Chapter 10: Pipelined and Parallel Recursive and Adaptive Filters - PowerPoint PPT Presentation

Chapter 10: Pipelined and Parallel Recursive and Adaptive Filters Keshab K. Parhi

Outline • Introduction • Pipelining in 1 st -Order IIR Digital Filters • Pipelining in Higher-Order IIR Digital Filters • Parallel Processing for IIR Filters • Combined Pipelining and Parallel Processing for IIR Filters Chapter 10 2

Look-Ahead Computation First-Order IIR Filter Consider a 1 st -order linear time-invariant recursion (see Fig. 1) • + = ⋅ + ⋅ (10.1) y ( n 1 ) a y ( n ) b u ( n ) { } { } T + • The iteration period of this filter is , where represent m T T , m T a a word-level multiplication time and addition time • In look-ahead transformation , the linear recursion is first iterated a few times to create additional concurrency. • By recasting this recursion, we can express y(n+2) as a function of y(n) to obtain the following expression (see Fig. 2(a)) [ ] + = + + + (10.2) y ( n 2 ) a ay ( n ) bu ( n ) bu ( n 1 ) ( ) 2 T + • The iteration bound of this recursion is , the same as the 2 T m a original version, because the amount of computation and the number of logical delays inside the recursive loop have both doubled Chapter 10 3

• Another recursion equivalent to (10.2) is (10.3). Shown on Fig.2(b), its ( ) 2 T + T iteration bound is , a factor of 2 lower than before. m a + = ⋅ + ⋅ + ⋅ + 2 y ( n 2 ) a y ( n ) ab u ( n ) b u ( n 1 ) (10.3) • Applying (M-1) steps of look-ahead to the iteration of (10.1), we can obtain an equivalent implementation described by (see Fig. 3) − M 1 ∑ + = ⋅ + ⋅ ⋅ + − − M i y ( n M ) a y ( n ) a b u ( n M 1 i ) (10.4) = i 0 z − − M 1 z – Note: the loop delay is instead of , which means that the loop computation must be completed in M clock cycles (not 1 clock cycle) . The ( ) M m + T T iteration bound of this computation is , which corresponds a to a sample rate M times higher than that of the original filter { } ⋅ ⋅ ⋅ − 2 M 1 M ab , a b , , a b , a – The terms in (10.4) can be pre-computed (referred to as pre-computation terms ). The second term in RHS of (10.4) is the look-ahead computation term (referred to as the look-ahead complexity ); it is non-recursive and can be easily pipelined Chapter 10 4

u(n) b a Fig. 1 y(n) D y(n+1) u(n+1) u(n) D a b y(n+1) Fig.2.(a) a b 2D y(n) y(n+2) u(n+1) u(n) D 2 b ab a Fig.2.(b) 2D y(n) y(n+2) Chapter 10 5

u(n+M-1) u(n+M-2) u(n) D D M M-1 a b ab a b y(n) MD y(n+M) Fig. 3: M-stage Pipelinable 1st-Order IIR Filter • Look-ahead computation has allowed a single serial computation to be transformed into M independent concurrent computations, and to pipeline the feedback loop to achieve high speed filtering of a single time series while maintaining full hardware utilization. • Provided the multiplier and the adder can be conveniently pipelined, the iteration bound can be achieved by retiming or cutset transformation (see Chapter 4) Chapter 10 6

Pipelining in 1 st -Order IIR Digital Filters – Example: Consider the 1 st -order IIR filter transfer function 1 = (10.5) H ( z ) − − ⋅ 1 1 a z • The output sample y(n) can be computed using the input sample u(n) and the past output sample as follows: (10.6) = ⋅ − + y ( n ) a y ( n 1 ) u ( n ) • The sample rate of this recursive filter is limited by the computation time of one multiply-add operation – Look-ahead techniques add canceling poles and zeros with equal angular spacing at a distance from the origin which is same as that of the original pole. The pipelined filters are always stable provided that the original filter is stable – The pipelined realizations require a linear increase in complexity but decomposition techniques can be used to obtain an implementation with logarithmic increase in hardware with respect to the number of loop pipeline stages Chapter 10 7

Pipelining in 1 st -Order IIR Digital Filters (continued) 1. Look-Ahead Pipelining for 1 st -Order IIR Filters • Look-ahead pipelining adds canceling poles and zeroes to the transfer ( ) } − ⋅ ⋅ ⋅ − − 1 M 1 { z , , z function such that the coefficients of in the denominator of the transfer function are zero. Then, the output sample y(n) can be computed using the inputs and the output sample y(n-M) such that there are M delay elements in the critical loop, which in turn can be used to pipeline the critical loop by M stages and the sample rate can be increased by a factor M ( ) − = − ⋅ 1 H ( z ) 1 1 a z Example: Consider the 1 st -order filter, , which • has a pole at z=a (a ≤ 1). A 3-stage pipelined equivalent stable filter can ( ) = ± 2 π j 3 be derived by adding poles and zeroes at , and is given z ae by + ⋅ − + ⋅ − 1 2 2 1 a z a z = ( ) H z − ⋅ − 3 3 1 a z Chapter 10 8

Pipelining in 1 st -Order IIR Digital Filters (continued) 2. Look-Ahead Pipelining with Power-of-2 Decomposition • With power-of-2 decomposition, an M-stage (for power-of-2 M) pipelined implementation for 1 st -order IIR filter can be obtained by log M sets of transformations 2 Example: Consider a 1 st -order recursive filter transfer function • ( ) ( ) − − = ⋅ − ⋅ 1 1 described by . The equivalent pipelined ( ) 1 H z b z a z transfer function can be described using the decomposition technique ( ) as follows ∏ − ⋅ − log M 1 + ⋅ − i i 1 2 2 2 1 b z a z = = i 0 ( ) H z (10.7) − − ⋅ M M 1 a z – This pipelined implementation is derived by adding (M-1) poles and zeros at identical locations. z = a – The original transfer function has a single pole at (see Fig.4(a)). Chapter 10 9

– The pipelined transfer function has poles at the following locations (see Fig.4(b) for M=8): { } ( ) ( ) π ⋅ ⋅ ⋅ − ⋅ π j 2 M j M 1 2 M a , ae , , ae – The decomposition of the canceling zeros is shown in Fig.4(c). The i-th i 2 stage of the decomposed non-recursive portion implements zeros ( ) ( ) ) , located at: ( ( ) = + π i = ⋅ ⋅ ⋅ − z a exp j 2 n 1 2 i n 0 , 1 , , 2 1 (10.8) – The i-th stage of the decomposed non-recursive portion requires a single pipelined multiplication operation independent of the stage number i – The multiplication complexity of the pipelined implementation is ( ) + log 2 M 2 – The finite-precision pipelined filters suffer from inexact pole-zero cancellation, which leads to magnitude and phase error. These errors can be reduced by increasing the wordlength (see p. 323) Chapter 10 10

Chapter 10 Fig. 4 Pole representation of a 1 ST -order recursive filter • Pole/zero representation of a 1 ST -order IIR with 8 loop stages • • Decomposition based on pipelined IIR for M = 8 11

Pipelining in 1 st -Order IIR Digital Filters (continued) 3. Look-Ahead Pipelining with General Decomposition • The idea of decomposition can be extended to any arbitrary number of = ⋅ ⋅ ⋅ M M M M ( ) loop pipelining stages M. If , then the non-recursive ( ) ( ) 1 2 p ⋅ ⋅⋅ ⋅ ⋅ ⋅ − 1 − − , M M M p M 1 M 1 M M 1 stages implement , , zeros, − 1 2 1 p 1 2 respectively, totaling (M-1) zeros Example (Example 10.3.3, p.325) Consider the 1 st -order IIR • 1 = H ( z ) − − ⋅ 1 1 a z – A 12-stage pipelined decomposed implementation is given by ( )( )( ) = ∑ 11 − ⋅ i i + − + − + − + − a z 1 2 2 4 4 6 6 1 az 1 a z a z 1 a z = = i 0 H ( z ) − ⋅ − − ⋅ − 12 12 12 12 1 a z 1 a z × × 2 3 2 - This implementation is based on a decomposition (see Fig. 5) Chapter 10 12