g13cd calculates the smoothed sample cross spectrum of a bivariate time series using spectral smoothing by the trapezium frequency (Daniell) window.


public static void g13cd(
	int nxy,
	int mtxy,
	double pxy,
	int mw,
	int ish,
	double pw,
	int l,
	int kc,
	double[] xg,
	double[] yg,
	out int ng,
	out int ifail
Visual Basic
Public Shared Sub g13cd ( _
	nxy As Integer, _
	mtxy As Integer, _
	pxy As Double, _
	mw As Integer, _
	ish As Integer, _
	pw As Double, _
	l As Integer, _
	kc As Integer, _
	xg As Double(), _
	yg As Double(), _
	<OutAttribute> ByRef ng As Integer, _
	<OutAttribute> ByRef ifail As Integer _
Visual C++
static void g13cd(
	int nxy, 
	int mtxy, 
	double pxy, 
	int mw, 
	int ish, 
	double pw, 
	int l, 
	int kc, 
	array<double>^ xg, 
	array<double>^ yg, 
	[OutAttribute] int% ng, 
	[OutAttribute] int% ifail
static member g13cd : 
        nxy : int * 
        mtxy : int * 
        pxy : float * 
        mw : int * 
        ish : int * 
        pw : float * 
        l : int * 
        kc : int * 
        xg : float[] * 
        yg : float[] * 
        ng : int byref * 
        ifail : int byref -> unit 


Type: System..::..Int32
On entry: n, the length of the time series x and y.
Constraint: nxy1.
Type: System..::..Int32
On entry: whether the data is to be initially mean or trend corrected.
For no correction.
For mean correction.
For trend correction.
Constraint: 0mtxy2.
Type: System..::..Double
On entry: the proportion of the data (totalled over both ends) to be initially tapered by the split cosine bell taper.
A value of 0.0 implies no tapering.
Constraint: 0.0pxy1.0.
Type: System..::..Int32
On entry: M, the frequency width of the smoothing window as 2πM.
A value of n implies that no smoothing is to be carried out.
Constraint: 1mwnxy.
Type: System..::..Int32
On entry: S, the alignment shift between the x and y series. If x leads y, the shift is positive.
Constraint: -l<ish<l.
Type: System..::..Double
On entry: p, the shape parameter of the trapezium frequency window.
A value of 0.0 gives a triangular window, and a value of 1.0 a rectangular window.
If mw=nxy (i.e., no smoothing is carried out) then pw is not used.
Constraint: if mwnxy, 0.0pw1.0.
Type: System..::..Int32
On entry: L, the frequency division of smoothed cross spectral estimates as 2πL.
  • l1;
  • l must be a factor of kc.
Type: System..::..Int32
On entry: the order of the fast Fourier transform (FFT) used to calculate the spectral estimates. kc should be a product of small primes such as 2m where m is the smallest integer such that 2m2n, provided m20.
  • kc2×nxy;
  • kc must be a multiple of l. The largest prime factor of kc must not exceed 19, and the total number of prime factors of kc, counting repetitions, must not exceed 20. These two restrictions are imposed by the internal FFT algorithm used.
Type: array<System..::..Double>[]()[][]
An array of size [kc]
On entry: the nxy data points of the x series.
On exit: the real parts of the ng cross spectral estimates in elements xg[0] to xg[ng-1], and xg[ng] to xg[kc-1] contain 0.0. The y series leads the x series.
Type: array<System..::..Double>[]()[][]
An array of size [kc]
On entry: the nxy data points of the y series.
On exit: the imaginary parts of the ng cross spectral estimates in elements yg[0] to yg[ng-1], and yg[ng] to yg[kc-1] contain 0.0. The y series leads the x series.
Type: System..::..Int32%
On exit: the number of spectral estimates, L/2+1, whose separate parts are held in xg and yg.
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).


The supplied time series may be mean and trend corrected and tapered as in the description of g13cb before calculation of the unsmoothed sample cross-spectrum
for frequency values ωj=2πjK, 0ωjπ.
A correction is made for bias due to any tapering.
As in the description of g13cb for univariate frequency window smoothing, the smoothed spectrum is returned as a subset of these frequencies,
νl=2πlL,  l=0,1,,L/2
where [ ] denotes the integer part.
Its real part or co-spectrum cfνl, and imaginary part or quadrature spectrum qfνl are defined by
where the weights w~k are similar to the weights wk defined for g13cb, but allow for an implicit alignment shift S between the series:
It is recommended that S is chosen as the lag k at which the cross-covariances cxyk peak, so as to minimize bias.
If no smoothing is required, the integer M, which determines the frequency window width 2πM, should be set to n.
The bandwidth of the estimates will normally have been calculated in a previous call of g13cb for estimating the univariate spectra of yt and xt.


Bloomfield P (1976) Fourier Analysis of Time Series: An Introduction Wiley
Jenkins G M and Watts D G (1968) Spectral Analysis and its Applications Holden–Day

Error Indicators and Warnings

Errors or warnings detected by the method:
On entry,nxy<1,
orpw<0.0 and mwnxy,
orpw>1.0 and mwnxy,
On entry,kc<2×nxy,
orkc is not a multiple of l,
orkc has a prime factor exceeding 19,
orkc has more than 20 prime factors, counting repetitions.
This indicates that a serious error has occurred. Check all array subscripts in calls to g13cd. Seek expert help.
An error occured, see message report.
Negative dimension for array value
Invalid Parameters value


The FFT is a numerically stable process, and any errors introduced during the computation will normally be insignificant compared with uncertainty in the data.

Parallelism and Performance


Further Comments

g13cd carries out an FFT of length kc to calculate the sample cross spectrum. The time taken by the method for this is approximately proportional to kc×logkc (but see method document (C06PAF not in this release) for further details).


This example reads two time series of length 296. It selects mean correction and a 10% tapering proportion. It selects a 2π/16 frequency width of smoothing window, a window shape parameter of 0.5 and an alignment shift of 3. It then calls g13cd to calculate the smoothed sample cross spectrum and prints the results.

Example program (C#): g13cde.cs

Example program data: g13cde.d

Example program results: g13cde.r

See Also