292 Investigating Economic Trends and Cycles
Whenτ=0, this becomes an ordinary sinc function that is a continuous function
oft, and which is the Fourier transform of the following frequency function:
φd(ω)=⎧
⎨
⎩1, if|ω|∈[0,ωd];0, otherwise.(6.162)Whenτ =0, it represents a sinc function that has been displaced in time by
τintervals of lengthπ/ωd. The set of such displaced sinc functions constitutes
an orthogonal basis for all continuous functions that are band-limited to the
frequency interval[−ωd,ωd].
In the case of a stationary stochastic process, the sampled sequence is not square
summable and, therefore, in a strict sense, this proof of the interpolation via the
Nyquist–Shannon theory does not apply. However, the convergence of the inter-
polation formula of (6.160), whenx(τ)={xτ;τ=0,±1,±2,...}is a stationary
sequence, can be confirmed by considering a sum withτ∈[−N,N]for some finite
integerN. The variance of the sum of discarded terms can be made arbitrarily small
by increasing the value ofN.
The reconstruction of a continuous function from its sampled ordinates in the
manner suggested by the sampling theorem is not possible in practice, because
it requires forming a weighted sum of an infinite number of sinc functions, each
of which is supported on the entire real line. Nevertheless, a continuous band-
limited periodic function defined on a finite interval – which corresponds to the
circumference of a circle – can be reconstituted from a finite number of wrapped or
periodic sinc functions, which are Dirichlet kernels by another name. However, the
most practical means of reconstituting the function is by a simple Fourier synthesis
of the sort described by equation (6.14).
Example The analysis of the example following (6.14) suggests that the business
cycle of the detrended logarithmic consumption data fits within the frequency
band[0,π/ 8 ]. If this structure can be isolated and thereafter mapped into the fre-
quency interval[0,π], then it will be capable of being described by an ordinary
linear stochastic model of the ARMA variety. For this purpose, the spectral elements
that fall outside the frequency range of the business cycle must first be removed.
This operation, which constitutes an anti-alias filtering, may be carried out either
in the time domain or in the frequency domain.
Given the availability of the spectral ordinates of the data, it is straightforward to
operate in the frequency domain by setting the rejected ordinates to zeros. Then, a
continuous low-frequency function can be synthesized from the selected ordinates.
An example is provided by the interpolated function in Figure 6.5. The synthesized
function can be resampled at a rate that corresponds to the maximum frequency
within the spectral structure of the business cycle.
There is some advantage in fitting a trend function that is more flexible than the
straight line of Figure 6.5. Therefore, a fourth degree polynomial has been fitted to