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Summary of Fourier Representations

Module by: Richard Baraniuk. E-mail the author

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Summary: A summary of the fourier representations used in ELEC 301.

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Key Ideas

  • Each Hilbert space has a natural Fourier representation in terms of sines and cosines
  • Why sines and cosines? Becuase they are eigenfunctions of LTI systems:
    Figure 1
    Figure 1 (fig1.png)
Figure 2
Figure 2 (fig2.png)

Fourier Transforms and Convolution

Each Fourier representation turns the natural convolution for the Hilbert space into multiplication of Fourier representation.

Table 1
x x space convo transform
l 2 0N−1 l 2 0 N 1 DT circular DFT
l 2 ℤ l 2 DT linear DTFT
L 2 0T L 2 0 T DT circular CTFS
L 2 ℝ L 2 DT linear CTFT
Figure 3
Figure 3 (fig3.png)

Example 1

Figure 4
Figure 4 (fig4.png)

Example 2

Figure 5
Figure 5 (fig5.png)

Example 3

Combnine to get:

Figure 6
Figure 6 (fig6.png)

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