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Is there a name for the result of summing the bins of a DFT?
I don't mean to sum the squares of the bins, but to simply add the magnitude of the frequency bins together to get a single result.
Is there a special name that describes how this result would relate to the signal in the time-domain?
The best I can think of would be the total power across the DFT bandwidth, but I believe that is calculated by taking the root of the sum of the squares of the bins, per Parseval's theorem.
It has the name "first element of the time domain vector times the length of the vector".
reason:
With the DFT of a $N$ long sequence $y[n]$ being
$$Y[k] = \sum_{n=0}^{N -1}y[n]e^{-j2\pi k\frac{n}N}\text,$$
your sum of FFT bins is
\begin{align} \sum_{k=0}^{N -1}Y[k] &= \sum_{k=0}^{N -1}\sum_{n=0}^{N -1}y[n]e^{-j2\pi k\frac{n}N}&\text{finite sums can be exchanged in order}\\ &=\sum_{n=0}^{N -1}\sum_{k=0}^{N -1}y[n]e^{-j2\pi k\frac{n}N} &\text{$y[n]$ is constant over all $k$}\\ &=\sum_{n=0}^{N -1}y[n]\underbrace{\sum_{k=0}^{N -1}e^{-j2\pi k\frac{n}N}}_{=\begin{cases}N\cdot1 &n=0\\0&\text{else}\end{cases}}& \text{sum-orthonogonality of the DFT}\\ &=y[0]N \end{align}
