More on the Fourier Transforms

Rotational Properties (and basic properties in transforming sinusoidals)

In the next portion, the effect of rotating the object on the Fourier Transform is investigated. For this portion, a sine wave viewed from the top will be the object, and its Fourier Transform will be calculated. Then, the sine wave shall be rotated, and the new Fourier Transform will be calculated.

Left – Sine Wave along the horizontal axis; Right – Fourier Transform of the sine wave. (Two dots that represent the frequency of the sine wave.

The image above shows the fourier transform of the normal sine wave along the x-axis. Its Fourier Transform consists of two dots along the horizontal axis whose distance from the center is directly proportional to the frequency of the sine wave. For the next portion, since sine waves have positive and negative values, while images only have positive values, we investigate the FFT of sine wave biased such that it has no negative values.

FFT of the biased sine wave. Note the addition of a brighter third dot in the middle. This represents the zero-frequency biasing function.

For the biased sine wave, the most noticeable change is the addition of a brighter third dot in the center. This represents the totally flat, zero-frequency constant used to bias the sine wave. An application of this is in determining the frequency of a wave using Young’s Double Slit experiment. It is important to perform a high pass filter to remove the low-frequency noise added in the image-recording process. Next, the effect of rotating the sine wave is investigated.

FT of the rotated sinusoidal. Note that the FT and sine wave is rotated by the same angle.

What’s amazing here is that the Fourier Transform is rotated at the same angle at which the sine wave is rotated. This is because the two dots must be located along the line in which the sine wave varies the most. Next, the result of multiplying sinusoidals is investigated.

FT of the product of sinusoids along x and y. Note that the FT is the convolution of the individual FTs.

The image above shows the results when sinusoids along the horizontal and vertical are multiplied (left), and then Fourier transformed(right). Note that the Fourier Transform is a convolution of the individual FT’s of both sinusoids. This is further proof that multiplication in real space is a convolution in Fourier space. All the properties discussed are combined in the following picture.

Fourier Transform of a combination of 3 sine waves. sin(x)*sin(y)*sine rotated by some angle.

The left figure in the above image shows the result of multiplying 3 sine waves: one along the horizontal, one along the vertical, and one along a certain rotated axis. Note that it’s FT is the convolution of all three, as expected.