Is there a name for the result of summing the bins of an FFT?

Question

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.

Answer 1

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}