1. ESPL 1 Wordlength Optimization with Complexity-and-Distortion Measure and Its Application to Broadband Wireless Demodulator Design Kyungtae Han and Brian L. Evans Embedded Signal Processing Laboratory Wireless Networking and Communications Group The University of Texas at Austin

2. ESPL 2 Fixed-Point Design Digital signal processing algorithms –Often developed in floating point –Later mapped into fixed point for digital hardware realizationfixed point Fixed-point digital hardware –Lower area –Lower power –Lower per unit production cost Introduction

3. ESPL 3 Fixed-Point Design Float-to-fixed point conversion required to target –ASIC and fixed-point digital signal processor core –FPGA and fixed-point microprocessor core All variables have to be annotated manually –Avoid overflow –Minimize quantization effects –Find optimum wordlength Manual process supported by simulation –Time-consuming –Error prone Introduction

4. ESPL 4 Optimum Wordlength Longer wordlength –May improve application performance –Increases hardware cost Shorter wordlength –May increase quantization errors and overflows –Reduces hardware cost Optimum wordlength –Maximize application performance or minimize quantization error –Minimize hardware cost Wordlength (w) Cost c(w) Distortion d(w) [1/performance] Optimum wordlength Background

5. ESPL 5 Wordlength Optimization Express wordlengths in digital system as vector Wordlength range for kth wordlength Cost function c Background

6. ESPL 6 Wordlength Optimization Application performance function p Wordlength optimization problem Iterative update equation Good choice of update direction can reduce number of iterations to find optimum wordlength Background

7. ESPL 7 Sequential Search [K. Han et al. 2001] Greedy search based on sensitivity information (gradient) Example –Minimum wordlengths {2,2} –Direction of sequential search –Optimum wordlengths {5,5} –12 iterations Advantage: Fewer trials Disadvantage: Could miss global optimum point Search Methods

8. ESPL 8 Measures for Optimum Wordlength Complexity measure method [W.Sung and K.Kum 1995]Complexity measure method [W.Sung and K.Kum 1995] –Minimize complexity c(w) subject to constraint on distortion d(w) –Update direction uses complexity sensitivity information Distortion measure [K. Han et al. 2001]Distortion measure [K. Han et al. 2001] –Minimize distortion d(w) subject to constraint on complexity c(w) –Update direction uses distortion sensitivity information ? w0w0 w1w1 wnwn … Complexity ? w0w0 w1w1 wnwn … Distortion Measures

9. ESPL 9 Combine complexity measure with distortion measure by weighting factor (0≤α≤1) Tradeoffs between measures by changing weighting factor Update direction uses both sources of sensitivity information Complexity-and-Distortion Measure Update direction Objective function Optimization problem ? w0w0 w1w1 wnwn … α Complexity Distortion Measures

10. ESPL 10 Broadband Wireless Access (IEEE 802.16a) Demodulator w 0 : Input wordlength of orthogonal frequency division multiplex (OFDM) demodulator which performs a fast Fourier transform (FFT) w 1 : Input wordlength of equalizer w 2 : Input wordlength of channel estimator w 3 : Output wordlength of channel estimator Case Study

11. ESPL 11 Simulations Assumptions –Internal wordlengths of blocks have been decided –Complexity increases linearly as wordlength increases Required application performance –Bit error rate of 1.5 x 10 -3 (without error correcting codes) Simulation tool –LabVIEW 7.0LabVIEW 7.0 Case Study InputWeight FFT1024 Equalizer (right) 1 Estimator128 Equalizer (upper) 2

12. ESPL 12 Minimum Wordlengths Change one wordlength variable while keeping other variables at high precision {1,16,16,16},{2,16,16,16},... {16,1,16,16},{16,2,16,16},... … …{16,16,16,15},{16,16,16,16} Minimum wordlength vector is {5,4,4,4} Case Study

13. ESPL 13 Number of Trials Start at {5,4,4,4} wordlength Next wordlength combination for complexity measure (α = 1.0) {5,4,4,4}, {5,5,4,4}, … Increase wordlength one-by- one until satisfying required application performance Case Study

14. ESPL 14 Complexity and Number of Iterations Each iteration computes complexity & distortion measures Distortion measure: high cost, low iterations Complexity-distortion: medium cost, fewer iterations Complexity measure: low cost, more iterations Full search: low cost, more iterations MethodαwComplexityIterations Distortion Only Complexity-Distortion Complexity Only Full Search 0 0.5 1 n/a {10,9,4,10} {7,10,4,6} {7,7,4,6} 10781 7702 7699 16 15 69 210 Case Study

15. ESPL 15 Conclusion Summary –Fixed-point conversion requires wordlength optimization –Develop complexity-and-distortion measure –Complexity-and-distortion method finds optimal solution in one-third the time that full search takes for case study Future extensions for wordlength optimization –Automate selection of wordlength range –Combine simulation-based and analytical approaches –Employ genetic algorithms

16. ESPL 16 Fixed-Point Representation Fixed point type –Wordlength –Integer wordlength Quantization modes –Round –Truncation Overflow modes –Saturation –Saturation to zero –Wrap-around SXXXXX Wordlength Integer wordlength SystemC format www.systemc.org Introduction XXXXX Wordlength Integer wordlength =  2 Back

17. ESPL 17 Full Search [W. Sung and K. Kum 1995] Exhaustive search of all possible wordlengths Advantages –Does not miss optimum points –Simple algorithm Disadvantage –Many trials (=experiments) Distance Expected number of iterations Direction of full search: minimum wordlengths {2,2} optimum wordlengths = {5,5} d = 6 trials = 24 Search Methods Back

18. ESPL 18 FFT Cost N Tap FFT cost 256 Tap FFT cost Back

19. ESPL 19 Uses complexity sensitivity information as direction to search for optimum wordlength Advantage: minimizes complexity Disadvantage: demands large number of iterations Complexity Measure [W.Sung and K.Kum 1995] Update direction Objective function Optimization problem Measures Back

20. ESPL 20 Applies the application performance information to search for the optimum wordlengths Advantage: Fewer number of iterations Disadvantage: Not guaranteed to yield optimum wordlength for complexity Distortion Measure [K. Han et al. 2001] Update direction Objective function Optimization problem Measures Back

21. ESPL 21 Broadband Wireless Access Demodulator Simulation Case Study

22. ESPL 22 Top-Level Simulation Case Study Back

23. ESPL 23 Tools for Fixed-Point Simulation gFix (Seoul National University) –Using C++, operator overloading Simulink (Mathworks) –Fixed-point block set 4.0 SPW (Cadence) –Hardware design system CoCentric (Synopsys) –Fixed-point designer gFix a(12,1); gFix b(12,1); gFix c(13,2); c = a + b; float a; float b; float c; c = a + b; Introduction Wordlengths determined manually Wordlength optimization tool needed

24. ESPL 24 Wordlength Optimization Methods Analytical approach –Quantization error model –Overestimates signal wordlength –For feedback systems, instability and limit cycles can occur –Difficult to develop analytical quantization error model of adaptive or non-linear systems Simulation-based approach –Wordlengths chosen while observing error criteria –Repeated until wordlengths converge –Long simulation time Background

25. ESPL 25 Optimum Wordlength Search Methods Full search [W. Sung and K. Kum 1995] Min + b bit search [W. Sung and K. Kum 1995] Max – b bit search [M. Cantin et al. 2002] Hybrid search [M. Cantin et al. 2002] Sequential search [K. Han et al. 2001] Preplanned search [K. Han et al. 2001] Branch and bound search [H. Choi and W.P.Burleson 1994] Simulated annealing search [P.D. Fiore and L. Lee 1999] Search Methods

26. ESPL 26 Procedure 1.Range estimation –Find maximum and minimum values for each 2.Find minimum wordlengths –Defines starting wordlength values to use 3.Iterative search –Increase/decrease wordlengths one-by-one until meeting specification using one of the measures Case Study