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 type of return value: 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 amplitudes (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
- The Cooley-Tukey radix-2 decimation-in-time fast Fourier transform (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-indicied 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-indicied 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.
Examples
Example 1:
|
|
| Formula | Description (Result) |
|---|---|
| Amplitude for frequency 1 (1.171) | Phase for frequency 1 (2.497) |
| Amplitude for frequency 2 (2.267) | Phase for frequency 2 (3.708) |
| Amplitude for frequency 3 (1.136) | Phase for frequency 3 (0.097) |
| Amplitude for frequency 4 (2.325) | Phase for frequency 4 (0.067) |
| Amplitude for frequency 5 (1.408) | Phase for frequency 5 (1.839) |
Files Examples
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
Comments
Article is closed for comments.