Signal Processing - November 2016 - 145

(13) is a repeating function of a. As
a result, a global minimum of C h ^ah
exists in the range 61, 2h . Thus, the
optimization is easily solved numerically by sweeping a within this range.
For the filter considered, Figure 4
shows the resulting cost function and
how its value is improved by the optimal choice of a
f ht i ^ a h =

ah i - \
ah i
a

s i ^ah
t i ^ah@
= li $ 2
$ 6m i ^ah - m
a
(16)
f ^ah = 26log 2 ^a h i h@ a .
(17)

Summary of approach
Below, the specific procedure for applying the optimization generally to minimize quantization error is provided.
1) Select a precision requirement for
h i (based on preferred tradeoff of
computational accuracy and hardware complexity), e.g., 6 bits in our
application.
2) Perform floating-point quantization of h i [assigning bits to significand and exponent based on
the corresponding dynamic range
^max h i h ^min h i h ].
3) Determine the scaling parameter a .
a) Model the statistical distribution
of inputs X i .

b) Compute the distribution of output error f yt due to quantization
of h i [see (10)].
c) Define a cost function based
on the distribution of output
error f yt .
d) Minimize the cost function to
find optimal a.
4) Redetermine the quantization for h i
with the scaling parameter a.
5) Only if the absolute value of output
y is critical, apply multiplication by
1/a to the outputs.
Note that, when required, the energy
of Step 5 is greatly amortized since
it involves one multiplication, compared to many in the preceding convolution operation.

Experimental demonstration
For demonstration, the presented approach is applied to the EEG-based
seizure-detection system of Figure 2.
For analysis and demonstration of the
approach, three hardware implementations of filters are compared, each
employing 6-10 bits for representing h i .
The first implementation is a conventional fixed-point FIR filter. The second
implementation is the mixed fixed-/floating-point FIR filter without any a optimizaton. For this, s i is assigned 4 b for
the decimation filter and 3 b for the bandpass filter to address the required range

150
100
50
0

Occurrence

300

200

40
35
30
25
20
15
10
5
0

0

0

0.5
1
1.5
Quantization Error (LSB)

1
2
3
4
Quantization Error (LSB)

Value
(Output Sample or Output-Sample Error)

Occurrence

250

of h i ^^max h i h ^min h i h . 37 k for
the decimation filter, and 500 for the
band-pass filter). m i is assigned correspondingly for 6-10 bit overall precision
of h i . The third implementation is the
presented mixed fixed-/floating-point
FIR filter with a optimization. Simulation results are presented using 210 seconds of EEG data (sampled at 256 Hz).
All implementations are developed in
both MATLAB and Verilog RTL. For
analysis of quantization error, the MATLAB implementations are used. For
hardware energy and area analysis, the
Verilog implementations are synthesized
to standard cells and laid out in a 32-nm
complementary metal-oxide-semiconductor technology, with transistor-level
postlayout simulations performed using
NanoSim.
The results in Figure 5 focus on quantization errors. The histograms show the
quantization error distributions for the
decimation filter and band-pass filter
after applying the optimization. Comparing Figure 5 to Figure 2, we can see
that with a optimization applied to the
mixed fixed-/floating-point approach,
the quantization errors are more densely
clustered close to zero. The output error
resulting from the three implementations for 6-bit coefficient representation are also compared. The samples
are ordered based on the magnitude of

Y (Double)
εY (Fixed Point)
εY (Mixed, Optimization)
εY (Mixed, No Optimization)

200

100

0

-100
-200
-300

0

2,500
5,000
7,500
Output Sample Index (Sorted)

10,000

FIgure 5. The reduction of quantization error by applying the presented optimization.
IEEE Signal Processing Magazine

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November 2016

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145
Table of Contents for the Digital Edition of Signal Processing - November 2016
