Palgrave Handbook of Econometrics: Applied Econometrics

286 Investigating Economic Trends and Cycles

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Figure 6.13 The central coefficients of the Fourier transform of the frequency response of
an ideal lowpass filter with a cut-off point atω=π/2. The sequence of coefficients extends
indefinitely in both directions. The coefficients are the sampled ordinates of a sinc function

The coefficients of the filter are given by the discrete-time sinc function, which is
the (inverse) Fourier transform of the periodic frequency response function:

βk=

1

2 π

∫ωd

−ωd

eiωkdω=

⎧

⎪⎨

⎪⎩

ωd

π

,ifk=0;

sin(kωd)

πk

,ifk=0.

(6.150)

Such a frequency response presupposes a doubly-infinite data sequence, insofar as it
represents the relative amplification and attenuation of trigonometrical functions
that are defined over the entire real line.
The coefficients of (6.150) form a doubly infinite sequence, of which a central
part is illustrated in Figure 6.13. In order to obtain a practical filter, it seems that
one must truncate the sequence, retaining only a limited number of its central
elements. This truncation gives rise to a filter of which the frequency response has
certain undesirable characteristics. (See Figure 6.19 for an example.)
In particular, there is a ripple effect whereby the gain of the filter fluctuates
within the pass band, where it should be constant with a unit value, and within
the stop band, where it should be zero-valued. Within the stop band, there is a
corresponding problem of leakage whereby the truncated filter transmits elements
that ought to be blocked.
However, it is clear that an ideal filter can be implemented in the frequency
domain by preserving the ordinates of the Fourier transform of the data that are
associated with frequencies less thanωdand by setting all other ordinates to zero.
This is a matter of applying the following set of weights to the Fourier ordinates:

λj=

⎧

⎨

⎩

1, ifj∈{−d,...,d},

0, otherwise.

(6.151)