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I would like to compute the inverse of the function

$$f(w)=\frac{e^{iw}(-1+e^{imw})}{(-1+e^{iw})}+\frac{e^{-imw}(-1+e^{imw})}{(-1+e^{iw})}.$$

What could be a possible approach to find an expression of $w$?

1 Answers 1

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$\dfrac{e^{iw}(-1+e^{imw})}{(-1+e^{iw})}+\dfrac{e^{-imw}(-1+e^{imw})}{(-1+e^{iw})} = \dfrac{-e^{iw}+e^{i(m+1)w}-e^{-imw}+1}{(-1+e^{iw})}= -1+\dfrac{e^\frac{iw}{2}(2\cos((\frac{1}{2}+m)\omega))}{e^\frac{iw}{2}(2\cos(\omega))}=-1+\dfrac{(\cos((\frac{1}{2}+m)\omega))}{(\cos(\omega))}$.

This function (in signal processing) is known as discrete Sinc function (because DFT of discrete rect function a.k.a. moving average results in discrete Sinc in digital frequency domain) and is not an injective function hence there is no inverse. However by restricting its domain you can define an inverse for it.

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    Thank you. I am new in the filed of signal processing, but I am working on that now. In fact I am trying to compute the Fourier transform of a sequence $\{t_{k}\}$ given as $f(w)= \sum\limits_{\substack{k=-m}}^{m} {t_{k} exp(iwk)}$ but I already have the information that $t_{k}=0$ for k=0 and $t_{k}$ is a fixed number for each k. Then, computing the resulting sum I got what I published before and I have the information that $|w| <= \pi$. Do you think with this bound I can find the inverse easily ? – 2017-02-21
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    I think you want to calculate it's inverse transform and not it's inverse. am I correct? – 2017-02-22
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    Now that I am progressing on this, I would like to find the expression of $w$ for which $f(w) \leq x $ – 2017-02-22
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    sorry for delay. really busy these days. finding an interval that gives you $f(\omega) \le x$ may have a unique solution, infinite solution or no solution based on the value of x. just plot the function and you can see it. so solving the inequality in reverse may not be possible. I think this case must be solved numerically. – 2017-02-25
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    There is a mistake in your answer. we can not have $exp(i w/2)$ because we have $-sin(w/2)+i cos(w/2)$ and not $cos(w/2)+i sin(w/2)$. – 2017-03-01