I have got a discrete input signal of length $N$, purely real. Is there any research on how the DFT and inverse DFT results depend on number of Fourier modes? I mean, function of number of frequencies showing the error between real input and recreated signal after DFT by inverse DFT.
If the sequence of DFT coefficients can be calculated as follows:
$$X_k = \sum_{n=0}^{N-1}x_n\cdot e^{-\frac{i2\pi}{N}kn}$$
then what is the dependency of error between input signal and signal recreated by inverse DFT and total number $k$ of coefficients?
If there is a research on this, it must have been done long time ago because the knowledge is almost mainstream in the industry now.
Parseval's Theorem
The input signal is given by linear combination of orthogonal basis:
$$ x_{n} = \frac{1}{N} \sum_{k=0}^{N-1} X_{k} \cdot e^{\frac{i2\pi}{N}kn} $$
The sum of squares of the input signal's elements is given by the following expression:
$$ \sum_{n=0}^{N-1} \|x_{n}\|_{2}^{2} = \frac{1}{N}\sum_{k=0}^{N-1} \|X_{k}\|_{2}^{2} $$
Truncated DFT Result
Suppose you approximate the discrete input signal using only the first $m$ coefficients:
$$ \hat{x}_{n}=\sum_{k=0}^{m-1}X_{k}\cdot e^{\frac{i2\pi}{N}kn} $$
The error is then given by the following:
$$ x_{n}-\hat{x}_{n}=\sum_{k=m}^{N-1}X_{k}\cdot e^{\frac{i2\pi}{N}kn} $$
and the sum of squared error is given by (using Parseval's theorem):
$$ \sum_{n=0}^{N-1} \|x_{n}-\hat{x}_{n}\|_{2}^{2} = \frac{1}{N}\sum_{k=m}^{N-1} \|X_{k}\|_{2}^{2} $$
Remarks
As you add more and more coefficient, the error may not necessarily decrease but cannot increase. However, the error also depends on which coefficients you use i.e. using the first $m$ coefficients may give you lower or higher error compared to if you use the last $m$ coefficients, although the number of coefficients used is the same.
Discrete Fourier transform is nothing more than decomposition of input vector (the input signal) into a linear combination of orthogonal basis. Unfortunately, in many engineering schools, connection between DFT and linear algebra is not taught properly.