Instrumentation & Measurement Magazine 24-4 - 13

Fig. 2. The spectrum of signal Vl (f) bandlimited to (-B, B) is represented in terms of M = 6 contiguous sub-spectra, Vlh
and χh(f) is the characteristic function of the interval (-B+(h-1)fs
rate, fs 
2B
M, its spectrum is replicated at a pace fs, such that, in any interval (-B+(h-1)fs
shifted and overlapped to each other, as shown on the right side for the portion related to the interval (-B, -B+fs

  t nTs ,

n
and interpolation as convolution (here represented by means
of asterisk) with the interpolating function:
u t    
   t nT y t
M

 
ln
1


v t
l
By Fourier transforming (2):

1


l
 
s

sl
U f H f U f f f kf Y f (3) 
lk
   
M

sl
where Hl (f)U(f) is the output of the l-th channel, namely Vl
corresponding in the time domain to vl (t), and  is the
f f kf

ss
k


comb-line spectrum characterizing the ideal pulse train. Since
convolution with the considered comb-line spectrum produces
replication with pace fs
1
, (3) is substituted with:
Uf f H f kf Uf kf Y f


   
 

s
11
MM
l
l kM 
where the range of the index k has been limited to the interval
(-M +1, M -1), since the functions U(f - kfs
) with |k| ≥ M
represent spectral replicas that fall outside the interval (-B,
B). These replicas can be neglected because the left side of (4)
is zero in this interval, and the required band limited character
of the interpolating functions yl
side (-B, B).
Moreover, since sampling produces replication with pace fs
the attention can be restricted to the sub-intervals (-B + (h - 1)fs
-B + hfs
) with k < -M + h and k > h - 1 are equal to zero in (-B + (h - 1)fs
-B + hfs
U f f fs  
 
) h = 1, ..., M, such that it holds:
 
h
where, with respect to (4):
June 2021
  (5)
k Mh l 1
1 
hM


l sl
 
 H f kf Y f U f kfs
 
s
sl

(4)

(f),
(2)
(f), h = 1, ..., M, where Vlh
, -B+hfs), equal to 1 inside the interval and zero outside. As Vl
(f) = Hl
(f)U(f)χh
(f),
(f) undergoes sampling at sub-Nyquist
, -B+hfs), the spectrum of its sampled version includes all the M sub-spectra
).
◗ the characteristic function h f of the interval (-B +
(h - 1)fs, -B + hfs
), equal to 1 inside the interval and zero
outside, has been introduced in order to emphasize that
the solution is valid in a limited interval;
◗ the range of the index k has been limited from -M + h up
to h - 1;
◗ the inner sum has been swapped with the outer one.
For the sake of clarity, Fig. 2 highlights the effects of subNyquist
sampling by illustrating how the amplitude spectrum
available on the l-th channel after filtering U(f) with Hl
represented by function Vl (f), is modified by the sampling op(f),
eration;
in particular, to better highlight sampling effects, the
spectrum is conveniently represented in terms of M contiguous
sub-spectra, referred to as Vlh
(f), h = 1, ..., M.
From (5) it is deduced that the reconstruction filters have
to satisfy:
M
 H f kf Y f 

l
l1
lh
s lh
 l
where   
h
      
fs
1
Y f Yf f
h f k M h h,, 1 (6)

 and δk
k
represents the Kronecher
(t) assures that also
all the Yl(f) on the right side of the equation are zero out1h

,
,
)
h = 1, ..., M that divide (-B, B) into M uniform and adjacent
intervals with identical spectral contents: it is easily found
that, due to their band limited nature, even the functions U(f -
kfs
,
where δMh 1 is an M-size column vector whose (M - h +1)th
component is equal to 1 and the remaining ones are zero.
For instance, the reconstruction filters in the interval (-B,
-B + fs
given by:
IEEE Instrumentation & Measurement Magazine
13
), corresponding to h = 1 and selected by 1 f , are






H f M hf H f M hf Yf
Yf
1  sM s
 
H fh f H fh f
f
 
 
 
fs
 δ 
h Mh 1
1 11
1
sM Mh
  

 s



(7)
delta, equal to 1 if k = 0 and zero otherwise. Equation (6) assures
that all aliases due to the sub-Nyquist sampling are
deleted, whereas the baseband term is saved. Letting k range
from -M + h, up to h - 1, (6) spans a set of systems, parameterized
by h, each one made up of M equations:

Instrumentation & Measurement Magazine 24-4

Table of Contents for the Digital Edition of Instrumentation & Measurement Magazine 24-4

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