By Simon J. Godsill Phd, Peter J. W. Rayner Phd (auth.)

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6 and we see that there are two effects. 1. Smearing. c. and wo) have become 'smeared' into a band of frequencies. 2. Spectral leakage. c. 'leaks' into the Wo component as a result of the sidelobes of W (I). This can be very troublesome when trying to measure the amplitude of a small component in the presence of a large component at a nearby frequency. 7. The two effects are not independent but roughly speaking the width of the lobes (or smearing) is an effect of the window duration but the sidelobes are due to the discontinuous nature of the window.

5 Example: Continuous random variable The random noise voltage V measured across a resistor R is a continuous random variable. e. V(w) = w, the random variable V is termed 'directly given'. Alternatively, we may wish to consider the instantaneous power V 2 / R across the resistor, in which case we can define a RV W(w) = w2 /R. 6 The probability distribution of a random variable The distribution Px of a RV X is simply a probability measure which assigns probabilities to events on the real line. ' .

7 Data windows 31 Hanning window . 6 Q. 9 Spectrum of Hanning/rectangular window ~ ::> . e . 2 ...... 0 -4 -- ..... -3 " "'-- ..... , -1 -2 0 1 Normalized frequency 2 " ...... -- 3 ..... 8. 2 Discrete-time signals Consider now the discrete case. We can go through the calculations for the windowed spectrum in a manner analogous to the continuous-time case: L g(pT) e- jpwT L {g(PT) w(PT)} e- jpwT 00 p=-oo 00 p=-oo 1 211" 1211" W(e j8 ) 0 L 00 g(pT) e- jp (wT-8) dO p=-oo ~ 1211" W(e j8 ) G(e j (wT-8)) dO 211" 0 Once again, the spectrum of the windowed signal is the convolution of the infinite duration signal's spectrum and the window's spectrum.

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