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DFT
Calculates the discrete fast Excel Fourier transformation for amplitude and phase.
Syntax
DFT^{i}(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.
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.
 The first value in the input time series must correspond to the earliest observation.
 The frequency component order, , must be a positive number less than , or the error (#VALUE!) is returned.
 The DFT returns the phase angle in radians, i.e. .

The discrete Excel Fourier transformation (DFT) is defined as follows:
.
Where:
 is the frequency component
 are the values of the input time series
 is the number of nonmissing values in the input time series

The CooleyTukey radix2 decimationintime fast Excek Fourier transform (FFT^{i}) algorithm divides a DFT of size N into two overlapping DFTs of size at each of its stages using the following formula:
Where:
 is the DFT of the evenindicied values of the input time series,
 is the DFT of the oddindicied values of the input time series,
 ,
 is the number of nonmissing values in the time series data.
 The unit frequency of the DFT is , where is the number of nonmissing observations.
Examples
Example 1:
A  B  

1  Date  Data 
2  January 10, 2008  0.66 
3  January 11, 2008  0.02 
4  January 12, 2008  0.54 
5  January 13, 2008  0.21 
6  January 14, 2008  0.73 
7  January 15, 2008  0.37 
8  January 16, 2008  1.00 
9  January 17, 2008  0.42 
10  January 18, 2008  0.99 
11  January 19, 2008  0.04 
12  January 20, 2008  0.23 
13  January 21, 2008  0.31 
14  January 22, 2008  0.69 
15  January 23, 2008  0.37 
16  January 24, 2008  0.78 
17  January 25, 2008  0.30 
18  January 26, 2008  0.97 
19  January 27, 2008  0.91 
20  January 28, 2008  0.92 
21  January 29, 2008  0.88 
22  January 30, 2008  0.14 
23  January 31, 2008  0.06 
24  February 1, 2008  0.19 
25  February 2, 2008  0.61 
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