In Physics, the Fourier Transform is widely used in various applications, including signal processing and analysis, modeling, and many more. The 2D Fourier Transform, in particular, is very useful in optics; calculations in the inverse space utilizes the Fourier Transform. In this activity we further improve our familiarity with the Fast Fourier Transform functions of Scilab and also the properties of the 2D Fourier Transform.
For the first part of this activity we generate some common shapes with familiar Fourier Transforms; figure 1 shows these shapes together with their 2D Fourier Transform. Note that the FT of a circle is an airy disk, while the FT of a double slit look very much like Young's double slit experiment. As for the square, we know that the Ft of a rectangular wave is the sinc function so the Ft of a square will resemble sinc functions along the x axis and y axis. Finally we can see that the FT of two dots is very much like a sinusoid (or cosine) pattern which is expected since the FT of a cosine is 2 peaks.
For the first part of this activity we generate some common shapes with familiar Fourier Transforms; figure 1 shows these shapes together with their 2D Fourier Transform. Note that the FT of a circle is an airy disk, while the FT of a double slit look very much like Young's double slit experiment. As for the square, we know that the Ft of a rectangular wave is the sinc function so the Ft of a square will resemble sinc functions along the x axis and y axis. Finally we can see that the FT of two dots is very much like a sinusoid (or cosine) pattern which is expected since the FT of a cosine is 2 peaks.
Figure 1. Different shapes or patterns generated using Scilab
and their corresponding Fourier Transform
The second part of the activity is aimed to familiarize ourselves with the properties of the 2D Fourier Transform using 2D sinusoid patterns. The sine function was chosen since, as I have said, the absolute value of the FT of a sinusoid (or cosine) is known to be 2 peaks.
Figure 2 below shows the 2D Fourier Transform of sinusoid patterns with different frequencies. As expected, the FT images contain 2 peaks vertically symmetric about the center. Also as the frequency of the sinusoid is increased we observe that the peaks move away from the center. This is also the expected result since low frequencies are concentrated on the center. Finally we see that with higher frequency the FT deviates slightly from a spot. For the higher frequencies the spots of the FT is less distinct and is slightly spread, which is probably due to aliasing.
Figure 2 below shows the 2D Fourier Transform of sinusoid patterns with different frequencies. As expected, the FT images contain 2 peaks vertically symmetric about the center. Also as the frequency of the sinusoid is increased we observe that the peaks move away from the center. This is also the expected result since low frequencies are concentrated on the center. Finally we see that with higher frequency the FT deviates slightly from a spot. For the higher frequencies the spots of the FT is less distinct and is slightly spread, which is probably due to aliasing.
Figure 2. Sinusoid patterns of different frequencies with their Fourier Transforms.
The frequencies are 1, 4, 40, from top to bottom.
For the next part we explore the effect of combining different functions, specifically a sine with an added constant bias and 2 sine functions with different frequencies. The base sinusoid we used has a frequency of 4 (figure 2 mid row) and we see that by adding a constant bias (figure 3 top row) that the FT only differs by the appearance of a central spot. This central spot corresponds to a 0 frequency signal which is exactly the constant bias. As for the 2 sinusoids, we simply added the sinusoid with frequencies 4 and 1. And again we see that the resulting FT is simply just the addition or superposition of their FT seen on figure 2. There for in general we can say that in 2D Fourier Transform the FT of an addition of 2 functions is just the addition of their FT.
Real life signals obtained from actual sensors usually contain a constant bias. For example, an interferogram as detected by a photodiode most likely rides above a bias. In trying to recover the frequencies of such a signal by taking its Fourier Transform, you would sure obtain a high valued zero order or zero frequency term just like in the top row of figure 3. If we want to recover the signal we can just ignore the zero order. But if it is too strong that it floods out the signal it would be better to just center your signal at zero.
Real life signals obtained from actual sensors usually contain a constant bias. For example, an interferogram as detected by a photodiode most likely rides above a bias. In trying to recover the frequencies of such a signal by taking its Fourier Transform, you would sure obtain a high valued zero order or zero frequency term just like in the top row of figure 3. If we want to recover the signal we can just ignore the zero order. But if it is too strong that it floods out the signal it would be better to just center your signal at zero.
Figure 3. (top) A sinusoid of frequency 4 with a constant bias and
(bottom) addition of sinusoids with frequencies 1 and 4. Beside them are their FT.
In figure 4 we see the effect of rotating the sinusoid pattern. Looking at the images we see that the FT is still 2 peaks except that it is also rotated. And just like in figure 2 the location of the peaks are symmetric about the center, also the line joining the 2 peaks is perpendicular to the pattern (same with figure 2).
Figure 4. Rotated sinusoid and its Fourier Transform.
We know look at the effect of combining a sinusoid pattern along the horizontal and the vertical. The first result is a multiplication of the two patterns, which resulted in a checkered pattern. The resulting FT of this pattern have 4 peaks, still symmetric about the center, but looks to be located at the corners of a smaller square. This makes sense since the pattern actually looks like an addition of 2 perpendicular sinusoids (horizontal and vertical) rotated by 45 degrees. In other words we see a pattern with a periodicity that is angled by 45 degrees. The next set of images is just an addition of a vertical and a horizontal sinusoid patterns.
Figure 5. (top) Multiplication and (bottom) addition of a vertical
and horizontal sinusoid with their Fourier Transforms.
Finally we make a pattern by adding multiple sinusoid patterns each rotated by an arbitrary angle. The resulting pattern doesn't seem to mimic the previous pattern we have presented but still looks to have some periodicities and symmetries. Looking at the FT of this pattern we can say that it is just like an addition or superposition of the FTs of the different rotated sinusoids. This is observation is confirmed by the circular appearance of the FT. So over all we can say this final image just reaffirm what we have already demonstrated in figures 3 and 4.
Figure 6. Addition of arbitrarily rotated sinusoid patterns and its Fourier Transform.
I give my self a grade of 10 in this activity. I also want to thank master, jica, and jaya for staying with me in the lab and for reminding me to do the blog.
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