lecture-5

# lecture-5 - EE378 Statistical Signal Processing Lecture 5...

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EE378 Statistical Signal Processing Lecture 5 - 04/18/2007 Spectral Density and Distribution Lecturer: Tsachy Weissman Scribe: Stefan Schuet, Qun Feng Tan, Kamakshi S In this lecture we review the concept of Spectral Densities familiar from EE278 and introduce the idea of the Spectral Distribution Function (SDF). The material of this lecture can be found in [Porat, 94]. 1 Spectral Density Throughout we will be dealing with a Wide Sense Stationary (WSS) random process. Recall that this means: E[ X ( n )] = μ Var[ X ( n )] < Cov[ X ( n ) , X ( n + k )] = r ( k ) is a function of k only Note that the condition Var[ X ( n )] < is among the requirements for a process to be considered WSS. Processes with infinite variance are sometimes referred to as ‘Heavy Tailed processes’, and the concept of a spectral density for these types of signals is meaningless. In addition to wide sense stationarity, we start by assuming that the covariance sequence has finite energy k =0 r 2 ( k ) < . This guarantees the existence of the Fourier transform of r ( k ) which allows us to define its spectral density as follows: Definition 1. The Spectral Density S xx ( e jw ) of a WSS process X ( n ) is the Fourier transform of its autocovariance function r xx ( k ) as defined below: S xx ( e jw ) = F { r xx ( k ) } k = -∞ r xx ( k ) e - jwk Note that we can also view this in terms of the z-transform of r ( k ) evaluated at z = e jw . Recall that the z-transform of some function r ( k ) is defined as S ( z ) = k = -∞ r ( k ) z - k . Of course, the autocovariance function can be recovered with the inverse Fourier transform: r xx ( k ) = F - 1 { S xx ( e jw ) } 1 2 π π - π S xx ( e jw ) e jwk dw In the homework you will show that if, further, the covariance sequence is absolutely summable then the spectral density is uniformly continuous, i.e.: k = -∞ | r xx ( k ) | < S xx ( e jw ) is uniformly continuous. As a side remark, the following properties regarding the summability of r ( k ) should be kept in mind: k = -∞ | r ( k ) | < ∞ ⇒ k = -∞ r 2 ( k ) < but not the other way around (see example 1 below).

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