248 Investigating Economic Trends and Cycles
We define[T/ 2 ]to be the integer part toT/2, which will ben=T/2, ifTis even,
or(T− 1 )/2, ifTis odd. Then:
xt=[T∑/ 2 ]j= 0{
αjcos(ωjt)+βjsin(ωjt)}=[T∑/ 2 ]j= 0ρjcos(ωjt−θj).(6.6)Here,ρj^2 =α^2 j+β^2 j andθj=tan−^1 (βj/αj), whilstαj=ρjcos(θj)andβj=ρjsin(θj).
The equality of (6.6) follows in view of the trigonometrical identity:
cos(A−B)=cos(A)cos(B)+sin(A)sin(B). (6.7)The frequencyωj= 2 πj/Tis a multiple of the fundamental frequencyω 1 = 2 π/T.
The latter belongs to a sine and a cosine function that complete a single cycle in the
time spanned by the data. The zero frequencyω 0 is associated with the constant
function cos(ω 0 t)=cos( 0 )=1, whereas sin(ω 0 t)=sin( 0 )=0.
IfT= 2 nis an even number, then the highest frequency isωn=π; and, within
(6.6), there are cos(ωnt)=cos(πt)=(− 1 )tand sin(ωnt)=sin(πt)=0. IfTis an
odd number, then the highest frequency isπ(T− 1 )/T, and there is both a sine and
a cosine function at this frequency. Counting the number of non-zero functions
in both cases shows that they are equal in number to the data points. Therefore,
there is a one-to-one correspondence between the data points and the coefficients
of the non-zero functions in the Fourier expression of (6.6).
In equation (6.6), the temporal indext∈{0, 1,...,T− 1 }assumes integer values.
However, by allowingt∈[0,T)to vary continuously, one can generate a contin-
uous function that interpolates theTdata points. This method of generating the
continuous function from sampled values may be described as Fourier interpola-
tion. It is notable that the interpolated function is analytic in the sense that it
possesses derivatives of all orders.
Although the process generating the data may contain components of frequen-
cies higher than the Nyquist frequency, these will not be detected when it is
sampled regularly at unit intervals of time. In fact, the effects on the process of
components of frequencies in excess of the Nyquist value will be confounded with
those of frequencies that fall below it.
To demonstrate this, consider the case where the process contains a component
that is a pure cosine wave of unit amplitude and zero phase and of a frequencyω
that lies in the intervalπ<ω< 2 π. Letω∗= 2 π−ω. Then:
cos(ωt)=cos{
( 2 π−ω∗)t}=cos( 2 π)cos(ω∗t)+sin( 2 π)sin(ω∗t)
=cos(ω∗t),(6.8)which indicates thatωandω∗are observationally indistinguishable. Here,ω∗<π
is described as the alias ofω>π.