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FFT algorithm

创建日期：2024-08-22 16:34:05

所属分类：计算机computer science

标签： FFT

FFT algorithm

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_

This programming assignment is about using the Fast Fourier Transform

(FFT) algorithm to clean-up/process financial data. I am only going to cover the

computational aspects of this procedure. To learn about the theory behind what

we are going to do, you might want to look at Oppenheim and Schafer’s classic

text. We will use the implementation of the FFT algorithm in NEWMAT.

I am going to give you the required “functional basics” for what an FFT

is etc. There is a lot more to this topic than what I am going to cover here.

The easiest way to get to the foundations of FFT is this – any periodic signal

x(t) with period T (i.e. x(t + T) = x(t) for any t) can be written as a sum-ofsinewaves

(click this wikipedia page for an illustrative example). That is,

x(t) = C0 +X∞i=1

. (1)

The terms Ai and Bi denote the amplitude (i.e. strength) of the sinewave/-

cosinewave whose frequency is i

T Hz – the larger these numbers are, the more

“significant” is the “contribution” of this frequency-component to the value of

x(?). To be a little more precise, the larger the value of pA2i + B2i

, there is a

larger contribution from the i

T Hz component.

In essence, we are are going to take the ticker data for a given duration (in

our case, it is for a day), and we are going to assume that it is a single-period

of a (fictitious) periodic waveform. Consequently, we can write this ticker data

“pattern” as a sum of sinewaves and cosinewaves. The summation is over a

large number of such terms, but some of the sinewaves/cosinewaves might have

a small contribution (i.e. pA2i + B2i

is small, compared to the rest) – we just

drop them – and we get a (smoother) “approximation” of the ticker data pattern.

Such filtered/smoothened version of the data can be used in numerous ways –

one of them is to do short-term prediction, possibly. Another, could be to come

up with triggers for automated-trades, but that requires some experimentation

on your part.

If the ticker data has n-many entries over a time period of T, it turns out we

can generate the values of Ai and Bi (0 ≤ i ≤ n) from the Fast-Fourier Transform

(FFT) routine. The version in NEWMAT (click here for the NEWMAT

documentation) that we will use has the format RealFFT(X, F, G);, where X is

the (real-valued) ticker data array, and F (resp. G) is the real-part (imaginarypart)

of the FFT of X. With reference to equation 1, it can be shown that for

1 ≤ i ≤n2 + 1,

C0 =1n

F(1); Ai =2n

F(i + 1); Bi =2n

G(i + 1).

To reconstruct the filtered tick data, you pick those components of F and G

that contribute significantly (compared to the others) to the original signal.

You compute a magnitude array whose i-th entry is p

F(i)

2 + G(i)

2. Following

this, you sort the entries of magnitude and pick the top-5, top-10, top-15 values

of magnitude (and these are the sinewave and cosinewave components that are

the “strongest” in the original tick data). After you picked the top, stronger

components, in the F and G arrays, you zero-out the remaining entries of the F

and G arrays, as their contributions are insignificant. You reconstruct the signal

using the inverse of the FFT, which is RealFFTI(F, G, X); in the NEWMAT

library. This is how the red, black, blue curves of figure 2 are generated.

In this programming assignment you are going to reconstruct a filteredversion

of a (highly noisy/fluctuating) price-path of an instrument1

Using the hint.cpp provided on Compass, and the video instructions, I want

you to write a C++ program that takes the ticker/price-path data and takes as

command-line inputs: (1) the #significant-terms in the fourier-series/fourierexpansion

of the price-path, (2) the #data-points in the ticker-data, (3) input-

file name that contains the ticker-data, and (4) the output-file that should contain

the filtered-data (after the program completes). A sample output is shown

in figure 1. The video instructions also tell you how to generate a sample output

as shown in figure 2.

Here is what I want from you for this programming assignment

1. A short write-up that explains what your code does. No more than a page.

2. C++ code with appropriate Class definitions along the lines of hint.cpp

that accomplishes what is required.

1You can get the second-by-second price-path of GOOG and MSF T recorded at COB

on 23 August, 2017. The GOOG data has 378 data-points, while the MSF T data has 390

data-points.

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