Instrumentation & Measurement Magazine 23-2 - 62

Demonstration of Ill-Posedness and the
Concept of Regularization

Regularization of Deconvolution in the
Frequency Domain

First, let us investigate the compensation possibility of the limited bandwidth effect of the measurement system. If a system
can be considered to be linear and time shift invariant, its inputoutput relationship can be described by convolution integral:

In the case of compensation of limited bandwidth (deconvolution), there are many different ways to suppress this noise
(output smoothing + naïve inverse filtering, Tikhonov regularization, model fitting in time domain, iterative deconvolution
based on Van Cittert's method, etc.). They all need to keep a
balance between the noise amplification and bias of the useful
signal to simultaneously fulfil accuracy and precision requirements. Here, we will show just one method, that is, Tikhonov's
regularization [3]. Tikhonov suggested introducing regularization operators beside the output error criterion (that leads
to the naïve inverse filter). As he was a mathematician, he
proposed infinite number of operators. In practice-because
of lack of a priori knowledge about the signal to be measured-only a couple of regularization operators are usually
introduced. The inverse filter with one free parameter takes
the following form:

∞

y ( t ) =  h (τ ) x ( t − τ ) dτ ,	(1)

	

−∞

where x(t) is the physical quantity to be measured, as a function of time, h(t) is the impulse response of the measurement
system, describing the effect of limited bandwidth, and y(t) is
the response of the system, that is, the distorted output of the
measurement device. Taking into account that today's measurement devices provide the information in sampled and
digitized form, the convolution integral becomes a sum:
N −1

y ( i ) = h ( j ) x ( i − j ) .	(2)

	

j =0

Transforming the signals into the frequency domain by
means of Digital Fourier Transform (DFT), or its fast representation (FFT), convolution becomes multiplication:
Y ( f ) = H ( f ) X ( f ) ,	(3)

	

where capital letters mean the DFT of corresponding time
domain signals. Inverse filtering means post processing the
measured signal with an additional filter to reconstruct it. Naïve inverse filtering (in this case, deconvolution, i.e., inversion
of convolution) is the one that compensates for the errors in a
noiseless case:
	

K( f ) =

Y( f )
H( f )
=Y( f )
= X ( f ) ,	(4)
,     Xˆ ( f ) =
2
H( f )
H( f )
H( f )
1

∗

where K(f) is the inverse filter, and Xˆ ( f ) is the spectrum of the
reconstructed signal. As measurement noise is always present,
it needs to be modeled as well. Let us assume an output noise,
or transform all noise sources to the output of the system:
	

	

K( f ) =

H( f )

∗

H( f ) +λ
2

.	(7)

The λ term comes from taking into account that the estimate
has finite energy. Comparing (4) with (7), it can be observed
that regularization puts a lower limit to the denominator of
the inverse filter; λ term do not let the denominator to become
zero, thus omitting the division by zero. Let us see this effect
on a measurement accomplished by two different accelerometers excited by a shaker. The Device Under Test (DUT) is a
low bandwidth sensor. The reference measurement is accomplished by a high bandwidth sensor (we will assume that the
reference measurement is perfect), depicted in Fig. 1.
The distortion can be compensated by inverse filtering (deconvolution), applying different levels of noise reduction (and
as a consequence, bias of the useful signal). Fig. 2 shows the
result of reconstruction without any noise reduction (naïve
deconvolution), the under-, over- and optimally-regularized

z ( i ) = y ( i ) + n ( i ) ,	(5)

where z(i) stands for the noisy observation, and n(i) denotes
the noise sequence. Applying the same naïve inverse filter for
the noisy observation we get:
	

Z( f ) X( f )H ( f ) + N ( f )
N( f )
Xˆ ( f ) =
=
= X( f )+
.	(6)
H( f )
H( f )
H( f )

One can observe that the estimate provides the spectrum
of the original signal (physical quantity to be measured) and
an additional noise term that is an amplified version of the
stochastic disturbances. At those frequencies at which the
measurement system has a large suppression (stop band), the
noise spectrum is divided by nearly zero. This is a good visualization of how the problem gets ill-posed.
62	

Fig. 1. Acceleration signal measured by a low- (DUT) and high-bandwidth
accelerometer (reference sensor).

IEEE Instrumentation & Measurement Magazine	

April 2020



Instrumentation & Measurement Magazine 23-2

Table of Contents for the Digital Edition of Instrumentation & Measurement Magazine 23-2

No label
Instrumentation & Measurement Magazine 23-2 - No label
Instrumentation & Measurement Magazine 23-2 - Cover2
Instrumentation & Measurement Magazine 23-2 - 1
Instrumentation & Measurement Magazine 23-2 - 2
Instrumentation & Measurement Magazine 23-2 - 3
Instrumentation & Measurement Magazine 23-2 - 4
Instrumentation & Measurement Magazine 23-2 - 5
Instrumentation & Measurement Magazine 23-2 - 6
Instrumentation & Measurement Magazine 23-2 - 7
Instrumentation & Measurement Magazine 23-2 - 8
Instrumentation & Measurement Magazine 23-2 - 9
Instrumentation & Measurement Magazine 23-2 - 10
Instrumentation & Measurement Magazine 23-2 - 11
Instrumentation & Measurement Magazine 23-2 - 12
Instrumentation & Measurement Magazine 23-2 - 13
Instrumentation & Measurement Magazine 23-2 - 14
Instrumentation & Measurement Magazine 23-2 - 15
Instrumentation & Measurement Magazine 23-2 - 16
Instrumentation & Measurement Magazine 23-2 - 17
Instrumentation & Measurement Magazine 23-2 - 18
Instrumentation & Measurement Magazine 23-2 - 19
Instrumentation & Measurement Magazine 23-2 - 20
Instrumentation & Measurement Magazine 23-2 - 21
Instrumentation & Measurement Magazine 23-2 - 22
Instrumentation & Measurement Magazine 23-2 - 23
Instrumentation & Measurement Magazine 23-2 - 24
Instrumentation & Measurement Magazine 23-2 - 25
Instrumentation & Measurement Magazine 23-2 - 26
Instrumentation & Measurement Magazine 23-2 - 27
Instrumentation & Measurement Magazine 23-2 - 28
Instrumentation & Measurement Magazine 23-2 - 29
Instrumentation & Measurement Magazine 23-2 - 30
Instrumentation & Measurement Magazine 23-2 - 31
Instrumentation & Measurement Magazine 23-2 - 32
Instrumentation & Measurement Magazine 23-2 - 33
Instrumentation & Measurement Magazine 23-2 - 34
Instrumentation & Measurement Magazine 23-2 - 35
Instrumentation & Measurement Magazine 23-2 - 36
Instrumentation & Measurement Magazine 23-2 - 37
Instrumentation & Measurement Magazine 23-2 - 38
Instrumentation & Measurement Magazine 23-2 - 39
Instrumentation & Measurement Magazine 23-2 - 40
Instrumentation & Measurement Magazine 23-2 - 41
Instrumentation & Measurement Magazine 23-2 - 42
Instrumentation & Measurement Magazine 23-2 - 43
Instrumentation & Measurement Magazine 23-2 - 44
Instrumentation & Measurement Magazine 23-2 - 45
Instrumentation & Measurement Magazine 23-2 - 46
Instrumentation & Measurement Magazine 23-2 - 47
Instrumentation & Measurement Magazine 23-2 - 48
Instrumentation & Measurement Magazine 23-2 - 49
Instrumentation & Measurement Magazine 23-2 - 50
Instrumentation & Measurement Magazine 23-2 - 51
Instrumentation & Measurement Magazine 23-2 - 52
Instrumentation & Measurement Magazine 23-2 - 53
Instrumentation & Measurement Magazine 23-2 - 54
Instrumentation & Measurement Magazine 23-2 - 55
Instrumentation & Measurement Magazine 23-2 - 56
Instrumentation & Measurement Magazine 23-2 - 57
Instrumentation & Measurement Magazine 23-2 - 58
Instrumentation & Measurement Magazine 23-2 - 59
Instrumentation & Measurement Magazine 23-2 - 60
Instrumentation & Measurement Magazine 23-2 - 61
Instrumentation & Measurement Magazine 23-2 - 62
Instrumentation & Measurement Magazine 23-2 - 63
Instrumentation & Measurement Magazine 23-2 - 64
Instrumentation & Measurement Magazine 23-2 - 65
Instrumentation & Measurement Magazine 23-2 - 66
Instrumentation & Measurement Magazine 23-2 - 67
Instrumentation & Measurement Magazine 23-2 - 68
Instrumentation & Measurement Magazine 23-2 - 69
Instrumentation & Measurement Magazine 23-2 - 70
Instrumentation & Measurement Magazine 23-2 - 71
Instrumentation & Measurement Magazine 23-2 - 72
Instrumentation & Measurement Magazine 23-2 - 73
Instrumentation & Measurement Magazine 23-2 - 74
Instrumentation & Measurement Magazine 23-2 - 75
Instrumentation & Measurement Magazine 23-2 - 76
Instrumentation & Measurement Magazine 23-2 - 77
Instrumentation & Measurement Magazine 23-2 - 78
Instrumentation & Measurement Magazine 23-2 - 79
Instrumentation & Measurement Magazine 23-2 - 80
Instrumentation & Measurement Magazine 23-2 - 81
Instrumentation & Measurement Magazine 23-2 - 82
Instrumentation & Measurement Magazine 23-2 - 83
Instrumentation & Measurement Magazine 23-2 - 84
Instrumentation & Measurement Magazine 23-2 - 85
Instrumentation & Measurement Magazine 23-2 - 86
Instrumentation & Measurement Magazine 23-2 - 87
Instrumentation & Measurement Magazine 23-2 - 88
Instrumentation & Measurement Magazine 23-2 - 89
Instrumentation & Measurement Magazine 23-2 - 90
Instrumentation & Measurement Magazine 23-2 - 91
Instrumentation & Measurement Magazine 23-2 - 92
Instrumentation & Measurement Magazine 23-2 - 93
Instrumentation & Measurement Magazine 23-2 - 94
Instrumentation & Measurement Magazine 23-2 - 95
Instrumentation & Measurement Magazine 23-2 - 96
Instrumentation & Measurement Magazine 23-2 - 97
Instrumentation & Measurement Magazine 23-2 - 98
Instrumentation & Measurement Magazine 23-2 - 99
Instrumentation & Measurement Magazine 23-2 - 100
Instrumentation & Measurement Magazine 23-2 - 101
Instrumentation & Measurement Magazine 23-2 - 102
Instrumentation & Measurement Magazine 23-2 - 103
Instrumentation & Measurement Magazine 23-2 - 104
Instrumentation & Measurement Magazine 23-2 - 105
Instrumentation & Measurement Magazine 23-2 - 106
Instrumentation & Measurement Magazine 23-2 - 107
Instrumentation & Measurement Magazine 23-2 - 108
Instrumentation & Measurement Magazine 23-2 - 109
Instrumentation & Measurement Magazine 23-2 - 110
Instrumentation & Measurement Magazine 23-2 - 111
Instrumentation & Measurement Magazine 23-2 - 112
Instrumentation & Measurement Magazine 23-2 - 113
Instrumentation & Measurement Magazine 23-2 - 114
Instrumentation & Measurement Magazine 23-2 - 115
Instrumentation & Measurement Magazine 23-2 - 116
Instrumentation & Measurement Magazine 23-2 - 117
Instrumentation & Measurement Magazine 23-2 - 118
Instrumentation & Measurement Magazine 23-2 - 119
Instrumentation & Measurement Magazine 23-2 - 120
Instrumentation & Measurement Magazine 23-2 - Cover3
Instrumentation & Measurement Magazine 23-2 - Cover4
https://www.nxtbook.com/allen/iamm/24-6
https://www.nxtbook.com/allen/iamm/24-5
https://www.nxtbook.com/allen/iamm/24-4
https://www.nxtbook.com/allen/iamm/24-3
https://www.nxtbook.com/allen/iamm/24-2
https://www.nxtbook.com/allen/iamm/24-1
https://www.nxtbook.com/allen/iamm/23-9
https://www.nxtbook.com/allen/iamm/23-8
https://www.nxtbook.com/allen/iamm/23-6
https://www.nxtbook.com/allen/iamm/23-5
https://www.nxtbook.com/allen/iamm/23-2
https://www.nxtbook.com/allen/iamm/23-3
https://www.nxtbook.com/allen/iamm/23-4
https://www.nxtbookmedia.com