Linear Signal Models Overview Introduction • Introduction h ( n ) w ( n ) x ( n ) • Linear nonparametric vs. parametric models • Many researchers use signal models to analyze stationary • Equivalent representations univariate time series • Spectral flatness measure • Goal: estimate the process from which the signal was generated • PZ vs. ARMA models • Called signal modeling • Wold decomposition • Related to, but different from, system identification • Popular assumptions: – x ( n ) is ergodic and WSS – The system is LTI and stable – The input signal is WN – The input signal is Gaussian – The system is a finite-order PZ system J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 1 J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 2 Terminology Nonparametric and Parametric Defined • This text distinguishes between systems and the sequences w ( n ) h ( n ) x ( n ) w ( n ) ˜ h min ( n ) x ( n ) (processes) that result when a WN input is applied • Systems: AZ, AP, PZ • Nonparametric Models : LTI system models that are specified by • Processes the impulse response – Moving Average (MA) – System is completely specified by h ( n ) – Autoregressive Moving-Average (ARMA) – Even if the system is causal, h ( n ) may be infinitely long in general – Autoregressive (AR) – Requires an infinite number of parameters to specify • The processes are assumed to have a WN input signal: x ( n ) ∼ WN(0 , σ 2 w ) • Parametric Models : system models that can be specified with a finite number of parameters • In some cases, the input is assumed to be a sum of sinusoids – Almost always finite-order AP, AZ, or PZ • In this case, the output consists of line spectra – Easier to deal with in practical applications • Goal: estimate frequencies and magnitudes of spectral components – Constrains h ( n ) and H ( z ) J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 3 J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 4
Nonparametric versus Parametric Why Assume Minimum-Phase? • We will focus on nonparametric estimators h ( n ) w ( n ) x ( n ) • These generally have greater variability but less bias than parametric estimators. Why? • Recall that from R x ( z ) alone we cannot distinguish between minimum and non-minimum-phase systems • However, they do not give a compact representation of the process • This is true in general • Will discuss parametric models in detail next term • For any stable, finite-order ARMA process, { H ( z ) , w ( n ) } where H ( z ) has no zeros or poles on the unit circle, there exists a white w ( n ) such that X ( z ) = H min ( z ) ˜ noise process ˜ W ( z ) and H min ( z ) is minimum-phase • If we have no other information but can only observe the signal, there is no reason not to assume h ( n ) is stable and minimum-phase! • Very important assumption • If w ( n ) is IID, it is not true in general that ˜ w ( n ) is IID J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 5 J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 6 Nonrecursive Representation Nonrecursive Representation Continued h ( n ) h ( n ) w ( n ) x ( n ) w ( n ) x ( n ) ∞ � ∞ x ( n ) = h ( k ) w ( n − k ) � r x ( ℓ ) = σ 2 x ( n ) = h ( k ) w ( n − k ) w r h ( ℓ ) k =0 k = −∞ • Since we cannot distinguish between signals produced by causal R x ( z ) = σ 2 R x (e jω ) = σ 2 w | H (e jω ) | 2 w H ( z ) H ∗ ( z −∗ ) and non-causal systems, we can assume the system is causal • This is identical to an infinite-order ( Q = ∞ ) AZ model! • Note that this is a non-recursive representation • We cannot distinguish between the system gain and the white noise process power: • Any LTI system can be written in this form • The shape of r x ( ℓ ) and R x (e jω ) are completely determined by the αh ( n ) ≡ αw ( n − k ) system • Without loss of generality, we can assume h (0) = 1 J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 7 J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 8
Recursive Representation Innovations Representation If we assume H ( z ) is minimum-phase (reasonable), both h ( n ) and x ( n ) h ( n ) h I ( n ) w ( n ) w ( n ) h I ( n ) exist and both are causal and stable. ∞ ∞ n � � ∞ � � w ( n ) = h I ( k ) x ( n − k ) = x ( n ) + h I ( k ) x ( n − k ) x ( n ) = h ( k ) w ( n − k ) = h ( n − k ) w ( k ) k =0 k =1 k =0 k = −∞ ∞ � n � x ( n ) = w ( n ) − h I ( k ) x ( n − k ) x ( n + 1) = w ( n + 1) + h ( n + 1 − k ) w ( k ) k =1 k = −∞ • Let us choose (without loss of generality) that h (0) = 1 x ( n + 1) = w ( n + 1) � �� � • If we assume the inverse system is causal and stable, then New Information h I (0) = 1 (why?) ⎛ ⎞ n n � � • In this case, the output is a function of the current (unknown) + h ( n + 1 − k ) h I ( k − j ) x ( j ) ⎝ ⎠ input and all past values of x ( n ) k = −∞ j = −∞ � �� � • This is identical to an infinite-order ( P = ∞ ) AP model! Past values of x ( n ) • Equivalent representation of the nonrecursive form J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 9 J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 10 Comments on Innovations Representation PZ, AZ, and AP Representations ⎛ ⎞ • We just saw that a nonparametric model can be represented as n n � � either white noise driving an AZ( ∞ ) system or a PZ( ∞ ) system x ( n + 1) = w ( n + 1) + h ( n + 1 − k ) h I ( k − j ) x ( j ) ⎝ ⎠ � �� � j = −∞ k = −∞ • Any causal PZ, AZ, or PZ system with finite order can be New Information � �� � represented as either a causal AZ( ∞ ) or PZ( ∞ ) system Past values of x ( n ) • If the system is stable, then h ( n ) → 0 as n → ∞ • If the system generating x ( n ) is minimum-phase, w ( n + 1) carries all the new information needed to generate x ( n + 1) • Thus, if the order of the system is sufficiently large, we can represent any of these systems accurately with an AZ( Q ) or • Thus, w ( n + 1) is sometimes called the innovation AP( P ) system • All other information can be predicted from past observations of • We’ll see next term that AP( P ) are much easier to estimate than the output PZ( Q, P ) or AZ( Q ) systems • Only holds if h ( n ) is minimum-phase • Thus, it is very good news that AP( P ) systems can represent any • In some contexts, h ( n ) is called the synthesis or coloring filter PZ( Q, P ) or AZ( Q ) system if P is large enough • The inverse is called the analysis or whitening filter • See Example 4.2.1 in the text and discussion in preceding paragraph J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 11 J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 12
Spectral Factorization Cepstrum The cepstrum is the inverse Fourier transform of ln R x (e jω ) R x (e jω ) σ 2 w | H min (e jω ) | 2 = � π c ( k ) � 1 � � R x (e jω ) e jkω d ω ln 2 π − π • The minimum-phase component of the spectrum is the causal part • Regular : Processes that satisfy the Paley-Wiener condition c + ( k ) � 1 � π 2 c (0) + c ( k ) u ( k − 1) | ln R x (e jω ) | d ω < ∞ h min ( n ) = F − 1 { exp F { c + ( k ) }} − π • Regular processes can be factored as • The maximum-phase component is the anticausal part c − ( k ) � 1 R x ( z ) = σ 2 2 c (0) + c ( k ) u ( − k − 1) w H min ( z ) H ∗ min ( z −∗ ) � 1 h max ( n ) = F − 1 { exp F { c − ( k ) }} � π � � � σ 2 R x (e jω ) w = exp ln d ω 2 π − π • This is rarely used in practice. • If R x ( z ) is a rational function, spectral factorization is straightforward J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 13 J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 14 Spectral Flatness Parametric Signal Models � � � π � � 1 R x (e jω ) exp − π ln d ω Q P = σ 2 2 π � � SFM x � w a k x ( n − k ) = b k w ( n − k ) � π 1 σ 2 − π R x (e jω ) d ω x 2 π k =0 k =0 � Q k =0 b k z − k H ( z ) = X ( z ) k =0 a k z − k = B ( z ) W ( z ) = • Single measure of the spectral flatness � P A ( z ) • Bounded: 0 ≤ SFM x ≤ 1 • Parametric models have rational (finite-order) system functions • If SFM x = 1 , then x ( n ) is a white process • Each can be specified by a linear constant-coefficient difference • Numerator is the geometric mean, denominator is the arithmetic equation mean • To make the specifications unique, we always set a 0 = 1 and usually b 0 = 1 • The model is then defined by { a 1 , a 2 , . . . , a P , b 1 , . . . , b Q , σ 2 w } J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 15 J. McNames Portland State University ECE 538/638 Signal Models Ver. 1.02 16
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