Finite Impulse Response (FIR) Digital Filters

1. Finite Impulse Response (FIR) Digital Filters • Digital filters are rapidly replacing classic analog filters. • Programmable DSP with MAC can be used to implement digital filters. • For high-bandwidth signal processing applications, FPGA technology can provide multiple MACs to achieve the desired thoughput. Digital Kommunikationselektronik TNE027 Lecture 4

2. FIR Theory • An FIR with constant coefficient is an Linear Time-Invariant (LTI) filter. • The ouput of an FIR of order (or length) L, to an input time-series x[n], is given by a finite version of the convolution sum: • y[n] = f[n]*x[n] = Σf[k] x[n-k] • where f[0]  0 through f[L-1]  0 are the filter’s L coefficients. They also correspond to the FIR’s impulse response. L-1 k=0 Digital Kommunikationselektronik TNE027 Lecture 4

3. LTI system expressed in the z-domain: Y(z) = F(z) X(z) where F(z) is the FIR’s transfer function defined in z-domain by F(z) = Σf[k] z–k The roots of polynomial F(z) define the zeros of the filter. FIRs are also called all zero filters. L-1 k=0 Digital Kommunikationselektronik TNE027 Lecture 4

4. Direct Form FIR Filter Tapped delay line Fig. 3.1. Tapped weight Digital Kommunikationselektronik TNE027 Lecture 4

5. FIR Filter with Transposed Structure A variation of the direct FIR model is called the transposed FIR filter. It can be constructed from the direct form FIR filter by • Exchanging the input and output • Inverting the direction of signal flow • Substituting an adder by a fork, and vice versa Digital Kommunikationselektronik TNE027 Lecture 4

6. FIR Filter in the Transposed Structure Fig. 3.3. See Example 3.1: Programmable FIR Filter Digital Kommunikationselektronik TNE027 Lecture 4

7. Comparison of the two forms of the FIR filter • The direct form FIR filter needs extra pipeline registers between the adders to reduce the delay of the adder tree and to achieve high throughput. • The FIR filter with transposed structure has registers between the adders and can achieve high throughput without adding any extra pineline registers. Digital Kommunikationselektronik TNE027 Lecture 4

8. Symmetry in FIR Filters • Define the center point of an odd-order FIR’s impulse response as the 0th sample: F(z) = Σf[k] z–k (An even-order FIR can be similarly defined.) • Linear-phase FIR filter • Linear-phase is achieved if the filter is symmetric or antisymmetric. See Table 3.1. k=(L-1)/2 k=-(L-1)/2 Digital Kommunikationselektronik TNE027 Lecture 4

9. Linear-phase even-order filter with reduced number of multipliers f[L’-2] f[L’-1] Fig. 3.5. L’ = L/2, where L is an even number. Digital Kommunikationselektronik TNE027 Lecture 4

10. Constant Coefficient FIR Design • There are only a few applications, e.g., adaptive filters, where we need a general programmable filter. • In many applications, the filters are Linear Time Invariant (LTI) and the coefficients do not change over time. • The hardware effort can be reduced for constant coefficient FIR. Digital Kommunikationselektronik TNE027 Lecture 4

11. Direct FIR implementation • In a practical situation, the FIR coefficients are obtained from a computer design tool and presented to the designer as floating point numbers. • The performance of a fixed-point FIR, based on the floating-point coefficients, should be verified using simulation or algebraic analysis to ensure that design specifications remain satisfied. Digital Kommunikationselektronik TNE027 Lecture 4

12. Dynamic range overflow should be avoided. • The worst-case dynamic range growth G of an Lth- order FIR is G  log2(Σ|f[k]|) See Example 3.2: Four-tap direct FIR filter. L-1 k=0 Digital Kommunikationselektronik TNE027 Lecture 4

13. Improve the direct FIR design • Realize each filter coefficient with an optimal CSD code. • Increase effective multiplier speed by pipelining. • For FIR with symmetric coefficients, the number of multipliers can be reduced. See Table 3.3 and Example 3.3. Digital Kommunikationselektronik TNE027 Lecture 4

14. Rephasing pipelined multiplier in FIR filter Add a positive delay f[n]=z d f[n] z -d Digital Kommunikationselektronik TNE027 Lecture 4

15. FIR Filter with Transposed Structure • If the transposed filter has constant coefficients, two improved designs should be considered: • Multiple use of the repeated coefficients using the reduced adder graph (RAG) algorithm • Pipeline adders Digital Kommunikationselektronik TNE027 Lecture 4

16. Reduced Adder Graph Algorithm 3.4: Reduced Adder Graph • Remove the sign of the coefficient. • Remove all coefficients and factors that are a power of two. • Realize all cost “1” coefficients. • Use cost “1” coefficients in building the multiplier of higher cost. (Use Table 2.3.) See Example 3.5. Digital Kommunikationselektronik TNE027 Lecture 4

17. Reduced Adder Graph for F6 Half-band Filter Fig. 3.11. Realization of F6 using RAG algorithm Digital Kommunikationselektronik TNE027 Lecture 4

18. FIR Filter Using Distributed Arithmetic • Distributed Arithmetic Using Logic Cells See Example 3.6: Distributed Arithmetic Filter as State Machine. • Logic cells are used to implement small look-up tables for low-order filters. • The outputs of a collection of low-order filters can be added together to define the output of a high-order FIR. See Example 3.7: Five-input DA Table. Digital Kommunikationselektronik TNE027 Lecture 4

19. DA Using Embedded Array Blocks • It is not economical to use the 2-kbit EABs for a short FIR filter, mainly because the number of available EABs is limited. • The maximum registered speed of an EAB is 76 MHz, and an LC table implementation may be faster for a short FIR filter. • For long filters, EABs have registered throughput at a constant 76 MHz and routing effort is reduced. See Example 3.8: Distributed Arithmetic Filter using EABs. Digital Kommunikationselektronik TNE027 Lecture 4

20. Example 3.10: Loop Unrolling for DA FIR Filter Fig. 3.16: Parallel implementation of a distributed arithmetic FIR filter Digital Kommunikationselektronik TNE027 Lecture 4