D.S.G. Pollock 285wherein the diagonal matricesγδ(D)andγκ(D)contain the ordinates of the
spectral density functions of the component processes. By replacing the disper-
sion matrices of (6.131) and (6.132) by their circulant counterparts, we derive the
following formulae:
d=U ̄γδ(D){γδ(D)+γκ(D)}−^1 Ug=Pδg, (6.147)k=U ̄γκ(D){γδ(D)+γκ(D)}−^1 Ug=Pκg. (6.148)We may note thatPδandPκare circulant matrices.
The filtering formulae may be implemented in the following way. First, a Fourier
transform is applied to the (differenced) data vectorgto giveUg, which resides in
the frequency domain. Then, the elements of the transformed vector are multiplied
by those of the diagonal weighting matricesJδ=γδ(D){γδ(D)+γκ(D)}−^1 andJκ=
γκ(D){γδ(D)+γκ(D)}−^1. Finally, the products are carried back into the time domain
by the inverse Fourier transform, which is represented by the matrixU ̄. (An efficient
implementation of a mixed-radix fast Fourier transform, which is designed to cope
with samples of arbitrary sizes, has been provided by Pollock, 1999. The usual
algorithms demand a sample size ofT= 2 n.)
An advantage of the Fourier method is that it is possible to effect a total sup-
pression of the elements within the stop band of the desired frequency response.
Also, the transition between the pass band and the stop band can be confined
to the interval between adjacent Fourier frequencies, which means that it can be
perfectly abrupt.
Neither of these features are available to the ordinary finite-sample WK filters.
Nevertheless, it is possible to achieve both of these effects by working in the time
domain. This fact is manifest in the formulae of (6.147) and (6.148) which entail
the equationsd=Pδgandk=Pκgrespectively.
In effect, a pair of wrapped filters can be applied to the data in the time domain via
processes of circular convolution. If we can imagine the leading rows of the matrices
PδandPκdisposed around a circle of circumferenceT, then each of the succeeding
rows is derived from its predecessor via an anticlockwise rotation through an angle
of 2π/Tradians.
Example It is commonly believed that, in the case of samples of finite lengthT,
it is impossible to design a filter that will preserve completely all elements within
a specified range of frequencies and that will remove all elements outside it. A
filter that would achieve such an objective is described as an ideal filter. The ideal
lowpass filter with a cut-off frequency ofωd= 2 πd/Thas the following frequency
response over the interval[−π,π]:
φ(ω)=⎧
⎨
⎩1, ifω∈[−ωd,ωd],0, otherwise.(6.149)