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An FIR digital filter of length N with impulse response ${b(n)}, n=0,1,...,N$. Therefore, the complex-valued, multiplicative function $H(f)$ is the filter's frequency response.It is defined by a Fourier series: 01. $$ H(f)=\sum_{n=0}^{N}b(n)e^{-in2\pi f} \tag{1} $$

What are the areas of application of the equation (1) with examples?

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closed as unclear what you're asking by Stanley Pawlukiewicz, Fat32, Marcus Müller, MBaz, lennon310 Feb 11 at 4:17

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Hi! It's a bit unclear what you are asking... And also the first and second equations and paragraphs have nothing in common. Please be very clear , be to the point, and describe very well your backgound in signal processing, the problem's origination and anything else related to it with extreme clarity. Please do not force us into puzzles. This is not sunday puzzle place. $\endgroup$ – Fat32 Feb 10 at 12:00
  • $\begingroup$ @Fat32 i edited the question ,i hope it is understandabl. $\endgroup$ – N.na91 Feb 10 at 13:28
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    $\begingroup$ no. even worse now. $\endgroup$ – Fat32 Feb 10 at 13:59
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    $\begingroup$ I think you're mixing multiple different forms of notation and then are trying to compare something – not sure this is productive. What is definitely counter-productive is using two-letters "$gf$" to denote a function, and what's worse, forgetting to what that function is a function of. Your problem can probably be answered by "please write your equation $(1)$ properly". Your formulas $(2)$ and $(4)$ make no sense either – there's no $n$ in the summands, $(3)$ simply looks wrong, and uses two new variables that you didn't define anywhere, and your last paragraph is self-contradicting. $\endgroup$ – Marcus Müller Feb 10 at 15:45
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    $\begingroup$ We try to understand what you mean, but this notation is just too sloppy for us to even guess. For example "length of $a_k$": $a_k$ is, probably, a real number, so it has no "length". So: sit down. Start your question with a clear definition of all the symbols you're going to use. Whenever you use a term such as "general form", be very sure that it actually means what you think it means. And check twice whether you're not introducing things that you're not using anywhere – for example, your question seems to revolve around a $P$ that appears nowhere in any of your formulas. $\endgroup$ – Marcus Müller Feb 10 at 15:49