Discrete Fourier Analysis by M. W. Wong

By M. W. Wong

This textbook provides easy notions and strategies of Fourier research in discrete settings. Written in a concise type, it really is interlaced with feedback, discussions and motivations from sign analysis.

The first half is devoted to themes regarding the Fourier rework, together with discrete time-frequency research and discrete wavelet research. uncomplicated wisdom of linear algebra and calculus is the one prerequisite. the second one half is outfitted on Hilbert areas and Fourier sequence and culminates in a piece on pseudo-differential operators, supplying a lucid advent to this complicated subject in research. a few degree concept language is used, even if so much of this half is offered to scholars conversant in an undergraduate direction in genuine research.

Discrete Fourier Analysis is geared toward complex undergraduate and graduate scholars in arithmetic and utilized arithmetic. more advantageous with workouts, will probably be an exceptional source for the school room in addition to for self-study.

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Let z ∈ L2 (ZN ). 7, Cb z = b ∗ z = (b ∗ z)∨ = (ˆbˆ z )∨ = (σ zˆ)∨ = Tσ z. 7 again, we get Tσ z = (σ zˆ)∨ = (ˆbˆ z )∨ = (b ∗ z)∨ = b ∗ z = Cb z. 4. Let A : L2 (ZN ) → L2 (ZN ) be a linear operator. Then A is a Fourier multiplier if and only if the matrix (A)F of A with respect to the Fourier basis F is diagonal. Moreover, if A is the Fourier multiplier Tσ associated to the symbol σ in L2 (ZN ), then   σ(0) 0 0 ··· 0  0  σ(1) 0 ··· 0    0  0 σ(2) · · · 0 (A)F = (Tσ )F =  .  ..  .. ..

By the principle of mathematical induction, the proof is complete. 2. In order to appreciate the power of the FFT, let l = 10. Then N = 210 = 1024. Direct matrix multiplication without the FFT requires (1024)2 complex multiplications. We need to perform more than a million complex multiplications. Using the FFT, we need at most 12 × 1024 × 10 = 5120 complex multiplications. The save in computer time is enormous. 3. Convolutions can also be computed rapidly with the FFT. 7 and the Fourier inversion formula, we get z ∗ w = (z ∗ w)∨ = (ˆ z w) ˆ ∨ , z, w ∈ L2 (ZN ).

The filter A is also a Fourier multiplier Tσ with symbol σ and σ = ˆb. The matrix (A)F of the filter A with respect to the Fourier basis F for L2 (ZN ) is diagonal, and is given by     (A)F =    λ0 0 0 .. 0 λ1 0 .. 0 0 λ2 .. 0 0 0 ··· ··· ··· ··· ··· 0 0 0 ..     ,   λN −1 where λm is the eigenvalue of A corresponding to the eigenfunction Fm , m = 0, 1, . . , N − 1. We can now give an explicit formula for the eigenvalues λ0 , λ1 , . . , λN −1 . We want the formula to be so tractable that it can be used in computation.

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