Lecture 6 Windowing of DFT.pptVIP

Fall 2008 * Lecture 6: Windowing of DFT Instructor: Dr. Gleb V. Tcheslavski Contact: gleb@ Office Hours: Room 2030 Class web site: /gleb/dsp/index.htm by Steve Higgins The idea of windowing (6.2.1) - not really practical What if we observe an infinitely long sequence over a finite length time window? 0 D-1 Than we don’t see the rest of the signal. Let D is the number of samples (data points) observed. Windowing; what it causes (6.3.1) where: (6.3.2) yn is a D- sequence (finite length), therefore, we can evaluate its DFT. periodic convolution property Therefore, is a “dispersed/convoluted/obscured view of (6.3.3) - a windowed view of xn zero-padded length (6.3.4) Windowing; what it causes (cont) (6.4.1) If ?0 ? [-?, ?] (6.4.2) (6.4.3) (6.4.4) delay due to a center of the window Leakage Let D = 100 (6.5.1) (6.5.2) Spectrum of a rectangular window This is what causes so called “leakage” – observation of frequency components that do not exist in the spectrum of the signal – Due to observation only! Frequency sampling DFT means “frequency sampling”! We can only observe specific frequency components of the sinc curve, and, therefore, can see only the specific frequency components of our signal! Zero-padding increases number of observed frequency samples. BUT! our sinusoid leaves at ?0 and this is the only frequency where we expect it to be! Therefore, by “smart” choice of P we can observe the signal ONLY at ?0 and at the zero-crossings, i.e. at (6.5.1) Characteristics of window functions Main lobe width (MLW) at xx dB Windows characteristics Peak Side Lobe level (PSL), dB Side Lobe Roll-off (SLR), dB/octave Characteristics of window functions Windows characteristics We want: MLW – narrow for better spectral resolution PSL – lower to have less masking for nearby components SLR – “better” (faster) to have less masking for far away components Multiple sinusoids (6.9.1) Assume for simplicity that ?m = 0 for every m. As a consequenc