Minimum-Bandwidth Nyquist-Rate Signaling

Minimun-Bandwidth
Nyquist-Rate Signaling
Dae Young KIM
dykim6@gmail.com
13APR17 @ETRI

Nyquist Sampling Theorem
t f0 0-W’
T 1/2T
f
0 W 2W-2W
W’
W’ ≦ W
3
t
0 1/2W-1/2W-1/W 1/W
-T 0-2T 2T 1/T-1/T 0-1/2T

Nyquist Criteria
for Bandlimited Pulse Shaping
T-T 0 2T-2T
1st
2nd1
1/2
[Bennett]
a1
a2
3rd
area: a1 = a2
4

Nyquist 1st Criterion:
ISI-free Pulse Transmission
T0 2T 3T 4T-T-2T-3T-4T
t
1
ɑ = 0
0.5
1
T
0 1/2T 1/T
f
-1/2T-1/T
Geq(f) = Σn G(f-n/T)
= T f ≦1/2T
= 0 elsewhere
5

Sync Pulse
T-T 0 2T-2T
1
1/2
7

Minimum-Bandwidth
Transmission
sync(t/T)
xn yn
t = nT
C(D)
line coding
C-1(D)
xn
*yn
*
decoding
x(t) = Σ xn 𝜹(t-nT)
y(t) = Σ yn 𝜹(t-nT)
s(t)
s(t) = Σ yn sync(t-nT) D = z-1
8

Pulse Sequence / Eye Width
T-T 0 2T-2T
1
1/2
9

Alternate Sum Variation
ASV ≜ max ❘Σ (-1)n yn ❘
ASV be Finite!
Eye Opening Condition
[dyKim83/85]
10

Binary: ASV = 1
11
T-T 0 2T-2T
1/2
-1/2

Ternary: ASV = 1
12
T-T 0 2T-2T
1
-1

ASV: Frequency View
13
y(t) = Σ yn 𝜹(t-nT)
Y(f) = Σ yn exp(-j2π f nT)
Y(1/2T) = Σ yn exp(-j2π nT / 2T)
= Σ yn exp(-jπ n)
= Σ yn exp(-jπ )n
= Σ yn (-1)n
ASV = max ❘Y(1/2T)❘
[dyKim83/85]

Eye Opening Condition
15
ASV be Finite
P(1/2T) = 0
or
[dyKim83/85]

Worst-case Sequences
16
Yn
-2
-4:I---
(a) (b)
Fig. 1 Eye numberingand eyeboundarynotations for
(a)even lcvel systems, (b) odd level systems.
Reference symbol
= +I
= +I
n
--
-1
-3
-54I
I I I I I I I I I I I I I n
-6 4 -2 0 2 6
-5 -3 -1 1 3 5
I I
I I
I I
-n
-6 -4 -2 0 2 6
-5 -3 -1 1 3 5
Fig.2Worst-case symbolsequencesfor
(a) U,, M=5, ASV= 4,
(b) L,, M=6, ASV= 6.
0
Yn
-2
-4:I---
Fig. 1 Eye numberingand eyeboundary
(a)even lcvel systems, (b) odd le
Reference symbol
= +I
= +I
n
--
-1
-3
-54I
-6 4 -2 0 2 6
-5 -3 -1 1 3 5
-n
Fig.2Worst-case symbolsequenc
(a) U,, M=5, ASV= 4,
(b) L,, M=6, ASV= 6.
2
1
-1
-2
5/2
3/2
1/2
-1/2
-3/2
-5/2
Yn
-2
-4:I---
-2 -1
-1
I
TI2
I
0
I
-TI2
I
(a) (b)
TI2
I
0
I
-TI2
Reference symbol
= +I
= +I
n
--
-1
-3
-54I
-6 4 -2 0 2 6
-5 -3 -1 1 3 5
I I
I I
I I
-n
-6 -4 -2 0 2 6
-5 -3 -1 1 3 5
Fig.2Worst-case symbolsequencesfor
(a) U,, M=5, ASV= 4,
(b) L,, M=6, ASV= 6.
-1
[dyKim95]

TABLEI
LOWER-BOUNDEYEWIDTHS OF~~NIMUM-BANDWIDTHSYSTEMS
-
1
3
3
3
4
4
4
5
5
5
6
6
6
7
7
7
8
8
8
9
9
9
0
0
0
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
--
-
-
-
I
--
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
1
3
5
7
1
3
5
7
1
3
5
7
1
3
5
7
-
-
-
bEW
182
222
286
148
167
200
129
138
154
114
121
129
105
110
116
098
102
,106
,092
.095
.098
.088
.090
.093
.133
.154
.182
.222
.114
.125
.143
.167
.102
.108
.118
.133
.093
.097
.102
.111
--
-
-
-
4
1
2
3
4
5
6
7
8
9
0
1
2
3
4
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
--
-
-
-
--
)
)
)
)
)
1
)
3
3
3
3
3
3
3
D
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
-
-
-
bEW
813
506
418
373
345
324
309
297
287
278
271
265
260
255
--
-
4
3
3
3
4
4
4
5
5
5
6
6
6
7
7
7
8
8
8
9
9
9
10
10
10
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
--
-
-
-
--
1
1
I
1
2
1
3
2
4
D
2
4
D
2
4
0
2
4
0
2
4
0
2
4
0
2
4
6
0
2
4
6
0
2
4
6
0
2
4
6
-
-
-
4
1
2
3
4
5
6
7
8
9
0
1
2
13
' 4
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
-
-
-
-
I
I
I
I
1
I
I
I
1
1
1
1
1
1
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
-
7
-
lbEW
,204
,222
,286
,170
.173
,200
,140
.148
.159
.126
.132
.141
.116
.120
.126
.110
.111
.115
.lo4
.IO5
.lo8
.098
.099
.lo2
-lbEW
667
357
283
,245
,223
,207
,195
.185
.173
.172
.166
.162
.158
.154
,343
,400
.242
.264
.201
.217
.I84
.187
.164
.I70
.153
.158
.147
.148
.138
.141
.132
.I35
.129
.130
.124
.125
,120
.122
.118
.I18
.286
.400
,209
.250
.173
.189
.153
.164
.139
.146
.129
.I35
.122
.125
.116
.119
.111
.113
.IO6
.lo9
.lo3
.lo5
.loo
.lo1
.097
.098
.145
.154
.I82
.222
.127
.129
.143
,167
.113
.114
.121
.133
.099
.lo2
,108
.114
M Number of levels.
lbEW Lower-bound eye width in T.
A :ASV. i :Absolute value of the eye number.
n
Eye
number
2
0
-2
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL 1995
.eve1
1
112
0
-112
-1
Eye
number
1
-1
I
TI2
I
0
I
-TI2
I
(a) (b)
TI2
I
0
I
-TI2
Reference symbol
= +I
= +I
n
--
-1
-3
I I
I I
ACTIONS ON COMMUNICATIONS, VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL 1995
.eve1
1
112
0
-112
-1
Eye
number
1
-1
I
TI2
I
0
I
-TI2
I
(a) (b)
TI2
Eye numberingand eyeboundarynotations for
[dyKim83/95]17

Partial-Response Signaling
- Linear MB Code -
• yn = Σ cnxn
• yn = xn + xn+1 C(D) = 1 + D ;DB
• yn = xn - xn+2 C(D) = 1 - D2 ;MDB 
= (1 - D)(1 + D) 
= (BP)(DB)
[Kabal75]
* DB: Duobinary MDB: Modiﬁed Duobinary BP: Bipolar
18

Parameters of Interest
• RAS (Running Alternate Sum) ≜ Σ (-1)n yn
• ASV (Alternate Sum Variation) ≜ max ❘Σ (-1)n yn ❘
• RDS (Running Digital Sum) ≜ Σ yn
• DSV (Digital Sum Variation) ≜ max ❘Σ yn ❘
19

lbEW Lower-bound eye width in T.
TABLEII
LOWER-BOUNDAND ACTUAL EYEWIDTHS OF SOMEPR SYSTEMS
-
)SV
1
-
CO
2
W
4
W
4
2
4
W
6
-
-
S-
CO
1
2
3
4
2
2
4
4
3
6
-
-
M
3
3
3
3
3
5
5
5
5
5
5
-
-
-
lbEW
-
&W
.O
.667
.357
__
.6S9
.363
.200
.249
.243
.l64
__
-
#
1
2
3
4
5
6
7
8
9
10
11
-
-
1 - D
1 + D
1 -Dz= (1-D)(l+D)
1 +ol
1 -D4= (1-D)(1+D)(1+I?)
1+2 0 +D 2 =(l+D)'
1 +D-D2-D3=(1-D)(1+D)z
(1+D)(1-D+@)
1-D -0' +O3 = (l-D)'( 1+D)
1- 20' + d = (l-D)z(l+D)z
2 +D -Dz = (2-D)(1+D)
2 -Dz-D' = (1-D)(1+D)(2+Dz)
.o
.667
.357
.288
.245
.286
.286
.173
.173
.209
.139
# :PR system number[4],[SI.
DSV:Digital Sum Variationdefined in [9], [6].which is equalto
M Numberofoutput levels withbinary input L=2 assumed.
lbEW :Lower-boundeye widthfmmTableI.
aEW :Actual eye width reportedin [SI.
(6) with (-1r omittedas mentionedat the end of Appendix.
[dyKim95]20

ODIFIED DUOBINARY 14:3
ith the coder shown in
ecodedto “ +1’s” or
”’ complyingwiththe
edbythe
y its absence. Except for
he coder of Fig. l(a) is
DB.
presented by thestate
to “0’s” and 1’s are to
ove,butwehave
s 010,1/ +1, and 1/ - 1
evity. Note that any two
ven number of“0’s” are
ighboring two separated
oppositepolarities.The
tionalityofthe “0”
duetothe self
tbound,resulting in an
, DB is not dc-free [4].
l(b) isby itself adequate
ough to show that DB is of
rsion ofthe state diagram
diagram is unfolded to
in encoding.Sincethe
RZS isofany
sen RZS to take on 0 and
PR&DER CONVERTER
iEVEL
ENCODER
MAIN
U U U
4 1
+
RIS
I
3 0
(C)
Fig. 1. Duobinary (DB). (a) Encoder. (b) State diagram. (c) Unfolded state
diagram.
ven number of“0’s” are
ghboring two separated
oppositepolarities.The
ionalityofthe “0”
uetothe self
tbound,resulting in an
DB is not dc-free [4].
l(b) isby itself adequate
ugh to show that DB is of
rsion ofthe state diagram
diagram is unfolded to
in encoding.Sincethe
RZS isofany
en RZS to take on 0 and
iagram may be easily
d by (2) in an encoding
We see thatIS Vdefined by
B code.
B and Fig. 2(b) its state
MDB can be interpreted
bipolars[4], [6]. Hence,
“ +” or “ - ” pulses as
y in each channel. With
tdifficulttonotifyone
from Fig. 2(b): whereas
y an even number of“0’s”
such two separated by
of opposite polarities.
tate diagram ofFig.. 2(c)
iorof RDS and RZS.
RIS
I
3 0
(C)
Fig. 1. Duobinary (DB). (a) Encoder. (b) State diagram. (c) Unfolded state
diagram.
PRECODER CONVERTER ENCCCER
MAINLEVEL
U U U
1+D
(DB)
1-D2
(MDB)
21

BUDA Cell
- Binary Unit DSV/ASV -
25
0 1/2-1/2
DSV = 1
0ASV = 1
RDS
RAS
0 0
1 1
-1/2
1/2
[dyKim89]

B Code
- BUDA Cell Stack -
26
ASV = 2
00
1 1
00
1 1
00
1 1
00
1 1
S1
S2 S3
S4
DSV = 2
[dyKim89]

B Code
- State Diagram -
27
0/11
0/00
0/11
0/00
1/01 1/10 1/01 1/10
S4
S3
S1
S2
[dyKim89]

WACX
28
00
1 1
00
1 1
00
1 1
00
1 1
S0
11/0110
S0
00/1001
10/1100
01/0011

K3B4
29
ASV = 4
DSV = 4
00
1 1
0
1 1
0
00
1 1
00
1 1
00
1 1
0
1 1
0
00
1 1
00
1 1
00
1 1
0
1 1
0
00
1 1
00
1 1
00
1 1
0
1 1
0
00
1 1
00
1 1
00
1 1
0
1 1
0
00
1 1
00
1 1
[dyKim89]

Power Spectrum: B code
30
2p(1-p)(2-4p-4p2-(1-2p)cosπfN-(1-2p)cos3πfN)sin2πfN
P(fN,p) = ————————————————————————
1-4p+12p2-16p3+8p4-(1-2p)2cos4πfN
[Alexandru03]

MB34_D4A4
- Coding -
32
[jhKim92]
93.5%

MB58_D3A4
- Power Spectrum -
33
[jhKim92]

MB58_D3A4
- Eye Pattern -
34
[jhKim92]
ɑ = 0

Set of Binary MB Codes
36
[jhKim92]

MB810+
37
tively, as depicted in Fig. 1. States are selected on the
cross-sections of the BUDA cells. For a coding state,
there exist more than 256 outward paths to reach the
other states, including return to a self-state. However,
each state in Fig. 1 has only 200 output paths in total;
for example, there are 100 output paths (that is, code
words) each from S1 to S3 and S1 to S2, respectively.
To reduce the number of encoding states and re-
quired code words, we statistically control one of the
Fig. 1 MB810+ BUDA cell stack.
outputs with a randomized input data stream, of which
the probability of being 0 or 1 is 0.5. The data rates
are assumed 12.5 Gb/s for comparison with the MB810
code proposed for a 10 Gigabit Ethernet line code. The
simulated results are shown in Fig. 3.
There are spectral nulls at 0 Hz and 6.25 GHz,
which means the line code is dc-free and of minimum-
bandwidth. The DSV and ASV of the MB810 are both
7, whereas the MB810+ values are 5 and 3, respec-
tively. Therefore, the widths of the spectral nulls of the
MB810+ are wider than those of the MB810, indicating
that the low pass ﬁlter for the MB810+ is easier to im-
plement than that of the MB810. We conﬁrmed that
(a) MB810+.
[cgLee03]

non-MB Bipolar Code
39 [dyKim83/85]

VDB5
42
[dyKim83/87]
IEEE TRANSACTIONSONCOMMUNICATIONS, VOL. COM-35, NO. 2, FEBRUARY 1987
DB’s
izing
ought
d S8
s, no
ects.
ways
The
orre-
of
vice
r DB,
able
of 1,
sider
-S3-
sing.
to 0,
2-S8
state
to 0
” is a
rated
data
RIS
1
0
0
-1
I - - - - - - - ,
---m+
RIS
bols on t = n T where n is even) to the odd channelor vice
a, if we assume that RIS takes on 1 and 0 in the upper DB,
it takes on 0 and - 1 in the lowerone; theinevitable
ease in ISVdue to V’s has been kept to a minimum of 1,
ISV resulting in 2. For a better understanding, consider
ollowing example. Let us travelalongthepath S4-S3-
S8-S7.Thenweget athree-symboloutputsequence
0+ ” with no symbol being emittedin the S2-S8crossing.
ming that RIS is 1at S4,the first “ +” decreases it to 0,
ext “0” does not change it, and neither does the S2-S8
sing, thelast “ +” decreases it to - I, andthe state
nates in S7.Hence, RIS changesstepwisefrom 1 to 0
hen to - 1. Furthermore, we note that the last “ +” is a
nce it has the same polarity as the first “ +” separated
itself by a single “0. ”
ow assume that the coder is in S4 and thenext r input data
all O’s, requiring a “0” SP.Inprinciple, anypattern
rated along any path starting from S4 and terminating at
of the states in the lower DB can be taken as,an SP; for
mple, “ +0+0- - ” (S4-S3-S2-SS-S7-S6-S5-S6),
0 - 0 +” (S4-Sl-S2-S3-S5-S6-S7-S8),etc., for the
of r = 6. This is also true of SP’s for all other states.
ously,choices should bemade and wehave takenthe
wing additional strategies in the selection of SP’s.
Every SPshould contain one andonly one V ; V‘s disturb
iginal encoding and so the statistical properties of the
nal code DB. Wewant this distubance to be minimum in
r to preserve theoriginalDB properties as faithfully as
ble.
Every SPshould end with this V.This isboth to keep the
ding delay to a minimum and to ease the decoding along
strategy,7).
)Every “0” SP is to startwith a “0,”every “ +” SP with
+,” and every ‘‘ - ” SP with a “ - .” This is to easethe
ding along with strategy.6). Every timea Vis detected at
Dora
Coded
Pulse
0
-1
I - - - - - - - ,
---m+
RIS
‘1
0
1 0 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 1 1 1
v
! l ! l , l , ~ l
l 7 ’ I 1V

Modiﬁed Duobinary
43
[dyKim83/87]

Power Spectra
46
VMDB5 TIB5
[Croisier70]
[dyKim83/87]

Pilot-Aided PAM/QAM
50
fT
1 -112 0 112 1
(C)
nceptual sketches of three different pilot insertion schemes with
power spectral density (PSD) notches provided by some power
on codes. T is the duration of an output symbol. (a) Single pilot
OFOl (or FJOl) coding for a carrier. (b)Dual pilots with the KFlO
r a clock and a carrier. (c)Triple pilots with the OF00 (or FJOO)
r a clock and an unambiguous carrier.
W W W 1
I Frame 1 I Frame 2 I Frame 3 I Flag II I I I I
1 Block = (3W + 1) symbols
ock structure of the KFlO and the OFOO codes for N = log, 8 =
3(8-PAM).
rted or not is encoded into a bit. N such bits
a flag symbol.
FlO coding, the RIS (running IS1 sum) [9] of thejth
he kth block
ed to the RIS accumulated up to the last frame:
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL.37, NO. 9, SEPTEMBER 1989
rrespondence
wer Suppression at the Nyquist Frequency for
Pilot-Aided PAM and QAM Systems
DAE YOUNG KIM AND KAMILO FEHER
act-Two low redundancy codes for in-band pilot insertion are
ed. One named KFlO suppresses the baseband PAM and the
ed QAM signal power at the Nyquist frequency so that the dual-
ne calibration technique of recent interest can be arranged to
both a symbol timing and a carrier referenceto the receiver. The
med OF00 suppresses the signal power at zero as well as the
frequency, and enables the utilization of a triple-pilot scheme
rovides, in addition to a symbol timing, a carrier reference with
phase ambiguity.
I. INTRODUCTION
ous pilot-aided synchronization techniques have been
red for digital microwave and mobile radio systems
wherein in-band pilot insertion is considered [141. To
n-band schemes, however, we need to suppress the data
power around the desired pilot frequencies. Fig. 1
es three in-band pilot schemes with appropriate PSD
spectral density) notches.
umber of coding techniques have been proposed to
e such PSD notches. For example, the FJOl coding of
a) indicating the FBC (feedback balanced code) [6]
es a PSD notch at dc. The interleaved FBC [7], or the
oding as we call it, generates a notch both at dc and at
quist frequency. The OFOl coding, also called LOFS
a recent improvement over the FJO1. In this paper, we
ce two new coding techniques: the KF10, and the
ved OFOl which we call OFOO.
PSD
:-112 1/2 fT
PSD
a-112 112
PSD
fT
1 -112 0 112 1
(C)
Fig. 1. Conceptual sketches of three different pilot insertion schemes with
associated power spectral density (PSD) notches provided by some power
suppression codes. T is the duration of an output symbol. (a) Single pilot
with the OFOl (or FJOl) coding for a carrier. (b)Dual pilots with the KFlO
coding for a clock and a carrier. (c)Triple pilots with the OF00 (or FJOO)
coding for a clock and an unambiguous carrier.
[dyKim/Feher89]
DSV
ASV
DSV & ASV
Interleave {X01}
= X00

Timing Wave
51
-
a
e
-
e
-
I,
-
m k j l
· qm(t − kT)ql(t − kT − jT) ⊗ h (t)
(3)
Fig. 1 A fourth-power clock recovery circuit.
ation is satisfied:
T) ⊗ h (t)
,
5).
⊗ h (t)
⊗ h (t). (5)
hich is useful in
n in terms of a
r
T
ej 2πrt
T , (6)
only the r = ±1 terms in Eq. (8) remain and the other
terms are eliminated, then for N( l
T ), we obtain
N ±
1
T
=
1
T
l
H ±
1
T
· H ±
1
T
− v
· G ±
1
T
− v − τ · G(τ) · H(v)
· G v+τ −
1
T
G
1
T
−τ ·dτ ·dv. (9)
Since the minimum-bandwidth signal is band-limited
to 1
2T , the spectral power beyond the Nyquist band-
width becomes zero, G(f) = 0 for |f| > 1
2T . Further-
more, since the convolution form of G(f)G( 1
T − f) be-
comes zero, Eq. (9) also becomes zero. Finally, as only
the M(± 1
T ) terms remain, Eq. (8) reduces to Eq. (10),
which is our desired result.
E{z(t)} =
r
Vrej2πrt/T
=
a4 − 3α2
0
T
H
1
T
Q0
1
T
− v
j2πrt/T 1
t − kT) ⊗ h (t)
− kT) ⊗ h (t). (5)
ula, which is useful in
function in terms of a
·
r
M
r
T
ej 2πrt
T , (6)
Q0(f)],
G(η)dη.
t)
T
, (7)
T
· G v+τ −
1
T
G
1
T
−τ ·dτ ·dv. (9)
to 1
2T . Further-
T − f) be-
the M(± 1
E{z(t)} =
r
Vrej2πrt/T
=
a4 − 3α2
0
T
H
1
T
Q0
1
T
− v
· Q0(v)dv · ej2πrt/T
+ H −
1
T
Q0
· −
1
T
− v Q0(v)dv · e−j2πrt/T
= 2 a4 − 3α2
0 · |u| · cos
2πt
T
+ Φ , (10)
where
u =
1
T
H
1
T
Q0
1
T
− v · Q0(v) · dv,
and Φ is the phase angle of u.
LETTER
where qm(t − kT)
and E{·} denotes
is statistically ind
i ̸= j. After elimi
obtain a new expr
E{z(t)} = a4 −
⊗ h
· q0(t −
+
k
⊗ h
· qm(t
where a2 = α0. Si
k j
q0(t −
ncy
HUN††
, and Dae Young KIM†††
, Nonmembers
een conducted on square-law as well as fourth-law rec-
fiers [1]–[4], there have not been any reports on the
mathematical derivation of the timing wave expression
or fourth-law rectifiers with PAM signals band-limited
o the Nyquist frequency. This is the subject of this
aper.
. Derivation of Timing Wave
he timing circuit under consideration for the mini-
mum bandwidth system is shown in Fig. 1. It consists
f a fourth-law circuit (FLC) followed by a narrow band
ass filter tuned to the pulse repetition frequency. We
onstruct the FLC in such a way that the incoming sig-
al is first squared, then spectral shaped by a band pass
lter H(f) centered around the Nyquist frequency, and
quared again by the second squarer.
The received PAM signal x(t) has a pulse shape
(t) at the input to the FLC. If the data sequence {ak}
stationary, then the PAM signal
x(t) =
k
akg(t − kT) (1)
cyclostationary with period T. Passing this sig-
al through the FLC and the narrow band pass filter
k j
=
k j
q2
j (t − kT) ⊗ h (t),
Equation (4) can be modified to Eq. (5).
E{z(t)}
=
k
a4 − 3α2
0 · q2
0(t − kT) ⊗ h (t)
+ 3
k m
α2
0 · q2
m(t − kT) ⊗ h (t). (5)
Using the Poisson sum formula, which is useful in
expressing the periodic mean function in terms of a
Fourier series, we have
k
q2
0(t−kT)⊗h (t)=
1
T
·
r
M
r
T
ej 2πrt
T , (6)
where
m(t)h (t) ⊗ q2
0(t);
M(f) = H (f)[Q0(f) ⊗ Q0(f)],
and
Q0(v) = H(v) G(v − η)G(η)dη.
For the second term in Eq. (5),
k m
q2
m(t − kT) ⊗ h (t)
=
1
T r
N
r
T
ej2πt/T
, (7)
where
n(t) = h (t) ⊗ q2
m;
N ±
1
T
=
1
T
l
H ±
1
T
· H ±
1
T
− v
· G ±
1
T
− v − τ · G(τ) · H(v)
· G v+τ −
1
T
G
1
T
−τ ·dτ ·dv. (9)
to 1
2T . Further-
T − f) be-
the M(± 1
E{z(t)} =
r
Vrej2πrt/T
=
a4 − 3α2
0
T
H
1
T
Q0
1
T
− v
+ H −
1
T
Q0
· −
1
T
= 2 a4 − 3α2
0 · |u| · cos
2πt
T
+ Φ , (10)
where
u =
1
T
H
1
T
Q0
1
T
− v · Q0(v) · dv,
and Φ is the phase angle of u.
We notice from Eq. (10) that a sinusoidal wave of
k m
⊗ h (t) +
k m
α2
0 · qm(t − kT)
· qm(t − kT) ⊗ h (t) (4)
where a2 = α0. Since the following relation is satisfied:
k j
q0(t − kT) · q0(t − kT − jT) ⊗ h (t)
=
k j
q2
j (t − kT) ⊗ h (t),
Equation (4) can be modified to Eq. (5).
E{z(t)}
=
k
a4 − 3α2
0 · q2
0(t − kT) ⊗ h (t)
+ 3
k m
α2
0 · q2
m(t − kT) ⊗ h (t). (5)
Using the Poisson sum formula, which is useful in
expressing the periodic mean function in terms of a
Fourier series, we have
k
q2
0(t−kT)⊗h (t)=
1
T
·
r
M
r
T
ej 2πrt
T , (6)
where
m(t)h (t) ⊗ q2
0(t);
M(f) = H (f)[Q0(f) ⊗ Q0(f)],
and
Q0(v) = H(v) G(v − η)G(η)dη.
For the second term in Eq. (5),
q2
m(t − kT) ⊗ h (t)
Vr =
T
· M
T
+
T
· N
T
.
If H (f) is ideally band-limited to 1/T such
H (f) = 0 for |f| −
1
T
>
1
2T
,
only the r = ±1 terms in Eq. (8) remain and t
terms are eliminated, then for N( l
T ), we obta
N ±
1
T
=
1
T
l
H ±
1
T
· H ±
1
T
· G ±
1
T
− v − τ · G(τ) · H(v)
· G v+τ −
1
T
G
1
T
−τ ·dτ ·d
Since the minimum-bandwidth signal is band
to 1
2T , the spectral power beyond the Nyqui
2T .
T
comes zero, Eq. (9) also becomes zero. Finally
the M(± 1
T ) terms remain, Eq. (8) reduces to
E{z(t)} =
r
Vrej2πrt/T
=
a4 − 3α2
0
T
H
1
T
Q0
1
T
−
+ H −
1
T
· −
1
T
= 2 a4 − 3α2
0 · |u| · cos
2πt
T
+ Φ[mtPark/kgJeon02]

Clock Recovery
52
which was evaluated together with others. In an
optical link containing non-linear components such as
Fig. 2. A simulation model for a 40 Gbit/s system.
lation Study on Clock Recovery
[kgJeon03]
Fig. 7. MB810 power spectrum.
Fig. 8. Power spectra at the APD output.APD output

Remarks
• Applications
• Transmission, Recording, Servo, …
• Electric, Optical, Wireless
• Duality with OFDM
• MB: sync in time, square in freq.
• OFDM: square in time, sync in freq.
• both: ISI-free, ICI-free
53

Duality with OFDM
54
0 T 2T 3T 4T-T-2T-3T-4T
1
t
t
f
f
T
-T/2-T 0 T/2 T
1/T
1
0 1/2T-1/2T 1/T-1/T
0 1/T-1/T 1/2T-1/2T-2/T 2/T
MB
OFDM

ubchannel. did not presume knowledge at t
Figure 5. Rectangular and full-cosine-rolloff pulses and resulting OFDM
Rectangular
pulse
t f
Pulse spectrum OFDM spectrum Full co
spectr
-T/2 T/2
Channel
Cyclic extension
IDFT
aO,...,aN-1
sO,...,sN-1
hO,...,hv
escribed (Fig. 7) is now called
DM-OQAM (offset QAM). In order
iminate intersymbol and inter-sub-
d interference (for a distortionless
smission channel), he showed that
iming of the in-phase and quadra-
data streams should be staggered
/2 and adjacent subbands staggered
other way, as Fig. 8 indicates. He
not presume knowledge at the
enhancements to OFDM/OQAM
ticularly much faster processing th
replacement of an N-point DFT w
N/2-point DFT, if the passband
frequency is chosen such that th
tional part of f1/Δf is 0.5, where f1
lowest subcarrier and Δf is the su
nel spacing. For digital signal pro
(DSP) implementation, he deter
that his OFDM/OQAM design
e-rolloff pulses and resulting OFDM spectra.
f
OFDM spectrum Full cosine pulse
spectrum
Full cosine OFDM
spectrum
0 T
0
1/T-1/T
0
1/T-1/T
DB
55

G-OFDM
56
ForPeerReview
ime domain signals of the columns of the filter matrix, and it is seen that the real parts are
ary parts are odd. As expected, all curves vanish at both ends because the abrupt jumps
the columns of the filter matrix.
n responses of columns of the filter matrix.
he frequency responses of the OFDM with 6 sub-channels and the G-OFDM with 6
200 400 600 800 1000
-2
-1
0
1
2
x 10
-3
n
real(q)
200 400 600 800 1000
-2
-1
0
1
2
x 10
-3
n
imag(q)
q0
q1
q2
q3
q4
q5
q0
q1
q2
q3
q4
q5
orPee
(a)
Fig. 6. Frequency response of (a) OFDM with 6
1024.
(a)
Fig. 7. (a) Power spectral densities of the OFDM w
Normalized Frequency ( rad/sample)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-150
-100
-50
0
Normalized Frequency ( rad/sample)
Power/frequency(dB/Hz)
OFDM
G-OFDM
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50 [msKim17]

References
• W. R. Bennet and J. R. Davey, Data Transmission, McGraw-Hill, 1965.
• R. W. Lucky, J. Salz, and E. J. Weldon, Jr., Principles of Data Communication, McGraw-Hill, 1968.
• J. G. Proakis, Digital Communications, McGraw-Hill, 1983.
• S. Benedetto, E. Biglieri, and V. Castellani, Digital Transmission Theory, Prentice-Hall, 1987.
• N. D. Alexandru and D. Y. Kim, Spectral Shaping via Coding, Cermi Iasi, Romania, 2003.
• A. Croisier, “Compatible high-density Bipolar codes: an unrestricted transmission plan for PCM carriers,” IEEE
Trans. Commun. Technol., vol. COM-18, pp. 265-268, June 1970.
• P. Kabal and S. Pasupathy, “Partial-response signaling,” IEEE Trans. Common., vol. COM-23, pp. 921-934,
Sep. 1975.
• D. Y. Kim and J. K. Kim, "New runlength-limited and DC-free line code of minimum bandwidth," IEE Electronics
Letters, vol. 18, no. 24, pp. 1040-1041, Nov. 25, 1982.
• D. Y. Kim, A Study on the Line Codes of Minimum Bandwidth, Ph. D. Dissertation, KAIST, Jan. 1983.
• D. Y. Kim and J. K. Kim, "A condition for stable minimum bandwidth line codes," IEEE Trans. on Commun., vol.
COM-33, no. 2, pp. 152-157, Feb. 1985.
57

• D. Y. Kim, J. I. Baek, J. K. Kim, S. K. Hyun, and Y. K. Park, "Runlength-limited variants of duobinary and
modiﬁed duobinary," IEEE Trans. on Commun., vol. COM-35, no. 2, pp. 142-150, Feb. 1987.
• D. Y. Kim, “K3B4 code for high-speed binary signaling over ﬁber-optic links,” Proc. 4th Int’l Joint Workshop on
Computer Communications (JWCC), pp. 397-407, 13-15 July 1989.
• D. Y. Kim and K. Feher, "Power suppression at the Nyquist frequency for pilot-aided PAM and QAM systems,"
IEEE Trans. on Commun., vol. 37, no. 9, pp. 984-986, Sept. 1989.
• D. Y. Kim, "Lower-bound eye widths of minimum-bandwidth systems," IEEE Trans. on Commun., vol. 43, no.
2/3/4, pp. 1235-1249, Feb./ Mar./Apr. 1995.
• M. T. Park, K. G. Jeon, and D. Y. Kim, "Derivation of timing wave expression on a PAM signal limited to the
Nyquist frequency," IEICE, Sept. 2002.
• K. G. Jeon and D. Y. Kim, "A simulation study on clock recovery of a minimum bandwidth signal," Photonic
Network Communications, vol. 6, no. 1, pp. 83-89, 2003.
• C. G. Lee and D. Y. Kim, “Experimental evaluation of a binary minimum-bandwidth line code MB34 for high-
speed optical transmission,” Transmission Systems and Equipment, vol. E86-B, no. 4, p. 1416, Apr. 2003.
• C. G. Lee and D. Y. Kim, “A design for a new minimum-bandwidth line code of MB810+,” Transmission
Systems and Equipment, vol. E86-B, no. 4, p. 1419, Apr. 2003.
• M. S. Kim and D. Y. Kwak, “Generalized OFDM for 5th generation mobile communications,” Proc. IEEE VTC
2017, 4–7 June 2017, Sydney, Australia.
58

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