Performance analysis of algorithmic noise-tolerance ... - IEEE Xplore
PERFORMANCE ANALYSIS OF ALGORITHMIC NOISE-TOLERANCE TECHNIQUES Byonghyo Shim and Naresh R. Shanbhag Coordinated Science Laboratory, ECE Dept. University of Illinois at Urbana-Champaign 1308 West Main Street, Urbana, IL 61801 Email: [bshim,shanbhag]@mail.icims.csl.uiuc.edu ABSTRACT
The output in the presence of soft errors is
In this paper, we present performance analysis of algorithmic noisetolerance (ANT) techniques. First, we analyze the predictor and 'RPR based ANT schemes. Next, we present a hybrid ANT scheme :which is resilient to burst errors usually occurring in a high softerror rate (Per)region. For a frequency selective FIR Pltering, it is shown that simulation results match well with the analytic bounds while providing about 40 dB improvement in the mean square erIt is also 'ror over a conventional DSP system at a Per = &shownthat the proposed hybrid ANT scheme maintains its robustness in noise mitigation even in the high soft-error rate region of upto P,, =
yo,n
We have shown in the past that algorithmic noise-tolerance (ANT) technique [3]-[5] are very effective in combating deep submicron (DSM) noise [1]-[2]. However, past work has focused on simulations to prove the effectiveness of ANT. This paper has two contributions: 1) performance analysis of two existing ANT techniques, the prediction based [3] and reduced-precision redundancy (RPR) [5], and 2) a new ANT technique that is referred to as hybrid ANT and its performance analysis. This paper is organized as follows. After discussing ANTscheme in Section 11, we present its performance analysis and 'propose hybrid ANT scheme in Section 111. Simulation results and discussion are given in Section IV.
2. PRELIMINARIES We Prst present the ANT based digital signal processing (DSP) system that ensures high reliability in the presence of DSM noise and briel3y discuss previous ANT schemes.
(4)
where Q[n]and q(n]are assumed to be uncorrelated. Equation (4) can be rewritten as SN&,t,,,, = SNRout,o,g- A where A =
10 logl0(l+
3).
By employing ANT, we reduce the noise power
Q
,
. +
=
The output of a DSP. system is represented by
+ Qn
(1)
(2)
lYo,a
VYO,"
- YANT,,~ I Th
- Y A N T . ~ ) > Th.
(5)
- YANT,nl.
(6)
In doing so, no false alarms can occur and therefore only three possible cases exist; (1) no error, (2) undetected error, (3) detected error, with P,,,, P,,,, and P d e r , being the corresponding probability of occurrence, respectively. The estimation error power is given by
= =
E[IYo
-
m
- yANT12] [Yo - Yo[' I (Yo - Y A N T 5 T h ) ]
p d e r E [ [Yo
+
PuerE[
+
PnerE[lyo - ~ o 1 ~ ]
(7)
Note that the third term in (7) is zero. Furthermore, we denote the , soft error power U:, as noise power due to ANT u ~ N T and
17
= E [ I YO - YANT url = E [ I Yo - Y a 1' I (Ya - YANT 2
(8)
5 Th)]
(9)
Substituting (8) and (9) in (7), we get
OQ
U;o-$
This work is supported by NSF grant CCR 99-79381, ITR 00-85929 and DARPA MSP grant
0-7803-7761-3/03/$17.0002003 IEEE
if [!/a,, if lYo,n
Ya,n YANT,n
Th = m m
where d, is desired output signal and Q, is the noise due to channe1 effects, ADC quantization noise, etc.. The output S N R is 0: SN%ut.org = 10 log,o( 7)
{
where Th is a precomputed threshold and PANT,, is the corrected output. In order to guarantee that yo,,, = yo,n in the absence of errors, the threshold becomes
2.1. ANT based DSB system
=d n
(3)
due to soft error U: and thereby guaranteeing that A is sumciently small. The decision rule for an ANT-based DSP system is given by
1.. INTRODUCTION
yo,n
+ Qn + 71,
= dn
where ~ [ n is ]the noise due to the soft error. In this case, the output SNR is
IV-113
=
Pder
'
+puer
U ~ N T
'
U:
(10)
................Main .....DSP .....
the forward predictor output is
-,,................ ........EC ..........................
N
k=O
:....... .........................................
where the prediction error ep,,, = y n - W'Y. 1, we can show the miniBy using the derivative of E[ mum MSE which corresponds to noise power of forward predictor ~ i ~is given r by , ~ ~ ~ ~
:
ilnl
= E[y:] - P'R-lP
( T ~ N T =, minE[e;,,] ~ ~ ~
(12)
where
P'r
.................................................................................................. . . ManLXP
. .
i
= E [ Y n y'] = E [ (y n y n - 1
'
=EIYYT]=E[(yn-l
Yn-N)
'
'
.
fir
"'
YnYn-N T (!/n-l"'Yn-N)]
In addition, by applying the Triangular inequality, we can obtain the upper bound on u;f as
= E [ IYa = E [ 1%
U :
................................................................................... Ec
5
- Yo12 I (Yo - Y p r e 5 T h , p r e ) ] - Y p r e + Y p r e - YoI2 1 (Ya - Y p r e 5 T h , p r e ) ] (13)
Th*pref c i N T , p r e
Inserting (12) and (13) into (lo), we get the upper bound on the minimum noise power for the prediction based ANT system as
aio-c 5 P d e r u i N T , p r e Figure 1: Algorithmic noise-tolerance schemes: (a) prediction based ANT scheme, and (b) RPR based ANT.
+Puer(Th,pre +u i N T , p r e )
(14)
Predictor based ANT works well in narrowband systems (typically Bz.In addition, we denote the quantization step size of an original DSP and that of RPR as A, = a,nd Ar = The quantization noise N, and Nh between t e onginal value and that of RPR z r and h, is dePned as N , = z - z r and Nh = h - h,, where z and h are the input and Plter coefPcients, respectively. With this assumption, one can show that the noise power of the i ~is given ~ by, ~ ~ ~ , RPR ANT scheme ~
+
$
2.3. RPR based ANT Figure l(b) shows the block diagram of an RPR based ANT scheme. The RPR is a replica of the original system with small precision operands [ 5 ] . Though RPR output yr,n is generally not equal to the original one yo,n due to the LSB quantization noise, it is a good estimate in case of an error event. When an error is detected using (3, yr+ is employed as the Pnal output.
&.
3. PERFORMANCE ANALYSIS OF ANT
where U: is the input signal power. By applying an analysis similar to the one in (13), we get the noise power U: as
In this section, we present an analysis of the predictor, RPR and the hybrid ANT technique which is composed of both techniques. Here, we assume that the error-control block is designed to be error free. Indeed, this assumption is reasonable since the complexity of the error control block is much lesser than that of MDSP and hence the likelihood of soft-error is signiPcantly smaller.
Employing (15). (16), and (IO) we can obtain the upper bound on the noise power for the RPR based ANT as
5
aio-c = 5
3.1. Predictor based ANT Let the output vector of an original system is Y = [yn-l, ....y n - N I T and the predictor coefPcient vector is W = [ w l , .... W N I T . Then,
Pder
'
Pder
'
Th2,Tpr
+ciNT,rpr
(16)
+ puer (17) u i N T , r p r + p u e r .(Th,rpr +b i N T , r p r )
ciNT,rpr
'
Note that the noise power term in (17) depends only on precision and quantization noise but not on bandwidth. While the RPR based ANT provides good performance for a wide range of bandwidths, it consumes more power than the predictor based ANT system.
IV-114
U:.(NP ...............................................
Halm DSP.Mp
Mu" DSP Ec .................................................................... . ,
I
I :
YJ"1
Y,'"
YJ"'
FIN
Figure 3: Simulation setup.
.
Er
............................
Figure 2: The hybrid ANT technique: (a) block diagram, and (b) the state machine.
3.3. Hybrid ANT The predictor based ANT, which produces good performance in low soft-error rate P e r , has a problem as Per increases. Once error is made, an erroneous output is fed back into the predictor, leading to an incorrect decision until the erroneous output is purged. On the other hand, the RPR based ANT has relatively high power consumption than the predictor based scheme. Figure 2(a) shows the block diagram of the proposed hybrid ANT scheme that overcomes this problem. The key idea in the hybrid ANT is to use a predictor when there are no errors (i.e., for error detection) and RPR when an error occurs (i.e., error correction). As illustrated in Fig. 2(b), the error control is transferred to ) a predictor state ( S p r eafter ) error detecan RPR state ( S r p r from tion. If RPR does not detect an error for N consecutive cycles, i.e., the error propagation is terminated, then the error control is passed back to the predictor. Assuming that the error event is independent, the probability of being in the RPR and predictor states are, respectively,
+ (1-
Prpr
=per
Ppre
= (1 -
+'.
Per)Pe,
= Prpr
4. SIMULATIONS AND DISCUSSION The setup used to measure the performance of the proposed scheme is shown in Fig. 3, where a frequency selective Plter is used to generate a bandlimited signal yo[.] from a wideband input with noise z[n] 4121.The S N R without ANT is given by
+
SNR,, = 10loglo
giNT,rpr
+ppre ciNT,pre
5
PderPpre
'
uiNT,pred
-k
puerppre
'
(Th,pre
PuerPrpr
.(Th,rpr
+
PdeTPTpr
'
SNRout = lOlog,,
+
(19)
d N T , ~ p r
+0:NT.pre) uiNT,rpr)
(21) gv,--Y,
(18)
Using (18), (19), and (14), we can obtain the upper bound on the noise power of the hybrid ANT as U:,-$
Since RPR is used only when error is detected, we can tum this block off in the predictor state ,S ,, and thereby save power while maintaining robustness when Per is high.
whereas, the S N R at the ANT output is given by
(1 - P e r ) N p - l P e r
where N p is the number of taps in the predictor. Clearly, Prpr PP,, = 1. Then, the noise power of the hybrid ANT system is giNT,hybrid
Figure 4: Analytic results of prediction and RPR ANT scheme for bandwidth variation.
(20)
(22) cvo-ai
where uio,is the signal power of original Plter output. In this simulation, we employ a 31-tap Plter with a 12x12 multipliers and a 26-bit accumulator. For a DSM noise model, we employ random Bipping of an output bit in the digital Plter with a speciPed soft-error probability P e r . Figure 4 shows the analysis results of predictor and RPR based ANT, where the main Plter is a low-pass Plter having a bandwidth from wb = 0 . 1 to ~ 0 . 5 ~ .For the RPR MAC, we used the half precision of the original Plter (i.e., 6 x 6 multiplier) and a three tap of predictor is employed in the prediction based ANT. While the performance of RPR is quite similar
IV-115
IO '
10"
.........................................
. . . . . . . \h . ................
Figure 5: Performance analysis and simulation results of prediction and RPR ANT scheme.
. . .
- P m 10' - - Pmd 10.2 -4 RPR IO '
......
Figure 7: Contour plot for Pep and power overhead variation.
ing additional power consumption, we show the contour plot for the power overhead as well as Pep, which is given by
Poverheod(%)=
PANT- PMDSP PMDSP
(23)
where PMDSPand PANTare the power of original DSP and ANT based DSP, respectively. As shown in Fig. 7,the power overhead of hybrid ANT is about 15 20 9% lesser than that of RPR in a similar MSE region. Thus, the hybrid ANT scheme is not only robust to soft-errors when Pep, is high, it is also energy-efPcient.
-
5. REFERENCES
[ l ] 'T.Kamik and S . Vangal, OSelective node engineering for chiplevel soft error rate improvement,OProc. of Symp. on VLSI circuits, vol. 4, pp. 132-135, 2002.
Figure 6: Performance analysis and simulation results of hybrid ANT scheme.
over the entire frequency range, the performance of the predictor deteriorates as the bandwidth increases. Figure 5 shows the analysis and simulation results of the predictor and RPR based ANT, where the bandwidth of main Plter is now set to W b = 0 . 2 ~ .While the conventional system has high mean square error regardless of Per,predictor ANT has MSE less than at Pep= resulting in 40 dB gain over conventional MDSP. As discussed in Section 1I.A. the performance of predictor degrades severely as Pep increases due to the error propagation. On the contrary, the performance of RPR is close to the analytic results even when Pepis high. Next, we considered the hybrid ANT scheme under the same simulation setup. As shown in Fig. 6, the performance of hybrid ANT is similar to the prediction scheme in low P,,region. However, unlike to prediction scheme, we observe that hybrid scheme maintains its performance even when Perbecomes high. Finally, in order to observe the comprehensive picture includ-
[2] N. Shanbhag, K. Soumyanath, and S. Martin, OReliab!e lowpower design in the presence of deep submicron noise,OProc. of Intl. Symp. on Law-Power Electronics and Design, pp 295302,2000.
[3] R. Hedge and N. Shanbhag, OSoft digital signal processing.0 IEEE Trans. VLSI, vol. 9, pp. 813-823, Dec. 2001. [4] L. Wang and N. Shanbhag, OLow-power signal processing via error-cancellation,OProc. of IEEE workshop on Signal Processing Systems, Lafayette Oct. 2000.
[SI B. Shim, and N. R. Shanbhag, OLow-power digital Pltering via reduced-precision redundancy,OProc. of Asilomar con$, Vol. I, pp. 148-152, NOV.2001.
Iv-116
The output in the presence of soft errors is
In this paper, we present performance analysis of algorithmic noisetolerance (ANT) techniques. First, we analyze the predictor and 'RPR based ANT schemes. Next, we present a hybrid ANT scheme :which is resilient to burst errors usually occurring in a high softerror rate (Per)region. For a frequency selective FIR Pltering, it is shown that simulation results match well with the analytic bounds while providing about 40 dB improvement in the mean square erIt is also 'ror over a conventional DSP system at a Per = &shownthat the proposed hybrid ANT scheme maintains its robustness in noise mitigation even in the high soft-error rate region of upto P,, =
yo,n
We have shown in the past that algorithmic noise-tolerance (ANT) technique [3]-[5] are very effective in combating deep submicron (DSM) noise [1]-[2]. However, past work has focused on simulations to prove the effectiveness of ANT. This paper has two contributions: 1) performance analysis of two existing ANT techniques, the prediction based [3] and reduced-precision redundancy (RPR) [5], and 2) a new ANT technique that is referred to as hybrid ANT and its performance analysis. This paper is organized as follows. After discussing ANTscheme in Section 11, we present its performance analysis and 'propose hybrid ANT scheme in Section 111. Simulation results and discussion are given in Section IV.
2. PRELIMINARIES We Prst present the ANT based digital signal processing (DSP) system that ensures high reliability in the presence of DSM noise and briel3y discuss previous ANT schemes.
(4)
where Q[n]and q(n]are assumed to be uncorrelated. Equation (4) can be rewritten as SN&,t,,,, = SNRout,o,g- A where A =
10 logl0(l+
3).
By employing ANT, we reduce the noise power
Q
,
. +
=
The output of a DSP. system is represented by
+ Qn
(1)
(2)
lYo,a
VYO,"
- YANT,,~ I Th
- Y A N T . ~ ) > Th.
(5)
- YANT,nl.
(6)
In doing so, no false alarms can occur and therefore only three possible cases exist; (1) no error, (2) undetected error, (3) detected error, with P,,,, P,,,, and P d e r , being the corresponding probability of occurrence, respectively. The estimation error power is given by
= =
E[IYo
-
m
- yANT12] [Yo - Yo[' I (Yo - Y A N T 5 T h ) ]
p d e r E [ [Yo
+
PuerE[
+
PnerE[lyo - ~ o 1 ~ ]
(7)
Note that the third term in (7) is zero. Furthermore, we denote the , soft error power U:, as noise power due to ANT u ~ N T and
17
= E [ I YO - YANT url = E [ I Yo - Y a 1' I (Ya - YANT 2
(8)
5 Th)]
(9)
Substituting (8) and (9) in (7), we get
OQ
U;o-$
This work is supported by NSF grant CCR 99-79381, ITR 00-85929 and DARPA MSP grant
0-7803-7761-3/03/$17.0002003 IEEE
if [!/a,, if lYo,n
Ya,n YANT,n
Th = m m
where d, is desired output signal and Q, is the noise due to channe1 effects, ADC quantization noise, etc.. The output S N R is 0: SN%ut.org = 10 log,o( 7)
{
where Th is a precomputed threshold and PANT,, is the corrected output. In order to guarantee that yo,,, = yo,n in the absence of errors, the threshold becomes
2.1. ANT based DSB system
=d n
(3)
due to soft error U: and thereby guaranteeing that A is sumciently small. The decision rule for an ANT-based DSP system is given by
1.. INTRODUCTION
yo,n
+ Qn + 71,
= dn
where ~ [ n is ]the noise due to the soft error. In this case, the output SNR is
IV-113
=
Pder
'
+puer
U ~ N T
'
U:
(10)
................Main .....DSP .....
the forward predictor output is
-,,................ ........EC ..........................
N
k=O
:....... .........................................
where the prediction error ep,,, = y n - W'Y. 1, we can show the miniBy using the derivative of E[ mum MSE which corresponds to noise power of forward predictor ~ i ~is given r by , ~ ~ ~ ~
:
ilnl
= E[y:] - P'R-lP
( T ~ N T =, minE[e;,,] ~ ~ ~
(12)
where
P'r
.................................................................................................. . . ManLXP
. .
i
= E [ Y n y'] = E [ (y n y n - 1
'
=EIYYT]=E[(yn-l
Yn-N)
'
'
.
fir
"'
YnYn-N T (!/n-l"'Yn-N)]
In addition, by applying the Triangular inequality, we can obtain the upper bound on u;f as
= E [ IYa = E [ 1%
U :
................................................................................... Ec
5
- Yo12 I (Yo - Y p r e 5 T h , p r e ) ] - Y p r e + Y p r e - YoI2 1 (Ya - Y p r e 5 T h , p r e ) ] (13)
Th*pref c i N T , p r e
Inserting (12) and (13) into (lo), we get the upper bound on the minimum noise power for the prediction based ANT system as
aio-c 5 P d e r u i N T , p r e Figure 1: Algorithmic noise-tolerance schemes: (a) prediction based ANT scheme, and (b) RPR based ANT.
+Puer(Th,pre +u i N T , p r e )
(14)
Predictor based ANT works well in narrowband systems (typically Bz.In addition, we denote the quantization step size of an original DSP and that of RPR as A, = a,nd Ar = The quantization noise N, and Nh between t e onginal value and that of RPR z r and h, is dePned as N , = z - z r and Nh = h - h,, where z and h are the input and Plter coefPcients, respectively. With this assumption, one can show that the noise power of the i ~is given ~ by, ~ ~ ~ , RPR ANT scheme ~
+
$
2.3. RPR based ANT Figure l(b) shows the block diagram of an RPR based ANT scheme. The RPR is a replica of the original system with small precision operands [ 5 ] . Though RPR output yr,n is generally not equal to the original one yo,n due to the LSB quantization noise, it is a good estimate in case of an error event. When an error is detected using (3, yr+ is employed as the Pnal output.
&.
3. PERFORMANCE ANALYSIS OF ANT
where U: is the input signal power. By applying an analysis similar to the one in (13), we get the noise power U: as
In this section, we present an analysis of the predictor, RPR and the hybrid ANT technique which is composed of both techniques. Here, we assume that the error-control block is designed to be error free. Indeed, this assumption is reasonable since the complexity of the error control block is much lesser than that of MDSP and hence the likelihood of soft-error is signiPcantly smaller.
Employing (15). (16), and (IO) we can obtain the upper bound on the noise power for the RPR based ANT as
5
aio-c = 5
3.1. Predictor based ANT Let the output vector of an original system is Y = [yn-l, ....y n - N I T and the predictor coefPcient vector is W = [ w l , .... W N I T . Then,
Pder
'
Pder
'
Th2,Tpr
+ciNT,rpr
(16)
+ puer (17) u i N T , r p r + p u e r .(Th,rpr +b i N T , r p r )
ciNT,rpr
'
Note that the noise power term in (17) depends only on precision and quantization noise but not on bandwidth. While the RPR based ANT provides good performance for a wide range of bandwidths, it consumes more power than the predictor based ANT system.
IV-114
U:.(NP ...............................................
Halm DSP.Mp
Mu" DSP Ec .................................................................... . ,
I
I :
YJ"1
Y,'"
YJ"'
FIN
Figure 3: Simulation setup.
.
Er
............................
Figure 2: The hybrid ANT technique: (a) block diagram, and (b) the state machine.
3.3. Hybrid ANT The predictor based ANT, which produces good performance in low soft-error rate P e r , has a problem as Per increases. Once error is made, an erroneous output is fed back into the predictor, leading to an incorrect decision until the erroneous output is purged. On the other hand, the RPR based ANT has relatively high power consumption than the predictor based scheme. Figure 2(a) shows the block diagram of the proposed hybrid ANT scheme that overcomes this problem. The key idea in the hybrid ANT is to use a predictor when there are no errors (i.e., for error detection) and RPR when an error occurs (i.e., error correction). As illustrated in Fig. 2(b), the error control is transferred to ) a predictor state ( S p r eafter ) error detecan RPR state ( S r p r from tion. If RPR does not detect an error for N consecutive cycles, i.e., the error propagation is terminated, then the error control is passed back to the predictor. Assuming that the error event is independent, the probability of being in the RPR and predictor states are, respectively,
+ (1-
Prpr
=per
Ppre
= (1 -
+'.
Per)Pe,
= Prpr
4. SIMULATIONS AND DISCUSSION The setup used to measure the performance of the proposed scheme is shown in Fig. 3, where a frequency selective Plter is used to generate a bandlimited signal yo[.] from a wideband input with noise z[n] 4121.The S N R without ANT is given by
+
SNR,, = 10loglo
giNT,rpr
+ppre ciNT,pre
5
PderPpre
'
uiNT,pred
-k
puerppre
'
(Th,pre
PuerPrpr
.(Th,rpr
+
PdeTPTpr
'
SNRout = lOlog,,
+
(19)
d N T , ~ p r
+0:NT.pre) uiNT,rpr)
(21) gv,--Y,
(18)
Using (18), (19), and (14), we can obtain the upper bound on the noise power of the hybrid ANT as U:,-$
Since RPR is used only when error is detected, we can tum this block off in the predictor state ,S ,, and thereby save power while maintaining robustness when Per is high.
whereas, the S N R at the ANT output is given by
(1 - P e r ) N p - l P e r
where N p is the number of taps in the predictor. Clearly, Prpr PP,, = 1. Then, the noise power of the hybrid ANT system is giNT,hybrid
Figure 4: Analytic results of prediction and RPR ANT scheme for bandwidth variation.
(20)
(22) cvo-ai
where uio,is the signal power of original Plter output. In this simulation, we employ a 31-tap Plter with a 12x12 multipliers and a 26-bit accumulator. For a DSM noise model, we employ random Bipping of an output bit in the digital Plter with a speciPed soft-error probability P e r . Figure 4 shows the analysis results of predictor and RPR based ANT, where the main Plter is a low-pass Plter having a bandwidth from wb = 0 . 1 to ~ 0 . 5 ~ .For the RPR MAC, we used the half precision of the original Plter (i.e., 6 x 6 multiplier) and a three tap of predictor is employed in the prediction based ANT. While the performance of RPR is quite similar
IV-115
IO '
10"
.........................................
. . . . . . . \h . ................
Figure 5: Performance analysis and simulation results of prediction and RPR ANT scheme.
. . .
- P m 10' - - Pmd 10.2 -4 RPR IO '
......
Figure 7: Contour plot for Pep and power overhead variation.
ing additional power consumption, we show the contour plot for the power overhead as well as Pep, which is given by
Poverheod(%)=
PANT- PMDSP PMDSP
(23)
where PMDSPand PANTare the power of original DSP and ANT based DSP, respectively. As shown in Fig. 7,the power overhead of hybrid ANT is about 15 20 9% lesser than that of RPR in a similar MSE region. Thus, the hybrid ANT scheme is not only robust to soft-errors when Pep, is high, it is also energy-efPcient.
-
5. REFERENCES
[ l ] 'T.Kamik and S . Vangal, OSelective node engineering for chiplevel soft error rate improvement,OProc. of Symp. on VLSI circuits, vol. 4, pp. 132-135, 2002.
Figure 6: Performance analysis and simulation results of hybrid ANT scheme.
over the entire frequency range, the performance of the predictor deteriorates as the bandwidth increases. Figure 5 shows the analysis and simulation results of the predictor and RPR based ANT, where the bandwidth of main Plter is now set to W b = 0 . 2 ~ .While the conventional system has high mean square error regardless of Per,predictor ANT has MSE less than at Pep= resulting in 40 dB gain over conventional MDSP. As discussed in Section 1I.A. the performance of predictor degrades severely as Pep increases due to the error propagation. On the contrary, the performance of RPR is close to the analytic results even when Pepis high. Next, we considered the hybrid ANT scheme under the same simulation setup. As shown in Fig. 6, the performance of hybrid ANT is similar to the prediction scheme in low P,,region. However, unlike to prediction scheme, we observe that hybrid scheme maintains its performance even when Perbecomes high. Finally, in order to observe the comprehensive picture includ-
[2] N. Shanbhag, K. Soumyanath, and S. Martin, OReliab!e lowpower design in the presence of deep submicron noise,OProc. of Intl. Symp. on Law-Power Electronics and Design, pp 295302,2000.
[3] R. Hedge and N. Shanbhag, OSoft digital signal processing.0 IEEE Trans. VLSI, vol. 9, pp. 813-823, Dec. 2001. [4] L. Wang and N. Shanbhag, OLow-power signal processing via error-cancellation,OProc. of IEEE workshop on Signal Processing Systems, Lafayette Oct. 2000.
[SI B. Shim, and N. R. Shanbhag, OLow-power digital Pltering via reduced-precision redundancy,OProc. of Asilomar con$, Vol. I, pp. 148-152, NOV.2001.
Iv-116
