D.S.G. POLLOCK: TOPICS IN TIMESERIES ANALYSIS
THE FOURIER DECOMPOSITION OF A TIME SERIES
In spite of the notion that a regular trigonometrical function is an inappropriate
means for modelling an economic cycle other than a seasonal ﬂuctuation, there
are good reasons for explaining a data sequence in terms of such functions.
The Fourier decomposition of a series is a matter of explaining the series
entirely as a composition of sinusoidal functions. Thus it is possible to represent
the generic element of the sample as
(1)
y
t
=
n
j
=0
α
j
cos(
ω
j
t
) +
β
j
sin(
ω
j
t
)
.
Assuming that
T
= 2
n
is even, this sum comprises
T
functions whose frequen
cies
(2)
ω
j
=
2
πj
T
,
j
= 0
, . . . , n
=
T
2
are at equally spaced points in the interval [0
, π
].
As we might infer from our analysis of a seasonal ﬂuctuation, there are
as many nonzeros elements in the sum under (1) as there are data points,
for the reason that two of the functions within the sum—namely sin(
ω
0
t
) =
sin(0) and sin(
ω
n
t
) = sin(
πt
)—are identically zero. It follows that the mapping
from the sample values to the coeﬃcients constitutes a onetoone invertible
transformation. The same conclusion arises in the slightly more complicated
case where
T
is odd.
The angular velocity
ω
j
= 2
πj/T
relates to a pair of trigonometrical com
ponents which accomplish
j
cycles in the
T
periods spanned by the data. The
highest velocity
ω
n
=
π
corresponds to the socalled Nyquist frequency. If a
component with a frequency in excess of
π
were included in the sum in (1), then
its effect would be indistinguishable from that of a component with a frequency
in the range [0
, π
]
To demonstrate this, consider the case of a pure cosine wave of unit am
plitude and zero phase whose frequency
ω
lies in the interval
π < ω <
2
π
. Let
ω
∗
= 2
π
−
ω
. Then
(3)
cos(
ωt
) = cos (2
π
−
ω
∗
)
t
= cos(2
π
)cos(
ω
∗
t
) + sin(2
π
)sin(
ω
∗
t
)
= cos(
ω
∗
t
);
which indicates that
ω
and
ω
∗
are observationally indistinguishable.
Here,
ω
∗
∈
[0
, π
] is described as the alias of
ω > π
.
1
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FOURIER DECOMPOSITION
For an illustration of the problem of aliasing, let us imagine that a person
observes the sea level at 6am. and 6pm. each day. He should notice a very
gradual recession and advance of the water level; the frequency of the cycle
being
f
= 1
/
28 which amounts to one tide in 14 days. In fact, the true frequency
is
f
= 1
−
1
/
28 which gives 27 tides in 14 days. Observing the sea level every
six hours should enable him to infer the correct frequency.
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 Fall '11
 D.S.G.Pollock
 Fourier Series, Variance

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