Discrete Fourier Transform ‐ II.pdf

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Discrete Fourier Transform ‐ II ‐1‐ © Spider Financial Corp, 2013

Discrete Fourier Transform ‐ II

This is the second tutorial in our ongoing series on time series spectral analysis. In this entry, we will

continue our discussion on discrete Fourier Transform in Excel, its interpretation and application in time

domain.

The DFT is basically a mathematical transformation and may be a bit dry, but we hope that this tutorial

will leave you with a deeper understanding and intuition through the use of NumXL functions and

wizards.

Background

There have been several inquiries since the time we released our first entry on DFT, especially about

using the DFT components to represent the input data set as the sum of the trigonometric sine‐cosine

functions. The

inquiries

were

motivated

by

using

this

representation

to

interpolate

intermediate

values,

and possibly extrapolate (aka forecast) beyond the input data set.

In principle, the DFT converts a discrete set of observations into a series of continuous trigonometric

(i.e. sine and cosine) functions. So the original signal can be represented as:

1

1( ) cos( )

N

o i i

i

x t A A i t N

Where

( ) x t is

the

value

of

the

observation

at

time

t.

t is the discrete time at which an observation was taken.

{0,1, 2,.., N 1}t

N is the number of observations in the input data set.

2 N

is the fundamental or principle frequency.

i i A is the amplitude and the phase of the i‐th discrete Fourier component.

Analysis

Examining the Fourier transform’s components (i.e. amplitude and phase) of a finite series closer, we

find the

following

observations:

OR

k k N k N k

k k N k k N

A A

A A




1. The amplitude series is symmetrical around the 2

N component.

2. The phase of the k component is the negative of the N K component.

In essence, we only need the 1st half of the DFT components to recover the original input data set. The

original time

is

represented

by

the

following

components:

2

1

1( ) 2 cos( )

N

o i i

i

x t A A i t N

Proof

2

2

2

2

2

1 1 1

1 1

1

1 1( ) cos( ) cos( ) cos( )

1( ) cos( ) cos( )

1( ) cos( )

N

N

N

N

N

N N

o i i o i i i i

i i i

N

o i i N i T i

i i

o i i

i

x t A A i t A A i t A i t N N

x t A A i t A i t N

x t A A i t N

2

2

2

2

1

1

1

1

cos( (N ) )

1( ) cos( ) cos( (N i) )

1( ) cos( ) cos(2 ( ))

1( ) 2 cos( )

N

N

N

N

i i

i

o i i i i

i

o i i i i

i

o i i

i

A i t

x t A A i t A t N

x t A A i t A t t i N

x t A A i t N

IMPORTANT: For an even‐sized input data set, the last DFT component (i.e. 2

N ) does not need to be

multiplied by 2. So the cosine representation of the input data is expressed as follows:

2

2 2

2

2 2

2

2 2

1

2

1

1

1

1

1

1( ) 2 cos( ) cos( )

1( ) 2 cos( ) cos( )

1( ) 2 cos( ) cos( ) cos( )

N

N N

N

N N

N

N N

N

o i i

i

o i i

i

o i i

i

x t A A i t A t N

x t A A i t A t N

x t A A i t A t N




Conclusion

Using the discrete Fourier transform, we represent the discrete input data set as the sum of

deterministic continuous trigonometric functions.

Dissimilar to

the

original

data,

which

is

defined

at

discrete

time

instances,

the

Fourier

representation

is

continuous and thus defined at all‐time values. Using this continuous representation, we can

interpolate any values in this range (but not for extrapolation/forecast).

Discrete Fourier Transform ‐ II.pdf

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