Bioen303_25January2010 - 1/25/2010 Odd-Even Symmetry If...

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1/25/2010 1 Odd-Even Symmetry • If x(n) is real and even , that is: x(n)=x*(n) and x(-n)=x(n), then: • X(m) is rea and even X(m) is real and . • If x(n) is real and odd , that is: x(n)=x*(n) and x(-n)=-x(n), then • X(m) is imaginary and odd . Discrete Fourier Transform • There are three possible phenomena that result in errors between the computed and the desired transform. These three phenomena are –(a) aliasing (Lyons 2.1) – (b) leakage (Lyons 3.8) –(c) scalloping loss (Lyons 3.10) Leakage and the Discrete Fourier Transform • The DFT gives exact frequency results for sinusoidal signals if the N-samples sequence contains an integer number of complete cycles of the sampled continuous signal x(t). • If it does not, the DFT produces a spread spectrum whose values are not exactly correct. We will discuss frequency leakage in detail later. DFT Leakage DFT Leakage Discrete Fourier Transform • (b) Leakage . This problem arises because we must limit observation of the signal to a finite interval. The process of terminating the signal after a finite number of terms is equivalent to multiplying the number of terms is equivalent to multiplying the signal by a window function . • The net effect is a distortion of the spectrum. There is a spreading or leakage of the spectral components away from the correct frequency, resulting in an undesirable modification of the total spectrum.
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1/25/2010 2 Discrete Fourier Transform • The leakage effect cannot always be isolated from the aliasing effect because leakage may also lead to aliasing. Since leakage results in a spreading of the spectrum, the upper frequency may move beyond the Nyquist frequency and aliasing may then result. • The best approach for alleviating the leakage effect is to choose a suitable window function that minimizes the spreading. Discrete Fourier Transform •( c ) Scalloping loss . This effect is produced by the inability of the DFT to observe the spectrum as a continuous function, since computation of the spectrum is limited to integer multiples of the
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This note was uploaded on 04/01/2010 for the course BIOEN 303 taught by Professor Staff during the Spring '08 term at University of Washington.

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Bioen303_25January2010 - 1/25/2010 Odd-Even Symmetry If...

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