IEEE Signal Processing - May 2018 - 33

value may not be monotonic in fs . That is, some sub-Nyquist
sampling rates may introduce more aliasing than sampling
rates that are lower. This phenomenon does not occur in sampling signals with a unimodal PSD.
The dependency of the passband of H(f) on the sampling
frequency fs comes from the aliasing-free property. In particular, this property restricts the Lebesgue measure of the
passband of any aliasing-free filter to be smaller than fs [52,
Prop. 2]. It follows from this that a lower bound on the function D SI (fs, R) is obtained by taking the part of the spectrum
of highest energy and overall Lebesgue measure not exceeding
fs . That is, we denote by F * (fs) the part of the spectrum that
maximizes # S X (f) df over all sets F of Lebesgue measure not
F
exceeding fs . The following expression bounds the function
D SI (fs, R) from below:
D (fs, R) = mmse (fs) + # *

F (fs)

R (i) = 1
2

min " S X (f), i ,

#F (f ) log 2+ 6S X (f) /i@df,
*

(13a)
(13b)

s

as a result of the aliasing-free requirement in a single SI filter,
leading to higher-energy frequency components in the resulting
signal representation before encoding and therefore lower distortion after encoding.
This intuition motivates replacing the SI sampler in Figure 11
with an array of such samplers, as shown in Figure 18. Within each branch, the presampling filter may pass only a narrow part of the signal's spectrum and apply passband sampling
[52]. This multibranch uniform sampler covers a wide class of
sampling systems used in practice, including single-branch SI
sampling, nonuniform periodic sampling, and multicoset sampling [7], [53].
The analysis of the system is greatly simplified if all of
the sampling branches have the same sampling rate. Thus, we
assume that the sampling rate at each branch equals fs /L , so
that the overall effective sampling rate is fs . Similar to the case
of a single SI sampler, the optimal selection of the presampling
filters across all branches leads to a collection of filters with the
aliasing-free property at each branch, such that the net energy
passed by these filters is maximal [51]. Since the measure of the
passband of each aliasing-free filter for sampling at rate fs /L is

where
mmse (fs) =

#-33 S X (f) df - #F (f ) S X (f) df.
)

(14)

s

X (f

)

A graphical water-filling interpretation of the prior expression is given in Figure 17. In the next section, we describe how
to attain this lower bound by extending SI samplers to an array
of such samplers.

θ

S

Multibranch sampling
In contrast to the case of a unimodal PSD, it is, in general,
impossible to attain the function D (fs, R) of (13) using a single
SI sampler. Indeed, once we fix a band, no other bands located
at integer multiples of the sampling rate are included in the
support of the optimal presampling filter because of the
aliasing-free property. This limitation implies that the support of
the optimal presampling filter does not necessarily consist of a
set of measured fs with the largest signal energy, as in the definition of D (fs, R) . By using more sampling branches, the global
aliasing-free property is relaxed to a local aliasing-free property
at each sampling branch. Therefore, while each branch has constraints on the position of the bands in the support of its filter to
avoid aliasing, the increment in sampling branches allows for
more freedom in selecting the overall part of the spectrum
passed by all filters. As a result, the union of the supports of an
optimal set of L filters that are aliasing-free with respect to fs /L
approximates the set of maximal energy of measure fs better
than is possible with a single filter that is aliasing-free with
respect to fs . This situation is shown in Figure 16. In particular,
components that needed to be eliminated in the single-branch
case because of aliasing with higher-energy components can
now be retained, as these two components can be preserved on
separate branches without interference with each other after
sampling. In other words, multibranch sampling reduces part of
the constraint on retaining desired signal components that arises

f
−3

−2

−1

0

1

2

3

Preserved Spectrum
Lossy Compression Distortion
Sampling Distortion

FIGURE 17. A water-filling interpretation of the fundamental minimal distortion D (fs, R) in ADX. The overall distortion is the sum of the sampling
distortion and the lossy compression distortion. The set F )(fs) defining
D (fs, R) is the support of the preserved spectrum.

H1(f )
H2(f )

X (t )

HL(f )

fs/L

Y1(n)

fs/L

Y2(n)

fs/L

YL(n)

Yn

FIGURE 18. A multibranch filter-bank uniform sampler.

IEEE Signal Processing Magazine

|

May 2018

|

33



Table of Contents for the Digital Edition of IEEE Signal Processing - May 2018

Contents
IEEE Signal Processing - May 2018 - Cover1
IEEE Signal Processing - May 2018 - Cover2
IEEE Signal Processing - May 2018 - Contents
IEEE Signal Processing - May 2018 - 2
IEEE Signal Processing - May 2018 - 3
IEEE Signal Processing - May 2018 - 4
IEEE Signal Processing - May 2018 - 5
IEEE Signal Processing - May 2018 - 6
IEEE Signal Processing - May 2018 - 7
IEEE Signal Processing - May 2018 - 8
IEEE Signal Processing - May 2018 - 9
IEEE Signal Processing - May 2018 - 10
IEEE Signal Processing - May 2018 - 11
IEEE Signal Processing - May 2018 - 12
IEEE Signal Processing - May 2018 - 13
IEEE Signal Processing - May 2018 - 14
IEEE Signal Processing - May 2018 - 15
IEEE Signal Processing - May 2018 - 16
IEEE Signal Processing - May 2018 - 17
IEEE Signal Processing - May 2018 - 18
IEEE Signal Processing - May 2018 - 19
IEEE Signal Processing - May 2018 - 20
IEEE Signal Processing - May 2018 - 21
IEEE Signal Processing - May 2018 - 22
IEEE Signal Processing - May 2018 - 23
IEEE Signal Processing - May 2018 - 24
IEEE Signal Processing - May 2018 - 25
IEEE Signal Processing - May 2018 - 26
IEEE Signal Processing - May 2018 - 27
IEEE Signal Processing - May 2018 - 28
IEEE Signal Processing - May 2018 - 29
IEEE Signal Processing - May 2018 - 30
IEEE Signal Processing - May 2018 - 31
IEEE Signal Processing - May 2018 - 32
IEEE Signal Processing - May 2018 - 33
IEEE Signal Processing - May 2018 - 34
IEEE Signal Processing - May 2018 - 35
IEEE Signal Processing - May 2018 - 36
IEEE Signal Processing - May 2018 - 37
IEEE Signal Processing - May 2018 - 38
IEEE Signal Processing - May 2018 - 39
IEEE Signal Processing - May 2018 - 40
IEEE Signal Processing - May 2018 - 41
IEEE Signal Processing - May 2018 - 42
IEEE Signal Processing - May 2018 - 43
IEEE Signal Processing - May 2018 - 44
IEEE Signal Processing - May 2018 - 45
IEEE Signal Processing - May 2018 - 46
IEEE Signal Processing - May 2018 - 47
IEEE Signal Processing - May 2018 - 48
IEEE Signal Processing - May 2018 - 49
IEEE Signal Processing - May 2018 - 50
IEEE Signal Processing - May 2018 - 51
IEEE Signal Processing - May 2018 - 52
IEEE Signal Processing - May 2018 - 53
IEEE Signal Processing - May 2018 - 54
IEEE Signal Processing - May 2018 - 55
IEEE Signal Processing - May 2018 - 56
IEEE Signal Processing - May 2018 - 57
IEEE Signal Processing - May 2018 - 58
IEEE Signal Processing - May 2018 - 59
IEEE Signal Processing - May 2018 - 60
IEEE Signal Processing - May 2018 - 61
IEEE Signal Processing - May 2018 - 62
IEEE Signal Processing - May 2018 - 63
IEEE Signal Processing - May 2018 - 64
IEEE Signal Processing - May 2018 - 65
IEEE Signal Processing - May 2018 - 66
IEEE Signal Processing - May 2018 - 67
IEEE Signal Processing - May 2018 - 68
IEEE Signal Processing - May 2018 - 69
IEEE Signal Processing - May 2018 - 70
IEEE Signal Processing - May 2018 - 71
IEEE Signal Processing - May 2018 - 72
IEEE Signal Processing - May 2018 - 73
IEEE Signal Processing - May 2018 - 74
IEEE Signal Processing - May 2018 - 75
IEEE Signal Processing - May 2018 - 76
IEEE Signal Processing - May 2018 - 77
IEEE Signal Processing - May 2018 - 78
IEEE Signal Processing - May 2018 - 79
IEEE Signal Processing - May 2018 - 80
IEEE Signal Processing - May 2018 - 81
IEEE Signal Processing - May 2018 - 82
IEEE Signal Processing - May 2018 - 83
IEEE Signal Processing - May 2018 - 84
IEEE Signal Processing - May 2018 - 85
IEEE Signal Processing - May 2018 - 86
IEEE Signal Processing - May 2018 - 87
IEEE Signal Processing - May 2018 - 88
IEEE Signal Processing - May 2018 - 89
IEEE Signal Processing - May 2018 - 90
IEEE Signal Processing - May 2018 - 91
IEEE Signal Processing - May 2018 - 92
IEEE Signal Processing - May 2018 - 93
IEEE Signal Processing - May 2018 - 94
IEEE Signal Processing - May 2018 - 95
IEEE Signal Processing - May 2018 - 96
IEEE Signal Processing - May 2018 - 97
IEEE Signal Processing - May 2018 - 98
IEEE Signal Processing - May 2018 - 99
IEEE Signal Processing - May 2018 - 100
IEEE Signal Processing - May 2018 - 101
IEEE Signal Processing - May 2018 - 102
IEEE Signal Processing - May 2018 - 103
IEEE Signal Processing - May 2018 - 104
IEEE Signal Processing - May 2018 - 105
IEEE Signal Processing - May 2018 - 106
IEEE Signal Processing - May 2018 - 107
IEEE Signal Processing - May 2018 - 108
IEEE Signal Processing - May 2018 - 109
IEEE Signal Processing - May 2018 - 110
IEEE Signal Processing - May 2018 - 111
IEEE Signal Processing - May 2018 - 112
IEEE Signal Processing - May 2018 - 113
IEEE Signal Processing - May 2018 - 114
IEEE Signal Processing - May 2018 - 115
IEEE Signal Processing - May 2018 - 116
IEEE Signal Processing - May 2018 - 117
IEEE Signal Processing - May 2018 - 118
IEEE Signal Processing - May 2018 - 119
IEEE Signal Processing - May 2018 - 120
IEEE Signal Processing - May 2018 - 121
IEEE Signal Processing - May 2018 - 122
IEEE Signal Processing - May 2018 - 123
IEEE Signal Processing - May 2018 - 124
IEEE Signal Processing - May 2018 - 125
IEEE Signal Processing - May 2018 - 126
IEEE Signal Processing - May 2018 - 127
IEEE Signal Processing - May 2018 - 128
IEEE Signal Processing - May 2018 - 129
IEEE Signal Processing - May 2018 - 130
IEEE Signal Processing - May 2018 - 131
IEEE Signal Processing - May 2018 - 132
IEEE Signal Processing - May 2018 - Cover3
IEEE Signal Processing - May 2018 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201809
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201807
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201805
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201803
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201801
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1117
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0917
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0717
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0517
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0317
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0117
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1116
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0916
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0716
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0516
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0316
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0116
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1115
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0915
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0715
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0515
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0315
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0115
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1114
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0914
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0714
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0514
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0314
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0114
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1113
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0913
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0713
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0513
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0313
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0113
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1112
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0912
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0712
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0512
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0312
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0112
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1111
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0911
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0711
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0511
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0311
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0111
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1110
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0910
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0710
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0510
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0310
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0110
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1109
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0909
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0709
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0509
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0309
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0109
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1108
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0908
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0708
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0508
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0308
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0108
https://www.nxtbookmedia.com