Instrumentation & Measurement Magazine 23-2 - 63

	

 h (0)
0
0

 h ( 1)
0
h (0)

...


H =  h ( M − 1) h ( M − 2 ) h ( M − 3 )

0
h ( M − 1) h ( M − 2 )


...

0
0
0


...
...
...
...
...




0



0
 ,	(8)

0



h ( M − 1) 

0

where H denotes column vector, H denotes matrix, T stands for
transpose, and M stands for the length of the impulse response.
Matrix H -containing the shifted impulse response parts-is
not quadratic; furthermore, the set of linear equations is perturbed by stochastic disturbances, thus, the solution is the
Moore-Penrose pseudoinverse that provides the best fit to the
measured values in least squares sense:

Fig. 2. Reconstruction of input signal with deconvolution (inverse filter). (a)
Result of naïve inverse filter (no noise reduction); (b) Under-regularized version;
(c) Optimal regularization; and (d) Over-regularized version.

version. The naïve inverse filter amplifies the noise in an extent that hides the useful signal (bad precision). This example
is one of the most moderate versions of ill-posedness, since the
system could be modeled with a first order lowpass filter. That
means the smallest roll off rate for the transfer function, thus,
the most moderate noise amplification during the inverse filtering. As the order of the system increases, the problem gets
more and more ill-posed. By introducing regularization, it
suppresses the noise, but after a certain amount, the distortion
of the signal gets significant (see over regularized version on
Fig. 2) and that spoils accuracy.
Yet another challenge is to find the optimal parameter set
(level of regularization) for the inverse filter, as in real life we
do not have too much information about the shape, sharpness,
and smoothness of the signal to adjust the balance of accuracy
and precision. In autonomous systems, parameter optimization needs to be accomplished automatically, without any user
interaction. There are many ad-hoc algorithms for this problem and a couple of systematic ones. We will not investigate its
possibilities within this paper. (We used the method described
in [4] and [5] to select model parameters automatically.)

Regularization of Deconvolution in the Time
Domain
Convolution sum (2) can be treated as a matrix equation:
	

z = H x + n	

z T =  z ( 0 ) , z ( 1) , ... z ( N − 1) 	

	

xT =  x ( 0 ) , x ( 1) , ... x ( P − 1) 	

	

nT = n ( 0 ) , n ( 1) , ...n ( N − 1) 	

April 2020	

)

−1

T

H z.	(9)

H is a lower Toeplitz matrix; however, we have not used
this special circumstance at the solution, thus, the above solution is general for any linear set of equations. Unfortunately,
this solution provides large noise amplification, similar to the
naïve inverse filter derived in the frequency domain. However, we can introduce regularization operators that improve
the condition number of non-quadratic matrix H :

	

	

(

T
T
xˆ = H  H + λ I + γ L L

1 0 0

 −2 1 0
 1 −2 1
L=
1 −2
0

...

 0 0 0

)

−1

T

H z	

0 ... 0

0 ... 0

0
	(10)
1 ... 0


0
1 

where I is a unity matrix, and L is the matrix of second order
backward difference operator.
Regularized matrix inversion can also be accomplished
using factorization (e.g., singular value decomposition) by
omitting singular vectors belonging to small singular values.

Regularization of the Inversion of Nonlinearity
The second most common type of distortions is nonlinearity.
Nonlinear distortion might have memory, if the distortion depends also on earlier states, not just on current sample. It can
be described by so called Volterra series:
	

	

(

T
xˆ = H  H

	

N b

b

n

n=1 a

a

j =1

(

)

y ( t ) = h0 +   ... hn (τ 1 ,τ 2 ,... ,τ n ) ∏x t − τ j dτ j ,	(11)

where hn (τ 1 ,τ 2 ,... ,τ n ) is the Nth order Volterra kernel [6]. It
is interesting to note that convolution (modeling the limited
bandwidth effect) is a special case of Volterra series consisting
of only first order kernel. For weakly nonlinear systems, one

IEEE Instrumentation & Measurement Magazine	63



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