1. “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

2. Outline Motivation Alternatives Related approaches methods used
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

3. Rationale for DSP Algorithms in Silicon
Function
TI C6416T at 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  Taps, Linear phase  bit data/coefficients
15.5 MSPS
300 MSPS 
Complex FFT  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
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

4. 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 => Cost
Fxied pt required for battery powered
Source: The MathWorks, 2003

5. Previous Work Rule-of-thumb methods Recent research
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, )
Algorithm described in graphical design tool
Optimization-based methodology
Requires Monte Carlo simulation runs

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

7. 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

8. Fixed-Point Quantization
Tool supports two fixed point datatypes
Fixed 2’s complement
Required for negative numbers
Much easier to multiply (don’t 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

9. 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

10. Fixed-Point Generation Process
Process is automated, yet requires designer’s knowledge…

11. 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

12. 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 -> 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}}

13. 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

14. 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);
01b = 1
10b = 2
11b = 3
A fidelity error is caused by either an overflow or an underflow
Overflow errors tend to be severe distortions
Underflow errors tend to be minor distortions
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”

15. 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

16. 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 can’t 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

17. 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., = <binary pt>011 and 0.3 = <binary pt>
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

18. 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 algorithmic non-implementation-specific)
“m” language (used by MATLAB)
More abstract, less implementation- specific
Untimed C Domain
(non-implementation-specific)
Standard C (used by Catapult C)
Timed C Domain
(implementation-specific)
Handel-C
SystemC
Less abstract, more implementation- specific
RTL Domain
(implementation-specific)
Verilog and/or VHDL
Different levels of synthesis abstraction [source: Mentor Graphics white paper, “Catapult C Synthesis-based Design Flow,” October 2003]