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Engineering Math

DFT (Discrete Fourier Transform)

Probably Fourier Transform would be a mathematical tool which is the most widely used in engineer area. I would not explain in details how Fourier Transform works mathematically, you can have a lot of information from internet or at least you would go through at least a couple of month for this topic during your university course.

When I think about the word "Fourier" (Actually a person's name), I have two images as shown below. One is a Fourier Series Expension and the other one is Fourier Transform. With Fourier Series expansion, you would have a continous function made up of very simple components (sin() + cos()) for any data whether the data is continous or non-continous. I am pretty sure that you would learn how to create a sequence of (sin() + cos()) to approximate computer clock-like rectangular pulse. (You could easily find tutorial or even many of Java application from internet).

Another image that pops up in my mind when I think about 'Fourier' is 'Fourier Transform'. One of the most typical usage (or purpose) of courier transform is to convert a time domain data into a frequency domain data and this section will mostly focused on this 'Fourier Transform'. (I think I am hearing those questions from many reader... saying ... "What is time domain data ? What is frequecy domain data ? why we want to convert the time domain data into frequency domain data ?". All of these are very good question. Try very hard until you get your own answers for these questions -:)).

I would not explain much of the theory about FFT, you would already get too much of it from text book or from on-line. As in other pages in my site, I will put as much examples and plots as possible just to make you "feel" about it. Of course, this pages would keep being extended as I find more examples that would help you with "intuitive" understanding of FFT.

What is Fourier Transform ?

Fourier Transform is a special kinds of mathematical series technology that can approximate a function or a data with summation of sin() and cos() function. There are two kind of Fourier transform, one is continuous fourier transform and the other is discrete fourier transform.

Basic principles are same, but in this section I will explain the concept of Fourier transform using a discrete transform since it is easy to visualize and less scary :)

A typical mathematical presentation of Discrete Fourier Transform looks as follows. It may look a little bit complicated, but if you wrap it up into proper chunks you would notice that this is also a kind of "Sum of Times" format.

As you learned in Correlation section (explained above), the most intuitive meaning of "Sum of Times" is a kind of indicator showing the correlation between two data sequences.

So Fourier Transform is also based on calculating the two specified data sequence as follows.

xi = input data sequence for Fourier Transform

yi = e^(-i 2pi k n/N)

It means that Fourier Transform is indication of correlation between a given data sequence and "e^(-i 2pi k n/N)". If you studied "Exponential Form" or any material about Euler's formula, you would know e^(-i 2pi k n/N) is a cyclic function. It is a cyclic function, the cycle of which is represented by the variable 'k'. Therefore, we can say "Fourier Transform is a tool to show the correlation between a specified data sequence and muliples of cyclic data sequence with different cycle".

If I represent my explanation, Fourier Transform can depicted as follows. (In the following graph, "x1" is the input data you want to do 'fourier transform', and the series of plots on right side is the multiple cyclic data sequence with different cycles. The stem plot at the left bottom is the graph shows the forrelation coefficient of each data pair, i.e (x1, s1), (x1,s2), (x1, s3) and so forth.

If you convert this description into a mathematical forumula, Fourier Transform can be depicted as follows.

Following is Matlab/Octave script generating the sequence of graphs shown above. Try change the data for x1 in various different way until you get your own intuition of this concept.

t=0:2*pi/400:3;

Nt = size(t,2);

x1 = sin(2*pi*2*t) + 0.3 * sin(2*pi*5*t) + 0.7*sin(2*pi*7*t);

s1=sin(2*pi*1*t);

s2=sin(2*pi*2*t);

s3=sin(2*pi*3*t);

s4=sin(2*pi*4*t);

s5=sin(2*pi*5*t);

s6=sin(2*pi*6*t);

s7=sin(2*pi*7*t);

s8=sin(2*pi*8*t);

s9=sin(2*pi*9*t);

s10=sin(2*pi*10*t);

corr_vec = [corrcoef(x1,s1) corrcoef(x1,s2) corrcoef(x1,s3) corrcoef(x1,s4) corrcoef(x1,s5) corrcoef(x1,s6) corrcoef(x1,s7) corrcoef(x1,s8) corrcoef(x1,s9) corrcoef(x1,s10)];

subplot(10,2,1);plot(x1);ylabel('x1');axis([0,Nt,-2.1,2.1]);

subplot(10,2,1*2);plot(s1);ylabel('s1(1Hz)');axis([0,Nt,-1.1,1.1]);

subplot(10,2,2*2);plot(s2);ylabel('s2(2Hz)');axis([0,Nt,-1.1,1.1]);

subplot(10,2,3*2);plot(s3);ylabel('s3(3Hz)');axis([0,Nt,-1.1,1.1]);

subplot(10,2,4*2);plot(s4);ylabel('s4(4Hz)');axis([0,Nt,-1.1,1.1]);

subplot(10,2,5*2);plot(s5);ylabel('s5(5Hz)');axis([0,Nt,-1.1,1.1]);

subplot(10,2,6*2);plot(s6);ylabel('s6(6Hz)');axis([0,Nt,-1.1,1.1]);

subplot(10,2,7*2);plot(s7);ylabel('s7(7Hz)');axis([0,Nt,-1.1,1.1]);

subplot(10,2,8*2);plot(s8);ylabel('s8(8Hz)');axis([0,Nt,-1.1,1.1]);

subplot(10,2,9*2);plot(s9);ylabel('s9(9Hz)');axis([0,Nt,-1.1,1.1]);

subplot(10,2,10*2);plot(s10);ylabel('s10(10Hz)');axis([0,Nt,-1.1,1.1]);

subplot(10,2,[13 15 17 19]);stem(abs(corr_vec));ylabel('corr');axis([0,10,0,1.0]);

Common Example of Fourier Transform

Followings are some of the common examples of Fourier Transform that you may often see in engineering math textbook. These examples are from my visual note (www.slide4math.com). The concept shown in these example is basically same as what I explained in previous section, but this visualization shows more strictly on Fourier Transform formula with complex numbers.

Example 01 > Cos(x)