IQ Math Virtual Floating Point Engine for DSP

2011/06/23, Advanced Motion Control TeamCopyright 2011 Ajinextek Co., Ltd.ⓒ

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IQ-Math Virtual Floating-Point Engine

Introduction & Application- 2nd Year Technical Competence -

06/23/2011Advanced Motion Control Team

2011/06/23, Advanced Motion Control TeamCopyright 2011 by Ajinextek Co., Ltd., All rights reserved.ⓒ

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1. Introduction

2. IQ-Math vs. Floating Math and Fixed Math (ANSI C)

3. Implementation

4. Summary

Introduction


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1. Introduction


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Floating implementation in embedded application

1. IQMath Introduction

Takes many days/weeks to convert (one way

process)

Fixed-Point Algorithm

(ASM, C, C++)

Fixed-Point DSP

Can be easily ported to floating-point

device

Floating-Point DSP

Floating-PointAlgorithm(C or C++)

Natural development starts with simulation in floating-

point

SimulationPlatform

(i.e. MatLab)


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Floating-point Processor Fixed-point Processor

- Hardware core unit support floating point operation which fol-lows IEEE 754 standarde.g: Intal’s Pentium Series, TI TMS320C6000, …

- Efficient when operation with floating-point data

- Not efficient in control task (bit manipulation, input/output con-trol, interrupt response)

- Expensive

- Internal hardware support inte-ger datae.g: all microcontroller families (Motorola HC68x, Infineon C166, …), TI TMS 430, TMS320C5000, C2000, …

- Arithmetic unit and hardware multiply unit (DSP) expect data to be in one of fixed-point type

- Inexpensive


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IQ-Math format IQxxx() or IQNxxx(): virtual floating-point engine+ Based on lookup tables available in BOOTROM of F281x devices + Variable range and resolution (present later)+ Achieve higher speed and precision, shorten DSP application devel-

opment time when porting floating point algorithm into fixed point code

Fixed-Point format (Q Math)- A value of a fixed-point data type is essentially an integer that is scaled by a specific

factor determined by the type. - The scaling factor is the same for all values of the same type, and does not change dur-

ing the entire computation.- The scaling factor: 10^x, 2^x, 1/3600, ….- Notation: Qf : f is number of fraction digits Qm.f : m is number of integer digits Precision loss and overflow (e.g. multiplying (x) and dividing(-))Floating-Point format - Describes a system for representing real numbers which supports a wide range of val-

ues- Numbers are in general represented approximately to a fixed number of

significant digits and scaled using an exponent- The base for the scaling is normally 2, 10 or 16 Significant digits × base exponent -The most common used format is IEEE standard 754 (discuss later) More storage to encode the position of the radix but larger precision

http://en.wikipedia.org/wiki/Integer

http://en.wikipedia.org/wiki/Real_number

http://en.wikipedia.org/wiki/Significant_figures

http://en.wikipedia.org/wiki/Exponentiation


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The TI IQmath library offers usage in C/CPP program and it consists of 5 parts:

1) IQmath header file:IQmathLib.h

2) IQmath object library containing all function & look-up tablesIQmath.lib

3) Linker Command FileIQmath.cmd

4) IQmath GEL file for debuggingIQmath.gel

5) Example programs



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2. IQ-math vs. Floating Pointand Fixed Point (ANSI C)


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2.1 IQMath vs. Floating Math

Z = -2I + 2I-1 + … + 21 + 20 . 2-1 + 2-2 + … + 2-Q

“IQ” – Format (or Fractional Format)“I” - INTEGER – Fraction“Q” - QUOTIENT – Fraction

IEEE Standard 754 Single Precision Floating Format

s eeeeeeee fffffffffffffffffffffff031 30 23 22

23 bit mantissa (fraction)8 bit exponent1 bit sign

Advantage Disadvantage

Floating Math

Large dynamic rangePrecision depends on exponent term

IQ Math Same precision for same IQ format

Limited dynamic range

1 . .2S E OffsetZ M

23

1

1 2 ii

i

M f


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2.1 IQMath vs. Floating Math

Limitation of single precision floating format

Improve by IQ-math format

Example: x = 10.0 (0x41200000)+ y = 0.000000238 (0x347F8CF1)

z = 10.000000238

0x412000000 = 10.00000000010.000000238 - can’t represent!

0x412000001 = 10.000000950

I8Q24 Example: x = 10.0 (0x0A000000)y = 0.000000238 (0x00000004)

z = 10.000000238 (0x0A000004)

The gap between two identified numbers in single floating-point format is about ten million times smaller than the numbers


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I8 . Q24

I16 . Q48

M

X

B

Y

>> 24 Align DecimalPoint Of Multiply

I8 . Q24

I8 . Q24

sssssssssssssssssI16 . Q24

I8 . Q24I8 . Q24

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2.2 IQMath vs. Fixed-Point Math (QMath)

Execute floating-point data or implement floating point algorithm on a fixed-point processor:

<< 24Align DecimalPoint for Add

I8 . Q24 M

X

B

Y

I8 . Q24

I8 . Q24

I16 . Q48

ssssssssssssssssssI8 . Q24

ssssI8 . Q48

I16 . Q48

sssssssssssssssssI16 . Q24 I8 . Q24

>> 24Align DecimalPoint for Store

Y = ((i64) M * (i64) X + (i64) B << Q) >> Q;in C: Y = ((i64) M * (i64) X) >> Q + B;in C:

*Y M X B

+ +

x xANSI C IQ-Math


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float Y, M, X, B;

Y = M * X + B;

Floating-Point

long Y, M, X, B;

Y = ((i64) M * (i64) X + (i64) B << Q)) >> Q;

TraditionalFix-Point Q

_iq Y, M, X, B; Y = _IQmpy(M, X) + B;

“IQmath”In C

iq Y, M, X, B; Y = M * X + B;

“IQmath”In C++

2.3 Programming Scope

Comparison of floating format implementation


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3. IQ-math Implementation


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3. Implementation

#define GLOBAL_Q 18 // define in “IQmathLib.h” file

_iq Y, M, X, B;

Y = _IQmpy(M,X) + B; // all values are in Q = 18

User selects “Global Q” value for the whole application

based on the required dynamic range or resolution, for example:

The user can also explicitly specify the Q value to use:

_iq20 Y, M, X, B;

Y = _IQ20mpy(M,X) + B; // all values are in Q = 20

IQ-Math application


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3. Implementation

IQ-Math range and resolution

Q range Stability Range

Q31 - Q27Unstable (not enough

Dynamic range)

Q26 - Q19 Stable

Q18 – Q1Unstable (low resolu-

tion, quantization problem)


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3. Implementation

IQ-Math supported functions: supports most of common floating-point func-tions


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Compile & Runusing floating-point math on

C3x, C67x,C28x (RTS), PC,..

Compile & Runusing “IQ-math” on

C28x

User selects target math type(in “IQmathLib.h” file)

Y = _IQmpy(M, X) + B;

#if MATH_TYPE == IQ_MATH #if MATH_TYPE == FLOAT_MATH

Y = (float)M * (float)X + (float)B;

3. Implementation

Flexible portability between Fixed-point and Floating-point with IQ-Math


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4. Summary


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Seamless portability of code between fixed and floating-point

devicesUser selects target math type in “IQmathLib.h” file

#if MATH_TYPE == IQ_MATH

#if MATH_TYPE == FLOAT_MATH

One source code set for simulation vs. target device Numerical resolution adjustability based on application

requirementSet in “IQmathLib.h” file

#define GLOBAL_Q 18

Explicitly specify Q value_iq20 X, Y, Z;

Numerical accuracy without sacrificing time and cycles Rapid conversion/porting and implementation of algorithms

“IQ-math” + 32-bit fixed-point processor

4. Summary

NOTICE: Proprietary and ConfidentialThis material is proprietary to Ajinextek Co., Ltd.. It contains trade secret and confidential information which is solely the property of Ajinextek Co., Ltd.This material is for client’s internal use only. This material shall not be used, reproduced, copied, disclosed, transmitted, in whole or in part, without the express consent of Ajinextek Co., Ltd.Copyright © 2011 by Ajinextek Co., Ltd. All rights reserved.

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IQ Math Virtual Floating Point Engine for DSP

Documents

point format

point data

point math

point processor

point algorithm

floating math

mathtype floatmath

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