AccelChip Presentation

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An Automated System for Floating- to Fixed-Point
Conversion of High Performance of MATLAB
Algorithms in FPGAs and ASICs Eric Cigan and
Robert Anderson September 2004
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Outline

• Motivation
• For FPGA / ASIC implementation
• For fixed-point arithmetic
• Alternatives
• Related approaches methods used
• Changshun Shi performance criteria and
  optimization
• Approach used
• MATLAB-based algorithmic synthesis
• Heuristic-based approach
• Advantages of approach
• Single design source
• Verification flow
• Example
• FIR filter

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Rationale for DSP Algorithms in Silicon
Function TI C6416Tat 1 GHz Xilinx Virtex-II Pro Platform
8x8 Multiply Accumulate (MAC) 8.0 Billion MAC/s 1 Trillion MACs/s fclk 300 MHz
FIR Filter  - 256 Taps, Linear phase  - 16-bit data/coefficients 15.5 MSPS 300 MSPS  fclk 300 MHz
Complex FFT  - 1024 point, 16-bit data 6 µs 1 µs fclk 150 MHz
Viterbi Decoding Throughput 600 channels at 7.95-Kbps for a total of 4.77 Mbps 155 Mbps (OC-3 rates)
Reed-Solomon Decoding Throughput 6.8 Mbps 10 Gbps (OC-192 rates) fclk 85 MHz
Turbo Convolutional Decoder Throughput Seven 2-Mbps data streams (6 iterations) 5.4 Mbps (6 iterations)
Dedicated coprocessor 16 cores running in parallel in a single device.
Source Xilinx website (1/2004) and AccelChip estimates
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Fixed-Point Implications

• Increases in performance (throughput, frequency,
  etc.)
• In communications a 1 dB increase in the link
  power budget results in a 25 increase in
  coverage
• Power consumption
• Word size silicon area/memory gt Cost

Source The MathWorks, 2003
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Previous Work

• Rule-of-thumb methods
• Ad-hoc methods such as rounding and truncation.
• Manual scaling to and from integer
  representations.
• Recoding the source code in a hardware
  description language and then verifying
  performance in the RTL code.
• Substitution of floating-point functions with
  fixed-point equivalents, such as in C fixed-point
  libraries.
• Recent research
• Keding, et al (RWTH Aachen, 1998)
• Algorithm described in C/C
• Oriented toward minimizing all wordlengths at
  same time
• Can require large number of iterations
• Shi and Brodersen (UC Berkeley, 2002-2004)
• Algorithm described in graphical design tool
• Optimization-based methodology
• Requires Monte Carlo simulation runs

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MATLAB-Based Algorithmic Synthesis Flow
Algorithm Exploration
Floating Point Design
DSP IP Libraries
Fixed Point Models
SignalProcessing
System-Level Verification
Automated Fixed Point Model Generation
Project Directory
Communications
Fixed-Point Model Generation Report
Model Export
Implementation Exploration
Verification Report
IC Design Tools
RTL Model
Place Route
Final IC Design Verification
Algorithmic Synthesis Environment
Implementation Optimization
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Synthesizable MATLAB Coding Basics

• MATLAB script file applies input vectors to the
  design function in a loop called the streaming
  loop
• The design to be synthesized needs to exist in a
  function called the design function
• Additional MATLAB files or external data files
  may exist

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Fixed-Point Quantization

• Tool supports two fixed point datatypes
• Fixed 2s complement
• Required for negative numbers
• Much easier to multiply (dont have to worry
  about sign bit)
• ufixed (unsigned)
• Provides a greater range with same hardware when
  numbers are all positive
• To optimize the dynamic range of a number
• Use a minimal of integer bits to accommodate
  the range of possible values
• Use a minimal of fraction bits to accommodate
  acceptable precision

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Automated Quantization Method

• Quantization performed on floating-point MATLAB
• Bit widths are derived from
• Stimulus waveforms
• Text files loaded to initialize constants
• Bit growth propagated from arithmetic operations
• MATLAB quantizer statements in M-file
• Explicit quantizer directives

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Fixed-Point Generation Process

• Process is automated, yet requires designers
  knowledge

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Generated Fixed-Point MATLAB Files

• Tool generates fixed-point M-files from source
  floating point M-files
• The MATLAB quantize and quantizer functions
  are inserted into the original source code
• Optimization of quantizer functions reduces
  simulation overhead of fixed-point simulation
  up to 100x faster

Floating Point MATLAB
Generated Fixed-Point Output
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The Quantize Directive

• The Quantize directive allows users to override
  the automatic quantization
• Applying directive allows use of unmodified
  MATLAB source
• Directives can be typed at the command line or
  added to an .add directives file
• AccelChip will create this file automatically
  when Save -gt Project is executed

Directives file located here
Quantize directives can be applied interactively
from the design browser
SetDirective Variables.indatabuf -quantize fixed
floor wrap 32 16
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Auto-Quantization Report
MATLAB Editor
All variables in a design are displayed in a
single, flat format
Directives Editor
Double-Click
Specifies the source of the directive Flags
directives that are set to maximum
Double-Click
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Overflow and Underflow

• An underflow is defined as a number that is
  nonzero before it is quantized, and zero after it
  is quantized.
• Example of an underflow
• X quantize(quantizer(ufixed,floor,wrap,2
  2), 0.1)
• 00b 0
• 01b 0.25
• 10b 0.50
• 11b 0.75
• An overflow occurs when the magnitude of a number
  exceeds the dynamic range allowable by the
  integer bits as defined by the quantizer
• Example of an overflow
• X quantize(quantizer(ufixed,floor,wrap,2
  0), 5)
• 00b 0

X will be zero because the smallest non-zero
number will be too large
X will be one because the binary representation
of 5, 101 will be wrapped to 01
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Managing Underflow and Overflow Issues

• Often a design has a large number of underflows
  and only a few overflows.
• Displaying overflow and underflow messages
  together often makes it difficult to locate the
  overflows
• Recommended Practice
• Begin the debug process by setting Show
  Underflows to False
• Once overflows are addressed then set Show
  Underflows to True and address any remaining
  fidelity issues

Project options are available to selectively
disable displaying overflow or underflow message
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Addressing Bit Growth Due to Constants
A constant introduced here
Will affect all downstream hardware
Will affect all downstream hardware

• Constants are represented in binary form with
  maximum accuracy
• This can lead to unnecessary bit growth
• Constants used directly in expressions cant be
  quantized directly
• Change in coding style allows bit growth
  management

Y x quantize( quantizer(ufixed','floor','wrap
', 10,9), 1.3 )
or
Y x 1.3
Const 1.3 Y x cost
Recommended
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Improving Hardware through Quantization

• Quantization can have a dramatic affect on
  performance and area
• Tool will attempt to auto-quantize to preserve
  signal fidelity
• A maximum of 53 bits will be used for internal
  variables
• Consistent with abscissa of MATLAB double
  number
• A maximum of 32 bits will be used for IO ports
• Start by reviewing the fractional bits of the
  default quantization assigned to constants and
  the input data stream
• Tool will use as much precision as necessary to
  represent number
• E.g., 0.375 ltbinary ptgt011 and 0.3 ltbinary
  ptgt01100110011001100110011001100110011001100110011
  001100
• Designer uses tool directives to trim bits

Example 16-tap FIR filter
Quantization on Filter Coefficients of Slices
Fixed, wrap, floor, 10 8 55
Fixed, wrap, floor, 53 51 169
Default quantization used 53 bits
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Conclusion

• Algorithmic synthesis tool enables MATLAB design
  to be design source throughout process
• Tool aids in automating process of converting
  floating point designs to fixed-point
• Provides design exploration to increase
  performance and reduce size/power

MATLAB Domain (Pure algorithmicnon-implementation
-specific)
m language (used by MATLAB)
More abstract, lessimplementation-specific
Untimed C Domain (non-implementation-specific)
Standard C (used by Catapult C)
Handel-C
Timed C Domain (implementation-specific)
SystemC
Less abstract, moreimplementation-specific
RTL Domain (implementation-specific)
Verilog and/or VHDL
Different levels of synthesis abstractionsource
Mentor Graphics white paper, Catapult C
Synthesis-based Design Flow, October 2003