Effects of error sources on the parallelism of an optical matrix-vector processor

Caroline J. Perlee; David P. Casasent

doi:10.1364/AO.29.002544

Applied Optics
Vol. 29,
Issue 17,
pp. 2544-2555
(1990)
•https://doi.org/10.1364/AO.29.002544

Effects of error sources on the parallelism of an optical matrix-vector processor

Caroline J. Perlee and David P. Casasent

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Caroline J. Perlee and David P. Casasent, "Effects of error sources on the parallelism of an optical matrix-vector processor," Appl. Opt. 29, 2544-2555 (1990)

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Abstract

The error sources in a high accuracy optical matrix-vector processor are analyzed by numerical simulation in terms of their effects on the parallelism and speed of the processor. These effects are detailed for radices −2, −4 and −8. Radix −4 is shown to provide maximum parallel processing capabilities under the effects of the system’s error sources. Processing speed is shown to be a function of matrix partitioning and the number of parallel processing channels. Consequently, radix −4 operation provides a higher processing speed than radix −2 and −8 for most matrix-vector multiplications when error source effects are considered.

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Radix	M	N	T₂ = NT₁ (ns)	MOPS	Number of A–D bits
−2	62	16	160	387	6
−4	125	8	80	1562	11
−8	83	6	60	1383	12

Radix	M	N	T₂ = NT₁ (ns)	MOPS	Number of A–D bits
−2	31	32	320	96	5
−4	62	16	160	387	10
−8	83	11	110	754	12

Plane	Error source	Error model
P₁	Spatial gain variations between laser diodes	( $1 + δ_{i}^{1 f}$ ): fixed multiplicative, same $δ_{i}^{1 t}$ for all time.
	Quantum noise	( $1 + δ_{i}^{1 t} (t)$ ): time-varying multiplicative, different $δ_{i}^{1 t}$ every T₁.
P₃	Spatial response variations in the detector elements	( $1 + δ_{j}^{3 f}$ ): fixed multiplicative, same $δ_{j}^{3 f}$ for all time.
	Shot noise	[ $1 + δ_{j}^{3 t} (t)$ ]: time-varying multiplicative, different $δ_{j}^{3 t}$ every T₁.
	Detector thermal noise	d_j(t): time-varying additive, different d_j(t) every T₁.
	A–D circuit noise	n_j(t): time-varying additive, different n_j(t) every T₁.

Error	Measured MPE_L or MPFSE_L	Achievable MPE_B or MPFSE_B
P₁ spatial gain	1.0%	0.01%
P₃ spatial gain	1.0%	0.01%
P₁ quantum noise	1.44%	0.04%
P₃ shot noise	0.02%	0.02%
Detector thermal noise	3.0% (MPFSE)	0.002% (MPFSE)
A–D circuit noise	0.42% (MPFSE)	0.01% (MPFSE)

	Nonlinear algorithm steps
Step 1	Set initial conditions. uⁿ^,^r = u^0,0
Step 2	Solve LAE below for uⁿ⁺¹ = uⁿ^+1,^r = uⁿ^+1,0:
	D(uⁿ)uⁿ⁺¹ = eⁿ Eq. (16)
	where, D(uⁿ⁺¹) in Eq. (15) is approximated by D(uⁿ) in Eq. (16).
Step 3	Refine solution (by Newton-Raphson).
	Solve LAE below for uⁿ^+1,^r⁺¹:
	Jⁿ^+1,^r(uⁿ^+1,^r⁺¹ − uⁿ^+1,^r) = −Rⁿ^+1,^r Eq. (17)
	where,
	Residual vector = Rⁿ^+1,^r = [D(uⁿ^+1,^r)uⁿ^+1,^r − eⁿ^,0]
	Jacobian = $J^{n + 1, r} = \frac{\partial R^{n + 1, r}}{\partial u^{n + 1, r}}$
	eⁿ^,0 is a function of the converged response from the previous time step uⁿ^,0.
Step 4	Update J and R and form new Eq. (17) r → r + 1.
	Solve new Eq. (17) for uⁿ^+1,^r⁺², etc.
	Repeat until convergence. \|uⁿ^+1,^r⁺¹ − uⁿ^+1,^r\| < ∊_NR
	Result is uⁿ⁺¹ at present time.
Step 5	Go to step 2.
	Use step 4 result uⁿ⁺¹ to solve for uⁿ⁺², etc.
	Repeat step 2 and iterative steps 3 and 4 until reach time t = 0.1.

Radix r	Errors	M
	MPE_L	7
−2	(0.1)MPE_L	72
	MPE_B	200*
	MPE_L	0
−4	(0.1)MPE_L	8
	MPE_B	60
	MPE_L	0
−8	(0.1)MPE_L	1
	MPE_B	11

Δ_{c_j}a	Radix	N	M	T₂ (ns)	MOPS	Processing time T₂N_vip
0.5	−2	32	31	320	96	320 N_vip
	−4	16	60	160	375	160 N_vip
	−8	11	11	110	100	110 N_vip
0.7	−2	32	31	320	96	320 N_vip
	−4	16	62	160	387	160 N_vip
	−8	11	15	110	136	110 N_vip

Radix	N	M	Number of matrix partitions (M_p)	T₂ (ns)	Processing time T₂M_pN_vip (ns)
−2	32	31	2	320	640 N_vip
−4	16	62	1	160	160 N_vip
−8	11	15	5	110	550 N_vip

Radix r	k in LU	k in Gauss-Seidel
−2	3	3
−4	1/3	1/3
−8	1/14	1/14

Radix r	k in LU	k in Gauss-Seidel
−2	3	3
−4	1/3	1/3
−8	1/14	1/14

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Applied Optics