ELEG 305: Digital Signal Processing
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1 ELEG 305: Digital Signal Processing Lecture 19: Lattice Filters Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 2008 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19 Outline 1 Review of Previous Lecture 2 Lecture Objectives 3 Lattice to Direct Form Translation K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19
2 Review of Previous Lecture Review of Previous Lecture Efficient FFT computation for real sequences For x 1 (n) and x 2 (n) real, set x(n) x 1 (n) + jx 2 (n) then X 1 (k) 1 2 [X(k) + X (N k)]; X 2 (k) 1 2j [X(k) X (N k)] Similar tricks hold for length 2N real sequences Linear filtering computation of the DFT the Goertzel algorithm y k (n) W k N y k(n 1) + x(n) gives X(k) y k (n) nn Implementations of FIR discrete time systems Direct, Cascade, and Lattice filter structures K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19 Lecture Objectives Lecture Objectives Objective Derive lattice structures for FIR filters; Develop a method for converting lattice filter coefficients to direct form (FIR) coefficients Reading Chapters 9 (9.2); Next lecture, complete lattice filters, structures for IIR systems (Chapter ); start filter design (Chapter ) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19
3 Objective: Developed a lattice structure realization of FIR filters Approach: Suppose we have a sequence of FIR filters H m (z) A m (z) m 0, 1,..., M 1 where by definition A m (z) 1 + α m (k)z k m 1 k1 Suppose m 1. Then the output of H 1 (z), for input x(n), is y(n) x(n) + α 1 (1)x(n 1) ( ) To build a cascade of 1 st order stages, generalize the notation. Let f 0 (n) x(n) g 0 (n) x(n) f 1 (n) f 0 (n) + K 1 g 0 (n 1) g 1 (n) K 1 f 0 (n) + g 0 (n 1) Question: For what K 1 value is ( ) realized, i.e., f 1 (n) y(n)? K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19 Single Stage Lattice Filter To realize ( ), let K 1 α 1 (1). Then f 1 (n) f 0 (n) + K 1 g 0 (n 1) x(n) + α 1 (1)x(n 1) y(n) [order m 1 filter output] Similarly, g 1 (n) α 1 (1)x(n) + x(n 1) Note: The stage 1 governing equations are f 1 (n) f 0 (n) + K 1 g 0 (n 1) g 1 (n) K 1 f 0 (n) + g 0 (n 1) where we set the reflection coefficient as K 1 α 1 (1) to realize ( ) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19
4 Objective: Add a second stage and equate to the m 2 filter Note that [m 2 filter] H 2 (z) A 2 (z) 1 + α 2 (1)z 1 + α 2 (2)z 2 y(n) x(n) + α 2 (1)x(n 1) + α 2 (2)x(n 2) ( ) The two stage lattice filter is given by The governing equations are f 1 (n) f 0 (n) + K 1 g 0 (n 1) f 2 (n) f 1 (n) + K 2 g 1 (n 1) g 1 (n) K 1 f 0 (n) + g 0 (n 1) g 2 (n) K 2 f 1 (n) + g 1 (n 1) Procedure: Set K 1 and K 2 such that ( ) is realized Note: f 0 (n) g 0 (n) x(n) [pipeline input] and f 2 (n) y(n) [output] K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19 [lattice f 1 (n) f 0 (n) + K 1 g 0 (n 1) f 2 (n) f 1 (n) + K 2 g 1 (n 1) equations] g 1 (n) K 1 f 0 (n) + g 0 (n 1) g 2 (n) K 2 f 1 (n) + g 1 (n 1) By substitution f 2 (n) f 1 (n) + K 2 g 1 (n 1) [f 0 (n) + K 1 g 0 (n 1)] + K 2 [K 1 f 0 (n 1) + g 0 (n 2)] [x(n) + K 1 x(n 1)] + K 2 [K 1 x(n 1) + x(n 2)] x(n) + K 1 (1 + K 2 )x(n 1) + K 2 x(n 2) ( ) Recalling ( ) [direct FIR form] y(n) x(n) + α 2 (1)x(n 1) + α 2 (2)x(n 2) and equating with ( ) [lattice form] yields or, equivalently, α 2 (2) K 2 and α 2 (1) K 1 (1 K 2 ) K 2 α 2 (2) and K 1 α 2(1) 1 + α 2 (2) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19 ( )
5 Process: This is repeated for M 1 stages, with general recursion f 0 (n) g 0 (n) x(n) f m (n) f m 1 (n) + K m g m 1 (n 1), m 1, 2,..., M 1 g m (n) K m f m 1 (n) + g m 1 (n 1), m 1, 2,..., M 1 At the final stage y(n) f m 1 (n) m 1 α m 1 (k)x(n k) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19 Observation: At each stage there are two outputs, f m (n) and g m (n) Consider g 2 (n). First & second stage equations: f 1 (n) f 0 (n) + K 1 g 0 (n 1) f 2 (n) f 1 (n) + K 2 g 1 (n 1) g 1 (n) K 1 f 0 (n) + g 0 (n 1) g 2 (n) K 2 f 1 (n) + g 1 (n 1) Apply substitution similarly to the previous case g 2 (n) K 2 f 1 (n) + g 1 (n 1) K 2 [f 0 (n) + K 1 g 0 (n 1)] + [K 1 f 0 (n 1) + g 0 (n 2)] K 2 x(n) + K 1 (1 + K 2 )x(n 1) + x(n 2) using the prior result K 2 α 2 (2) and K 1 (1 K 2 ) α 2 (1), g 2 (n) α 2 (2)x(n) + α 2 (1)x(n 1) + x(n 2) Result: The coefficients for g m (n) are reverse order those for f m (n) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19
6 Result: The two outputs at arbitrary stage m are expressed as f m (n) α m (k)x(n k) and g m (n) β m (k)x(n k) where β m (k) α m (m k), k 0, 1,..., m Note: f m (n) is the forward prediction; g m (n) the backward prediction Taking the z transform Similarly F m (z) A m (z)x(z) or A m (z) F m(z) X(z) F m(z) F 0 (z) G m (z) B m (z)x(z) or B m (z) G m(z) X(z) G m(z) G 0 (z) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19 Recall A m (z) α m (k)z k Question: How does B m (z) relate to A m (z)? B m (z) β m (k)z k [substitute β m (k) α m (m k)] α m (m k)z k [let l m k] α m (l)z l m l0 m z m α m (l)z l l0 z m A m (z 1 ) Result: B(z) has reciprocal zeros of A(z); B(z) is the reciprocal, or reverse, polynomial of A(z) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19
7 Lattice Filter Representation Summary Case 1: FIR filter representations f m (n) α m (k)x(n k) and g m (n) β m (k)x(n k) Case 2: z domain representations F m (z) A m (z)x(z) or A m (z) F m(z) X(z) G m (z) B m (z)x(z) or B m (z) G m(z) X(z) Also, B m (z) z m A m (z 1 ), m 1, 2,..., M 1 Case 3: Recursion lattice representations f 0 (n) g 0 (n) x(n) f m (n) f m 1 (n) + K m g m 1 (n 1), m 1, 2,..., M 1 g m (n) K m f m 1 (n) + g m 1 (n 1), m 1, 2,..., M 1 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19 Taking z transform of the recursion lattice representations F 0 (z) G 0 (z) X(z) F m (z) F m 1 (z) + K m z 1 G m 1 (z), m 1, 2,..., M 1 G m (z) K m F m 1 (z) + z 1 G m 1 (z), m 1, 2,..., M 1 Dividing by X(z) and using A m (z) Fm(z) X(z) and B m(z) Gm(z) X(z) Case 4: z domain recursion lattice representations A 0 (z) B 0 (z) 1 A m (z) A m 1 (z) + K m z 1 B m 1 (z), m 1, 2,..., M 1 B m (z) K m A m 1 (z) + z 1 B m 1 (z), m 1, 2,..., M 1 or in matrix notation [ ] [ ] [ ] Am (z) 1 Km A m 1 (z) B m (z) K m 1 z 1 B m 1 (z) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19
8 Lattice to Direct Form Translation Design Questions: How do we convert lattice coefficients (K terms) to direct-form filter coefficients (α terms)? How do we convert direct form coefficients to lattice coefficients? Case 1: Lattice to Direct Form Translation Approach: Utilize relations A 0 (z) B 0 (z) 1 A m (z) A m 1 (z) + K m z 1 B m 1 (z), m 1, 2,..., M 1 B m (z) z m A m (z 1 ), m 1, 2,..., M 1 Solve for α terms recursively, starting with m 1 Example Determine the direct form realization of a three stage (M 4) lattice filter with coefficients K 1 1/2, K 2 1/2, K 3 1/4. K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19 Lattice to Direct Form Translation Start recursion set m 1 A 1 (z) A 0 (z) + K 1 z 1 B 0 (z) z 1 ( ) Set ( ) equal to α 1 polynomial A 1 (z) 1 + α 1 (1)z 1 α 1 (1) 1 2 Note, B m (z) is the reverse polynomial of A m (z) B 1 (z) z 1 A 1 (z 1 ) z 1 ( z ) z 1 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19
9 Lattice to Direct Form Translation For m 2, use recursion A m (z) A m 1 (z) + K m z 1 B m 1 (z) A 2 (z) A 1 (z) + K 2 z 1 B 1 (z) [ ] [ ( )] 1 1 z z z 1 equating α terms yields z z 2 α 2 (1) 3 4 and α 2 (2) 1 2 Reversing the polynomial of A m (z) to get B m (z) B 2 (z) z 2 A 2 (z 1 ) z 1 + z 2 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19 For m 3, Thus A 3 (z) A 2 (z) + K 3 z 1 B 2 (z) [ z ] z z z z 3 Lattice to Direct Form Translation [ ( z )] 4 z 1 + z 2 α 3 (1) 7 8, α 3(2) 11 16, α 3(3) 1 4 Result: The system in direct-form: (where α 3 (0) 1) y(n) 3 α 3 (k)x(n k) x(n) x(n 1) x(n 2) + 1 x(n 3) 4 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19
10 Lecture Summary Lecture Summary Lattice Filters general recursion f 0 (n) g 0 (n) x(n) f m (n) f m 1 (n) + K m g m 1 (n 1) g m (n) K m f m 1 (n) + g m 1 (n 1), Lattice to Direct Form Translation utilize recursion on A 0 (z) B 0 (z) 1 A m (z) A m 1 (z) + K m z 1 B m 1 (z), m 1, 2,..., M 1 B m (z) z m A m (z 1 ), m 1, 2,..., M 1 Next lecture Complete lattice filters, structures for IIR systems (Chapter ); start filter design (Chapter ) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 19
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