Discrete Fourier Transform

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1 Discrete Fourier Transfor This is the first tutorial in our ongoing series on tie series spectral analysis. In this entry, we will closely exaine the discrete Fourier transfor (aa DFT) and its inverse, as well as data filtering using DFT outputs. The DFT is basically a atheatical transforation and ay be a bit dry, but we hope that this tutorial will leave you with a deeper understanding and intuition through the use of NuXL functions and wizards. In future entries, we will dedicate ore tie for discrete data filters, their construction, and off course, application. Bacground You have probably occasionally transfored your data to stabilize the variance (e.g. log transfor) or to iprove the values distribution in the saple data. xt { x1, x2..., xt} yt log(x t) y { y, y..., y } t 1 2 In atheatics, the discrete Fourier transfor (DFT) converts a finite list of equally spaced saples of a function into a list of coefficients of a finite cobination of coplex sinusoids, ordered by their frequencies, which have those sae saple values. DFT converts the sapled function fro its original doain (often tie or position along a line) to the frequency doain. In su, the Fourier transfor has the following properties: 1. The transfored data is no longer in the tie doain. 2. The transforation operates on the whole data set. It is not a point by point transforation as we have seen with earlier transforations in the tie doain. x { x1, x2..., xt } Y F({ x1, x2..., xt }) Y { y, x..., x } { x, x..., x } F ({ y, x..., x }) 1 2 K T The transfored data is coplex (not real valued). What is the DFT? In plain words, the discrete Fourier transfor decoposes the input tie series into a set of cosine functions. T K Discrete Fourier Transfor Tutorial 1 Spider Financial Corp, 2013

2 N 1 x A cos( ) N 1 So, you can thin of the th output of the DFT as the A. The and the as the phase (in radians). A is referred to as the aplitude, The input tie series can now be expressed either as a tie sequence of values, or as a frequencysequence of [ A ] pairs. Knowing the set of [ A ], we can recover the exact input tie series. What is? is the fundaental or the principal radian frequency. IT is expressed as follows: Where: 2 T T is the nuber of observations in the equally spaced input tie series. What is N? The nuber of [ A ] pairs we need to have, so we can recover the original input tie series with a floor value of 2 T. Note that the zero frequency coponent (i.e. =0) is always real value, and in the case of even sized tie series, the last frequency coponent is also real value, which brings the total nuber of values (aplitude and phase) to T. There is no gain or loss of inforation or storage requireent because of this transfor. Finally, note that A A A T T A T T In essence, only the values of the first 2 T frequency coponents are needed, whereas the rest can be easily iplied fro the. Furtherore, the DFT values are periodic with a cycle length of T. Discrete Fourier Transfor Tutorial 2 Spider Financial Corp, 2013

3 Why decopose tie series data into a series of cosine functions? Consider the following tie series ( x1, x2,..., x 25) : Now, let s copute the tie series using a subset of the frequency sequence: Case 1: Using a zero frequency coponent: x (0) A o Using the zero frequency, we get the long run average of the tie series. Discrete Fourier Transfor Tutorial 3 Spider Financial Corp, 2013

4 Case 2: Using first frequency coponent (=1) x (0) A o x A A cos( ) x A cos( ) (1) (0) o Note that the graph on the right is essentially the sae as the one on the left, but with x (1) plotted using the right hand side axis scale. Case 2: Using first and second frequency coponents (=2) x A (0) o (1) (0) o (2) (1) 2cos( 2 2 ) x A A cos( ) x A cos( ) x x A Note, soother. (2) x is closer to the original tie series than (1) x due to the added cosine function, but (1) x is Discrete Fourier Transfor Tutorial 4 Spider Financial Corp, 2013

5 In essence, the process of recovering the original tie series fro the subset is siilar to tie series soothing, but without the drawbac of the lag effect. Case 3: Using the first 8 frequency coponents (=8) x (0) x A A cos( ) x A cos( ) (1) (0) o x x A cos( 2 ) (2) (1) (3) (2) (8) (7) 2 2 x x A cos( 3 )... A o 3 3 x x A cos( 8 ) 8 8 In su, by decoposing the input tie series into cosine functions, we can separate the coponent(s) attributed to noise (high frequency), uncover periodicity, and find a long run value for the process. Discrete Fourier Transfor Tutorial 5 Spider Financial Corp, 2013

Process First, let s organize our input data.

Now we are ready to construct our DFT output table.

be generated, then locate and clic on the Fourier icon in the NuXL tab (or

6 Process First, let s organize our input data. We can start by placing the values of the saple data in a separate colun. Now we are ready to construct our DFT output table. First, select the epty cell in your worsheet where you wish the output table to be generated, then locate and clic on the Fourier icon in the NuXL tab (or toolbar). The DFT Wizard pops up. Select the cells range for the values of the input variable. Discrete Fourier Transfor Tutorial 6 Spider Financial Corp, 2013

7 Notes: 1. By default, the table cells range is set to the current selected cell in your worsheet. Finally, once we select the input data (X) cells range, the Options and Missing Values tabs becoe available (enabled). Next, select the Options tab: Initially, the tab is set to the following values: Frequency Coponent Output is checed. Leave this option checed. o The Aplitude and Phase options are checed. Leave those options checed as well. o The nuber of coponents corresponds to the size of the output table. Set this value to five (5) to generate the first five frequency coponents. On the right side, Input Variable Output is uncheced. Chec this option to generate bac the input tie series using a subset of the frequency coponents. o Under No. of Coponents, set this value to 4. You can change this value later on in the output table. Now, clic on the Missing Values tab. Discrete Fourier Transfor Tutorial 7 Spider Financial Corp, 2013

This treatent is a good approach for our analysis, so let s leave it unchanged.

8 In this tab, you can select an approach to handle issing values in the data set (X s). By default, DFT wizard does not allow any issing value in the analysis. This treatent is a good approach for our analysis, so let s leave it unchanged. Now, clic OK to generate the output tables. Discrete Fourier Transfor Tutorial 8 Spider Financial Corp, 2013

In the first table (on the left), it displays the aplitude and phase (in radians) for different frequency coponents (i.e. cosine functions). Note that coponent zero has zero phase.

9 In the first table (on the left), it displays the aplitude and phase (in radians) for different frequency coponents (i.e. cosine functions). Note that coponent zero has zero phase. In the second table, it carries on the inverse Fourier transfor using a subset of the frequencies. If you wish to change the nuber of coponents, siply edit the nuber in the cell table, and the values under the Fitted title will be recalculated. Conclusion In this tutorial, we presented the interpretation of the discrete Fourier transfor (DFT) and its inverse (IDFT), as well as the process to carry out the related calculation in Excel using NuXL s add in functions. Where do we go fro here? Using DFT, we constructed an analytical forula representation for the input tie series. One direct application that we can thin of is to copute values for the new interediate observation, or to alter the sapling frequency (i.e. up saple) and introduce a new tie series. But what about issing values? What if we don t have a fixed sapling rate? Different types of Fourier transfors are available (e.g. non unifor tie discrete Fourier transfor (NUT DFT)) to handle unequally spaced input tie series, which generate a finite discrete set of frequencies. This will prove to be useful for iputing interediate issing values using the dynaics of the whole data set, rather than adjacent observations, as is the case in interpolation or Gaussian bridge ethods. Discrete Fourier Transfor Tutorial 9 Spider Financial Corp, 2013

10 The tie series spectru contains a significant aount of inforation, which we barely scratched in this tutorial. In our next entry, we will loo at the discrete filter (operator) definition in both tie and frequency doain, and its application to our tie series analysis/odeling. Discrete Fourier Transfor Tutorial 10 Spider Financial Corp, 2013

COLOR HISTOGRAM AND DISCRETE COSINE TRANSFORM FOR COLOR IMAGE RETRIEVAL

COLOR HISTOGRAM AND DISCRETE COSINE TRANSFORM FOR COLOR IMAGE RETRIEVAL 1 Te-Wei Chiang ( 蔣德威 ), 2 Tienwei Tsai ( 蔡殿偉 ), 3 Jeng-Ping Lin ( 林正平 ) 1 Dept. of Accounting Inforation Systes, Chilee Institute