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Additional info for Adaptive Digital Filters, 2nd Edition (Signal Processing and Communications)
Several methods of analysis primarily aim at ﬁnding out that polynomial for a start. The above deterministic or predictable signals have discrete power spectra. To obtain continuous spectra, one must introduce random signals. They bring innovation in the processes. 3. RANDOM SIGNALS A random real signal xðnÞ is deﬁned by a probability law for its amplitude at each time n. 0 Prob½x 4 xðnÞ 4 x þ Áx Áx ð2:27Þ It is used to calculate, by ensemble averages, the statistics of the signal or process .
Xðnk Þ is Gaussian. 46), the probability law of that variable is completely deﬁned by the ACF rðpÞ of xðnÞ. The power spectral density Sð f Þ is obtained as the Fourier transform of the ACF: Sð f Þ ¼ 1 X rðpÞeÀj2pf p¼À1 or, since rðpÞ is an even function, ð2:47Þ 26 Chapter 2 Sð f Þ ¼ rð0Þ þ 2 1 X rðpÞ cosð2pf Þ ð2:48Þ p¼1 If the data in the sequence xðnÞ are independent, then rðpÞ reduces to rð0Þ and the spectrum Sð f Þ is ﬂat; the signal is then said to be white. An important aspect of the Gaussian probability laws is that they preserve their character under any linear operation, such as convolution, ﬁltering, differentiation, or integration.
5. 55 Evaluate the mean and variance associated with the uniform probability density function on the interval ½x1 ; x2 . Comment on the results. Consider the signal 0 n<0 xðnÞ ¼ 0:8xðn À 1Þ þ eðnÞ; n 51 assuming eðnÞ is a stationary zero mean random sequence with power e2 ¼ 0:5. The initial condition is deterministic with value xð0Þ ¼ 1. Calculate the mean sequence mn ¼ E½xðnÞ. Give the recursion, for the variance sequence. What is the stationary solution. Calculate the ACF of the stationary signal.