Signal Processing - November 2016 - 141
TIPS & TrIckS
Zhuo Wang, Jintao Zhang,
and Naveen Verma
Reducing Quantization Errors for Inner-Product Operations
in Embedded Digital Signal Processing Systems
within the quantization process, which
is shown to significantly improve accuracy at a given bit precision. By targeting
the way in which operands are represented, the approach incurs no added
hardware or computational overheads
and in fact is shown to reduce energy
and area thanks to simplified computation associated with the representation.
A distribution of operands
commonly encountered
An inner-product operation can be represented as follows:
y = / i hi xi .
(1)
Here, x i is input data, while h i is
typically predefined coefficients. For
example, in the case of a finite impulse
response (FIR) filter, x i are signal samples to be filtered, while h i are the filter
coefficients; in the case of a perceptron
(i.e., linear classifier), x i are input feature elements, while h i are the weights
and bias. Quite often, the distribution
of h i values can result in severe quantization errors when implementing the
hi
Histogram
i
Occurrence
(a)
Histogram
log2 |hi |
I
nner-product operations are used
extensively in embedded digital signal processing (DSP) systems. Their
applications range from signal processing (filtering/convolution) to inference
(classification). In embedded systems,
resources (energy, area, etc.) are typically
highly constrained, making tradeoffs
with computational precision a fundamental concern. Indeed, with increasing
re quirements on algorithmic performance, many systems are trending
toward higher computational precision to
ensure accuracy of results.
This article describes an approach
to significantly enhance accuracy for
inner-product operations at very low bit
precisions, significantly improving the
energy/area-versus-accuracy tradeoffs
traditionally incurred [1]. Low-energy
embedded systems often employ linear,
fixed-precision representation for operands due to the simplicity and relative
efficiency at lower dynamic range. We
focus on the use of a floating-point representation. For specific distributions
of operands very commonly observed,
such a representation substantially
enhances the accuracy of most operands. However, particularly at the low
bit precisions we focus on, it raises the
issue that many operands can incur a
large quantization error, much greater
than that with standard linear, fixedprecision representation. To address
this, a simple optimization is applied
Occurrence
i
(b)
Digital Object Identifier 10.1109/MSP.2016.2585278
FIgure 1. The distribution of filter coefficients for a low-pass FIR filter, with (a) standard linear fixed-
Date of publication: 4 November 2016
point quantization and (b) floating-point representation.
1053-5888/16©2016IEEE
IEEE Signal Processing Magazine
|
November 2016
|
141
Table of Contents for the Digital Edition of Signal Processing - November 2016
Signal Processing - November 2016 - Cover1
Signal Processing - November 2016 - Cover2
Signal Processing - November 2016 - 1
Signal Processing - November 2016 - 2
Signal Processing - November 2016 - 3
Signal Processing - November 2016 - 4
Signal Processing - November 2016 - 5
Signal Processing - November 2016 - 6
Signal Processing - November 2016 - 7
Signal Processing - November 2016 - 8
Signal Processing - November 2016 - 9
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Signal Processing - November 2016 - 28
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Signal Processing - November 2016 - 138
Signal Processing - November 2016 - 139
Signal Processing - November 2016 - 140
Signal Processing - November 2016 - 141
Signal Processing - November 2016 - 142
Signal Processing - November 2016 - 143
Signal Processing - November 2016 - 144
Signal Processing - November 2016 - 145
Signal Processing - November 2016 - 146
Signal Processing - November 2016 - 147
Signal Processing - November 2016 - 148
Signal Processing - November 2016 - Cover3
Signal Processing - November 2016 - Cover4
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