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Unformatted text preview: ENEE 241 02 * READING ASSIGNMENT 12 Tue 03/24 Lecture Topic: introduction to the discrete Fourier transform; basic examples Textbook References: sections 3.1, 3.2 Key Points: • The columns of the N × N matrix V , defined by V nk = e j (2 π/N ) kn , are sinusoidal vectors with frequencies which are (all the distinct) multiples of 2 π/N . These vectors are orthogonal and such that V H V = N I • Any signal vector s can be expressed as a linear combination of such sinusoids: s = Vc , where c = V H s /N . Time Domain Frequency Domain s DFT ←→ S s = 1 N VS ←→ S = V H s (Synthesis Equation) (Analysis Equation) s = ifft(S) S = fft(s) • The N-point vector s is a complex sinusoid of frequency ω = 2 kπ/N if and only its DFT (or spectrum) S contains all zeros except for S [ k ] 6 = 0. Theory and Examples: 1 . The discrete Fourier transform (DFT) is a powerful computational tool. It allows us to resolve finite-dimensional signal vectors into sinusoids of different frequencies, some of which may be more prominent than others. For example, the 200-point signal vector s constructed in n = (0:199).’; s = 4.7*cos(0.12*pi*n-1.3) + ... 3.8*cos(0.19*pi*n+0.8) + ... 5.1*cos(0.23*pi*n+2.4) + ... 2.0*randn(size(n)); plot(n,s) is the sum of three real-valued sinusoids plus noise. This is not at all that obvious from a plot of the signal. If, however, we compute S = fft(s); bar(abs(S)) then we obtain a symmetric graph with three clear peaks on either half. These peaks corre- spond to the main frequency components at ω = (0 . 12) π , (0 . 19) π and (0 . 23) π . Based on this information, we can characterize the signal s as a sum of three sinusoids plus noise....
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- Spring '08
- Complex number, Hilbert space