Filming is currently underway on a special online course based on this blog which will include videos, animations and work-throughs to illustrate, in a visual way, how the Fourier Transform works, what all the maths is all about and how it is applied in the real world.
Click here to reserve your free module
The module will be emailed to you the moment the course goes live.
The Discrete Fourier Transform
In the last post we derived a set of mathematical instructions, an equation, for calculating the Fourier Transform of a signal, or in other words, separating our signal out into its constituent frequencies. The problem with the equation was that it assumed we could cope with things that were infinite. For example, it assumed that our signal was of infinite length, it had no beginning and no end, but just kept going on forever. It assumed that we can test every frequency in existence (an infinite number of frequencies). It also assumed that we multiply (convolve) our signal with our test frequencies at every signal point in time (an infinite number of points).
This is clearly ridiculous. In our finite lives, we cannot cope with things which are infinite. Such God like powers are beyond our capabilities. Therefore, we have to look at things in a more discrete way. To look at something discretely means to look only at certain parts of it.
If we apply this to our signal:
- We only look at a part of the signal, we divide our signal up into byte sized chunks and do the Fourier Transform on one chunk before proceeding to the next.
- We cannot look at our signal at every single point in time as there are an infinite number of points in time. Therefore we look at our signal so many times a second. This is known as sampling the signal. How often we look at the signal is known as the sampling rate.
- We only try a limited frequency range with our signal, not every single frequency in existence.
- Within that frequency range we only try out frequencies at certain intervals.
By definition, the need to be discrete means that we are missing out data and we have to be careful that we do not miss out on so much data that we miss something important in our signal. Therefore, we have to be careful when deciding precisely how discrete we are going to be. However, there are two things we can be sure of.
Sampling Rate – In order to ensure that we don’t lose important information from our signal, we must sample the signal at least twice as many times a second as the maximum frequency that exists in the signal. “How do I know what the maximum frequency that exists in my signal is?” I hear you ask. After all, you are doing the Fourier Transform specifically in order to find all the frequencies. The answer to that questions depends on the type of signal you are testing. I am writing this blog from the viewpoint of a sound engineer. We tend to work with signals that the human ear can pick up and in that case, the maximum frequency of human hearing is 20kHz. This means that the minimum sample rate should be at least 40kHz (40,000 times per second).
Frequency Range – The range of frequencies we try out with our signal should cover the entire range of frequencies that appear in the signal. Again this depends on the type of signal you are testing. If it is a sound signal intended for the human ear then the sort of range we should be testing is from 20Hz to 20kHz, the range of human hearing.
So now let’s start putting all this into mathematical terms.
We’ll start with the Fourier Transform Equation we met in Part 6:
The Sampled Signal
In the Discrete Fourier Transform, we are no longer looking at our signal as a continuous signal of time. We sample it at regular intervals. Each of those samples has an index: 0, 1, 2, 3… etc. Therefore we no longer express our signal as a continuous function of time x(t), we express it as a function of its sample index which we’ll call n, so our signal is now called x(n).
x(n) is simply the list of the amplitudes of each sample. An example of a signal x(n) is shown below. This could be any signal. What is important to notice is that the signal is no longer a smooth line (the grey line in the graph below), rather a collection of dots (the purple diamonds in the graph below), each dot is one sample.
n is the index of the sample and is not in itself a measure of time. To know exactly when each sample was taken, we need to know the sample rate (let’s call it R). The time can then be calculated using the following formula.
Equation 1
The above signal has been sampled at 40Hz (40 samples per second) so R = 40. The first sample at index 0 is taken at time=0 seconds, the next at 1/40 seconds, the next at 2/40 seconds and so on.
Discrete Frequencies
Just as our signal x(t) is no longer a continuous signal and so the term has become x(n), so too, as we are only trying discrete frequencies with our signal, the frequency term f, now becomes k to denote the index of the frequency we are trying (0, 1, 2, 3… etc). Again, as k is the index of the frequency and not the frequency itself, if we ever wanted to convert our index into an actual frequency value, we need the following equation.
Equation 2
where:
N is the total number of samples in the section of our signal on which we are performing the transform
k is the index of the frequency we are testing
R is the sample rate at which we sampled the signal
This means that the Fourier Transform of our discrete signal is no longer a continuous function of frequency, X(f), but a function of the index of each discrete frequency we tried X(k). Again, if we use the same sample rate (R=40) and number of sample N=40 as we did above, then the first frequency we try, k=0 is 0Hz, the second frequency we try, k=1 is 1Hz and so on.
A signal that doesn’t go on for ever
As we are only performing the Fourier Transform on sections of our finite signal, not a signal that goes on forever, we don’t have to integrate our signal over the whole of time, so the ∞ and -∞ signs above and below the integration sign become finite numbers, usually from the first sample index at n=0 to the last sample index at n=N-1. (The first sample has an index of 0 so that last sample of N samples will have an index of N-1)
A summation, not an integration
The final change to the Fourier Transform equation is to realize that as we are now adding together a list of finite values, we are no longer integrating an infinite number of values. This is expressed by changing the curly integral sign to a summation sign denoted by the Greek symbol sigma.
All of the above gives rise to:
The Discrete Fourier Transform Equation
where:
- n is the time index of the sample (0, 1, 2, 3, …, N-1)
- k is the frequency index of the frequency being tested (0, 1, 2, 3, …, N-1)
- N is the total number of samples
This can be written in polar form as:
It is still doing the same thing as the Fourier Transform we met before: Multiplying the signal by sine and cosine waves at known frequencies and adding together the result, only this time it is doing so discretely by both only looking at the signal at certain points in time and only testing the signal with a limited number of frequencies over a limited time.
Now that we have something that we can actually calculate in the real world, let’s go ahead and try and calculate the Fourier Transform for the signal at the top of the page, or in other words, find out which frequencies, their amplitudes and phases, are present in our signal.
Example
We start with a 2-dimensional array of values. The first dimension (the left hand column) is a list of the sample indexes (n) of each point on the graph above, the x-coordinate of each point on the graph. The second dimension (the right hand column) is a list of the amplitudes of each point, or to put it another way x(n) – the y-coordinate of each point on the graph.
| Sample Index - n | Amplitude - x(n) |
|---|---|
| 0 | -0.4964 |
| 1 | 0.9290 |
| 2 | 0.3500 |
| 3 | -0.0805 |
| 4 | 0.1964 |
| 5 | -0.4094 |
| 6 | -0.0500 |
| 7 | -0.4391 |
| 8 | -0.4964 |
| 9 | 0.9290 |
| 10 | 0.3500 |
| 11 | -0.0805 |
| 12 | 0.1964 |
| 13 | -0.4094 |
| 14 | -0.0500 |
| 15 | -0.4391 |
| 16 | -0.4964 |
| 17 | 0.9290 |
| 18 | 0.3500 |
| 19 | -0.0805 |
| 20 | 0.1964 |
| 21 | -0.4094 |
| 22 | -0.0500 |
| 23 | -0.4391 |
| 24 | -0.4964 |
| 25 | 0.9290 |
| 26 | 0.3500 |
| 27 | -0.0805 |
| 28 | 0.1964 |
| 29 | -0.4094 |
| 30 | -0.0500 |
| 31 | -0.4391 |
| 32 | -0.4964 |
| 33 | 0.9290 |
| 34 | 0.3500 |
| 35 | -0.0805 |
| 36 | 0.1964 |
| 37 | -0.4094 |
| 38 | -0.0500 |
| 39 | -0.4391 |
If this was an audio signal, I would be sure to sample it at least 40,000 times per second in order to ensure that every frequency within the human hearing range was adequately represented. However, as I don’t want this blog post to be too long, the signal in the graph above is a much simpler signal. The highest frequency in the signal is 15Hz. Therefore, I need to sample the signal at least 30 times per second. I have allowed a little headroom and actually sampled the signal at 40Hz (40 times per second). I have sampled it for 40 samples which means that the actual duration of the signal is 1 second.
Now I’m going to multiply each sample of my signal by a cosine wave at my first frequency index, k=0. This correlates to an actual frequency of 0Hz, but what does a cosine wave with a frequency of zero actually look like? If we take the cosine part of the discrete Fourier transform equation:
…and plug in k=0, the zero term of k will make the whole expression inside the brackets equal to zero all the time meaning that no matter what the sample index (n) is for this first cosine wave, the amplitude of the wave will just end up being cos(0) which is equal to 1. Therefore a cosine wave with a frequency 0Hz is simple a straight line with amplitude 1. If we then multiply this by our signal, all we are doing is multiplying the whole signal by 1, which just gives us back the original signal.
The next stage is to add together all the points we have just calculated, Well even without doing the maths, if you look at the graph above, you might notice that there are as many points above zero as there are below with exactly the same amplitudes. This means the result of our summation operation is going to be zero. This tells us DC offset of our signal, or in other words, the amplitude around which our signal is oscillating. In the case of the signal in the graph above, its DC offset is zero.
The next stage is to multiply each sample of my signal by a sine wave at my first frequency index, k=0. If we take the sine part of the discrete Fourier transform equation:
…again k=0 means that the entire term inside the brackets will be zero. Therefore no matter what the sample index, sin(0) = 0. We get a straight line with an amplitude of zero. If we multiply that by our signal, it is simply going to come out at zero, so I won’t even bother!
We continue to multiply our signal by cosine and sine waves with every frequency index, k. As there are 40 points on the graph (N=40), we multiply our signal by cosine waves and sine waves at 40 different frequencies. This gives us a table of results with 40 rows (one for each sample) and 80 columns, (40 columns containing the cosine multiplication result and 40 columns containing the sine multiplication result). We then take the sum of each column. The result of summing each column is shown by the bold row marked “summation” at the bottom of the table below.
| Sample Index - n | Amplitude - x(n) | Cosine (k=0) | Sine (k=0) | Cosine (k=1) | Sine (k=1) | Cosine (k=2) | Sine (k=2) | Cosine (k=3) | Sine (k=3) | Cosine (k=4) | Sine (k=4) | Cosine (k=5) | Sine (k=5) | Cosine (k=6) | Sine (k=6) | Cosine (k=7) | Sine (k=7) | Cosine (k=8) | Sine (k=8) | Cosine (k=9) | Sine (k=9) | Cosine (k=10) | Sine (k=10) | Cosine (k=11) | Sine (k=11) | Cosine (k=12) | Sine (k=12) | Cosine (k=13) | Sine (k=13) | Cosine (k=14) | Sine (k=14) | Cosine (k=15) | Sine (k=15) | Cosine (k=16) | Sine (k=16) | Cosine (k=17) | Sine (k=17) | Cosine (k=18) | Sine (k=18) | Cosine (k=19) | Sine (k=19) | Cosine (k=20) | Sine (k=20) | Cosine (k=21) | Sine (k=21) | Cosine (k=22) | Sine (k=22) | Cosine (k=23) | Sine (k=23) | Cosine (k=24) | Sine (k=24) | Cosine (k=25) | Sine (k=25) | Cosine (k=26) | Sine (k=26) | Cosine (k=27) | Sine (k=27) | Cosine (k=28) | Sine (k=28) | Cosine (k=29) | Sine (k=29) | Cosine (k=30) | Sine (k=30) | Cosine (k=31) | Sine (k=31) | Cosine (k=32) | Sine (k=32) | Cosine (k=33) | Sine (k=33) | Cosine (k=34) | Sine (k=34) | Cosine (k=35) | Sine (k=35) | Cosine (k=36) | Sine (k=36) | Cosine (k=37) | Sine (k=37) | Cosine (k=38) | Sine (k=38) | Cosine (k=39) | Sine (k=39) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Summation | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 8.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | -3.0000 | 5.1962 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | -6.9282 | 4.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | -6.9282 | -4.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | -3.0000 | -5.1962 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | -8.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |
| 0 | -0.4964 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 | -0.4964 | 0.0000 |
| 1 | 0.9290 | 0.9176 | 0.0000 | 0.9176 | 0.1453 | 0.8836 | 0.2871 | 0.8278 | 0.4218 | 0.7516 | 0.5461 | 0.6569 | 0.6569 | 0.5461 | 0.7516 | 0.4218 | 0.8278 | 0.2871 | 0.8836 | 0.1453 | 0.9176 | 0.0000 | 0.9290 | -0.1453 | 0.9176 | -0.2871 | 0.8836 | -0.4218 | 0.8278 | -0.5461 | 0.7516 | -0.6569 | 0.6569 | -0.7516 | 0.5461 | -0.8278 | 0.4218 | -0.8836 | 0.2871 | -0.9176 | 0.1453 | -0.9290 | 0.0000 | -0.9176 | -0.1453 | -0.8836 | -0.2871 | -0.8278 | -0.4218 | -0.7516 | -0.5461 | -0.6569 | -0.6569 | -0.5461 | -0.7516 | -0.4218 | -0.8278 | -0.2871 | -0.8836 | -0.1453 | -0.9176 | 0.0000 | -0.9290 | 0.1453 | -0.9176 | 0.2871 | -0.8836 | 0.4218 | -0.8278 | 0.5461 | -0.7516 | 0.6569 | -0.6569 | 0.7516 | -0.5461 | 0.8278 | -0.4218 | 0.8836 | -0.2871 | 0.9176 | -0.1453 |
| 2 | 0.3500 | 0.3329 | 0.0000 | 0.3329 | 0.1082 | 0.2832 | 0.2057 | 0.2057 | 0.2832 | 0.1082 | 0.3329 | 0.0000 | 0.3500 | -0.1082 | 0.3329 | -0.2057 | 0.2832 | -0.2832 | 0.2057 | -0.3329 | 0.1082 | -0.3500 | 0.0000 | -0.3329 | -0.1082 | -0.2832 | -0.2057 | -0.2057 | -0.2832 | -0.1082 | -0.3329 | 0.0000 | -0.3500 | 0.1082 | -0.3329 | 0.2057 | -0.2832 | 0.2832 | -0.2057 | 0.3329 | -0.1082 | 0.3500 | 0.0000 | 0.3329 | 0.1082 | 0.2832 | 0.2057 | 0.2057 | 0.2832 | 0.1082 | 0.3329 | 0.0000 | 0.3500 | -0.1082 | 0.3329 | -0.2057 | 0.2832 | -0.2832 | 0.2057 | -0.3329 | 0.1082 | -0.3500 | 0.0000 | -0.3329 | -0.1082 | -0.2832 | -0.2057 | -0.2057 | -0.2832 | -0.1082 | -0.3329 | 0.0000 | -0.3500 | 0.1082 | -0.3329 | 0.2057 | -0.2832 | 0.2832 | -0.2057 | 0.3329 | -0.1082 |
| 3 | -0.0805 | -0.0717 | 0.0000 | -0.0717 | -0.0365 | -0.0473 | -0.0651 | -0.0126 | -0.0795 | 0.0249 | -0.0766 | 0.0569 | -0.0569 | 0.0766 | -0.0249 | 0.0795 | 0.0126 | 0.0651 | 0.0473 | 0.0365 | 0.0717 | 0.0000 | 0.0805 | -0.0365 | 0.0717 | -0.0651 | 0.0473 | -0.0795 | 0.0126 | -0.0766 | -0.0249 | -0.0569 | -0.0569 | -0.0249 | -0.0766 | 0.0126 | -0.0795 | 0.0473 | -0.0651 | 0.0717 | -0.0365 | 0.0805 | 0.0000 | 0.0717 | 0.0365 | 0.0473 | 0.0651 | 0.0126 | 0.0795 | -0.0249 | 0.0766 | -0.0569 | 0.0569 | -0.0766 | 0.0249 | -0.0795 | -0.0126 | -0.0651 | -0.0473 | -0.0365 | -0.0717 | 0.0000 | -0.0805 | 0.0365 | -0.0717 | 0.0651 | -0.0473 | 0.0795 | -0.0126 | 0.0766 | 0.0249 | 0.0569 | 0.0569 | 0.0249 | 0.0766 | -0.0126 | 0.0795 | -0.0473 | 0.0651 | -0.0717 | 0.0365 |
| 4 | 0.1964 | 0.1589 | 0.0000 | 0.1589 | 0.1154 | 0.0607 | 0.1868 | -0.0607 | 0.1868 | -0.1589 | 0.1154 | -0.1964 | 0.0000 | -0.1589 | -0.1154 | -0.0607 | -0.1868 | 0.0607 | -0.1868 | 0.1589 | -0.1154 | 0.1964 | 0.0000 | 0.1589 | 0.1154 | 0.0607 | 0.1868 | -0.0607 | 0.1868 | -0.1589 | 0.1154 | -0.1964 | 0.0000 | -0.1589 | -0.1154 | -0.0607 | -0.1868 | 0.0607 | -0.1868 | 0.1589 | -0.1154 | 0.1964 | 0.0000 | 0.1589 | 0.1154 | 0.0607 | 0.1868 | -0.0607 | 0.1868 | -0.1589 | 0.1154 | -0.1964 | 0.0000 | -0.1589 | -0.1154 | -0.0607 | -0.1868 | 0.0607 | -0.1868 | 0.1589 | -0.1154 | 0.1964 | 0.0000 | 0.1589 | 0.1154 | 0.0607 | 0.1868 | -0.0607 | 0.1868 | -0.1589 | 0.1154 | -0.1964 | 0.0000 | -0.1589 | -0.1154 | -0.0607 | -0.1868 | 0.0607 | -0.1868 | 0.1589 | -0.1154 |
| 5 | -0.4094 | -0.2895 | 0.0000 | -0.2895 | -0.2895 | 0.0000 | -0.4094 | 0.2895 | -0.2895 | 0.4094 | 0.0000 | 0.2895 | 0.2895 | 0.0000 | 0.4094 | -0.2895 | 0.2895 | -0.4094 | 0.0000 | -0.2895 | -0.2895 | 0.0000 | -0.4094 | 0.2895 | -0.2895 | 0.4094 | 0.0000 | 0.2895 | 0.2895 | 0.0000 | 0.4094 | -0.2895 | 0.2895 | -0.4094 | 0.0000 | -0.2895 | -0.2895 | 0.0000 | -0.4094 | 0.2895 | -0.2895 | 0.4094 | 0.0000 | 0.2895 | 0.2895 | 0.0000 | 0.4094 | -0.2895 | 0.2895 | -0.4094 | 0.0000 | -0.2895 | -0.2895 | 0.0000 | -0.4094 | 0.2895 | -0.2895 | 0.4094 | 0.0000 | 0.2895 | 0.2895 | 0.0000 | 0.4094 | -0.2895 | 0.2895 | -0.4094 | 0.0000 | -0.2895 | -0.2895 | 0.0000 | -0.4094 | 0.2895 | -0.2895 | 0.4094 | 0.0000 | 0.2895 | 0.2895 | 0.0000 | 0.4094 | -0.2895 | 0.2895 |
| 6 | -0.0500 | -0.0294 | 0.0000 | -0.0294 | -0.0405 | 0.0155 | -0.0476 | 0.0476 | -0.0155 | 0.0405 | 0.0294 | 0.0000 | 0.0500 | -0.0405 | 0.0294 | -0.0476 | -0.0155 | -0.0155 | -0.0476 | 0.0294 | -0.0405 | 0.0500 | 0.0000 | 0.0294 | 0.0405 | -0.0155 | 0.0476 | -0.0476 | 0.0155 | -0.0405 | -0.0294 | 0.0000 | -0.0500 | 0.0405 | -0.0294 | 0.0476 | 0.0155 | 0.0155 | 0.0476 | -0.0294 | 0.0405 | -0.0500 | 0.0000 | -0.0294 | -0.0405 | 0.0155 | -0.0476 | 0.0476 | -0.0155 | 0.0405 | 0.0294 | 0.0000 | 0.0500 | -0.0405 | 0.0294 | -0.0476 | -0.0155 | -0.0155 | -0.0476 | 0.0294 | -0.0405 | 0.0500 | 0.0000 | 0.0294 | 0.0405 | -0.0155 | 0.0476 | -0.0476 | 0.0155 | -0.0405 | -0.0294 | 0.0000 | -0.0500 | 0.0405 | -0.0294 | 0.0476 | 0.0155 | 0.0155 | 0.0476 | -0.0294 | 0.0405 |
| 7 | -0.4391 | -0.1994 | 0.0000 | -0.1994 | -0.3913 | 0.2581 | -0.3553 | 0.4337 | 0.0687 | 0.1357 | 0.4176 | -0.3105 | 0.3105 | -0.4176 | -0.1357 | -0.0687 | -0.4337 | 0.3553 | -0.2581 | 0.3913 | 0.1994 | 0.0000 | 0.4391 | -0.3913 | 0.1994 | -0.3553 | -0.2581 | 0.0687 | -0.4337 | 0.4176 | -0.1357 | 0.3105 | 0.3105 | -0.1357 | 0.4176 | -0.4337 | 0.0687 | -0.2581 | -0.3553 | 0.1994 | -0.3913 | 0.4391 | 0.0000 | 0.1994 | 0.3913 | -0.2581 | 0.3553 | -0.4337 | -0.0687 | -0.1357 | -0.4176 | 0.3105 | -0.3105 | 0.4176 | 0.1357 | 0.0687 | 0.4337 | -0.3553 | 0.2581 | -0.3913 | -0.1994 | 0.0000 | -0.4391 | 0.3913 | -0.1994 | 0.3553 | 0.2581 | -0.0687 | 0.4337 | -0.4176 | 0.1357 | -0.3105 | -0.3105 | 0.1357 | -0.4176 | 0.4337 | -0.0687 | 0.2581 | 0.3553 | -0.1994 | 0.3913 |
| 8 | -0.4964 | -0.1534 | 0.0000 | -0.1534 | -0.4721 | 0.4016 | -0.2918 | 0.4016 | 0.2918 | -0.1534 | 0.4721 | -0.4964 | 0.0000 | -0.1534 | -0.4721 | 0.4016 | -0.2918 | 0.4016 | 0.2918 | -0.1534 | 0.4721 | -0.4964 | 0.0000 | -0.1534 | -0.4721 | 0.4016 | -0.2918 | 0.4016 | 0.2918 | -0.1534 | 0.4721 | -0.4964 | 0.0000 | -0.1534 | -0.4721 | 0.4016 | -0.2918 | 0.4016 | 0.2918 | -0.1534 | 0.4721 | -0.4964 | 0.0000 | -0.1534 | -0.4721 | 0.4016 | -0.2918 | 0.4016 | 0.2918 | -0.1534 | 0.4721 | -0.4964 | 0.0000 | -0.1534 | -0.4721 | 0.4016 | -0.2918 | 0.4016 | 0.2918 | -0.1534 | 0.4721 | -0.4964 | 0.0000 | -0.1534 | -0.4721 | 0.4016 | -0.2918 | 0.4016 | 0.2918 | -0.1534 | 0.4721 | -0.4964 | 0.0000 | -0.1534 | -0.4721 | 0.4016 | -0.2918 | 0.4016 | 0.2918 | -0.1534 | 0.4721 |
| 9 | 0.9290 | 0.1453 | 0.0000 | 0.1453 | 0.9176 | -0.8836 | 0.2871 | -0.4218 | -0.8278 | 0.7516 | -0.5461 | 0.6569 | 0.6569 | -0.5461 | 0.7516 | -0.8278 | -0.4218 | 0.2871 | -0.8836 | 0.9176 | 0.1453 | 0.0000 | 0.9290 | -0.9176 | 0.1453 | -0.2871 | -0.8836 | 0.8278 | -0.4218 | 0.5461 | 0.7516 | -0.6569 | 0.6569 | -0.7516 | -0.5461 | 0.4218 | -0.8278 | 0.8836 | 0.2871 | -0.1453 | 0.9176 | -0.9290 | 0.0000 | -0.1453 | -0.9176 | 0.8836 | -0.2871 | 0.4218 | 0.8278 | -0.7516 | 0.5461 | -0.6569 | -0.6569 | 0.5461 | -0.7516 | 0.8278 | 0.4218 | -0.2871 | 0.8836 | -0.9176 | -0.1453 | 0.0000 | -0.9290 | 0.9176 | -0.1453 | 0.2871 | 0.8836 | -0.8278 | 0.4218 | -0.5461 | -0.7516 | 0.6569 | -0.6569 | 0.7516 | 0.5461 | -0.4218 | 0.8278 | -0.8836 | -0.2871 | 0.1453 | -0.9176 |
| 10 | 0.3500 | 0.0000 | 0.0000 | 0.0000 | 0.3500 | -0.3500 | 0.0000 | 0.0000 | -0.3500 | 0.3500 | 0.0000 | 0.0000 | 0.3500 | -0.3500 | 0.0000 | 0.0000 | -0.3500 | 0.3500 | 0.0000 | 0.0000 | 0.3500 | -0.3500 | 0.0000 | 0.0000 | -0.3500 | 0.3500 | 0.0000 | 0.0000 | 0.3500 | -0.3500 | 0.0000 | 0.0000 | -0.3500 | 0.3500 | 0.0000 | 0.0000 | 0.3500 | -0.3500 | 0.0000 | 0.0000 | -0.3500 | 0.3500 | 0.0000 | 0.0000 | 0.3500 | -0.3500 | 0.0000 | 0.0000 | -0.3500 | 0.3500 | 0.0000 | 0.0000 | 0.3500 | -0.3500 | 0.0000 | 0.0000 | -0.3500 | 0.3500 | 0.0000 | 0.0000 | 0.3500 | -0.3500 | 0.0000 | 0.0000 | -0.3500 | 0.3500 | 0.0000 | 0.0000 | 0.3500 | -0.3500 | 0.0000 | 0.0000 | -0.3500 | 0.3500 | 0.0000 | 0.0000 | 0.3500 | -0.3500 | 0.0000 | 0.0000 | -0.3500 |
| 11 | -0.0805 | 0.0126 | 0.0000 | 0.0126 | -0.0795 | 0.0766 | 0.0249 | -0.0365 | 0.0717 | -0.0651 | -0.0473 | 0.0569 | -0.0569 | 0.0473 | 0.0651 | -0.0717 | 0.0365 | -0.0249 | -0.0766 | 0.0795 | -0.0126 | 0.0000 | 0.0805 | -0.0795 | -0.0126 | 0.0249 | -0.0766 | 0.0717 | 0.0365 | -0.0473 | 0.0651 | -0.0569 | -0.0569 | 0.0651 | -0.0473 | 0.0365 | 0.0717 | -0.0766 | 0.0249 | -0.0126 | -0.0795 | 0.0805 | 0.0000 | -0.0126 | 0.0795 | -0.0766 | -0.0249 | 0.0365 | -0.0717 | 0.0651 | 0.0473 | -0.0569 | 0.0569 | -0.0473 | -0.0651 | 0.0717 | -0.0365 | 0.0249 | 0.0766 | -0.0795 | 0.0126 | 0.0000 | -0.0805 | 0.0795 | 0.0126 | -0.0249 | 0.0766 | -0.0717 | -0.0365 | 0.0473 | -0.0651 | 0.0569 | 0.0569 | -0.0651 | 0.0473 | -0.0365 | -0.0717 | 0.0766 | -0.0249 | 0.0126 | 0.0795 |
| 12 | 0.1964 | -0.0607 | 0.0000 | -0.0607 | 0.1868 | -0.1589 | -0.1154 | 0.1589 | -0.1154 | 0.0607 | 0.1868 | -0.1964 | 0.0000 | 0.0607 | -0.1868 | 0.1589 | 0.1154 | -0.1589 | 0.1154 | -0.0607 | -0.1868 | 0.1964 | 0.0000 | -0.0607 | 0.1868 | -0.1589 | -0.1154 | 0.1589 | -0.1154 | 0.0607 | 0.1868 | -0.1964 | 0.0000 | 0.0607 | -0.1868 | 0.1589 | 0.1154 | -0.1589 | 0.1154 | -0.0607 | -0.1868 | 0.1964 | 0.0000 | -0.0607 | 0.1868 | -0.1589 | -0.1154 | 0.1589 | -0.1154 | 0.0607 | 0.1868 | -0.1964 | 0.0000 | 0.0607 | -0.1868 | 0.1589 | 0.1154 | -0.1589 | 0.1154 | -0.0607 | -0.1868 | 0.1964 | 0.0000 | -0.0607 | 0.1868 | -0.1589 | -0.1154 | 0.1589 | -0.1154 | 0.0607 | 0.1868 | -0.1964 | 0.0000 | 0.0607 | -0.1868 | 0.1589 | 0.1154 | -0.1589 | 0.1154 | -0.0607 | -0.1868 |
| 13 | -0.4094 | 0.1859 | 0.0000 | 0.1859 | -0.3648 | 0.2406 | 0.3312 | -0.4044 | 0.0640 | 0.1265 | -0.3894 | 0.2895 | 0.2895 | -0.3894 | 0.1265 | 0.0640 | -0.4044 | 0.3312 | 0.2406 | -0.3648 | 0.1859 | 0.0000 | -0.4094 | 0.3648 | 0.1859 | -0.3312 | 0.2406 | -0.0640 | -0.4044 | 0.3894 | 0.1265 | -0.2895 | 0.2895 | -0.1265 | -0.3894 | 0.4044 | 0.0640 | -0.2406 | 0.3312 | -0.1859 | -0.3648 | 0.4094 | 0.0000 | -0.1859 | 0.3648 | -0.2406 | -0.3312 | 0.4044 | -0.0640 | -0.1265 | 0.3894 | -0.2895 | -0.2895 | 0.3894 | -0.1265 | -0.0640 | 0.4044 | -0.3312 | -0.2406 | 0.3648 | -0.1859 | 0.0000 | 0.4094 | -0.3648 | -0.1859 | 0.3312 | -0.2406 | 0.0640 | 0.4044 | -0.3894 | -0.1265 | 0.2895 | -0.2895 | 0.1265 | 0.3894 | -0.4044 | -0.0640 | 0.2406 | -0.3312 | 0.1859 | 0.3648 |
| 14 | -0.0500 | 0.0294 | 0.0000 | 0.0294 | -0.0405 | 0.0155 | 0.0476 | -0.0476 | -0.0155 | 0.0405 | -0.0294 | 0.0000 | 0.0500 | -0.0405 | -0.0294 | 0.0476 | -0.0155 | -0.0155 | 0.0476 | -0.0294 | -0.0405 | 0.0500 | 0.0000 | -0.0294 | 0.0405 | -0.0155 | -0.0476 | 0.0476 | 0.0155 | -0.0405 | 0.0294 | 0.0000 | -0.0500 | 0.0405 | 0.0294 | -0.0476 | 0.0155 | 0.0155 | -0.0476 | 0.0294 | 0.0405 | -0.0500 | 0.0000 | 0.0294 | -0.0405 | 0.0155 | 0.0476 | -0.0476 | -0.0155 | 0.0405 | -0.0294 | 0.0000 | 0.0500 | -0.0405 | -0.0294 | 0.0476 | -0.0155 | -0.0155 | 0.0476 | -0.0294 | -0.0405 | 0.0500 | 0.0000 | -0.0294 | 0.0405 | -0.0155 | -0.0476 | 0.0476 | 0.0155 | -0.0405 | 0.0294 | 0.0000 | -0.0500 | 0.0405 | 0.0294 | -0.0476 | 0.0155 | 0.0155 | -0.0476 | 0.0294 | 0.0405 |
| 15 | -0.4391 | 0.3105 | 0.0000 | 0.3105 | -0.3105 | 0.0000 | 0.4391 | -0.3105 | -0.3105 | 0.4391 | 0.0000 | -0.3105 | 0.3105 | 0.0000 | -0.4391 | 0.3105 | 0.3105 | -0.4391 | 0.0000 | 0.3105 | -0.3105 | 0.0000 | 0.4391 | -0.3105 | -0.3105 | 0.4391 | 0.0000 | -0.3105 | 0.3105 | 0.0000 | -0.4391 | 0.3105 | 0.3105 | -0.4391 | 0.0000 | 0.3105 | -0.3105 | 0.0000 | 0.4391 | -0.3105 | -0.3105 | 0.4391 | 0.0000 | -0.3105 | 0.3105 | 0.0000 | -0.4391 | 0.3105 | 0.3105 | -0.4391 | 0.0000 | 0.3105 | -0.3105 | 0.0000 | 0.4391 | -0.3105 | -0.3105 | 0.4391 | 0.0000 | -0.3105 | 0.3105 | 0.0000 | -0.4391 | 0.3105 | 0.3105 | -0.4391 | 0.0000 | 0.3105 | -0.3105 | 0.0000 | 0.4391 | -0.3105 | -0.3105 | 0.4391 | 0.0000 | -0.3105 | 0.3105 | 0.0000 | -0.4391 | 0.3105 | 0.3105 |
| 16 | -0.4964 | 0.4016 | 0.0000 | 0.4016 | -0.2918 | -0.1534 | 0.4721 | -0.1534 | -0.4721 | 0.4016 | 0.2918 | -0.4964 | 0.0000 | 0.4016 | -0.2918 | -0.1534 | 0.4721 | -0.1534 | -0.4721 | 0.4016 | 0.2918 | -0.4964 | 0.0000 | 0.4016 | -0.2918 | -0.1534 | 0.4721 | -0.1534 | -0.4721 | 0.4016 | 0.2918 | -0.4964 | 0.0000 | 0.4016 | -0.2918 | -0.1534 | 0.4721 | -0.1534 | -0.4721 | 0.4016 | 0.2918 | -0.4964 | 0.0000 | 0.4016 | -0.2918 | -0.1534 | 0.4721 | -0.1534 | -0.4721 | 0.4016 | 0.2918 | -0.4964 | 0.0000 | 0.4016 | -0.2918 | -0.1534 | 0.4721 | -0.1534 | -0.4721 | 0.4016 | 0.2918 | -0.4964 | 0.0000 | 0.4016 | -0.2918 | -0.1534 | 0.4721 | -0.1534 | -0.4721 | 0.4016 | 0.2918 | -0.4964 | 0.0000 | 0.4016 | -0.2918 | -0.1534 | 0.4721 | -0.1534 | -0.4721 | 0.4016 | 0.2918 |
| 17 | 0.9290 | -0.8278 | 0.0000 | -0.8278 | 0.4218 | 0.5461 | -0.7516 | -0.1453 | 0.9176 | -0.2871 | -0.8836 | 0.6569 | 0.6569 | -0.8836 | -0.2871 | 0.9176 | -0.1453 | -0.7516 | 0.5461 | 0.4218 | -0.8278 | 0.0000 | 0.9290 | -0.4218 | -0.8278 | 0.7516 | 0.5461 | -0.9176 | -0.1453 | 0.8836 | -0.2871 | -0.6569 | 0.6569 | 0.2871 | -0.8836 | 0.1453 | 0.9176 | -0.5461 | -0.7516 | 0.8278 | 0.4218 | -0.9290 | 0.0000 | 0.8278 | -0.4218 | -0.5461 | 0.7516 | 0.1453 | -0.9176 | 0.2871 | 0.8836 | -0.6569 | -0.6569 | 0.8836 | 0.2871 | -0.9176 | 0.1453 | 0.7516 | -0.5461 | -0.4218 | 0.8278 | 0.0000 | -0.9290 | 0.4218 | 0.8278 | -0.7516 | -0.5461 | 0.9176 | 0.1453 | -0.8836 | 0.2871 | 0.6569 | -0.6569 | -0.2871 | 0.8836 | -0.1453 | -0.9176 | 0.5461 | 0.7516 | -0.8278 | -0.4218 |
| 18 | 0.3500 | -0.3329 | 0.0000 | -0.3329 | 0.1082 | 0.2832 | -0.2057 | -0.2057 | 0.2832 | 0.1082 | -0.3329 | 0.0000 | 0.3500 | -0.1082 | -0.3329 | 0.2057 | 0.2832 | -0.2832 | -0.2057 | 0.3329 | 0.1082 | -0.3500 | 0.0000 | 0.3329 | -0.1082 | -0.2832 | 0.2057 | 0.2057 | -0.2832 | -0.1082 | 0.3329 | 0.0000 | -0.3500 | 0.1082 | 0.3329 | -0.2057 | -0.2832 | 0.2832 | 0.2057 | -0.3329 | -0.1082 | 0.3500 | 0.0000 | -0.3329 | 0.1082 | 0.2832 | -0.2057 | -0.2057 | 0.2832 | 0.1082 | -0.3329 | 0.0000 | 0.3500 | -0.1082 | -0.3329 | 0.2057 | 0.2832 | -0.2832 | -0.2057 | 0.3329 | 0.1082 | -0.3500 | 0.0000 | 0.3329 | -0.1082 | -0.2832 | 0.2057 | 0.2057 | -0.2832 | -0.1082 | 0.3329 | 0.0000 | -0.3500 | 0.1082 | 0.3329 | -0.2057 | -0.2832 | 0.2832 | 0.2057 | -0.3329 | -0.1082 |
| 19 | -0.0805 | 0.0795 | 0.0000 | 0.0795 | -0.0126 | -0.0766 | 0.0249 | 0.0717 | -0.0365 | -0.0651 | 0.0473 | 0.0569 | -0.0569 | -0.0473 | 0.0651 | 0.0365 | -0.0717 | -0.0249 | 0.0766 | 0.0126 | -0.0795 | 0.0000 | 0.0805 | -0.0126 | -0.0795 | 0.0249 | 0.0766 | -0.0365 | -0.0717 | 0.0473 | 0.0651 | -0.0569 | -0.0569 | 0.0651 | 0.0473 | -0.0717 | -0.0365 | 0.0766 | 0.0249 | -0.0795 | -0.0126 | 0.0805 | 0.0000 | -0.0795 | 0.0126 | 0.0766 | -0.0249 | -0.0717 | 0.0365 | 0.0651 | -0.0473 | -0.0569 | 0.0569 | 0.0473 | -0.0651 | -0.0365 | 0.0717 | 0.0249 | -0.0766 | -0.0126 | 0.0795 | 0.0000 | -0.0805 | 0.0126 | 0.0795 | -0.0249 | -0.0766 | 0.0365 | 0.0717 | -0.0473 | -0.0651 | 0.0569 | 0.0569 | -0.0651 | -0.0473 | 0.0717 | 0.0365 | -0.0766 | -0.0249 | 0.0795 | 0.0126 |
| 20 | 0.1964 | -0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 | 0.1964 | 0.0000 | -0.1964 | 0.0000 |
| 21 | -0.4094 | 0.4044 | 0.0000 | 0.4044 | 0.0640 | -0.3894 | -0.1265 | 0.3648 | 0.1859 | -0.3312 | -0.2406 | 0.2895 | 0.2895 | -0.2406 | -0.3312 | 0.1859 | 0.3648 | -0.1265 | -0.3894 | 0.0640 | 0.4044 | 0.0000 | -0.4094 | -0.0640 | 0.4044 | 0.1265 | -0.3894 | -0.1859 | 0.3648 | 0.2406 | -0.3312 | -0.2895 | 0.2895 | 0.3312 | -0.2406 | -0.3648 | 0.1859 | 0.3894 | -0.1265 | -0.4044 | 0.0640 | 0.4094 | 0.0000 | -0.4044 | -0.0640 | 0.3894 | 0.1265 | -0.3648 | -0.1859 | 0.3312 | 0.2406 | -0.2895 | -0.2895 | 0.2406 | 0.3312 | -0.1859 | -0.3648 | 0.1265 | 0.3894 | -0.0640 | -0.4044 | 0.0000 | 0.4094 | 0.0640 | -0.4044 | -0.1265 | 0.3894 | 0.1859 | -0.3648 | -0.2406 | 0.3312 | 0.2895 | -0.2895 | -0.3312 | 0.2406 | 0.3648 | -0.1859 | -0.3894 | 0.1265 | 0.4044 | -0.0640 |
| 22 | -0.0500 | 0.0476 | 0.0000 | 0.0476 | 0.0155 | -0.0405 | -0.0294 | 0.0294 | 0.0405 | -0.0155 | -0.0476 | 0.0000 | 0.0500 | 0.0155 | -0.0476 | -0.0294 | 0.0405 | 0.0405 | -0.0294 | -0.0476 | 0.0155 | 0.0500 | 0.0000 | -0.0476 | -0.0155 | 0.0405 | 0.0294 | -0.0294 | -0.0405 | 0.0155 | 0.0476 | 0.0000 | -0.0500 | -0.0155 | 0.0476 | 0.0294 | -0.0405 | -0.0405 | 0.0294 | 0.0476 | -0.0155 | -0.0500 | 0.0000 | 0.0476 | 0.0155 | -0.0405 | -0.0294 | 0.0294 | 0.0405 | -0.0155 | -0.0476 | 0.0000 | 0.0500 | 0.0155 | -0.0476 | -0.0294 | 0.0405 | 0.0405 | -0.0294 | -0.0476 | 0.0155 | 0.0500 | 0.0000 | -0.0476 | -0.0155 | 0.0405 | 0.0294 | -0.0294 | -0.0405 | 0.0155 | 0.0476 | 0.0000 | -0.0500 | -0.0155 | 0.0476 | 0.0294 | -0.0405 | -0.0405 | 0.0294 | 0.0476 | -0.0155 |
| 23 | -0.4391 | 0.3913 | 0.0000 | 0.3913 | 0.1994 | -0.2581 | -0.3553 | 0.0687 | 0.4337 | 0.1357 | -0.4176 | -0.3105 | 0.3105 | 0.4176 | -0.1357 | -0.4337 | -0.0687 | 0.3553 | 0.2581 | -0.1994 | -0.3913 | 0.0000 | 0.4391 | 0.1994 | -0.3913 | -0.3553 | 0.2581 | 0.4337 | -0.0687 | -0.4176 | -0.1357 | 0.3105 | 0.3105 | -0.1357 | -0.4176 | -0.0687 | 0.4337 | 0.2581 | -0.3553 | -0.3913 | 0.1994 | 0.4391 | 0.0000 | -0.3913 | -0.1994 | 0.2581 | 0.3553 | -0.0687 | -0.4337 | -0.1357 | 0.4176 | 0.3105 | -0.3105 | -0.4176 | 0.1357 | 0.4337 | 0.0687 | -0.3553 | -0.2581 | 0.1994 | 0.3913 | 0.0000 | -0.4391 | -0.1994 | 0.3913 | 0.3553 | -0.2581 | -0.4337 | 0.0687 | 0.4176 | 0.1357 | -0.3105 | -0.3105 | 0.1357 | 0.4176 | 0.0687 | -0.4337 | -0.2581 | 0.3553 | 0.3913 | -0.1994 |
| 24 | -0.4964 | 0.4016 | 0.0000 | 0.4016 | 0.2918 | -0.1534 | -0.4721 | -0.1534 | 0.4721 | 0.4016 | -0.2918 | -0.4964 | 0.0000 | 0.4016 | 0.2918 | -0.1534 | -0.4721 | -0.1534 | 0.4721 | 0.4016 | -0.2918 | -0.4964 | 0.0000 | 0.4016 | 0.2918 | -0.1534 | -0.4721 | -0.1534 | 0.4721 | 0.4016 | -0.2918 | -0.4964 | 0.0000 | 0.4016 | 0.2918 | -0.1534 | -0.4721 | -0.1534 | 0.4721 | 0.4016 | -0.2918 | -0.4964 | 0.0000 | 0.4016 | 0.2918 | -0.1534 | -0.4721 | -0.1534 | 0.4721 | 0.4016 | -0.2918 | -0.4964 | 0.0000 | 0.4016 | 0.2918 | -0.1534 | -0.4721 | -0.1534 | 0.4721 | 0.4016 | -0.2918 | -0.4964 | 0.0000 | 0.4016 | 0.2918 | -0.1534 | -0.4721 | -0.1534 | 0.4721 | 0.4016 | -0.2918 | -0.4964 | 0.0000 | 0.4016 | 0.2918 | -0.1534 | -0.4721 | -0.1534 | 0.4721 | 0.4016 | -0.2918 |
| 25 | 0.9290 | -0.6569 | 0.0000 | -0.6569 | -0.6569 | 0.0000 | 0.9290 | 0.6569 | -0.6569 | -0.9290 | 0.0000 | 0.6569 | 0.6569 | 0.0000 | -0.9290 | -0.6569 | 0.6569 | 0.9290 | 0.0000 | -0.6569 | -0.6569 | 0.0000 | 0.9290 | 0.6569 | -0.6569 | -0.9290 | 0.0000 | 0.6569 | 0.6569 | 0.0000 | -0.9290 | -0.6569 | 0.6569 | 0.9290 | 0.0000 | -0.6569 | -0.6569 | 0.0000 | 0.9290 | 0.6569 | -0.6569 | -0.9290 | 0.0000 | 0.6569 | 0.6569 | 0.0000 | -0.9290 | -0.6569 | 0.6569 | 0.9290 | 0.0000 | -0.6569 | -0.6569 | 0.0000 | 0.9290 | 0.6569 | -0.6569 | -0.9290 | 0.0000 | 0.6569 | 0.6569 | 0.0000 | -0.9290 | -0.6569 | 0.6569 | 0.9290 | 0.0000 | -0.6569 | -0.6569 | 0.0000 | 0.9290 | 0.6569 | -0.6569 | -0.9290 | 0.0000 | 0.6569 | 0.6569 | 0.0000 | -0.9290 | -0.6569 | 0.6569 |
| 26 | 0.3500 | -0.2057 | 0.0000 | -0.2057 | -0.2832 | -0.1082 | 0.3329 | 0.3329 | -0.1082 | -0.2832 | -0.2057 | 0.0000 | 0.3500 | 0.2832 | -0.2057 | -0.3329 | -0.1082 | 0.1082 | 0.3329 | 0.2057 | -0.2832 | -0.3500 | 0.0000 | 0.2057 | 0.2832 | 0.1082 | -0.3329 | -0.3329 | 0.1082 | 0.2832 | 0.2057 | 0.0000 | -0.3500 | -0.2832 | 0.2057 | 0.3329 | 0.1082 | -0.1082 | -0.3329 | -0.2057 | 0.2832 | 0.3500 | 0.0000 | -0.2057 | -0.2832 | -0.1082 | 0.3329 | 0.3329 | -0.1082 | -0.2832 | -0.2057 | 0.0000 | 0.3500 | 0.2832 | -0.2057 | -0.3329 | -0.1082 | 0.1082 | 0.3329 | 0.2057 | -0.2832 | -0.3500 | 0.0000 | 0.2057 | 0.2832 | 0.1082 | -0.3329 | -0.3329 | 0.1082 | 0.2832 | 0.2057 | 0.0000 | -0.3500 | -0.2832 | 0.2057 | 0.3329 | 0.1082 | -0.1082 | -0.3329 | -0.2057 | 0.2832 |
| 27 | -0.0805 | 0.0365 | 0.0000 | 0.0365 | 0.0717 | 0.0473 | -0.0651 | -0.0795 | -0.0126 | 0.0249 | 0.0766 | 0.0569 | -0.0569 | -0.0766 | -0.0249 | 0.0126 | 0.0795 | 0.0651 | -0.0473 | -0.0717 | -0.0365 | 0.0000 | 0.0805 | 0.0717 | -0.0365 | -0.0651 | -0.0473 | -0.0126 | 0.0795 | 0.0766 | -0.0249 | -0.0569 | -0.0569 | -0.0249 | 0.0766 | 0.0795 | -0.0126 | -0.0473 | -0.0651 | -0.0365 | 0.0717 | 0.0805 | 0.0000 | -0.0365 | -0.0717 | -0.0473 | 0.0651 | 0.0795 | 0.0126 | -0.0249 | -0.0766 | -0.0569 | 0.0569 | 0.0766 | 0.0249 | -0.0126 | -0.0795 | -0.0651 | 0.0473 | 0.0717 | 0.0365 | 0.0000 | -0.0805 | -0.0717 | 0.0365 | 0.0651 | 0.0473 | 0.0126 | -0.0795 | -0.0766 | 0.0249 | 0.0569 | 0.0569 | 0.0249 | -0.0766 | -0.0795 | 0.0126 | 0.0473 | 0.0651 | 0.0365 | -0.0717 |
| 28 | 0.1964 | -0.0607 | 0.0000 | -0.0607 | -0.1868 | -0.1589 | 0.1154 | 0.1589 | 0.1154 | 0.0607 | -0.1868 | -0.1964 | 0.0000 | 0.0607 | 0.1868 | 0.1589 | -0.1154 | -0.1589 | -0.1154 | -0.0607 | 0.1868 | 0.1964 | 0.0000 | -0.0607 | -0.1868 | -0.1589 | 0.1154 | 0.1589 | 0.1154 | 0.0607 | -0.1868 | -0.1964 | 0.0000 | 0.0607 | 0.1868 | 0.1589 | -0.1154 | -0.1589 | -0.1154 | -0.0607 | 0.1868 | 0.1964 | 0.0000 | -0.0607 | -0.1868 | -0.1589 | 0.1154 | 0.1589 | 0.1154 | 0.0607 | -0.1868 | -0.1964 | 0.0000 | 0.0607 | 0.1868 | 0.1589 | -0.1154 | -0.1589 | -0.1154 | -0.0607 | 0.1868 | 0.1964 | 0.0000 | -0.0607 | -0.1868 | -0.1589 | 0.1154 | 0.1589 | 0.1154 | 0.0607 | -0.1868 | -0.1964 | 0.0000 | 0.0607 | 0.1868 | 0.1589 | -0.1154 | -0.1589 | -0.1154 | -0.0607 | 0.1868 |
| 29 | -0.4094 | 0.0640 | 0.0000 | 0.0640 | 0.4044 | 0.3894 | -0.1265 | -0.1859 | -0.3648 | -0.3312 | 0.2406 | 0.2895 | 0.2895 | 0.2406 | -0.3312 | -0.3648 | -0.1859 | -0.1265 | 0.3894 | 0.4044 | 0.0640 | 0.0000 | -0.4094 | -0.4044 | 0.0640 | 0.1265 | 0.3894 | 0.3648 | -0.1859 | -0.2406 | -0.3312 | -0.2895 | 0.2895 | 0.3312 | 0.2406 | 0.1859 | -0.3648 | -0.3894 | -0.1265 | -0.0640 | 0.4044 | 0.4094 | 0.0000 | -0.0640 | -0.4044 | -0.3894 | 0.1265 | 0.1859 | 0.3648 | 0.3312 | -0.2406 | -0.2895 | -0.2895 | -0.2406 | 0.3312 | 0.3648 | 0.1859 | 0.1265 | -0.3894 | -0.4044 | -0.0640 | 0.0000 | 0.4094 | 0.4044 | -0.0640 | -0.1265 | -0.3894 | -0.3648 | 0.1859 | 0.2406 | 0.3312 | 0.2895 | -0.2895 | -0.3312 | -0.2406 | -0.1859 | 0.3648 | 0.3894 | 0.1265 | 0.0640 | -0.4044 |
| 30 | -0.0500 | 0.0000 | 0.0000 | 0.0000 | 0.0500 | 0.0500 | 0.0000 | 0.0000 | -0.0500 | -0.0500 | 0.0000 | 0.0000 | 0.0500 | 0.0500 | 0.0000 | 0.0000 | -0.0500 | -0.0500 | 0.0000 | 0.0000 | 0.0500 | 0.0500 | 0.0000 | 0.0000 | -0.0500 | -0.0500 | 0.0000 | 0.0000 | 0.0500 | 0.0500 | 0.0000 | 0.0000 | -0.0500 | -0.0500 | 0.0000 | 0.0000 | 0.0500 | 0.0500 | 0.0000 | 0.0000 | -0.0500 | -0.0500 | 0.0000 | 0.0000 | 0.0500 | 0.0500 | 0.0000 | 0.0000 | -0.0500 | -0.0500 | 0.0000 | 0.0000 | 0.0500 | 0.0500 | 0.0000 | 0.0000 | -0.0500 | -0.0500 | 0.0000 | 0.0000 | 0.0500 | 0.0500 | 0.0000 | 0.0000 | -0.0500 | -0.0500 | 0.0000 | 0.0000 | 0.0500 | 0.0500 | 0.0000 | 0.0000 | -0.0500 | -0.0500 | 0.0000 | 0.0000 | 0.0500 | 0.0500 | 0.0000 | 0.0000 | -0.0500 |
| 31 | -0.4391 | -0.0687 | 0.0000 | -0.0687 | 0.4337 | 0.4176 | 0.1357 | 0.1994 | -0.3913 | -0.3553 | -0.2581 | -0.3105 | 0.3105 | 0.2581 | 0.3553 | 0.3913 | -0.1994 | -0.1357 | -0.4176 | -0.4337 | 0.0687 | 0.0000 | 0.4391 | 0.4337 | 0.0687 | 0.1357 | -0.4176 | -0.3913 | -0.1994 | -0.2581 | 0.3553 | 0.3105 | 0.3105 | 0.3553 | -0.2581 | -0.1994 | -0.3913 | -0.4176 | 0.1357 | 0.0687 | 0.4337 | 0.4391 | 0.0000 | 0.0687 | -0.4337 | -0.4176 | -0.1357 | -0.1994 | 0.3913 | 0.3553 | 0.2581 | 0.3105 | -0.3105 | -0.2581 | -0.3553 | -0.3913 | 0.1994 | 0.1357 | 0.4176 | 0.4337 | -0.0687 | 0.0000 | -0.4391 | -0.4337 | -0.0687 | -0.1357 | 0.4176 | 0.3913 | 0.1994 | 0.2581 | -0.3553 | -0.3105 | -0.3105 | -0.3553 | 0.2581 | 0.1994 | 0.3913 | 0.4176 | -0.1357 | -0.0687 | -0.4337 |
| 32 | -0.4964 | -0.1534 | 0.0000 | -0.1534 | 0.4721 | 0.4016 | 0.2918 | 0.4016 | -0.2918 | -0.1534 | -0.4721 | -0.4964 | 0.0000 | -0.1534 | 0.4721 | 0.4016 | 0.2918 | 0.4016 | -0.2918 | -0.1534 | -0.4721 | -0.4964 | 0.0000 | -0.1534 | 0.4721 | 0.4016 | 0.2918 | 0.4016 | -0.2918 | -0.1534 | -0.4721 | -0.4964 | 0.0000 | -0.1534 | 0.4721 | 0.4016 | 0.2918 | 0.4016 | -0.2918 | -0.1534 | -0.4721 | -0.4964 | 0.0000 | -0.1534 | 0.4721 | 0.4016 | 0.2918 | 0.4016 | -0.2918 | -0.1534 | -0.4721 | -0.4964 | 0.0000 | -0.1534 | 0.4721 | 0.4016 | 0.2918 | 0.4016 | -0.2918 | -0.1534 | -0.4721 | -0.4964 | 0.0000 | -0.1534 | 0.4721 | 0.4016 | 0.2918 | 0.4016 | -0.2918 | -0.1534 | -0.4721 | -0.4964 | 0.0000 | -0.1534 | 0.4721 | 0.4016 | 0.2918 | 0.4016 | -0.2918 | -0.1534 | -0.4721 |
| 33 | 0.9290 | 0.4218 | 0.0000 | 0.4218 | -0.8278 | -0.5461 | -0.7516 | -0.9176 | 0.1453 | -0.2871 | 0.8836 | 0.6569 | 0.6569 | 0.8836 | -0.2871 | 0.1453 | -0.9176 | -0.7516 | -0.5461 | -0.8278 | 0.4218 | 0.0000 | 0.9290 | 0.8278 | 0.4218 | 0.7516 | -0.5461 | -0.1453 | -0.9176 | -0.8836 | -0.2871 | -0.6569 | 0.6569 | 0.2871 | 0.8836 | 0.9176 | 0.1453 | 0.5461 | -0.7516 | -0.4218 | -0.8278 | -0.9290 | 0.0000 | -0.4218 | 0.8278 | 0.5461 | 0.7516 | 0.9176 | -0.1453 | 0.2871 | -0.8836 | -0.6569 | -0.6569 | -0.8836 | 0.2871 | -0.1453 | 0.9176 | 0.7516 | 0.5461 | 0.8278 | -0.4218 | 0.0000 | -0.9290 | -0.8278 | -0.4218 | -0.7516 | 0.5461 | 0.1453 | 0.9176 | 0.8836 | 0.2871 | 0.6569 | -0.6569 | -0.2871 | -0.8836 | -0.9176 | -0.1453 | -0.5461 | 0.7516 | 0.4218 | 0.8278 |
| 34 | 0.3500 | 0.2057 | 0.0000 | 0.2057 | -0.2832 | -0.1082 | -0.3329 | -0.3329 | -0.1082 | -0.2832 | 0.2057 | 0.0000 | 0.3500 | 0.2832 | 0.2057 | 0.3329 | -0.1082 | 0.1082 | -0.3329 | -0.2057 | -0.2832 | -0.3500 | 0.0000 | -0.2057 | 0.2832 | 0.1082 | 0.3329 | 0.3329 | 0.1082 | 0.2832 | -0.2057 | 0.0000 | -0.3500 | -0.2832 | -0.2057 | -0.3329 | 0.1082 | -0.1082 | 0.3329 | 0.2057 | 0.2832 | 0.3500 | 0.0000 | 0.2057 | -0.2832 | -0.1082 | -0.3329 | -0.3329 | -0.1082 | -0.2832 | 0.2057 | 0.0000 | 0.3500 | 0.2832 | 0.2057 | 0.3329 | -0.1082 | 0.1082 | -0.3329 | -0.2057 | -0.2832 | -0.3500 | 0.0000 | -0.2057 | 0.2832 | 0.1082 | 0.3329 | 0.3329 | 0.1082 | 0.2832 | -0.2057 | 0.0000 | -0.3500 | -0.2832 | -0.2057 | -0.3329 | 0.1082 | -0.1082 | 0.3329 | 0.2057 | 0.2832 |
| 35 | -0.0805 | -0.0569 | 0.0000 | -0.0569 | 0.0569 | 0.0000 | 0.0805 | 0.0569 | 0.0569 | 0.0805 | 0.0000 | 0.0569 | -0.0569 | 0.0000 | -0.0805 | -0.0569 | -0.0569 | -0.0805 | 0.0000 | -0.0569 | 0.0569 | 0.0000 | 0.0805 | 0.0569 | 0.0569 | 0.0805 | 0.0000 | 0.0569 | -0.0569 | 0.0000 | -0.0805 | -0.0569 | -0.0569 | -0.0805 | 0.0000 | -0.0569 | 0.0569 | 0.0000 | 0.0805 | 0.0569 | 0.0569 | 0.0805 | 0.0000 | 0.0569 | -0.0569 | 0.0000 | -0.0805 | -0.0569 | -0.0569 | -0.0805 | 0.0000 | -0.0569 | 0.0569 | 0.0000 | 0.0805 | 0.0569 | 0.0569 | 0.0805 | 0.0000 | 0.0569 | -0.0569 | 0.0000 | -0.0805 | -0.0569 | -0.0569 | -0.0805 | 0.0000 | -0.0569 | 0.0569 | 0.0000 | 0.0805 | 0.0569 | 0.0569 | 0.0805 | 0.0000 | 0.0569 | -0.0569 | 0.0000 | -0.0805 | -0.0569 | -0.0569 |
| 36 | 0.1964 | 0.1589 | 0.0000 | 0.1589 | -0.1154 | 0.0607 | -0.1868 | -0.0607 | -0.1868 | -0.1589 | -0.1154 | -0.1964 | 0.0000 | -0.1589 | 0.1154 | -0.0607 | 0.1868 | 0.0607 | 0.1868 | 0.1589 | 0.1154 | 0.1964 | 0.0000 | 0.1589 | -0.1154 | 0.0607 | -0.1868 | -0.0607 | -0.1868 | -0.1589 | -0.1154 | -0.1964 | 0.0000 | -0.1589 | 0.1154 | -0.0607 | 0.1868 | 0.0607 | 0.1868 | 0.1589 | 0.1154 | 0.1964 | 0.0000 | 0.1589 | -0.1154 | 0.0607 | -0.1868 | -0.0607 | -0.1868 | -0.1589 | -0.1154 | -0.1964 | 0.0000 | -0.1589 | 0.1154 | -0.0607 | 0.1868 | 0.0607 | 0.1868 | 0.1589 | 0.1154 | 0.1964 | 0.0000 | 0.1589 | -0.1154 | 0.0607 | -0.1868 | -0.0607 | -0.1868 | -0.1589 | -0.1154 | -0.1964 | 0.0000 | -0.1589 | 0.1154 | -0.0607 | 0.1868 | 0.0607 | 0.1868 | 0.1589 | 0.1154 |
| 37 | -0.4094 | -0.3648 | 0.0000 | -0.3648 | 0.1859 | -0.2406 | 0.3312 | -0.0640 | 0.4044 | 0.1265 | 0.3894 | 0.2895 | 0.2895 | 0.3894 | 0.1265 | 0.4044 | -0.0640 | 0.3312 | -0.2406 | 0.1859 | -0.3648 | 0.0000 | -0.4094 | -0.1859 | -0.3648 | -0.3312 | -0.2406 | -0.4044 | -0.0640 | -0.3894 | 0.1265 | -0.2895 | 0.2895 | -0.1265 | 0.3894 | 0.0640 | 0.4044 | 0.2406 | 0.3312 | 0.3648 | 0.1859 | 0.4094 | 0.0000 | 0.3648 | -0.1859 | 0.2406 | -0.3312 | 0.0640 | -0.4044 | -0.1265 | -0.3894 | -0.2895 | -0.2895 | -0.3894 | -0.1265 | -0.4044 | 0.0640 | -0.3312 | 0.2406 | -0.1859 | 0.3648 | 0.0000 | 0.4094 | 0.1859 | 0.3648 | 0.3312 | 0.2406 | 0.4044 | 0.0640 | 0.3894 | -0.1265 | 0.2895 | -0.2895 | 0.1265 | -0.3894 | -0.0640 | -0.4044 | -0.2406 | -0.3312 | -0.3648 | -0.1859 |
| 38 | -0.0500 | -0.0476 | 0.0000 | -0.0476 | 0.0155 | -0.0405 | 0.0294 | -0.0294 | 0.0405 | -0.0155 | 0.0476 | 0.0000 | 0.0500 | 0.0155 | 0.0476 | 0.0294 | 0.0405 | 0.0405 | 0.0294 | 0.0476 | 0.0155 | 0.0500 | 0.0000 | 0.0476 | -0.0155 | 0.0405 | -0.0294 | 0.0294 | -0.0405 | 0.0155 | -0.0476 | 0.0000 | -0.0500 | -0.0155 | -0.0476 | -0.0294 | -0.0405 | -0.0405 | -0.0294 | -0.0476 | -0.0155 | -0.0500 | 0.0000 | -0.0476 | 0.0155 | -0.0405 | 0.0294 | -0.0294 | 0.0405 | -0.0155 | 0.0476 | 0.0000 | 0.0500 | 0.0155 | 0.0476 | 0.0294 | 0.0405 | 0.0405 | 0.0294 | 0.0476 | 0.0155 | 0.0500 | 0.0000 | 0.0476 | -0.0155 | 0.0405 | -0.0294 | 0.0294 | -0.0405 | 0.0155 | -0.0476 | 0.0000 | -0.0500 | -0.0155 | -0.0476 | -0.0294 | -0.0405 | -0.0405 | -0.0294 | -0.0476 | -0.0155 |
| 39 | -0.4391 | -0.4337 | 0.0000 | -0.4337 | 0.0687 | -0.4176 | 0.1357 | -0.3913 | 0.1994 | -0.3553 | 0.2581 | -0.3105 | 0.3105 | -0.2581 | 0.3553 | -0.1994 | 0.3913 | -0.1357 | 0.4176 | -0.0687 | 0.4337 | 0.0000 | 0.4391 | 0.0687 | 0.4337 | 0.1357 | 0.4176 | 0.1994 | 0.3913 | 0.2581 | 0.3553 | 0.3105 | 0.3105 | 0.3553 | 0.2581 | 0.3913 | 0.1994 | 0.4176 | 0.1357 | 0.4337 | 0.0687 | 0.4391 | 0.0000 | 0.4337 | -0.0687 | 0.4176 | -0.1357 | 0.3913 | -0.1994 | 0.3553 | -0.2581 | 0.3105 | -0.3105 | 0.2581 | -0.3553 | 0.1994 | -0.3913 | 0.1357 | -0.4176 | 0.0687 | -0.4337 | 0.0000 | -0.4391 | -0.0687 | -0.4337 | -0.1357 | -0.4176 | -0.1994 | -0.3913 | -0.2581 | -0.3553 | -0.3105 | -0.3105 | -0.3553 | -0.2581 | -0.3913 | -0.1994 | -0.4176 | -0.1357 | -0.4337 | -0.0687 |
The row of values marked “summation” is the result of the Discrete Fourier Transform X(k) for our signal, a list of 80 values, 40 for the cosine term and 40 for the sine term. We can now take those values and calculate the magnitudes, M(k), of each frequency index in a table by using Pythagoras’ theorem to combine the cosine term, let’s call it ℜ (the real part) and the sine term, let’s call it ℑ (the imaginary part):
| Frequency Index - k | Result of Cosine Summations | Result of Sine Summations | Magnitude - M(k) |
|---|---|---|---|
| 0 | 0.0000 | 0.0000 | 0.0000 |
| 1 | 0.0000 | 0.0000 | 0.0000 |
| 2 | 0.0000 | 0.0000 | 0.0000 |
| 3 | 0.0000 | 0.0000 | 0.0000 |
| 4 | 0.0000 | 0.0000 | 0.0000 |
| 5 | 0.0000 | 8.0000 | 8.0000 |
| 6 | 0.0000 | 0.0000 | 0.0000 |
| 7 | 0.0000 | 0.0000 | 0.0000 |
| 8 | 0.0000 | 0.0000 | 0.0000 |
| 9 | 0.0000 | 0.0000 | 0.0000 |
| 10 | -3.0000 | 5.1962 | 6.0000 |
| 11 | 0.0000 | 0.0000 | 0.0000 |
| 12 | 0.0000 | 0.0000 | 0.0000 |
| 13 | 0.0000 | 0.0000 | 0.0000 |
| 14 | 0.0000 | 0.0000 | 0.0000 |
| 15 | -6.9282 | 4.0000 | 8.0000 |
| 16 | 0.0000 | 0.0000 | 0.0000 |
| 17 | 0.0000 | 0.0000 | 0.0000 |
| 18 | 0.0000 | 0.0000 | 0.0000 |
| 19 | 0.0000 | 0.0000 | 0.0000 |
| 20 | 0.0000 | 0.0000 | 0.0000 |
| 21 | 0.0000 | 0.0000 | 0.0000 |
| 22 | 0.0000 | 0.0000 | 0.0000 |
| 23 | 0.0000 | 0.0000 | 0.0000 |
| 24 | 0.0000 | 0.0000 | 0.0000 |
| 25 | -6.9282 | -4.0000 | 8.0000 |
| 26 | 0.0000 | 0.0000 | 0.0000 |
| 27 | 0.0000 | 0.0000 | 0.0000 |
| 28 | 0.0000 | 0.0000 | 0.0000 |
| 29 | 0.0000 | 0.0000 | 0.0000 |
| 30 | -3.0000 | -5.1962 | 6.0000 |
| 31 | 0.0000 | 0.0000 | 0.0000 |
| 32 | 0.0000 | 0.0000 | 0.0000 |
| 33 | 0.0000 | 0.0000 | 0.0000 |
| 34 | 0.0000 | 0.0000 | 0.0000 |
| 35 | 0.0000 | -8.0000 | 8.0000 |
| 36 | 0.0000 | 0.0000 | 0.0000 |
| 37 | 0.0000 | 0.0000 | 0.0000 |
| 38 | 0.0000 | 0.0000 | 0.0000 |
| 39 | 0.0000 | 0.0000 | 0.0000 |
These results are little more understandable if we plot them on a graph of Magnitude, M(k), against frequency index, k.
Aliasing
Looking at the above graph, you might be tricked into thinking that there were 6 frequencies making up our signal. This is not the case. There are actually only 3. Remember I said that the highest frequency in my signal was 15Hz? Well we can see that we have 3 frequencies, 5, 10 and 15Hz that have non-zero values on the graph. I purposely chose the number of samples, N, to equal the sample rate, R, of my signal so that the frequency index, k, is equal the actual frequency being tested to make the above graph clearer.
If we continue to increase in frequency, another 3 frequencies 25, 30 and 35Hz also have non-zero values. This is a phenomenon known as aliasing. The upper 3 non-zero frequencies are a reflection of the lower 3. It is like placing a mirror at the 20Hz point on the x-axis. 20Hz being half the sampling rate, R. They are not actually present in our original signal, but are artifacts generated by the sampling process. It is easy to get rid of these artifacts as they are easy to filter out. We simply disregard all the results above 20Hz. If we were to sample the signal at a rate that was too low, say 20Hz for example, the reflected frequencies generated by the sampling process would appear in the same frequency range as our signal and therefore be much more difficult to get rid of. This is why, when sampling a signal, we must sample at least twice as many times per second as the maximum frequency in our signal.
To hear what aliasing sounds like listen to the following recording:
Phase
Just as we used the list of Cosine and Sine terms in the results table to calculate the magnitude, M(k), of each frequency component, we can use the inverse tangent function to find the phase, P(k), of each frequency present in our signal.
where ℑ is the result of the Sine term and ℜ is result of the Cosine term.
The result we get from the inverse tangent function is in radians so I have converted it into degrees using the following formula:
| Frequency Index - k | Result of Cosine Summations | Result of Sine Summations | Phase - P(k) |
|---|---|---|---|
| 0 | 0.0000 | 0.0000 | |
| 1 | 0.0000 | 0.0000 | |
| 2 | 0.0000 | 0.0000 | |
| 3 | 0.0000 | 0.0000 | |
| 4 | 0.0000 | 0.0000 | |
| 5 | 0.0000 | 8.0000 | 90.0000 |
| 6 | 0.0000 | 0.0000 | |
| 7 | 0.0000 | 0.0000 | |
| 8 | 0.0000 | 0.0000 | |
| 9 | 0.0000 | 0.0000 | |
| 10 | -3.0000 | 5.1962 | -60.0000 |
| 11 | 0.0000 | 0.0000 | |
| 12 | 0.0000 | 0.0000 | |
| 13 | 0.0000 | 0.0000 | |
| 14 | 0.0000 | 0.0000 | |
| 15 | -6.9282 | 4.0000 | -30.0000 |
| 16 | 0.0000 | 0.0000 | |
| 17 | 0.0000 | 0.0000 | |
| 18 | 0.0000 | 0.0000 | |
| 19 | 0.0000 | 0.0000 | |
| 20 | 0.0000 | 0.0000 | |
| 21 | 0.0000 | 0.0000 | |
| 22 | 0.0000 | 0.0000 | |
| 23 | 0.0000 | 0.0000 | |
| 24 | 0.0000 | 0.0000 | |
| 25 | -6.9282 | -4.0000 | 30.0000 |
| 26 | 0.0000 | 0.0000 | |
| 27 | 0.0000 | 0.0000 | |
| 28 | 0.0000 | 0.0000 | |
| 29 | 0.0000 | 0.0000 | |
| 30 | -3.0000 | -5.1962 | 60.0000 |
| 31 | 0.0000 | 0.0000 | |
| 32 | 0.0000 | 0.0000 | |
| 33 | 0.0000 | 0.0000 | |
| 34 | 0.0000 | 0.0000 | |
| 35 | 0.0000 | -8.0000 | 90.0000 |
| 36 | 0.0000 | 0.0000 | |
| 37 | 0.0000 | 0.0000 | |
| 38 | 0.0000 | 0.0000 | |
| 39 | 0.0000 | 0.0000 | |
| 40 | 0.0000 | 0.0000 |
As you can see from the results table above, a lot of the Cosine terms are zero. This is going to make life awkward as dividing by zero is never a good idea. However, it doesn’t matter as the frequencies for which that happens (apart from one) don’t exist in our signal. Only the phase of the 3 contributing frequencies (5, 10 and 15Hz) interest us because these are the 3 frequencies that make up our signal. The one frequency that does have a zero Cosine component is the 5Hz frequency. We know that it has a non-zero Sine component though. This means that it is a pure sine wave, or to put it another way, it is a cosine wave with a phase shift of 90º.
Therefore,
- The 5Hz component has a phase of 90º
- The 10Hz component has a phase of -60º
- The 15Hz component has a phase of -30º.
Conclusion
So now we have used the Discrete Fourier Transform to extract the frequency, magnitude and phase information for a real sampled signal, we are still left with a problem. My signal is quite a simple, repeating signal and I’ve made sure that I have performed the Fourier Transform on 5 complete cycles of it. What would happen if I were to try a signal that didn’t repeat itself? Most useful signals in the real world are non-repetitive. To put it another way, if I was talking to you and kept repeating the same word over and over again, we wouldn’t be having much of a conversation. This highlights a whole new problem, a problem we are going to look at and try to solve as best we can next time when we take a look at “Windowing”.
