Calculates the discrete fast Fourier transformation for amplitude and phase.

## Syntax

**DFT**(

**X**,

**Order**,

**Component**,

**Return_type**)

- X
- is the univariate time series data (one-dimensional array of cells (e.g., rows or columns)).
- Order
- is the time order in the data series (i.e., the first data point's corresponding date (earliest date=1 (default), latest date=0)).
Order Description 1 ascending (the first data point corresponds to the earliest date) (default) 0 descending (the first data point corresponds to the latest date) - Component
- is the input frequency component order. If missing, component 0 is assumed. If negative, an array of all components is assumed.
- Return_type
- is a number that determines the return value type: 1 (or missing) = Amplitude, 2 = Phase.
RETURN_TYPE NUMBER RETURNED 1 or omitted Amplitude 2 Phase

## 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.
- If the component value is negative, the DFT function returns an array of amplitude (or phase) values for all components in the sample data.
- The first value in the input time series must correspond to the earliest observation.
- The frequency component order, $k$, must be a positive number less than $N$, or the 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

- A fast Fourier transform (FFT) is a highly optimized implementation of the discrete Fourier transform (DFT), which converts discrete signals from the time domain to the frequency domain.
- The most commonly used FFT algorithm is the Cooley-Tukey algorithm, which reduces a large DFT into smaller DFTs to increase computation speed and reduce complexity.
- A radix-2 decimation-in-time (DIT) FFT is the simplest and most common form of the Cooley–Tukey algorithm that 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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