20190117 Ph.D. Dissertation Oral Defense

Introduction
Proposed CPS-OFDM Waveform and Its Optimization Design
Implementation and Application of CPS-OFDM
Conclusions and Future Work
Circularly Pulse-Shaped Precoding for OFDM:
A New Waveform and Its Transceiver
Optimization Design
Yenming Huang 黃彥銘
Advanced Digital Communication Systems Lab,
Graduate Institute of Communication Engineering,
National Taiwan University, Taipei, Taiwan
January 17, 2019
Yenming Huang 黃黃黃彥彥彥銘銘銘 Ph.D. Dissertation Defense 1/68

Introduction
Outline of This Presentation
1 Introduction
5G Physical-Layer Signal Requirements
State-of-the-Art of 5G New Waveform Development
Problem Statement and Literature Review
2 Proposed CPS-OFDM Waveform and Its Optimization Design
Key Ideas of CPS Precoding
Formulations of the Transmit Signal, OSBEP, VIP, and NEP
Optimization Design via Majorization-Minimization (MM)
Performance Evaluations in 5G NR
3 Implementation and Application of CPS-OFDM
Analysis on CPS Precoder Structure and Its Complexity
Reducing CM of CPS-OFDM by Constellation Shaping
Heterogeneous Spectrum Access Using CPS-OFDM
4 Conclusions and Future Work

Introduction
5G New Radio New Waveforms Based on OFDM
5G New Radio (NR) highlights three main use cases:1
enhanced mobile broadband (eMBB),
ultra-reliable and low latency communications (URLLC),
massive machine type communications (mMTC),
operated in a wide range of frequencies and various deployments.
New waveforms, based on OFDM, are developed to2
enable diverse 5G new usage scenarios and applications,
ﬂexibly address manifold physical-layer signal requirements
by providing diﬀerent transmission properties.
Support of DFT-S-OFDM (a.k.a. SC-FDMA)3
based waveforms is
mandatory for user equipments in view of link budget.2
1
3GPP, “Study on scenarios and requirements for next generation access technologies,”
TR 38.913, V15.0.0, June 2018.
2
3GPP, “Study on New Radio (NR) access technology,” TR 38.912, V15.0.0, June 2018.
3
abbreviations for discrete Fourier transform spread OFDM and single-carrier frequency division multiple access

Introduction
Two Additional PHY Demands in 5G NR: Low OSBE and Low PAPR
On top of preserving most merits of OFDM, lowering
out-of-subband emission (OSBE) and
peak-to-average power ratio (PAPR)
are two additional demands, since 5G further concerns1234
- asynchronous transmissions with TO and CFO5,
- mixed numerologies (different subcarrier spacings) in a carrier,
- battery life and coverage range of a low-cost machine,
- communications at high carrier frequencies (e.g., mmWave).
1
G. Wunder et al., “5GNOW: non-orthogonal, asynchronous waveforms for future mobile applications,”
IEEE Communications Magazine, vol. 52, no.2, pp. 97-105, Feb. 2014.
2
A. A. Zaidi et al., “Waveform and numerology to support 5G services and requirements,”
IEEE Communications Magazine, vol. 54, no. 11, pp. 90-98, Nov. 2016.
3
P. Guan et al., “5G field trials: OFDM-based waveforms and mixed numerologies,”
IEEE Journal on Selected Areas in Communications, vol. 35, no. 6, pp. 1234-1243, June 2017.
4
C. Sexton et al., “Enabling asynchronous machine-type D2D communication using multiple waveforms in
5G,” IEEE Internet of Things Journal, vol. 5, no. 2, pp. 1307-1322, Apr. 2018.
5
abbreviations for timing offset and carrier frequency offset, respectively

Introduction
Out-of-Subband Emission (OSBE) Issue
OSBE
The spectral sidelobe leakage of a user assigned to a portion of
OFDM subcarriers in a carrier.1
Normalized Frequency (MHz)
-6 -4 -2 0 2 4 6
PSD(dBm/30kHz)
-50
-40
-30
-20
-10
0
10
20
Mask
One user transmission
Out-of-subband emission
(OSBE)
Out-of-subband emission
(OSBE)
Sampling bandwidth
Out-of-band
emission
(OOBE)
Out-of-band
emission
(OOBE)
Channel bandwidth
Guard
band
Guard
band
-6 -4 -2 0 2 4 6
PSD(dBm/30kHz)
-40
-20
0
Each User With High OSBE
-6 -4 -2 0 2 4 6
PSD(dBm/30kHz)
-40
-20
0
Each User With Low OSBE
1
M.-F. Tang and B. Su, “Filter optimization of low out-of-subband emission for universal-ﬁltered multicarrier
systems,” in Proc. ICC Workshop on 5G RAN Design, May 2016.
2
3GPP, “User Equipment (UE) radio transmission and reception,” TS 36.101, V15.1.0, Dec. 2017.

Introduction
Peak-to-Average Power Ratio (PAPR) Issue
PAPR
The ratio of the peak power to the average power during an
interval to quantify the signal envelope fluctuation.
Input Power (dB)
-20 -15 -10 -5 0 5 10
OutputPower(dB)
-20
-15
-10
-5
0
5
10
PA function
Saturation output power
Linear region
Input backoff (IBO)
Saturation region
Actual
operating
point
Ideal
operating
point
Sample (in one OFDM block duration)
0 100 200 300 400 500 600 700 800 900 1000
Square-Magnitude(dB)
-10
-8
-6
-4
-2
0
2
4
6
8
10
Before PA
After PA
Clipping effect
(nonlinear distortion)
Peak
Average power
Note: There are other metrics to quantify the signal envelope fluctuation.
1
Y. Rahmatallah and S. Mohan, “Peak-to-average power ratio reduction in OFDM systems: a survey and
taxonomy,” IEEE Communications Surveys & Tutorials, vol. 15, no. 4, Fourth Quarter 2013.

Introduction
Spectral Regrowth (Actual OSBE Related to PAPR)
Spectral Regrowth
Power amplifier (PA) nonlinearity imposing on the signal such that
mixing products between individual frequency components occur.
-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
PSD(dBm/30kHz)
-60
-50
-40
-30
-20
-10
0
10 Mask
After PA, IBO = 3 dB
After PA, IBO = 8 dB
Before PA
Spectral
regrowth
Spectral
regrowth
1
C.-Y. Lin, Y. Huang, and B. Su, “Prototype filter design in GFDM systems in presence of PA nonlinearity,”
in Proc. IEEE 23rd International Conference on DSP, Nov, 2018.
2
Skyworks Solutions Inc., “R1-165035: NR candidate waveforms: UL performance issues for PAPR,
out-of-channel emissions, and RF front-end linearity/efficiency,” in 3GPP RAN1 Meeting #85, May, 2016.

Introduction
Summary of 5G Waveform Design Aspects
Design Aspect 1
Pure OFDM waveform and its OFDMA1 usage, known to possess
severe OSBE and rather high PAPR, need to be improved.
Design Aspect 2
A joint consideration of lowering OSBE and PAPR together is
essential, since PA2 nonlinearity actually causes spectral regrowth
that may deteriorate expected low-OSBE virtue.
Design Aspect 3
The resulting signal detection reliability at the receiver and overall
spectral eﬃciency (SE) in the system must be satisfactory.
1
abbreviation for orthogonal frequency division multiple access
2
abbreviation for power ampliﬁer

Introduction
Precoding, Windowing, and Filtering Techniques Aided OFDM
To deal with the OSBE and PAPR issues, several new waveforms have
been studied in recent years.1
This research focuses on subband precoding since it prevents block extension
that causes extra inter-block interference (IBI) and extra envelope ﬂuctuation
1
S.-Y. Lien, S.-L. Shieh, Y. Huang, B. Su, Y.-L. Hsu, and H.-Y. Wei, “5G new radio: waveform, frame
structure, multiple access, and initial access,” IEEE Communications Magazine, vol. 55, pp. 64-71, June 2017.

Introduction
Review of Other OSBE and PAPR Reduction Techniques
OSBE Reduction Techniques:1
PAPR Reduction Techniques:2
Frequency domain:
- Cancellation carrier
- Subcarrier weighting
- Multiple choice sequence
Time domain:
- Adaptive symbol transition
Signal distortion:
- Clipping and ﬁltering
- Peak cancellation
Multiple signaling:
- Selective mapping
- Partial transmit sequences
Constrained constellation shaping
Linear subband precoding prevents imposing distortion on the transmitted
signal, additional power consumption at the transmitter, side information
required by the receiver, and real-time computation for every block.
1
Z. You, J. Fang, and I-T. Lu, “Out-of-band emission suppression techniques based on a generalized OFDM
framework,” EURASIP Journal on Advances in Signal Processing, May 2014.
2
Y. Rahmatallah and S. Mohan, “Peak-to-average power ratio reduction in OFDM systems: a survey and
taxonomy,” IEEE Communications Surveys & Tutorials, vol. 15, no. 4, Fourth Quarter 2013.

Introduction
System Model and Subband Precoder Design Problem
Problem Statement
Design an S × S precoding matrix P such that the resulting OSBE and
PAPR can be simultaneously reduced1
with linearithmic-order
implementation complexity2
and support of P being unitary3
.
1
as compared to those of legacy OFDMA using P = IS
2
O(S log2 S) instead of a pure matrix-vector multiplication requiring S2
complex multiplications
3
PH
P = IS to prevent noise enhancement penalty (NEP) at the receiver

Introduction
Thoughts on Subband Precoder Design
In the most general form, there are S × S complex-valued coefficients of
P to be specified to meet OSBE and PAPR requirements.
However, a precoding matrix with arbitrarily chosen entries has undesired
quadratic-order implementation complexity, i.e., O(S2
).
Some constraints on the precoder structure may make the implementation
efficient in linearithmic-order complexity, i.e., O(S log2 S).
Unlike a DFT precoder (P = WS), we also expect that there are still S
degrees of freedom (DoF) to be optimized according to demands.
Support of P being unitary can prevent detection performance
degradation due to noise enhancement penalty (NEP) at the receiver.

Introduction
Review of Existing Subband Precoder Designs
Linearithmic Unitary Additional
Low Low Order Precoder DoF1 in
OSBE PAPR Complexity Support Precoder
SC-FDMA
√ √ √
SS-SC-FDMA2 √ √ √
ZT DFT-S-OFDM3 √
(
√
)
√ √
OP-OFDM4 √ √ √
UP-OFDM5 √ √ √
SP-OFDM6 √ √
CPS-OFDM
√ √ √ √ √
1
DoF means that there are some coeﬃcients to be ﬂexibly designed according to needs.
2
B. Benammar, N. Thomas, M.-L. Boucheret, C. Poulliat, and M. Dervin, “Analytical expressions of power
spectral density for general spectrally shaped SC-FDMA systems,” in Proc. 21st EUSIPCO, Sep. 2013.
3
G. Berardinelli, F. M. L. Tavares, T. B. Sorensen, P. Mogensen, and K. Pajukoski,
“Zero-tail DFT-spread-OFDM Signals,” in Proc. GLOBECOM, pp. 229-234, Dec. 2013.
4
M. Ma, X. Huang, B. Jiao, and Y. J. Guo, “Optimal orthogonal precoding for power leakage suppression in
DFT-based systems,” IEEE Trans. on Communications, vol. 59, no. 3, pp. 844-853, Mar. 2011.
5
R. Xu, L. Wang, Z. Geng, H. Deng, L. Peng, and L. Zhang, “A unitary precoder for optimizing spectrum
and PAPR characteristic of OFDMA signal,” IEEE Trans. on Broadcasting, Early Access Articles, 2017.
6
M. Mohamad, R. Nilsson, and J. Van De Beek, “A novel transmitter architecture for spectrally-precoded
OFDM,” IEEE Trans. on Circuits and Systems I: Regular Papers, (Early Access), Feb. 2018.

Introduction
Outline
1 Introduction

Introduction
Main Contributions of the Dissertation
Propose a new circularly pulse-shaped (CPS) precoding OFDM
waveform, called CPS-OFDM, and its optimization design.1
CPS-OFDM can be regarded as a generalized DFT-S-OFDM.
CPS-OFDM has two main advantages of low OSBE and low PAPR.
-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
PSD(dBm/30kHz)
-60
-50
-40
-30
-20
-10
0
10 Mask
OFDMA (after PA)
OFDMA (before PA)
CPS-OFDM (after PA)
CPS-OFDM (before PA)
PAPR0 (dB)
0 2 4 6 8 10 12
CCDF:Pr{PAPR>PAPR0}
10-3
10-2
10-1
100
OFDMA
CPS-OFDM
1
Y. Huang and B. Su, “Circularly pulse-shaped precoding for OFDM: a new waveform and its
optimization design for 5G New Radio,” IEEE Access, vol. 6, pp. 44129-44146, Aug. 2018.

Introduction
Proposed CPS-OFDM Waveform and Its Parameters
The S × S CPS precoding matrix is deﬁned as
[P]i,kM+m = [p] i−kM S
e−j2π m
M
i
,
where
- p: an S × 1 prototype shaping vector
producing all entries of P,
- S = KM: number of used contiguous
OFDM subcarriers,
i ∈ ZS = {0, 1, · · · , S − 1},
- K: number of available shaped
subcarriers in the subband,
k ∈ ZK = {0, 1, · · · , K − 1},
- M: number of available subblocks
in the subband,
m ∈ ZM = {0, 1, · · · , M − 1}.

Introduction
Optimal Prototype Shaping Vector Design Problem
Problem Statement (Given K, M, and G)1
Find the optimal complex-valued S × 1 prototype shaping vector p
such that the resulting OSBE and PAPR can be simultaneously
reduced with the consideration of NEP.
1
The eﬀects of choosing diﬀerent K, M, and G will be discussed later in the simulation results.

Introduction
CPS-OFDM Transmit Signal Formulation
At the transmitter, the N × S synthesis matrix Φ is deﬁned as
Φ = G WH
N
I
P , I = {η, η + 1, · · · , η + S − 1} , N = N + G. (1)
For the proposed P, the (kM + m)th column of Φ can be written as
[Φ]kM+m = G WH
N
I
CkM diag (p) R [WM ]m . (2)
Each entry of Φ is obtained by (Note that for G = Gzp, set G = 0)
[Φ]n ,kM+m =
1
√
NM
S−1
i=0
[CkM p]i e−j 2π
M
im
ej 2π
N
(η+i)(n −G )
. (3)
The baseband discrete-time transmitted signal can be formulated as
xn =
k∈K
m∈M
[Φ]n ,kM+m · [d]kM+m , n = 0, 1, · · · , N − 1, (4)
where K ⊆ ZK and M ⊆ ZM are two sets of indices corresponding to
the shaped subcarriers and subblocks occupied by data symbols, respectively.

Introduction
Quantifying OSBE Using OSBE Power (OSBEP)
To quantify the OSBE by its power, we ﬁrst deﬁne a transfer function
Φk,m ejω
=
N −1
n =0
[Φ]n ,kM+m e−jωn
. (5)
The square-magnitude response is
Φk,m ejω
2
= wH
m ejω
CkM p
2
, (6)
where
wm ejω
i
=
1
√
NM
e
j2πi m
M
+ G
N W∗
ej[ω− 2π
N
(η+i)] ,
W ej[ω− 2π
N
(η+i)] =
N+G −1
n =0
e−j[ω− 2π
N
(η+i)]n
, ∀i ∈ ZS.
Assumption: The data symbols in the data vector d are
zero-mean,
independent and identically distributed (i.i.d.),
symbol power Es.

Introduction
PSD and OSBEP of CPS-OFDM Transmission
The power spectral density (PSD) can be derived as12
Sx ejω
=
Es
N k∈K
m∈M
Φk,m ejω 2
= pH




Es
N k∈K
m∈M
wH
m ejω
CkM
2




Ψ(ejω)
p. (7)
Thus, the OSBEP3
can be expressed as a quadratic form
γx (p) = pH
Ωp , (8)
where
Ω =
ω∈FOSB
Ψ ejω dω
2π
0S×S. (9)
1
Here we only consider the baseband PSD without an interpolation ﬁlter used in a digital-to-analog converter.
2
Y.-P. Lin, S.-M. Phoong, and P. P. Vaidyanathan, Filter Bank Transceivers for OFDM and DMT Systems,
Cambridge University Press, 2011.

Introduction
Handling Overall Signal Fluctuation Alternative to Limiting PAPR
The effective signal of the bth block transmission is
xN [b] = WH
N
I
Pd[b]. (10)
Traditionally, the block-wise PAPR is defined as
PAPR[b] =
xN [b] 2
∞
1
N
xN [b] 2
2
or
xN [b] 2
∞
Pavg
. (11)
Remark:
A number of studies have been devoted to reducing this instantaneous
quantity so as to avoid PA nonlinearity.
Nevertheless, a radio transmitter often intends to keep its power efficiency
sufficiently high even with some nonlinear distortion caused by PA. 12
It is known that handling overall envelope fluctuations of xN [b] is more
practical than merely limiting the peak. 12
In addition, a data-independent design of P is required.
1
A. Behravan and T. Eriksson, “Tone reservation to reduce the envelope fluctuations of multicarrier signals,”
IEEE Trans. on Wireless Communications, vol. 8, no. 5, pp. 2417-2423, May 2009.
2
A. Behravan and T. Eriksson, “Some statistical properties of multicarrier signals and related measures,”
in Proc. IEEE 63rd Vehicular Technology Conference, pp. 1854-1858, May 2006.

Introduction
Quantifying Signal Envelope Fluctuation Using VIP
Variance of instantaneous power (VIP) is a measure closely related to the
nonlinear distortion caused by PA, which is a more practical metric than PAPR.
Let xn[b] = [xN [b]]n, the VIP is deﬁned as12
¯σ2
x =
1
N
N−1
n=0
E |xn[b]|2
− ¯µx
2
=
1
N
N−1
n=0
E |xn[b]|4
− ¯µ2
x, (12)
where ¯µx is the mean of instantaneous power (MIP) given by
¯µx =
1
N
N−1
n=0
E |xn[b]|2
. (13)
With the assumptions of zero-mean and i.i.d. data symbols, we further deﬁne
σ4
d = E{|[d[b]]i|4
}, ∀i ∈ D, ∀b. Then, the VIP and MIP are independent of
the block index “[b]” which can be omitted.12
1
C. H. Yuen and B. Farhang-Boroujeny, “Analysis of the optimum precoder in SC-FDMA,”
IEEE Trans. on Wireless Communications, vol. 11, no. 11, pp. 4096-4107, Nov. 2012.
2
D. D. Falconer, “Linear precoding of OFDMA signals to minimize their instantaneous power variance,”
IEEE Trans. on Communications, vol. 59, no. 4, pp. 1154-1162, Apr. 2011.

Introduction
VIP of CPS-OFDM Signal (1/2)
The nth sample of the eﬀective CPS-OFDM signal xN is expressed as
xn = ej 2π
N ηn
k∈K
m∈M
dk,meH
m,nCkM p, (14)
where dk,m = [d]kM+m and [em,n]i = 1√
NM
ej2πi( m
M − n
N )
, ∀i ∈ ZS.
The MIP of CPS-OFDM signals
¯µx =
1
N
N−1
n=0
E |xn|
2
=
|K| |M| Es
NM
p
2
2 =
DEs
NM
ρ (15)
is a constant due to
N−1
n=0 em,neH
m,n = 1
M IS and CT
kM CkM = IS.

Introduction
VIP of CPS-OFDM Signal (2/2)
Hence, the VIP of CPS-OFDM signals shown as
¯σ2
x(p) =
1
N
N−1
n=0
E |xn|
4
f(p)
−¯µ2
x, (16)
can be suﬃciently described by the quartic function f(p) whose
E |xn|4
= σ4
d
k∈K
m∈M
eH
m,nCkM p
4
+ 2E2
s
k∈K
m∈M
eH
m,nCkM p
2
k ∈K
m ∈M
(k ,m )=(k,m)
eH
m ,nCk M p
2
. (17)

Introduction
NEP Related to CPS-OFDM Detection Reliability (1/2)
The unitarity of P can be easily identified by the characteristic matrix Γ,1
i.e.,
P = (WK ⊗ IM ) diag (vec (Γ)) WH
K ⊗ WM is unitary iff [Γ]m,k = 1, ∀k, m.
The noise enhancement penalty (NEP) can be written as2
1
S
K−1
k=0
M−1
m=0
1
| [Γ]m,k |2
≤ 1 + , (18)
where ≥ 0 is a noise tolerance factor.
1
P.-C. Chen, B. Su, and Y. Huang, “Matrix characterization for GFDM: low complexity MMSE receivers and
optimal filters”, IEEE Transactions on Signal Processing, vol. 65, no. 18, pp. 4940-4955, Sep. 2017.
2
P.-C. Chen and B. Su, “Filter optimization of out-of-band radiation with performance constraints for GFDM
systems,” in Proc. IEEE 18th International Workshop on SPAWC, July 2017.

Introduction
NEP Related to CPS-OFDM Detection Reliability (2/2)
The NEP constraint of CPS-OFDM can then be derived as1
ζ(p) = tr WH
K ⊗ IM ppH
WH
K ⊗ IM
H ◦−1
≤ (1 + )
S2
ρ
, (19)
Note:
When = 0, the CPS precoding matrix P is unitary.2
The matrix P is invertible if and only if Γ has no zero entries.2
1
Tip: vec(Γ) = S/ρ vec([p0 p1 · · · pK−1]WH
K ) = S/ρ(WH
K ⊗ IM )p
2

Introduction
Proposed Optimal Prototype Shaping Vector Design Problem (1/3)
Problem Formulation
minimize
p
¯σ2
x(p) . . . (VIP) (20a)
subject to γx (p) ≤ U . . . (OSBEP) (20b)
ζ (p) ≤ (1 + )
S2
ρ
. . . (NEP) (20c)
p 2
2 = ρ . . . (Fixed energy). (20d)
Remark:
The VIP function ¯σ2
x(p) is a fourth-order polynomial of p, which
generally makes the problem NP-hard1
and diﬃcult to be analyzed.2
The choice of U has to be a positive number no less than the
minimum value Umin that guarantees the existence of p in (20b).
1
abbreviation for non-deterministic polynomial-time hardness
2
B. Jiang Z. Li, and S. Zhang, “Approximation methods for complex polynomial optimization,”
Springer Computational Optimization and Applications, vol. 59, pp. 219–248, Oct. 2014.

Introduction
Problem Formulation
minimize
p
¯σ2
x(p) . . . (VIP) (21a)
subject to γx (p) ≤ U . . . (OSBEP) (21b)
ζ (p) ≤ (1 + )
S2
ρ
. . . (NEP) (21c)
p 2
2 = ρ . . . (Fixed energy). (21d)
The key idea of handling the problem is to introduce a lifting matrix X = ppH
to transform the order of ¯σ2
x(p) from quartic to quadratic1
. We thus rewrite
the VIP ¯σ2
x(p) as ¯σ2
x(X) = f(X) − ¯µ2
x∝ f(X),
the OSBEP γx(p) as tr(ΩX),
the NEP ζ(p) as ζ(X) = tr([(WH
K ⊗ IM )X(WH
K ⊗ IM )H ]◦−1),
the energy p 2
2 as tr (X).
1
Y. Sun, P. Babu, and D. P. Palomar, “Majorization-minimization algorithms in signal processing,
communications, and machine learning,” IEEE Trans. on Signal Processing, vol. 65, no. 3, pp. 794-816, Feb. 2017.

Introduction
Convexity and Matrix Quadratic Form of f(X)
In the dissertation, Section 2.6.2 has derived in detail that
f(X) is convex in X,
f(X) can be expressed as a matrix quadratic form, i.e.,
f(X) = vec (X)
H
Tvec (X), (22)
where T is an S2
× S2
Hermitian positive deﬁnite matrix
depending on N, S, M, K, M, Es, and σ4
d.
Note:
The matrix T, independent of incoming data, can be oﬄine
computed and stored before performing optimization procedures.
We can actually replace the objective function ¯σ2
x(X) by (22).

Introduction
Problem Reformulation
minimize
X
vec (X)H
Tvec (X) (23a)
subject to tr (ΩX) ≤ U (23b)
ζ (X) ≤ (1 + )
S2
ρ
(23c)
tr(X) = ρ (23d)
X 0S×S (23e)
rank(X) = 1, (23f)
Remark:
The rank-one constraint (23f) is nonconvex to be relaxed.
Traditional methods (e.g., convex iteration algorithm1
) used for solving
SDP2
problems are not applicable to this problem with dropped (23f).
1
J. Dattorro, Convex Optimization & Euclidean Distance Geometry, 2nd ed., Meboo Publishing, Sep. 2017.
2
abbreviation for semideﬁnite programming

Introduction
Solving the Problem by Majorization-Minimization (MM)
To solve the problem (23), we adopt the majorization-minimization
(MM) method whose key idea is to convert a difficult problem into a
series of simple problems with convergence guarantee12
.
Preview:
With our proof of the original objective VIP function being convex,
the globally optimal solution of the original problem is guaranteed
to be attained via MM iteration process.
We introduce a method of choosing the initial point, which must be
in the feasible set of the first minimization problem.
1
2
M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergence analysis of block successive minimization
methods for nonsmooth optimization,” SIAM Journal of Optimization, vol. 23, no. 2, pp. 1126-1153, June 2013.

Introduction
Majorization-Minimization (MM) Algorithmic Framework
minimize
x
f(x)
subject to x ∈ X
The MM procedure consists of
1 Majorization step:
Construct a surrogate function g(x|x( )
)
satisfying the upperbound properties
g(x|x( )
) ≥ f(x), ∀x ∈ X,
g(x( )
|x( )
) = f(x( )
).
2 Minimization step:
Update x as
x( +1)
∈ arg min
x∈X
g(x|x( )
).
The objective value is monotonically decreasing
at every iteration, i.e.,
f(x( +1)
) ≤ g(x( +1)
|x( )
) ≤ g(x( )
|x( )
) = f(x( )
).
1

Introduction
Majorization Step for Solving the Problem (1/2)
Construct a surrogate function g(X|X( )
) of f(X) = vec (X)H
Tvec (X) as1
g X|X( )
= vec (X)H
Dvec (X)
+ 2 vec (X)H
[T − D] vec X( )
+ vec X( )
H
[D − T] vec X( )
(25)
where
- D is a Hermitian matrix chosen to be D T,
- X( )
is the point at the th iteration.
Note:
g(X|X( )
) ≥ f(X) for any X as long as D T.
g(X( )
|X( )
) = f(X( )
).
1
Tip: xH
Lx ≤ xH
Mx + 2 {xH
(L − M)x( )
} + x( )H
(M − L)x( )
, where M L = LH

Introduction
Majorization Step for Solving the Problem (2/2)
Choose D = λmax (T) IS2 and simplify the surrogate function (25) as1
g X|X( )
= 2 vec (X)H
Jvec X( )
+ c
= 2 tr E( )
X + c
= tr V( )
X + c, (26)
where
- J = T − λmax(T)IS2 ,
- c is a constant term independent of X,
- E( )
= reshape(Jvec(X( )
), S, S),
and the following matrix will be updated for every iteration
V( )
=
1
2
E( )
+ E( )H
. (27)
1
One may accelerate the convergence rate of the MM process by ﬁnding a more suitable surrogate function.

Introduction
Minimization Step for Solving the Problem
Perform the following semideﬁnite programming (SDP) with multiple iterations:
minimize
X
tr V( )
X (28a)
subject to tr (ΩX) ≤ U (28b)
ζ (X) ≤ (1 + )
S2
ρ
(28c)
tr(X) = ρ (28d)
X 0S×S. (28e)
Note:
Although the rank-one constraint has been relaxed, from empirical results
we observe that rank(X( )
) = 1 always holds.
The optimal prototype shaping vector popt to the original problem (21)
can therefore be obtained by ﬁnding Xopt = poptpH
opt to (28).
The choices of an OSBEP upper bound U and an initial point X(0)
have not been discussed yet.

Introduction
Choices of OSBEP Upper Bound and Initial Point (1/2)
Choosing U = βUmin is proposed, where
- Umin > 0 is the minimum value of the OSBEP,
- β ≥ 1 is a factor we can decide.
To ﬁnd Umin, the following optimization problem is ﬁrst solved under the
convex-iteration (CI) algorithmic framework1
minimize
Y
w · tr YB(ϕ)
+ tr (ΩY)
OSBEP
(29a)
subject to ζ (Y) ≤ (1 + )
S2
ρ
. . . (NEP) (29b)
tr(Y) = ρ . . . (Fixed energy) (29c)
Y 0S×S, (29d)
1
J. Dattorro, Convex Optimization & Euclidean Distance Geometry, 2nd ed., Meboo Publishing, Sep. 2017.
2
To avoid confusion with X in the MM algorithm, here the variable is renamed as Y.

Introduction
Choices of OSBEP Upper Bound and Initial Point (2/2)
Perform the following CI programming with multiple iterations:
minimize
Y
w · tr YB(ϕ)
+ tr (ΩY) (30a)
S2
ρ
(30b)
tr(Y) = ρ (30c)
Y 0S×S, (30d)
where
- w > 0 is an empirical weighting factor,
- the matrix B(ϕ)
is derived from the SVD of Y = ˜U˜Σ ˜V
at the (ϕ − 1)th iteration.
B(ϕ)
= [ ˜U]{1,2,··· ,S−1}[ ˜U]H
{1,2,··· ,S−1} (31)
After the algorithm converges, we obtain Umin = tr(ΩYmin), where the
optimal rank-one point Ymin = pminpH
min also serves as the initial point
X(0)
locating in the feasible set of the ﬁrst minimization problem (28).

Introduction
Proposed MM-CI-Based Optimization Algorithm
Input: S, Ω, T, ρ, , w, β, εCI, εMM.
Output: Optimal S × 1 prototype shaping vector popt.
CI Process:
1: Initialize B(0)
= 0S×S , set ϕ = 0.
2: repeat
3: Solve Problem (30) to get the optimal solution Y(ϕ)
.
4: Calculate the OSBEP U(ϕ)
= tr ΩY(ϕ)
.
5: Obtain B(ϕ+1)
according to (31).
6: ϕ ← ϕ + 1
7: until convergence, i.e., |U(ϕ+1)
− U(ϕ)
| ≤ εCI.
8: Obtain Ymin = Y(ϕ)
and Umin = U(ϕ)
.
MM Process:
9: Initialize X(0)
= Ymin and V(0)
according to (27).
10: Set U = βUmin and = 0.
11: repeat
12: Solve Problem (28) to get the optimal solution X( +1)
.
13: Calculate the objective value g( )
= tr V( )
X( +1)
.
14: Obtain V( +1)
according to (27).
15: ← + 1
16: until convergence, i.e., |g( +1)
− g( )
| ≤ εMM.
17: Obtain popt = p( +1)
from X( +1)
= p( +1)
p( +1)H
.
Problem (30)
minimize
Y
w · tr YB
(ϕ)
+ tr (ΩY)
S2
ρ
tr(Y) = ρ
Y 0S×S ,
Problem (28)
minimize
X
tr V
( )
X
subject to tr (ΩX) ≤ U
ζ (X) ≤ (1 + )
S2
ρ
tr(X) = ρ
X 0S×S .
1
The tolerant precision for the stopping criteria of the CI and the MM processes are denoted by
εCI > 0 and εMM > 0, respectively. The two convex optimization problems can be solved by CVX.

Introduction
Simulation Parameters and Assumptions
The system parameters are practically chosen according to the agreements in
3GPP standardization meetings for 5G NR waveform evaluation1
.
Parameter Notation Value
Carrier frequency 4 GHz
Sampling rate 15.36 MHz
FFT size N 1024
OFDM subcarrier spacing 15 kHz
Guard interval (GI) length G 72
Number of allocated subcarriers per UE S 48 (i.e., UE bandwidth is 720 kHz)
Power amplifier (PA) model 9-order polynomial model 2
PA phase compensation 76.3 degrees
Input backoff (IBO) 3 dB
Modulation scheme 16-QAM with Es = 1 and σ4
d = 1.32
Coding scheme None
Number of antennas 1T1R
Multipath channel model TDL-C-300 for 3 km/hr3
Channel equalization MMSE-FDE (with perfect channel estimation)
1
3GPP, “Study on New Radio access technology physical layer aspects,” Technical Report (TR) 38.802,
V14.2.0, Annex A.1.1, Sep. 2017.
2
3GPP, “R1-166004: Response LS on realistic power amplifier model for NR waveform evaluation ,” in 3GPP
TSG RAN WG1 Meeting #85, Nanjing, China, May 23-27, 2016.
3
3GPP, “Study on channel model for frequencies from 0.5 to 100 GHz ,” Technical Report (TR) 38.901,
V14.3.0, Dec. 2017.

Introduction
Simulation Case 1b: Single User Communication
All uplink cases recorded in 3GPP TR 38.802, namely,
Case 1b, Case 3, and Case 4, are taken into consideration.12
Case 1b:
- One target user allocated on
the subcarriers indexed by
I = {212, 213, · · · , 259}.
- The required guard band
size ∆ is to make the OSBE
lower than -18dBm/30kHz
deﬁned by the SEM.34
1
V14.2.0, Sec. 7.1.1, Sep. 2017.
2
Case 1a and Case 2 are downlink cases and out of the scope of this work
3
Huawei and HiSilicon, “R1-166093: Waveform evaluation updates for case 1a and case 1b,”
in 3GPP TSG RAN WG1 Meeting #86, Gothenburg, Sweden, Aug. 22-26, 2016.
4
abbreviation for spectral emission mask

Introduction
Simulation Case 3: Asynchronous Transmissions
Case 3
- One target user and two asynchronous interfering users are assigned to
the subcarriers indexed by I = {488, 489, · · · , 535},
I1 = {540, 541, · · · , 587}, and I2 = {436, 437, · · · , 483}, respectively.
- The timing oﬀset (TO) of the target user is assumed to be perfectly
estimated and compensated at the receiver.
- The relative TOs of the interfering users in terms of 128 delayed samples
incur multiuser interference.
- The required guard band between two users is ﬁxed to be ∆ = 60 kHz.

Introduction
Simulation Case 4: Mixed Numerologies
Case 4
- One target user and two interfering users in synchronism but in diﬀerent
numerology.
- The target user complies with the same setting as in Case 3 in terms of
the basic 15 kHz subcarrier spacing.
- The two interfering users adopting 30 kHz subcarrier spacing are assigned
to the subcarriers indexed by I1 = {270, · · · , 293}, I2 = {218, · · · , 241},
with FFT size and GI length being 512 and 36, respectively.
- The required guard band between two users is ﬁxed to be ∆ = 60 kHz.

Introduction
Setting of Existing Waveforms for Comparisons
A generalized simulation flowchart of different waveform generation:
OFDMA: P = IS, G = Gcp, Z = 0
SC-FDMA: P = WS, G = Gcp, Z = 0
WOLA-OFDM: P = IS, G = Gcp, Z = 0, 2G RC window, α = 1
UF-OFDM: P = Ppreeq, G = Gzp, Z = 0, Chebyshev filter, Lf = G
f-OFDM: P = IS, G = Gcp, Z = 0, RC-windowed tone offset sinc, Lf = G, N
2
OP-OFDM: P = [0S×Z Po], G = Gcp, Z = 2
ZT DFT-S-OFDM: P = WS, G = IN , Z = 1 + SG/N
SS-SC-FDMA: P = [0S×Z/2 Pss 0S×Z/2], G = Gcp, Z = S/2, RRC, α = 1
1
Z is the number of zeros in a data vector; α is the roll-off factor; Lf is the filter order
2
RC and RRC are abbreviations of raised cosine and root raised cosine, respectively.

Introduction
Settings of the Proposed CPS-OFDM Waveform
CPS-OFDM possesses a ﬂexible DFT-based precoder parameterized by
1 (K, M, Z, ) = (2, S/2, 2, 0) and (3, S/3, 3, 0) for the comparisons with
OFDMA, WOLA-OFDM, UF-OFDM, f-OFDM, and OP-OFDM without
NEP, K = ZK , M = {1, 2, · · · , M − 1}.
2 (K, M, Z, ) = (1, S, 2, 0.2) for the comparison with SC-FDMA,
K = Z1, M = {1, 2, · · · , M − Z}.
3 (K, M, Z, ) = (1, S, 1 + SG/N , 0.5) and G = IN for the comparison
with ZT DFT-S-OFDM, K = Z1, M = {1, 2, · · · , M − Z}.
4 (K, M, Z, ) = (2, S/2, M + 1, 0.2) for the comparison with
SS-SC-FDMA, K = {0}, M = {1, 2, · · · , M − 1}.
Diﬀerent GI types: G = Gcp or Gzp or IN are to be considered.

Introduction
Settings of the Proposed Optimization Framework
The optimal prototype shaping vector p is then determined by
the proposed MM-CI-based optimization algorithm.
- The OSBEP upper bound factor β is 10.
- The weighting factor w is 1000.
- The energy of p is ρ = M.
- The maximum number of CI and MM iterations are 104
and 105
with εCI = 10−8
and εMM = 10−10
, respectively.

Introduction
Simulated PSD and OSBE Results
-5 -4 -3 -2 -1 0
PSD(dBm/30kHz)
-60
-50
-40
-30
-20
-10
0
10
OFDMA
WOLA-OFDM: RC, α = 1
UF-OFDM: Chebyshev, Lf = 72
f-OFDM: SincRC, Lf = 512, NT O = 2.5
f-OFDM: SincRC, Lf = 72, NT O = 5
OP-OFDM: Z = 2
CPS-OFDM: (K, M, Z, ǫ) = (3, 16, 3, 0), NoGI
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), CP
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), ZP
SEM
CPS-OFDM with CP
CPS-OFDM with NoGI or ZP
-5 -4 -3 -2 -1 0
PSD(dBm/30kHz)
-60
-50
-40
-30
-20
-10
0
10
OFDMA
OP-OFDM: Z = 2
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), CP
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), ZP
SEM
-5 -4.9 -4.8 -4.7
-20
-18
-16
CPS-OFDM
Observation:
CPS-OFDM leads to the lowest amount of OSBE in adjacent bands,
when practical PA nonlinearity is considered (see the right ﬁgure).
The expected low-OSBE properties of WOLA-OFDM, UF-OFDM, and
f-OFDM severely deteriorate due to their high-PAPR drawbacks.

Introduction
PAPR and Signal User Detection Performance Results
PAPR0 (dB)
0 2 4 6 8 10 12
CCDF:Pr{PAPR>PAPR0}
10-3
10-2
10-1
100
OFDMA
OP-OFDM: Z = 2
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), CP
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), ZP
7.5 8 8.5
0.05
0.1
0.15
CPS-OFDM
Eb/N0 (dB)
0 5 10 15 20 25 30
BER
10-4
10-3
10-2
10-1
100
OFDMA
OP-OFDM: Z = 2
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), CP
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), ZP
24.98 24.99 25
×10-3
2.6
2.8
3
3.2
3.4
3.6
CPS-OFDM with CP
Observation:
CPS-OFDM yields much lower PAPR and so provides better PA eﬃciency.
NoGI is more friendly to the PA than ZP by avoiding on-oﬀ switch.
In the presence of GI, CPS-OFDM possesses better detection
performance than those of WOLA-OFDM, UF-OFDM, and f-OFDM.

Introduction
Target User Detection Performance Results for Case 3 and Case 4
Eb/N0 (dB)
0 5 10 15 20 25 30
BER
10-3
10-2
10-1
100
OFDMA
OP-OFDM: Z = 2
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), CP
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), ZP
24.95 25 25.05
×10-3
5
6
7
8
CPS-OFDM with CP
Eb/N0 (dB)
0 5 10 15 20 25 30
BER
10-3
10-2
10-1
100
OFDMA
OP-OFDM: Z = 2
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), CP
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), ZP
24.9 25
0.01
0.012
0.014
14.8 15 15.2
0.025
0.03
0.035
0.04
CPS-OFDM with NoGI or ZP
Observation:
CPS-OFDM is more robust to TO than WOLA-OFDM, UF-OFDM, f-OFDM,
and OFDMA. (see the left ﬁgure for Case 3)
The superiority of CPS-OFDM over the other low-OSBE waveforms
can be found. (see the right ﬁgure for Case 4)
Particularly, CPS-OFDM with ZP or NoGI exhibits robustness against
the inter-numerology interference (INI).

Introduction
Spectral Efficiency (SE) Analysis
The spectral efficiency (SE) in bit/s/Hz is defined as1
SE =
χ
T(BWUE + ∆)
,
where
T is the transmission time interval (TTI) which is 1 ms.
χ = NbitDNblock(1 − BER) is the number of correctly received bits by
the target user per TTI,
- Nbit = 4 is the number of bits per 16-QAM data symbol,
- D is the number of data symbols per block transmission,
- Nblock is the number of transmitted blocks per TTI
(Nblock = 14 for CP and ZP, Nblock = 15 for NoGI),
- BER can be obtained from the results at Eb/N0 = 25 dB,
BWUE = 720 kHz is the target user bandwidth,
∆ is the required guard band size of the target user.
1
V14.2.0, Sep. 2017.

Introduction
SE Results of Case 1b, Case 3, and Case 4
Case 1b Case 3 Case 4
SE(bit/s/Hz)
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
OFDMA
OP-OFDM: Z = 2
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), CP
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), ZP
CPS-OFDM
Sample index (n) in one block duration
0 100 200 300 400 500 600 700 800 900 1000
WaveformSquare-Magnitude(dB)
-5
-4
-3
-2
-1
0
1
2
3
4
5
Before PA
After PA
Observation:
CPS-OFDM with NoGI achieves the highest SE in the three cases,
even though there exists the IBI at the receiver.
The IBI may be tolerable because of the scalability of |M| ≤ M
and the smooth transition between any two blocks.

Introduction
CPS-OFDM Waveform Visualization and Discussion on K
n
0 1023
-6
-4
-2
0
2
4
6
(K, M, Z, ǫ) = (3, 16, 3, 0), NoGI
n
0 71 1095
-6
-4
-2
0
2
4
6
(K, M, Z, ǫ) = (2, 24, 2, 0), CP
n
0 1023
-6
-4
-2
0
2
4
6
(K, M, Z, ǫ) = (2, 24, 2, 0), NoGI
n
0 1024 1095
-6
-4
-2
0
2
4
6
(K, M, Z, ǫ) = (2, 24, 2, 0), ZP
Before PA After PA
ZP
CP
Discussion on increasing K:
The OSBE ideally decreases, but the PAPR increases.
(The multiuser interference due to non-orthogonality might be mitigated.)
The smooth transition range at the two edges of a block enlarges.
(The IBI tolerance level for the NoGI case might be enhanced.)

Introduction
Summary of Representative Simulation Results
We highlight the gains of CPS-OFDM with four diﬀerent transmission types:
T1: Multicarrier transmission (K = |K| ≥ 2) with the requirements of low OSBE and no NEP.
T2: Single-carrier transmission (K = |K| = 1) with the requirements of low PAPR and low OSBE.
T3: Single-carrier transmission (K = |K| = 1) using dedicated internal GI with the requirements of
low PAPR and low OSBE.
T4: Single-carrier transmission (K = 2, |K| = 1) with the requirements of extremely low PAPR and
low OSBE at the cost of SE.

Introduction
Outline
1 Introduction

Introduction
Proposed Subband-wise CPS Precoding for OFDM
Proposed Circularly Pulse-Shaped (CPS) Precoding: P = WSA
We choose the ith entry of the (kM + m)th column of the matrix A as
[ak,m]i = [a] i−mK S
ej2π k
K i
(circular pulse shaping),
where a = a0,0 is an S × 1 prototype pulse vector that can be designed
and optimized to meet the requirements adaptively, S = KM.
Remark:
The matrix A is known as GFDM modulation that possesses the
properties of spectral containment and eﬃcient implementation.1
The resulting transmitted CPS-OFDM signals can inherit the
desired waveform properties from the embedded GFDM.2
1
2
Y. Huang, B. Su, and I-K. Fu, “Heterogeneous LTE downlink spectrum access using embedded-GFDM,”
in Proc. International Conference on Communication (ICC) Workshop on 5G RAN Design, May 2016.

Introduction
Transform Domain Expressions of Prototype Pulse Vector
Proposed Circularly Pulse-Shaped (CPS) Precoding: P = WSA
We choose the ith entry of the (kM + m)th column of the matrix A as
[ak,m]i = [a] i−mK S
ej2π k
K i
(circular pulse shaping),
where a = a0,0 is an S × 1 prototype pulse vector that can be designed
and optimized to meet the requirements adaptively, S = KM.
Equivalently, we can design and further optimize
an S × 1 prototype shaping vector p = WSa, so that the ith entry
of the (kM + m)th column of P is [pk,m]i = [p] i−kM S
e−j2π m
M i
an M × K characteristic matrix Γ = S/ρ reshape (p, M, K) WH
K ,
so pk,m =
1
√
S
vec diag
√
M [WM ]m ΓWK
¯Ck , ¯Ck =
0 IK−k
Ik 0
,
where ρ = p
2
2 denotes the energy of p.

Introduction
Three Equivalent CPS Precoder Implementation Methods
The precoded data symbols obtained from s = Pd can be realized by:
1 P = WSA (direct matrix multiplication)
2 Pk = CkM diag (p) RWM (for the kth spectrally shaped subcarriers), where
CkM =
0 IkM
IS−kM 0
, R = 1K ⊗ IM , P = [P0, P1, · · · , PK−1]
3 P = (WK ⊗ IM ) diag (vec (Γ)) WH
K ⊗ WM (by the characteristic matrix)

Introduction
CPS Precoder Implementation Complexity Analysis
The complexity is evaluated by computing the number of
complex multiplications (CMs) required to send KM data symbols.
Direct implementation ineﬃciently takes KM(KM + log2 KM) CMs.
Frequency-domain implementation takes KM(log2 M + K) CMs.
Characteristic-matrix-domain implementation takes
KM log2 M + MK log2 K + KM + MK log2 K = KM(log2 K2M + 1) CMs.
- The complexity is lower than that of frequency-domain
implementation when K−1
2
> log2 K.
- The CPS precoder can be eﬃciently realized with
linearithmic-order complexity O(KM log2 KM).

Introduction
CPS-OFDM as a Generalized DFT-S-OFDM
By properly choosing the parameters K, M, K, M, p,
and G, the derivations of CPS-OFDM are applicable to
OFDMA: M = 1, K = ZK , p = [1 01×(K−1)]T
, G = Gcp
SC-FDMA: K = 1, M = ZM , p = 1M , G = Gcp
SS-SC-FDMA: K = 2, |K| = 1, M = ZM ,
p is a 2M × 1 root raise cosine (RRC) vector, G = Gcp
ZT DFT-S-OFDM: K = 1, |M| < M, p = 1M , G = IN
GFDM: N = S = KM, p is usually composed by
a 2M × 1 RRC vector and zeros, G = Gcp.

Introduction
Further Reducing Cubic Metric (CM) of CPS-OFDM
Case of Interest
Considering the case that demands high power amplifier (PA) efficiency
at the transmitter, we intend to further reduce the cubic metric (CM)
of CPS-OFDM signals through a constellation shaping technique.
Remark:
CM is known as a more accurate measure of envelope fluctuation to
predict the required input backoff (IBO) of the PA than PAPR.12
Constellation shaping is to introduce offset values to input
data symbols so that CM (or PAPR) can be greatly reduced.34
1
Motorola, “R1-060023: Cubic metric in 3GPP-LTE,” in 3GPP TSG RAN WG1 LTE Adhoc Meeting,
Helsinki, Finland, Jan. 23-26, 2006.
2
C. Ni and T. Jiang, “Minimizing the error vector magnitude with constrained cubic metric and spectral
sidelobe in NC-OFDM-based cognitive radio systems,” IEEE Trans. on VT, vol. 66, no. 1, pp. 358-363, Jan. 2017.
3
X. Zhu, H. Hu, Z. Meng, and J. Xia, “On minimizing the cubic metric of OFDM signals using convex
optimization,” IEEE Trans. on Broadcasting, vol. 60, no. 3, pp. 511-523, Sep. 2014.
4
A. Aggarwal and T. H. Meng, “Minimizing the peak-to-average power ratio of OFDM signals using convex
optimization,” IEEE Trans. on Signal Processing, vol. 54, no. 8, pp. 3099-3110, Aug. 2006.

Introduction
Offset Vector Design Problem for CPS-OFDM
Problem Statement
Design the offset vector e[b] such that the RCM can be further reduced for each
CPS-OFDM block transmission.
The EVM must be constrained to warrant satisfactory detection reliability.
The OSBEE shall be limited to preserve the property of low OSBE.
Remark: Since e[b] = c[b] − d[b] and d[b] is known at the transmitter,
we will directly deal with the offset data vector c[b].

Introduction
Performance Evaluation: Cubic Metric (CM)
CM performance is assessed by the empirical CCDF1
curve of RCM outcomes,
where the CCDF is defined as 1 − Pr {RCM ≤ RCM0}2
.
RCM0 (dB)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
CCDF:Pr{RCM>RCM0}
10-3
10-2
10-1
100
OFDM: CP, EVMmax = 0
CPS-OFDM: CP, EVMmax = 0
CPS-OFDM: NoGI, EVMmax = 0
OFDM: CP, EVMmax = −13 dB
CPS-OFDM: CP, EVMmax = −13 dB
CPS-OFDM: NoGI, EVMmax = −13 dB
Observation:
The proposed optimal offset data
vector design can further reduce
the RCM of CPS-OFDM signals
to meet the requirement of rather
high PA efficiency.
The setting of NoGI is basically
more friendly to the PA than the
setting of CP at the CPS-OFDM
transmitter.
1
abbreviation for complementary cumulative distribution function
2
Pr {RCM ≤ RCM0} means the probability of a RCM that does not exceed a given threshold RCM0.

Introduction
Performance Evaluation: OSBE (by Simulating PSD)
To present OSBE performance, we simulate the PSD1
by 1
B
B−1
b=0 |Xb(ejω
)|2
,
where B = 104
is the number of transmitted blocks.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
PSD(dBm/30kHz)
-20
-15
-10
-5
0
5
10
OFDM: CP, EVMmax = 0
CPS-OFDM: CP, EVMmax = 0
CPS-OFDM: NoGI, EVMmax = 0
SEM
-0.75 -0.7 -0.65 -0.6
-11
-10
-9
Observation:
RCM reduction is much helpful to
spectral regrowth mitigation.
The required guard band size ∆
to meet the PSD of −10 dBm per
30 kHz can be reduced as well.
The proposed scheme can further
improve the spectral containment
property of CPS-OFDM.
1
abbreviation for power spectral density
2
Y. Huang, R. Yang, and B. Su, “Reducing cubic metric of circularly pulse-shaped OFDM signals through
constellation shaping optimization with performance constraints,” in Proc. IEEE 88th VTC Fall, Aug. 2018.

Introduction
Coexistence of 4G LTE and 5G NR Assisted by CPS-OFDM
Case of Interest
For smooth migration from 4G to 5G, a legacy LTE downlink band is
shared with 5G NR, but they adopt diﬀerent subcarrier spacing.
Many upcoming 5G applications may
require12
backhaul signaling provided by
widely-deployed LTE infrastructure,
physical characterizations of the
bands below 6 GHz for wide coverage
and mobility support.
No impact on incumbent 4G UE receivers
is needed.
1
MediaTek Inc., “RWS-150053: Technology & standardization for 5G,” 3GPP RAN Workshop on 5G,
Phoenix, USA, Sep. 17-18, 2015.
2
4G Americas, “5G spectrum recommendations,” Aug. 2015.

Introduction
Multiplexing CPS-OFDM With OFDMA Signaling
The downlink transmitter model and the PSD are shown as follows.
OFDM subcarrier index
0 100 200 300 400 500 600 700 800 900 1000
NormalizedPSD(dB/15kHz)
-40
-35
-30
-25
-20
-15
-10
-5
0
5
Using 144 OFDM subcarriers
Using 112 OFDM subcarriers
Using CPS-OFDM
Using guard band
Bridging
band
The spectral eﬃciency can be improved by adopting CPS-OFDM
transmission to reuse the guard band between 4G and 5G accesses.
1
Y. Huang, B. Su, and I-K. Fu, “Heterogeneous LTE downlink spectrum access using embedded-GFDM,”
in Proc. International Conference on Communications (ICC) Workshop on 5G RAN Design, May 2016.
2
Y. Huang and B. Su, “Device of handling block transmission in multicarrier system,”
US Patent Application, no. US20170070996 A1, Mar. 9, 2017.

Introduction
Outline
1 Introduction

Introduction
Conclusions
A new waveform called circularly pulse-shaped OFDM (CPS-OFDM), along
with its optimal prototype shaping vector design, is proposed for 5G NR.
Advantages of both low OSBE and low PAPR enable that the spectral
regrowth and signal distortion caused by PA nonlinearity can be alleviated.
The optimal prototype shaping vector is obtained by the proposed
optimization algorithm based on majorization-minimization (MM) and
convex-iteration (CI) iteration processes.
Simulation results demonstrate the performance gains in detection
reliability and spectral efficiency (SE) of applying the proposed schemes
to three practical sub-6 GHz uplink cases specified by 3GPP.
CPS precoder can be efficiently realized with linearithmic-order
complexity with support of being unitary to prevent NEP.
As a generalized DFT-S-OFDM, CPS-OFDM offers backward and
forward compatibility to wireless systems for 5G and beyond.

Introduction
Future Work
Communication Engineering Aspects:
Combining the CPS-OFDM waveform with 5G new non-orthogonal
multiple access (NOMA) schemes (e.g., SCMA) is to be studied.
Integrating the CPS-OFDM system with MIMO technologies is
strongly desirable and essential to 5G NR air-interface.
Developing dedicated synchronization and channel estimation
signal processing algorithms in CPS-OFDM based systems is needed.
Optimization Aspects:
A mathematical proof to the existence of the optimal rank-one
solution of Problem (28) is anticipated.
Accelerating the convergence rate of the MM process by ﬁnding a
more suitable surrogate function alternative to (25) is of interest.

Introduction
Thank you very much for your
time and advice!
Any questions or comments?
Contact me by:
E-mail: hymetalian@gmail.com; d01942015@ntu.edu.tw
Phone: +886 912-572-947

Related Publications and Main Achievements
5G OFDM-Based Waveform Candidates
Details of CPS Precoder Implementations
Others
Paper Publication and Patent Application
3 Journal Articles:
- Yenming Huang and Borching Su, “Circularly Pulse-Shaped Precoding for OFDM: A New Waveform
and Its Optimization Design for 5G New Radio,” IEEE Access, vol. 6, pp. 44129-44146, Aug. 2018.
- Shao-Yu Lien, Shin-Lin Shieh, Yenming Huang, Borching Su, Yung-Lin Hsu, and Hung-Yu Wei,
“5G New Radio: Waveform, Frame Structure, Multiple Access, and Initial Access,”
IEEE Communications Magazine, vol. 55, pp. 64-71, June 2017.
- Po-Chih Chen, Borching Su, and Yenming Huang, “Matrix Characterization for GFDM: Low Complexity
MMSE Receivers and Optimal Filters,” IEEE Trans. on Signal Processing, vol. 65, no. 18, Sep. 2017.
3 Conference Papers:
- Yenming Huang, Rueibin Yang, and Borching Su, “Reducing Cubic Metric of Circularly Pulse-Shaped
OFDM Signals Through Constellation Shaping Optimization With Performance Constraints,” in Proc.
IEEE 88th VTC Fall, Aug. 2018.
- Yenming Huang, Borching Su, and I-Kang Fu, “Heterogeneous LTE Downlink Spectrum Access Using
Embedded-GFDM,” in Proc. ICC Workshop on 5G RAN Design, May 2016.
- Chen-Yen Lin, Yenming Huang, and Borching Su, “Prototype Filter Design in GFDM Systems in
Presence of PA Nonlinearity,” in Proc. IEEE 23rd International Conference on DSP, Nov. 2018.
1 Patent Application:
- Yenming Huang and Borching Su, “Device of Handling Block Transmission in Multicarrier System,”
US 20170070996 A1, Mar. 9, 2017.

Others
3GPP RAN1 Technical Documents
In 2016, the ﬁrst phase of 5G standardization, I actively participated
3GPP RAN1#85, #86, and #86bis meetings as a delegate of National
Taiwan University to promote our ideas with 8 technical documents.
- R1-1610109: NR and LTE Coexistence Assisted by Embedded-GFDM,
in 3GPP RAN1 #86bis meeting, Oct. 10, 2016.
- R1-1610108: Evaluation Results of CPS-OFDM Waveform,
- R1-1610107: Precoder Flexibility and Forward Compatibility of CPS-OFDM Waveform,
- R1-167875: Preliminary Evaluation of CPS-OFDM Waveform in Case 2, Case 3, and Case 4,
in 3GPP RAN1 #86 meeting, Aug. 22, 2016.
- R1-167874: Evaluation Results of OFDM-Based Waveforms in Case 1a and Case 1b,
- R1-167820: Fundamentals of CPS-OFDM Waveform for 5G New Radio,
- R1-165114: Evaluation of OFDM-Based Waveforms,
in 3GPP RAN1 #85 meeting, May 23, 2016.
- R1-165113: OFDM-Based Waveform With Precoding Techniques,
in 3GPP RAN1 #85 meeting, May 23, 2016.

Others
Overview of the Dissertation
Chapter 1:
Introduction
Chapter 2:
Circularly Pulse-Shaped Precoding for OFDM and Its
Optimal Prototype Shaping Vector Design
Chapter 3:
Reducing Cubic Metric of CPS-OFDM Signals Through Constella-
tion Shaping Optimization With Performance Constraints
Chapter 4:
Heterogeneous LTE Downlink Spectrum Access Using CPS-OFDM
Chapter 5:

Main Achievements of This Work
1
VIP is the abbreviation for variance of instantaneous power that is to quantify the signal envelope ﬂuctuation.
2
OSBEP is the abbreviation for out-of-subband emission power that is to quantify the spectral leakage.
3
NEP is the abbreviation for noise enhancement penalty that is related to the matrix P being unitary.

Others
Graphical Abstract of 5G Waveform Research
A generalized OFDM-based transmitter structure for 5G
New Radio (NR) New Waveforms:1
Goal: Lower spectral leakage and envelope ﬂuctuation for the signal.
1
S.-Y. Lien, S.-L. Shieh, Y. Huang, B. Su, Y.-L. Hsu, and H.-Y. Wei, “5G New Radio: waveform, frame
structure, multiple access, and initial access,” IEEE Communications Magazine, vol. 55, pp. 64-71, June 2017.

Others
Orthogonal Frequency Division Multiplexing (OFDM)
OFDM has achieved great success in 4G Long Term Evolution (LTE) due to its
several merits such as1
robustness to channel frequency selectivity,
plain channel estimation,
flexibility in frequency-domain multiple access (e.g., OFDMA2
),
easy integration with MIMO3
technologies.
However, it has two undesirable drawbacks:
large out-of-subband emission (OSBE),
high peak-to-average power ratio (PAPR),
which make the synchronization and the realization of power amplifier (PA)
very stringent, respectively.
1
S. Sesia, I. Toufik, and M. Baker, LTE - The UMTS Long Term Evolution: From Theory to Practice,
2nd edition, John Wiley & Sons, 2011.
2
abbreviation for orthogonal frequency division multiple access
3
abbreviation for multiple-input multiple-output

Others
Background of OFDM Systems
An OFDM system with the guard interval (GI) length being larger than the
channel delay spread, i.e., no inter-block interference (IBI), can be shown as 1
1
Y.-P. Lin, S.-M. Phoong, and P. P. Vaidyanathan, Filter Bank Transceivers for OFDM and DMT Systems,
Cambridge University Press, 2011.

Others
List of 5G Waveform Candidates
Several waveform candidates adopting different techniques arose:123
WOLA-OFDM (weighted overlap-and-add OFDM)
UF-OFDM (universal-filtered OFDM)
f-OFDM (filtered-OFDM)
DFT-S-OFDM (discrete Fourier transform spread OFDM)
- a.k.a. SC-FDMA (signle-carrier frequency division multiple access)
- i.e., legacy 4G uplink transmission waveform
SS-SC-FDMA (spectrally-shaped SC-FDMA)
ZT DFT-S-OFDM (zero-tail DFT-S-OFDM)
GFDM (generalized frequency division multiplexing)
1
Qualcomm Inc., “R1-162199: Waveform candidates,” in 3GPP RAN1 #84bis, Apr. 2016.
2
X. Zhang et al., “On the waveform for 5G,” IEEE Comm. Magazine, vol. 54, no. 11, pp. 74-80, Nov. 2016.
3
R. Gerzaguet et al., “The 5G candidate waveform race: a comparison of complexity and performance,”
EURASIP Journal on Wireless Communications and Networking, 2017:13, Jan. 2017.

Others
Tradeoff of Applying New Waveforms
In practice, there are some pros and cons of adopting these waveforms:
WOLA-OFDM
- Pros: low OSBE and low complexity
- Cons: guard interval (GI) burden
UF-OFDM and f-OFDM
- Pros: low OSBE
- Cons: GI burden, high complexity, high PAPR, and spectral regrowth
SS-SC-FDMA
- Pros: low PAPR
- Cons: high OSBE and severe spectral efficiency (SE) loss
ZT DFT-S-OFDM
- Pros: low OSBE and low complexity
- Cons: PAPR increase when increasing the number of input zeros
GFDM
- Pros: design flexibility and low out-of-band emission (OOBE)
- Cons: difficulty in frequency domain multiple access support
We focus on subband precoding techniques, since they do not impose
GI burden that may lead to IBI and usually facilitate PAPR reduction.

Others
Review of Existing Subband Precoder Designs
Linearithmic Unitary Additional
Low Low Order Precoder DoF1 in
OSBE PAPR Complexity Support Precoder
SC-FDMA
√ √ √
SS-SC-FDMA2 √ √ √
ZT DFT-S-OFDM3 √
(
√
)
√ √
OP-OFDM4 √ √ √
UP-OFDM5 √ √ √
SP-OFDM6 √ √
Remark:
SS-SC-FDMA achieves extremely low PAPR at the cost of spectral
efficiency (SE) and noise enhancement penalty (NEP).
The PAPR of ZT DFT-S-OFDM significantly increases as the number of
input zeros increases, i.e., the PA efficiency becomes worse.
Apart from DFT-based precoding techniques, other precoder types such
as the precoders built in OP-OFDM, UP-OFDM, and SP-OFDM have
undesirable complexity and compatibility issues for 5G NR actualization.

Others
WOLA-OFDM Transceiver System
Remark:
OSBE is suppressed by prevents steep changes between two blocks.
The block extension and overlapping may eat up CP budget causing IBI.
1
Qualcomm Inc., “R1-162199: Waveform candidates,” in 3GPP RAN1 #84bis, Apr. 2016.
2
Y. Medjahdi et al., “WOLA processing: a useful tool for windowed waveforms in 5G with relaxed
synchronicity,” in Proc. IEEE ICC Workshops on the Main Trends in 5G Networks, pp.393-398, May 2017.

Others
UF-OFDM Transceiver System
Remark:
The subband-wise ﬁltering ideally results in extremely low OSBE but in
practice causes increased PAPR with signiﬁcant spectral regrowth.
The composite channel delay spread may eat up ZP budget causing IBI.
1
F. Schaich, T. Wild, and Y. Chen, “Waveform contenders for 5G - suitability for short packet and low latency
transmissions,” in Proc. IEEE 79th VTC, May 2014.

Others
f-OFDM Transceiver System
Remark:
The subband-wise ﬁltering ideally results in extremely low OSBE but in
practice causes increased PAPR with signiﬁcant spectral regrowth.
The composite channel delay spread may eat up CP budget causing IBI.
1
J. Abdoli, M. Jia, and J. Ma, “Filtered-OFDM: a new waveform for future wireless systems,”
in Proc. IEEE 16th SPAWC, pp. 66-70, June 2015.

Others
DFT-S-OFDM (a.k.a. SC-FDMA) Transceiver System
Remark:
SC-FDMA has been adopted as 4G LTE uplink scheme mainly due to its
low PAPR virtue.
1
H. G. Myung and D. J. Goodman, Single Carrier FDMA: A New Air Interface for Long Term Evolution,
John Wiley & Sons, Ltd., 2008.

Others
SS-SC-FDMA Transceiver System
Remark:
Traditionally, the S × 1 (root) raised cosine vector p is chosen after the
M-point DFT extended by repetition to reduce the PAPR, M < S.1
The vector p can further be designed to statistically reduce the PAPR. 23
1
T. Kawamura et al., “Investigations on optimal roll-oﬀ factor for DFT-spread OFDM based SC-FDMA radio
access in evolved UTRA uplink,” in Proc. ISWCS, Sep. 2006.
2
D. D. Falconer,“Linear Precoding of OFDMA signals to minimize their instantaneous power variance,”
IEEE Trans. on Communications, vol. 59, no. 4, pp. 1154-1162, Apr. 2011.
3
C. H. Yuen and B. Farhang-Boroujeny, “Analysis of the optimum precoder in SC-FDMA,”
IEEE Trans. on Wireless Communications, vol. 11, no. 11, pp. 4096-4107-4008, Nov. 2012.

Others
ZT DFT-S-OFDM Transceiver System
Remark:
Some input data symbols are replaced by zeros.
The OSBE can be reduced.
The PAPR increase depends on the number of used zeros.
The tail of a block (with small power) serves as an internal GI.
1
G. Berardinelli et al., “Zero-tail DFT-spread signals,” in Proc. Globecom Workshop on Broadband Wireless
Access, pp. 229-234, Dec. 2013.

Others
GFDM Transceiver System
Remark:
It offers design flexibility in modulation parameter adaptation.
The spectral leakage, envelope fluctuation, and detection reliability
depend on the choice of the prototype pulse vector built in A.
1
N. Michailow et al., “Generalized frequency division multiplexing for 5th generation cellular networks,”
IEEE Trans. on Communications, vol. 62. no. 9, pp. 3045-3061, Sep. 2014.

Others
GFDM Transmitter Structure
GFDM modulation can be represented by
an N × N square matrix: (N = KM)
A = a0,0, ..., aK−1,0, ..., ak,m, ..., a0,M−1, ..., aK−1,M−1
where
- K: Number of GFDM subcarriers.
- M: Number of GFDM subblocks in a block.
- a0,0: Prototype pulse vector.
- ak,m: Circularly shifted version of a0,0 as below
[ak,m]i = [a0,0] i−mK N
ej2π k
K
i
,
i = 0, 1, · · · , N − 1,
Commonly Known Issues:
A is in general not unitary.
K and M cannot be both even integers, otherwise A could be singular.
GFDM has diﬃculty in multiple access support unless otherwise speciﬁed.

Others
Illustration of GFDM Modulation
An illustration of the GFDM modulation matrix A using
K = 4,
M = 7,
Raised cosine prototype pulse vector with roll-oﬀ factor of 0.5.
0
0
5
10
Row index
15 30
0.1
25
20
20
Column index
15
25
10
Amplitude
5
30
0
Illustration of GFDM modulation matrix
0.2
0.3
0 1 2 3 4 5 6 7
Subblock index (n/K)
-0.05
0
0.05
0.1
0.15
0.2
0.25
Amplitude
K=4, M=7, N=28, Raised Cosine, =0.5
m=0
m=1
m=2
m=3
m=4
m=5
m=6
0 0.5 1 1.5 2 2.5 3 3.5 4
Subcarrier index (n/M)
0
0.2
0.4
0.6
0.8
1
1.2
Amplitude
k=0
k=1
k=2
k=3

Others
GFDM as A Generalized Form of OFDM and SC-FDE
Both OFDM (A = WH
N ) and SC-FDE1 (A = IN ) can be
considered as special cases of GFDM.
1
abbreviation for single-carrier frequency-domain equalization

Others
Frequency-Domain Shaping Approach for CPS Precoding
As a frequency-domain expression of the GFDM modulation matrix, we
can obtain the kth submatrix of P = [P0|P1| · · · |PK−1] as below.1
[ak,m]l = [a] l−mK S
ej2π k
K l
(34)
= ej2π kM
S l
· ([a]l S δ[l − mK]) (35)
[pk,m]i =
S−1
l=0
[ak,m]l e−j 2π
S il
(36)
= δ[i − kM] S [p]i · e−j 2π
M im
(37)
Pk =
0 IkM
IS−kM 0
CkM
(downshift permutation)
diag (p) [IM · · · IM ]
T
R
(repetition)
WM (38)
1
N. Michailow et al.. “Generalized frequency division multiplexing: analysis of an alternative multi-carrier
technique for next generation cellular systems,” in Proc. 9th ISWCS, pp. 171-175, Aug. 2012.

Others
Matrix Characterization for CPS Precoding
As a frequency-domain expression of the GFDM modulation matrix, we
can analogically prove1
P = (WK ⊗ IM ) diag (vec (Γ)) WH
K ⊗ WM .
pk,m =
1
√
S
vec diag
√
M [WM ]m ΓWK
¯Ck (39)
=
1
√
S
vec diag
√
M [WM ]m Γdiag
√
K WH
K
k
WK (40)
=
√
KM
√
S
vec

diag [WM ]m Γdiag WH
K
k
WK

 (41)
= (WK ⊗ IM ) vec

diag [WM ]m Γdiag WH
K
k

 (42)
= (WK ⊗ IM ) diag (vec (Γ)) WH
K ⊗ WM
kM+m
(43)
1

Others
CPS Precoder Complexity Analysis and Key Insights (1/3)
The complexity is evaluated by computing the number of complex
multiplications (CMs) required to send KM data symbols.
Direct implementation ineﬃciently takes KM(KM + log2 KM) CMs.
The time-frequency structure illustrations of OFDMA (A = WH
S ), CPS-OFDM
(A depending on K, M, and a), and SC-FDMA (A = IS) with S = KM = 12:

Others
Frequency-domain implementation takes KM(log2 M + K) CMs.
Remark:
- The repetition and permutation operations do not need
multiplier units in the circuit.
- It clearly reveals that the proposed CPS precoder can be
regarded as a generalized form of DFT-based precoders with
DoF (i.e., the prototype shaping vector p) to be designed.

Others
Characteristic-matrix-domain implementation takes
KM log2 M + MK log2 K + KM + MK log2 K = KM(log2 K2M + 1) CMs.
Remark:
- The complexity is lower than that of frequency-domain
implementation when K−1
2
> log2 K.
- The CPS precoder can be eﬃciently realized with
linearithmic-order complexity O(KM log2 KM).
- The unitarity can be easily identiﬁed1
, i.e.,
P is unitary if and only if [Γ]m,k = 1 (or a constant), ∀k, m.
1

Others
Summary of CPS Precoder Parameters
Parameters of CPS precoding:
S = KM: Number of used OFDM subcarriers in the subband.
K: Number of available shaped subcarriers in the CPS precoder.
M: Number of available subblocks in the CPS precoder.
K ⊆ ZK = {0, 1, · · · , K − 1}:
A set of indices corresponding to used shaped subcarriers, k ∈ K.
M ⊆ ZM = {0, 1, · · · , M − 1}:
A set of indices corresponding to used subblocks, m ∈ M.
D = |K| |M| ≤ S: Number of data symbols in one block transmission.
p: An S × 1 prototype shaping vector to be designed.

Others
Illustration of the Optimized Prototype Shaping Vector
Element index (i) of the optimized prototype shaping vector
0 5 10 15 20 25 30 35 40 45
Amplitude
0
0.5
1
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 2, 0), CP
0 5 10 15 20 25 30 35 40 45
Phase(radian)
-π
0
π
0 5 10 15 20 25 30 35 40 45
Amplitude
0
0.5
1
CPS-OFDM: (K, M, Z, ǫ) = (1, 48, 2, 0.2), CP
CPS-OFDM: (K, M, Z, ǫ) = (1, 48, 2, 0.2), NoGI
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 25, 0.2), CP
0 5 10 15 20 25 30 35 40 45
Phase(radian)
-π
0
π
Observation:
None of NEP ( = 0) may be reflected on flat amplitude of M consecutive
elements for the setting of NoGI. (see the left figure).
Although the use of CP leads to an irregular trend of amplitude, we will find
that the resulting PSD of the passband being flat.
The trends of amplitude look like “sun hat” signifying the existence of NEP
( > 0). (see the right figure)

Others
Simulated PSD and OSBE Results
-5 -4 -3 -2 -1 0
PSD(dBm/30kHz)
-60
-50
-40
-30
-20
-10
0
10
SC-FDMA
CPS-OFDM: (K, M, Z, ǫ) = (1, 48, 2, 0.2), CP
CPS-OFDM: (K, M, Z, ǫ) = (1, 48, 2, 0.2), ZP
ZT DFT-S-OFDM: Z=5
SS-SC-FDMA: RRC, α = 1, Z=24
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 25, 0.2), CP
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 25, 0.2), ZP
SEM
-5 -4 -3 -2 -1 0
PSD(dBm/30kHz)
-60
-50
-40
-30
-20
-10
0
10
SC-FDMA
CPS-OFDM: (K, M, Z, ǫ) = (1, 48, 2, 0.2), CP
CPS-OFDM: (K, M, Z, ǫ) = (1, 48, 2, 0.2), ZP
ZT DFT-S-OFDM: Z=5
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 25, 0.2), CP
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 25, 0.2), ZP
SEM
-4.8 -4.7 -4.6
-19
-18
-17
-3.21-3.2-3.19
-40.1
-40
-39.9

Others
PAPR and Signal User Detection Performance Results
PAPR0 (dB)
0 1 2 3 4 5 6 7 8 9
CCDF:Pr{PAPR>PAPR0}
10-3
10-2
10-1
100
SC-FDMA
CPS-OFDM: (K, M, Z, ǫ) = (1, 48, 2, 0.2), CP
CPS-OFDM: (K, M, Z, ǫ) = (1, 48, 2, 0.2), ZP
ZT DFT-S-OFDM: Z=5
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 25, 0.2), CP
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 25, 0.2), ZP
6.4 6.5 6.6
0.085
0.09
0.095
0.1
0.105
Eb/N0 (dB)
0 5 10 15 20 25 30
BER
10-4
10-3
10-2
10-1
100
SC-FDMA
CPS-OFDM: (K, M, Z, ǫ) = (1, 48, 2, 0.2), CP
CPS-OFDM: (K, M, Z, ǫ) = (1, 48, 2, 0.2), ZP
ZT DFT-S-OFDM: Z=5
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 25, 0.2), CP
CPS-OFDM: (K, M, Z, ǫ) = (2, 24, 25, 0.2), ZP
24.98 25
×10-3
1.5
2
2.5

20190117 Ph.D. Dissertation Oral Defense

20190117 Ph.D. Dissertation Oral Defense

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20190117 Ph.D. Dissertation Oral Defense