EECS20N: Signals and Systems

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Week 8

• Frequencies
• Musical scale
• Phase
• Timbre
• Images
• Periodicity
• Fourier series
• Examples
• Aperiodicity
• Images
• Coefficients
• Discrete signals
• DFS
• Examples

Fourier series approximations to images

Images are invariably finite signals. Given any image, it is possible to construct a periodic image by just tiling a plane with the image. Thus, there is again a close relationship between a periodic image and a finite one.

We have seen sinusoidal images, so it follows that it ought to be possible to construct a Fourier series representation of an image. The only hard part is that images have a two-dimensional domain, and thus are finite in two distinct dimensions. The sinusoidal images that we saw correspondingly could have a vertical frequency, a horizontal frequency, or both.

Suppose that the domain of an image is [a, b] × [c, d] ⊂ Reals × Reals. Let p_H = b − a, and p_V = d− c represent the horizontal and vertical "periods" for the equivalent periodic image. For constructing a Fourier series representation, we can define the horizontal and vertical fundamental frequencies as follows:

ω _H = 2π /p_H ω _V = 2π /p_V

The Fourier series representation of Image: [a, b] × [c, d] → Intensity is

For convenience, we have included the constant term A_0,0 in the summation, so we assume that Φ ₀ = φ ₀ = 0 (recall that cos(0) = 1).