Review Second-Order Linear Prediction Predictors Summary Lecture 16: Linear Prediction Mark Hasegawa-Johnson ECE 401: Signal and Image Analysis, Fall 2020
Review Second-Order Linear Prediction Predictors Summary Review: All-Pole Filters 1 Inverse Filtering 2 Linear Prediction 3 Finding the Linear Predictive Coefficients 4 Summary 5
Review Second-Order Linear Prediction Predictors Summary Outline Review: All-Pole Filters 1 Inverse Filtering 2 Linear Prediction 3 Finding the Linear Predictive Coefficients 4 Summary 5
Review Second-Order Linear Prediction Predictors Summary All-Pole Filter An all-pole filter has the system function: 1 1 H ( z ) = 1 z − 1 ) = 1 − a 1 z − 1 − a 2 z − 2 , (1 − p 1 z − 1 )(1 − p ∗ so it can be implemented as y [ n ] = x [ n ] + a 1 y [ n − 1] + a 2 y [ n − 2] where 1 ) = 2 e − σ 1 cos( ω 1 ) a 1 = ( p 1 + p ∗ a 2 = −| p 1 | 2 = − e − 2 σ 1
Review Second-Order Linear Prediction Predictors Summary Frequency Response of an All-Pole Filter We get the magnitude response by just plugging in z = e j ω , and taking absolute value: 1 | H ( ω ) | = | H ( z ) | z = e j ω = | e j ω − p 1 | × | e j ω − p ∗ 1 |
Review Second-Order Linear Prediction Predictors Summary Impulse Response of an All-Pole Filter We get the impulse response using partial fraction expansion: h [ n ] = ( C 1 p n 1 ) n ) u [ n ] 1 + C ∗ 1 ( p ∗ 1 sin( ω 1 ) e − σ 1 n sin ( ω 1 ( n + 1)) u [ n ] =
Review Second-Order Linear Prediction Predictors Summary Speech is made up of Damped Sinusoids Resonant systems, like speech, trumpets, and bells, are made up from the series combination of second-order all-pole filters.
Review Second-Order Linear Prediction Predictors Summary Outline Review: All-Pole Filters 1 Inverse Filtering 2 Linear Prediction 3 Finding the Linear Predictive Coefficients 4 Summary 5
Review Second-Order Linear Prediction Predictors Summary Speech Speech is made when we take a series of impulses, one every 5-10ms, and filter them through a resonant cavity (like a bell).
Review Second-Order Linear Prediction Predictors Summary Speech Speech is made when we take a series of impulses, one every 5-10ms, and filter them through a resonant cavity (like a bell). 1 S ( z ) = H ( z ) E ( z ) = A ( z ) E ( z ) where the excitation signal is a set of impulses, maybe only one per frame: e [ n ] = G δ [ n − n 0 ] The only thing we don’t know, really, is the amplitude of the impulse ( G ), and the time at which it occurs ( n 0 ). Can we find out?
Review Second-Order Linear Prediction Predictors Summary Speech: The Model
Review Second-Order Linear Prediction Predictors Summary Speech: The Real Thing
Review Second-Order Linear Prediction Predictors Summary Inverse Filtering If S ( z ) = E ( z ) / A ( z ), then we can get E ( z ) back again by doing something called an inverse filter: 1 IF: S ( z ) = A ( z ) E ( z ) THEN: E ( z ) = A ( z ) S ( z ) The inverse filter, A ( z ), has a form like this: p � a k z − k A ( z ) = 1 − k =1 where p is twice the number of resonant frequencies. So if speech has 4-5 resonances, then p ≈ 10.
Review Second-Order Linear Prediction Predictors Summary Inverse Filtering
Review Second-Order Linear Prediction Predictors Summary Inverse Filtering This one is an all-pole (feedback-only) filter: 1 S ( z ) = k =1 a k z − k E ( z ) 1 − � p That means this one is an all-zero (feedfoward only) filter: p � � � a k z − k E ( z ) = 1 − S ( z ) k =1 which we can implement just like this: p � e [ n ] = s [ n ] − a k s [ n − k ] k =1
Review Second-Order Linear Prediction Predictors Summary Outline Review: All-Pole Filters 1 Inverse Filtering 2 Linear Prediction 3 Finding the Linear Predictive Coefficients 4 Summary 5
Review Second-Order Linear Prediction Predictors Summary Linear Predictive Analysis This particular feedforward filter is called linear predictive analysis : p � e [ n ] = s [ n ] − a k s [ n − k ] k =1 It’s kind of like we’re trying to predict s [ n ] using a linear combination of its own past samples: p � s [ n ] = ˆ a k s [ n − k ] , k =1 and then e [ n ], the glottal excitation, is the part that can’t be predicted: e [ n ] = s [ n ] − ˆ s [ n ]
Review Second-Order Linear Prediction Predictors Summary Linear Predictive Analysis Actually, linear predictive analysis is used a lot more often in finance, these days, than in speech: In finance: detect important market movements = price changes that are not predictable from recent history. In health: detect EKG patterns that are not predictable from recent history. In geology: detect earthquakes = impulses that are not predictable from recent history. . . . you get the idea. . .
Review Second-Order Linear Prediction Predictors Summary Linear Predictive Analysis Filter s [ n ] e [ n ] z − 1 − a 1 z − 1 − a 2 z − 1 − a 3 z − 1 − a 4
Review Second-Order Linear Prediction Predictors Summary Linear Predictive Synthesis The corresponding feedback filter is called linear predictive synthesis . The idea is that, given e [ n ], we can resynthesize s [ n ] by adding feedback, because: 1 S ( z ) = k =1 a k z − k E ( z ) 1 − � p means that p � s [ n ] = e [ n ] + a k s [ n − k ] k =1
Review Second-Order Linear Prediction Predictors Summary Linear Predictive Synthesis Filter e [ n ] s [ n ] a 1 z − 1 z − 1 a 2 z − 1 a 3 z − 1 a 4
Review Second-Order Linear Prediction Predictors Summary Outline Review: All-Pole Filters 1 Inverse Filtering 2 Linear Prediction 3 Finding the Linear Predictive Coefficients 4 Summary 5
Review Second-Order Linear Prediction Predictors Summary Finding the Linear Predictive Coefficients Things we don’t know: The timing of the unpredictable event ( n 0 ), and its amplitude ( G ). The coefficients a k . It seems that, in order to find n 0 and G , we first need to know the predictor coefficients, a k . How can we find a k ?
Review Second-Order Linear Prediction Predictors Summary Finding the Linear Predictive Coefficients Let’s make the following assumption: Everything that can be predicted is part of ˆ s [ n ]. Only the unpredictable part is e [ n ].
Review Second-Order Linear Prediction Predictors Summary Finding the Linear Predictive Coefficients Let’s make the following assumption: Everything that can be predicted is part of ˆ s [ n ]. Only the unpredictable part is e [ n ]. So we define e [ n ] to be: p � e [ n ] = s [ n ] − a k s [ n − k ] k =1 . . . and then choose a k to make e [ n ] as small as possible. ∞ � e 2 [ n ] a k = argmin n = −∞
Review Second-Order Linear Prediction Predictors Summary Finding the Linear Predictive Coefficients So we’ve formulated the problem like this: we want to find a k in order to minimize: � 2 p � ∞ ∞ � � � e 2 [ n ] = E = s [ n ] − a m s [ n − m ] n = −∞ n = −∞ m =1
Review Second-Order Linear Prediction Predictors Summary Finding the Linear Predictive Coefficients We want to find the coefficients a k that minimize E . We can do that by differentiating, and setting the derivative equal to zero: p � � ∞ d E � � = 2 s [ n ] − a m s [ n − m ] s [ n − k ] , for all 1 ≤ k ≤ p da k n = −∞ m =1 p � � ∞ � � 0 = s [ n ] − a m s [ n − m ] s [ n − k ] , for all 1 ≤ k ≤ p n = −∞ m =1 This is a set of p different equations (for 1 ≤ k ≤ p ) in p different unknowns ( a k ). So it can be solved.
Review Second-Order Linear Prediction Predictors Summary Autocorrelation In order to write the solution more easily, let’s define something called the “autocorrelation,” R [ m ]: ∞ � R [ m ] = s [ n ] s [ n − m ] n = −∞ In terms of the autocorrelation, the derivative of the error is p � 0 = R [ k ] − a m R [ k − m ] ∀ 1 ≤ k ≤ p m =1 or we could write p � R [ k ] = a m R [ k − m ] ∀ 1 ≤ k ≤ p m =1
Review Second-Order Linear Prediction Predictors Summary Matrices Since we have p linear equations in p unknowns, let’s write this as a matrix equation: R [1] R [0] R [1] · · · R [ p − 1] a 1 R [2] R [1] R [0] · · · R [ p − 2] a 2 = . . . . . ... . . . . . . . . . . R [ p ] R [ p − 1] R [ p − 2] · · · R [0] a p where I’ve taken advantage of the fact that R [ m ] = R [ − m ]: ∞ � R [ m ] = s [ n ] s [ n − m ] n = −∞
Review Second-Order Linear Prediction Predictors Summary Matrices Since we have p linear equations in p unknowns, let’s write this as a matrix equation: � γ = R � a where R [1] R [0] R [1] · · · R [ p − 1] R [2] R [1] R [0] · · · R [ p − 2] � γ = , R = . . . . . ... . . . . . . . . R [ p ] R [ p − 1] R [ p − 2] · · · R [0]
Review Second-Order Linear Prediction Predictors Summary Matrices Since we have p linear equations in p unknowns, let’s write this as a matrix equation: � γ = R � a and therefore the solution is a = R − 1 � � γ
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