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Sorry about my unscientific and informal question but i have an exam $3$ days later and i need to learn this. Can anyone explain clearly, shortly and without formula Why do we need to find the frequency response of FIR filter?

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Here is a non-mathematical answer to your question. Knowing the frequency response of any FIR filter is important (and you need to know it, it is life-imperative!) because it characterises what how that system will eventually behave under any input. It really only makes sense when I give you an example:

Think of an opera concert hall as a "filter", and for an FIR filter, we're interested in the impulse response. Now imagine somewhere fires a gun, with a loud but short-lived bang - that is your impulse input. A microphone measures the audio signals in that opera hall, as your "filter" output. Now, what we have recorded on the microphone is basically the impulse response, $h(t)$ of the FIR filter that is your opera hall - which sounds like a loud, time-stretched and echoed version of a gunshot. Now we know how this system behaves, we can use the idea of convolution, where we input in a new signal, $x(t)$,

$$y(t)=\int_{-\infty}^{\infty}h(\tau)x(t-\tau)d\tau $$

$x(t)$ could be anything, say a bird chirp, or the sound of a whale or a dog bark, and because we already know how the impulse response of the system is like, we don't actually need to bring a bird, a dog or a whale into the opera hall to hear what it sounds like in the opera hall.

We convolve the sample sound, $x(t)$, with the impulse response, $h(t)$, of the opera hall, and we can hear how those animals sound like in an opera hall SIMPLY with convolution and mathematics.

That is the beauty of understanding the response of any FIR filter.

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  • $\begingroup$ Sam: Your explanation was beautiful but doesn't your integral represent the time response rather than the frequency response. I imagine the idea is the same but I'd be in seeing the mathematical formulation of the frequency response. Thanks. $\endgroup$
    – mark leeds
    Jan 10 '18 at 3:30
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You do not need to find the frequency response of an FIR filter.

Instead all LTI filters do have their own frequency responses as their associated property. And in order to distinguish different types of FIR filters from each other, which probably will be based on their frequency responses, you need to compute the frequency response of the filters and compare them.

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