencies multiples of the fundamental one (Fourier Series), non-periodic signals as well can be represented as a superposition of an infinite number of oscillations very close each other in frequency. In the limit, the series becomes a continuous sum, i.e., an integral (Fourier Integral).

Mathematically, this could be explained by the fact that non-periodic signals can be viewed as 'periodic' functions with 'infinite period' (which means infinitesimally small fundamental frequency, i.e., infinitesimally small frequency spacing between the spectral components).

Therefore, it is correct to say that human ear 'captures frequency components approximately between 20Hz and 20kHz', even if the sounds we hear are non-periodic signals.


What about filter implementation?

If you want to limit the frequency components of a signal with some filter, you just need to design it keeping in mind the periodic sinusoidal components! If you want to achieve a sinc() filter impulse response (i.e., a sharp rectangular-shaped magnitude response) you should better accept that only approximations are possible in the real world; your bandwidth will never have such sharp edges and flat in-band response : instead, there is a well-known tradeoff between edge sharpness (related to the filter's selectivity together with the bandwidth) and the in-band flatness.

For this reason, the bandwidth of filters is generally evaluated in terms of the -3dB frequency (i.e., the frequency corresponding to a gain reduction of 3dB with respect to the peak value).

Filter implementations in electronics design can be either passive (using just capacitors, inductors, resistors) and active (using active components like operational amplifiers). A popular example of active lowpass or highpass filter is the Sallen-Key topology.

Beyond these physical implementations, however, in most cases you start from some design specifications relating the bandwith, cut-off frequency, flatness, time-domain overshoot , center frequency and, according to them, a 'filter approximation' is chosen (this is true in particular for OpAmp-based active filters): these are just mathematical models which allow you to locate the poles of your filter starting from the desired frequency response. Popular examples are the Bessel, Čebyšëv and Butterworth filters; for the same model, different approximate implementations are possible.

There are online interactive tools which help in this design process (since sometimes there is a lot of nasty algebra behind it); these services are offered by famous analog IC technology companies; I don't know if I can link here, but just google 'filter design analog online' or something like that and you will find it. I hope this helps and... enjoy!!!

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  • \$\begingroup\$ Thanks for all the information. Mathematically the sum of sinusoids with an original signal period of infinity doesn't work. How does one really find the circuits response with respect to the fourier transform? Conceptually one could solve it in the time domain and then apply the fourier transform to the output but this doesn't make sense. Maybe the fourier transform can be applied to the system directly? Regardless thanks. \$\endgroup\$ – FourierFlux Feb 19 '17 at 18:39
  • \$\begingroup\$ The 'infinite period' is just a mathematical explanation... Just to make sense of why non-periodic signals have a continuous spectrum. If you have a circuit (I mean - a physical circuit) and you want to extract the Bode plots, this is actually done by applying a sinusoidal signal ('stimulus') whose frequency is swept over time from a minimum to a maximum value with a given frequency step. Meanwhile, you measure the output signal amplitude and phase delay (the output should also be a sinusoid if the system is linear)... and you obtain the Bode plots. That's it \$\endgroup\$ – NotANumber Feb 19 '17 at 19:14
  • \$\begingroup\$ Instead, if your circuit has not come into existence yet, you'd like to simulate it, okay? And PSPICE circuit simulators usually include 'ac' simulations which basically do the same thing- just simulated and not measured \$\endgroup\$ – NotANumber Feb 19 '17 at 19:16
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    \$\begingroup\$ The very basic thing to understand is that filters are designed taking into account sinusoidal signals. The great thing is that this is done without loss of generality. Since any finite-energy signal can be seen as a sum (continuous--->integral for nonperiodic signals, discrete---->series for periodic signals) of sinusoidal functions, the behaviour of any linear system (including the filter) that is excited with such bunch of summed signals can be described using the superposition principle. \$\endgroup\$ – NotANumber Feb 19 '17 at 19:29

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