Calculates the **inverse discrete Fourier transform**, recovering the time series.

## Syntax

**IDFT**(**Amp**, **Phase**, **N**)

- Amp
- is an array of the amplitudes of the Fourier transformation components (a one-dimensional array of cells (e.g., rows or columns)).
- Phase
- is an array of the phase angle (radian) of the Fourier transformation components (a one-dimensional array of cells (e.g., rows or columns)).
- N
- is the original number of observations used to calculate the Fourier transform. If missing, N is assumed to be twice the size of the amplitude/phase array.

## Remarks

- The input time series may include missing values (e.g., #N/A, #VALUE!, #NUM!, empty cell) at either end, but they will not be included in the calculations.
- The input time series must be homogeneous or equally spaced.
- The first value in the input time series must correspond to the lowest frequency component.
- The output time series is returned in ascending order, i.e., the first observation corresponds to the earliest date.
- The frequency component order, $k$, must be a positive number less than $N$, or an error (#VALUE!) is returned.
- The DFT returns the phase angle in radians, i.e., $0 \lt \phi \lt 2 \times \pi$.
- The discrete Fourier transformation (DFT) is defined as follows:

$$ X_k = \sum_{j=0}^{N-1} x_j e^{-\frac{2\pi i}{N} j k} $$.

Where:

- $k$ is the frequency component
- $x_0,...,x_{N-1}$ are the values of the input time series
- $N$ is the number of non-missing values in the input time series

- The Cooley-Tukey radix-2 decimation-in-time fast Fourier transformation (FFT) algorithm divides a DFT of size N into two overlapping DFTs of size $\frac{N}{2}$ at each of its stages using the following formula:

$$ X_{k} = \begin{cases} E_k + \alpha \cdot O_k & \text{ if } k \lt \dfrac{N}{2} \\ E_{\left (k-\frac{N}{2} \right )} - \ \alpha \cdot O_{\left (k-\frac{N}{2} \right )} & \text{ if } k \geq \dfrac{N}{2} \end{cases} $$

Where:

- $E_k$ is the DFT of the even-indices values of the input time series, $x_{2m} \left(x_0, x_2, \ldots, x_{N-2}\right)$
- $O_k$ is the DFT of the odd-indices values of the input time series, $x_{2m+1} \left(x_1, x_3, \ldots, x_{N-2}\right)$
- $\alpha = e^{ \left (-2 \pi i k /N \right )}$
- $N$ is the number of non-missing values in the time series data

- The unit frequency of the DFT is $\frac{2\pi}{N}$, where $N$ is the number of non-missing observations.

## Files Examples

## Related Links

## References

- Hamilton, J .D..
*Time Series Analysis*. Princeton University Press. (1994). ISBN 0-691-04289-6 - Tsay, Ruey S..
*Analysis of Financial Time Series*. John Wiley & SONS. (2005). ISBN 0-471-690740

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