NCL Home > Documentation > Functions > General applied math

ezfftf

Perform a Fourier analysis on a periodic sequence.

Prototype

	function ezfftf (
		x  : numeric   
	)

	return_val  :   typeof(x)

Arguments

A variable containing one or more periodic sequences to be transformed. For a multi-dimensioned array the rightmost dimension will be transformed. The size of the rightmost dimension need not be a power of 2.

Return value

A double array is returned if x is double, otherwise a float array is returned.

If npts is the size of the dimension to be transformed, and N represents all other variable dimensions, then the fourier coefficients are returned as an variable dimensioned 2 x N x (npts/2). The leftmost dimension of size 2 contains the real and imaginary coefficients.

It also returns the series mean(s) as an attribute called xbar. A second attribute, npts, specifies the length of the original series.

The examples clarify the above information.

Description

Given a real periodic sequence x, ezfftf performs a forward Fourier transform. This is often called Fourier Analysis.

If any missing values are encountered in one of the input arrays, then no calculations will be performed on that array, and the corresponding output array will be filled with missing values.

The user may wish to detrend [ dtrend ] and taper prior to performing the analysis. Consult a book on Fourier Analysis for details. A old but nice reference is:

        Peter Bloomfield
        Fourier analysis of time series : An introduction
        New York : John Wiley and Sons  , 1976

Examples

Example 1

Perform a fourier analysis of a one dimensional periodic sequence. Note the attributes npts and xbar are associated with the returned variable cf. The npts represents the number of values used while xbar is the series average.

The real and imaginary coefficients are accessed via the leftmost dimension. The 0-th and 1-th components refer to the real and imaginary components, respectively.

    x  = (/ 1002, 1017, 1018, 1020, 1018, 1027, \ 
            1028, 1030, 1012, 1012,  982, 1012, \
            1001,  996,  995, 1011, 1027, 1025, \ 
            1030, 1016,  996, 1006, 1002,  982  /)
    cf = ezfftf (x)
    printVarSummary(cf)

Variable: cf
Type: float
Total Size: 96 bytes
            24 values
Number of Dimensions: 2
Dimensions and sizes:   [2] x [12]       <=== 2 x (npts/2)
Coordinates: 
Number Of Attributes: 2
  npts :        24                       <=== # of values
  xbar :        1011.042                 <=== series mean

    print(cf(0,:) +"    "+cf(1,:))    ; real/imag 

(0)     1.33844    3.73528               <===   real     imaginary
(1)     -13.484    6.88814
(2)     2.16667    3.36294
(3)     3.29167    0.360844
(4)    -5.39841    3.01955
(5)    0.083333    1
(6)    -2.72477    4.11324
(7)     2.70833    1.51554
(8)     2.16667    2.52961
(9)   -0.349307   -2.63814
(10)    2.95141    2.80825
(11)   -1.79167    0

The frequency spacing is 1./npts. The frequencies that the cf components corresponds to are:

    cf(0,0) and cf(1,0) is 1./npts
    cf(0,1) and cf(1,1)    2./npts
    cf(0,2) and cf(1,2)    3./npts
            :
    cf(0,12)    cf(1,12)   12./npts  (=0.5 Nyquist Frequency)

Example 2

Let x(ntim,klvl,nlat,mlon). In the "Return Value" documentation, N corresponds to the dimension sizes (ntim,klvl,nlat) in this instance, and mlon is a number of longitude points:

    cf = ezfftf (x)       ; ==> cf(2,ntim,klvl,nlat,mlon/2)
                          ; ==> cf@npts = mlon
                          ; ==> cf@xbar ==> contains the means
                                            length=ntim*klvl*nlat

The attribute xbar is returned as a one dimensional array. The following will make this a three dimensional array: