Estimates the nonnormalized cross‑spectral density of two stationary time series using a weighted cross periodogram given the spectral densities and cross periodogram.

Routine CSWEP estimates the nonnormalized cross‑spectral density function of two jointly stationary time series using a fixed sequence of weights given the modified cross‑periodogram and spectral densities of the appropriately centered and padded data

and

for t = 1, …, N. The routine CPFFT may be used to compute the modified periodograms

and cross‑periodogram

over the discrete set of nonnegative frequencies

(Here, ⌊a⌋ means the greatest integer less than or equal to a.) Either routine SSWP or routine SWEP may be applied to the periodograms to obtain nonnormalized spectral density estimates

over the set of frequencies

where nƒ = NF. These frequencies are in the scale of radians per unit time. The time sampling interval Δt is assumed to be equal to one. Note that the spectral window or weight sequence used to compute

may differ from that used to compute

The cross‑spectral density function is complex‑valued in general and may be written as

The cospectrum is estimated by

and the quadrature spectrum is estimated by

where

and k(ω) is the integer such that ωk,0 is closest to ω. The sequence of m = NWT weights {wj} for j = ‑⌊m/2⌋, …, (m ‑ ⌊m/2⌋ ‑ 1) satisfies Σjwj = 1. These weights are fixed in the sense that they do not depend on the frequency ω at which to estimate hXY(ω). Usually, m is odd with the weights symmetric about the middle weight w0. If m is even, the weight to the right of the middle is considered w0. The argument WT may contain relative weights since they are normalized to sum to one in the actual computations. The nonnormalized cross‑spectral density estimate is computed over the same set of frequencies as the nonnormalized spectral density estimates. However, the particular weight sequence used to compute

need not correspond to either the weight sequence or spectral window used to compute either

An equivalent representation of hXY(ω) is the polar form defined by

The cross-amplitude spectrum is estimated by

and the phase spectrum is estimated by

Finally, the coherency spectrum is estimated by

The coherence or squared coherency is output.

Consider the Robinson Multichannel Time Series Data (Robinson 1967, page 204) where X is the Wölfer sunspot number and Y is the northern light activity for the years 1770 through 1869. Application of routine CSWEP to these data produces the following results.