R14 - ENEE 241 02 READING ASSIGNMENT 14 Thu 04/02 Lecture...

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* READING ASSIGNMENT 14 Thu 04/02 Lecture Topics: circular shift and reversal; circular conjugate symmetry of the DFT and IDFT matrices; implications of conjugation and circular reversal on the DFT Textbook References: sections 3.2.2, 3.3.2, 3.4 Key Points: Px is a circular shift (downward by one position) in the entries of x . Rx is a circular reversal in the entries of x . F k x is the result of multiplying x by the k th Fourier sinusoid, entry by entry. The columns and rows of V and W = V * exhibit circular conjugate symmetry. Time Domain Frequency Domain x = 1 N VX ←→ X = Wx x * ←→ RX * Rx ←→ RX Theory and Examples: 1 . A circular shift of an N -vector is the entry permutation described by [ x [0] x [1] ... x [ N - 2] x [ N - 1] ] T 7→ [ x [ N - 1] x [0] x [1] ... x [ N - 2] ] T and illustrated below. 0 1 2 3 4 5 6 7 8 9 9 0 1 2 3 4 5 6 7 8 P The associated permutation matrix P (acting on the column vector x to produce Px ) is the N × N matrix P = 0 0 ... 0 1 1 0 ... 0 0 0 1 ... 0 0 . . . . . . . . . . . . . . . 0 0 ... 1 0 P m ( m th power of P ) represents a circular shift by m positions, and P N = I . 1

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R14 - ENEE 241 02 READING ASSIGNMENT 14 Thu 04/02 Lecture...

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