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:
$$ X_k = \sum_{j=0}^{N1} x_j e^{\frac{2\pi i}{N} j k} $$
Where:
 $k$ is the frequency component
 $x_0,...,x_{N1}$ are the values of the input time series
 $N$ is the number of nonmissing values in the input time series
 The CooleyTukey radix2 decimationintime 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 evenindicied values of the input time series, $x_{2m} \left(x_0, x_2, \ldots, x_{N2}\right)$
 $O_k$ is the DFT of the oddindicied values of the input time series, $x_{2m+1} \left(x_1, x_3, \ldots, x_{N2}\right)$
 $\alpha = e^{ \left (2 \pi i k /N \right )}$,
 $N$ is the number of nonmissing values in the time series data.
 The unit frequency of the DFT is $\frac{2\pi}{N}$, where $N$ is the number of nonmissing 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 0691042896
 Tsay, Ruey S.. Analysis of Financial Time Series . John Wiley & SONS. (2005). ISBN 0471690740
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