Fourier Theorems

Goto: [ Symmetry | Shift | Flip | Convolution | Correlation | Power | Interpolation | Decimation | ]

Symmetry Theorem

[Symmetry theorem]

Shift Theorem

[ Shift Theorem: y(n-l) <-> e^(-jw_k l) Y(k)]

Shown below is a length 12 signal (an impulse) shifted 0, 1, 2, and 3 samples. The continuous lines represent the Discrete Time Fourier Transform (DTFT) which is generated by zero padding the 12 sample region used to calculate the Discrete Fourier Transform (DFT). No matter how much shifting occurs in a signal, the magnitude of its spectrum remain constant. The signal plots below were created with the SeqPlot function. The spectral phase plots were created with the PhasePlot function.

Notice that the first impulse shown below has even symmetry and is real. This means that its spectrum is also real and even, so the phase of the spectrum is only 0 (or π).

[example of the phase for various shifting]

Flip Theorem

Convolution Theorem

Correlation Theorem

Power Theorem

[ power theorem: <x, y> = 1/N <X, Y>]

Stretch Theorem

Stretch Theorem: stretch_L(y) <-> repeat_L(Y) ]

Decimation Theorem

dec_M(y) <-> 1/M alias_M(Y)
craig@ccrma.stanford.edu