Transcript Presentation by Tom Hummel (March 5th 2010)

Floating Point vs. Fixed Point for
FPGA
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Applications
Digital Signal Processing
- Encoders/Decoders
- Compression
- Encryption
Control
- Automotive/Aerospace
- Industrial
- Space
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Floating Point?
Data Structure
Simple Representation:
Mantissa: Numerical portion of
number
Exponent: Signed exponent to vary
range of mantissa
Sign: Sign of mantissa
-1sign * Mantissa * BaseExponent
IEEE 754 Representation (7 Digit Mantissa)
-1sign * 1.Mantissa * 2Exponent - 127
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What About Fixed Point?
Fixed point assumes constant scaling (radix)
-
No standard
Smaller range of numbers
Generally base 2 for fast radix conversion
Programmer must determine number ranges offline
Classic Fixed Point Bit Representation (Savich, 2007)
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Fixed Point Operations
Fixed-Point
Addition/Subtraction:
Sum = A + B
Multiplication:
Product = A x B
Precision Implications
-If the result is outside of the
expected format then overflow
can occur
-Programmer must account for
the potential ranges of operands
to avoid precision problems
Note:
- Numbers have to have same radix
- With base 2 scaling radix conversion is << or >>
- Programmer must account for radix differences
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Floating Point Operations
Floating Point (By Steps)
Addition/Subtraction:
- Normalize Exponents
- Fixed point add/subtract
- Round
Multiplication:
- Add Exponents
- Multiply Mantissa
Precision Implications
-If the result is outside of the
digits of the mantissa, the result
must be rounded
- Dynamic range means that
programmer has less control,
but easier to handle unknown
ranges of numbers
- Different options for rounding.
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Fixed Point or Floating Point?
Fixed Point
- Very fast when base 2
- No complicated logic
- Radix point not encoded
- Fixed Accuracy
- Can only represent small
number set
Floating Point
- Slower
- Accuracy Varies
- Represent very large
number set
- Radix point encoded
- Complex logic required
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FPGA Floating Point
Parallel Implementation
- HDTV needs 20 GFLOPS/Sec
- Current DSP’s cannot achieve this (Dido, 2002)
Optimized Format for Application
- Different bit formats optimize operation speed, accuracy
- If FPGA targets single application, IEEE does not need to be followed. (Connors,
1999)
Size vs Speed Issue
- Full feature FPGA units that are parallelized require many resources
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FPGA Floating Point vs. CPU
FPGA versus CPU Performance for 32 bit FP Addition Over Time (Underwood, 2004)
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FPGA Architectures
Standard and 2-Path Floating Point FPGA Adders (Liang, 2003)
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FPGA Architectures
LOP Floating Point FPGA Adder (Liang, 2003)
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Resources and Performance
FP Adder Area and Latency versus Mantissa Size and (Liang, 2003)
Spartan III Resource Table (Xilinx, 2009)
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Fixed vs. Floating Application
Neural Networks
- Use of log-sigmoid function
- Calculation of small error values
- Known number ranges
- Two inputs, two neuron hidden, one output
MLP Neural Network (Savich, 2007)
Parallel Neuron (Savich, 2007)
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Size Utilization
MLP-BP 2,2,1 NN with Parallel Neurons Design Size vs Manitssa Size (Savich, 2007)
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Fixed Point Performance
MLP-BP 2,2,1 NN with Parallel Neurons Fixed Point Training Performance (Savich, 2007)
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Floating Point Performance
MLP-BP 2,2,1 NN with Parallel Neurons Floating Point Training Performance (Savich, 2007)
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When do we NEED Floating Point?
1. Accuracy is paramount
- Accuracy at small numbers while operating on large
numbers
2. Range of numbers unpredictable
- Fixed point programs must anticipate number
ranges or errors will occur
3. Development time is very short
- Time must be spent to analyze algorithm on a low
level to determine number ranges
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Floating Point Application
Military Radar
- Compute complex integral at a high speed
- Accuracy is required due to obvious safety implications
- Floating point lowers noise introduction while executing FFT
High Performance DSP
- More favorable signal-to-noise ratio due to high accuracy at
low values
- Signal-to-noise for floating point is 30x106 to 1 versus 30,000
to 1 for fixed point
- High resolution ADC (20 bits plus) requires floating point,
fixed point registers are too small for accuracy
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Conclusions
Fixed Point is preferable for most applications
- Low Resources
- Low gate delays
- Simple implementation of HW components
Floating point is useful when:
- Accuracy over a large range of numbers is required
- Impossible or too hard to estimate number ranges
- Programming time is severely limited
- The floating point architecture is best customized via FPGA
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References
Dido, J., Geraudie, N., Loiseau, L., Payeur, O., Savaria, Y., & Poirier, D. (2002). A flexible floating-point
format for optimizing data-paths and operators in FPGA based DSPs. FPGA '02: Proceedings of the 2002
ACM/SIGDA Tenth International Symposium on Field-Programmable Gate Arrays, Monterey, California,
USA. 50-55.
Liang, J., Tessier, R., & Mencer, O. (2003). Floating point unit generation and evaluation for FPGAs.
FCCM '03: Proceedings of the 11th Annual IEEE Symposium on Field-Programmable Custom Computing
Machines, 185.
Savich, A. W., Moussa, M., & Areibi, S. (2007). The impact of arithmetic representation on implementing
MLP-BP on FPGAs: A study. Neural Networks, IEEE Transactions on, 18(1), 240-252
Underwood, K. (2004). FPGAs vs. CPUs: Trends in peak floating-point performance. FPGA '04:
Proceedings of the 2004 ACM/SIGDA 12th International Symposium on Field Programmable Gate Arrays,
Monterey, California, USA. 171-180.
Xilinx. (2009). Xilinx DS099 spartan-3 FPGA family data sheet. Retrieved 02/20, 2010, from
www.xilinx.com/support/documentation/data_sheets/ds099.pdf
Yoji, D. C., Connors, D. A., Yamada, Y., & Hwu, W. W. (1998). A software-oriented floating-point format for
enhancing automotive control systems. Control Systems, Workshop on Compiler and Architecture Support
for Embedded Computing Systems,
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Thank You
Questions?
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