1 N ∥x∥22.
1.6 Acronyms
The commonly used abbreviations in the dissertation are listed in Table 1.1 for readability.
3GPP The Third Generation Partnership Project 4G The Fourth Generation Wireless Systems 5G The Fifth Generation Wireless Systems
BER Bit error rate
CI Convex iteration
CP Cyclic prefix
CPS-OFDM Circularly pulse-shaped OFDM
DFT-S-OFDM Discrete Fourier transform spread OFDM
f-OFDM Filtered OFDM
GFDM Generalized frequency division multiplexing
GI Guard interval
IBI Inter-block interference
IBO Input backoff
LTE Long Term Evolution
MM Majorization-minimization
MMSE Minimum mean square error
NEP Noise enhancement penalty
NoGI Without GI
NR New Radio
OFDM Orthogonal frequency division multiplexing OFDMA Orthogonal frequency division multiple access
OSBE Out-of-subband emission
OSBEP OSBE power
PA Power amplifier
PAPR Peak-to-average power ratio
PSD Power spectral density
SC-FDMA Single-carrier frequency division multiple access SS-SC-FDMA Spectrally shaped SC-FDMA
SE Spectral efficiency
UF-OFDM Universal-filtered OFDM VIP Variance of instantaneous power WOLA-OFDM Weighted overlap-and-add OFDM
ZP Zero padding
ZT DFT-S-OFDM Zero-tail DFT-S-OFDM Table 1.1: List of abbreviations.
Chapter 2
Circularly Pulse-Shaped Precoding for OFDM and Its Optimal Prototype
Shaping Vector Design
A new waveform called circularly pulse-shaped OFDM (CPS-OFDM), along with its op-timal prototype shaping vector design, is fully introduced in this chapter. CPS-OFDM, characterized by user-specific DFT-based precoder flexibility, possesses the advantages of both low out-of-subband emission (OSBE) and low peak-to-average power ratio (PAPR).
CPS-OFDM prevents block extension causing extra inter-block interference (IBI) and en-velope fluctuation unfriendly to signal detection and power amplifier (PA) efficiency, re-spectively. An optimization problem of the prototype shaping vector is formulated to minimize the variance of instantaneous power (VIP) with controllable OSBE power (OS-BEP) and noise enhancement penalty (NEP). To solve the optimization problem involving a quartic objective function, we exploite the majorization-minimization (MM) algorithmic framework. The convexity of the proposed problem is proved so that the globally optimal solution invariant of incoming data is guaranteed to be attained. Simulation results show the merits of the proposed waveform scheme in terms of detection reliability and spec-tral efficiency (SE) for practical 5G cases such as asynchronous transmissions and mixed numerologies. Most of the contents in this chapter were published in [48].
2.1 System Model and Problem Statement
We consider a precoded OFDM system equipped with N OFDM subcarriers, among which each user in the system is assigned to a certain set of contiguous subcarriers. The baseband uplink transceiver model of the system in the view of a single user is schematized in Fig.
2.1. At the transmitter, the input S × 1 data vector of the bth block transmission d[b]
is first precoded by an S× S precoding matrix P to obtain s[b] = Pd[b]. The precoder P can be designed to obtain desired waveform properties. The precoded symbols s[b]
are then assigned to S contiguous OFDM subcarriers, whose indices are given inI = {η, η + 1, · · · , η + S − 1} with η ≥ 0 and η+S ≤ N. An N-point IFFT, characterized by WHN, is used for OFDM modulation. To prevent inter-block (IBI) stemmed from channel delay spread, a guard interval (GI) is added on each time-domain block signal xN[b] = [WHN]
Is[b]. The GI insertion can be represented by a matrix G chosen to be either
Gcp =
for cyclic prefix (CP) or zero padding (ZP) of length G, respectively. Thus, the transmitted signal of the bth block containing N′ = N + G samples is formulated as
x[b] = GxN[b] = G[ and call it a synthesis matrix. After parallel-to-serial conversion (P/S) of (2.2), the digital baseband transmit signal
is sent over a frequency-selective channel. The channel can be modeled as a linear time-invariant finite impulse response (FIR) filter H (ejω) =∑L
l=0h[l]e−jωlwith channel order L. The channel impulse response vector is h = [h[0] h[1]· · · h[L]]T ∈ C(L+1)×1.
Figure 2.1: Subband-wise precoded-OFDM baseband transceiver system model.
At the receiver, serial-to-parallel conversion (S/P) of sequentially incoming y[n] =
∑L
l=0h[l]x[n−l]+z[n] is first performed, where z[n] is a complex additive white Gaussian noise (AWGN) with variance N0. The received signal of the bth block can be written as [58]
y[b] = Tlowx[b] + T|upx[b{z− 1]}
IBI
+z[b], (2.4)
where Tlowis an N′×N′lower triangular Toeplitz matrix with its first column[
hT 0· · · 0]T
, Tupis an N′× N′upper triangular Toeplitz matrix with its first row [0· · · 0 h[L] · · · h[1]], and z[b] is a blocked noise vector with its covariance matrix N0IN′. For CP removal or overlap-add manipulation [58], a matrix ¯G is chosen as
G¯cp = [
0N×G IN ]
or ¯Gzp =
IG 0 IG 0 IN−G 0
, (2.5)
respectively. Under the assumption of G≥ L, we can extract the N × 1 received vector yN[b] from (2.4) without IBI, i.e.,
yN[b] = ¯Gy[b] = ¯GTlowx[b] + ¯Gz[b], (2.6)
where ¯G = ¯Gcpor ¯G = ¯Gzp, depending on the type of G used in the transmitter. Then, an
N -point FFT is applied to yN[b]. The received frequency-domain signal to be processed is expressed as [59, Ch. 7]
r[b] = HPd[b] + v[b], (2.7)
where H = diag([WN[hT 0T(N−L−1)×1]T]I) has diagonal elements corresponding to chan-nel frequency response on the occupied S subcarriers and v[b] =[
WTN]T
I Gz[b] is the noise¯ vector. Finally, the received data vector can be linearly obtained by
d[b] = Qr[b],ˆ (2.8)
where Q is a frequency-domain equalization (FDE) matrix in zero forcing (ZF) or mini-mum mean square error (MMSE) sense.
In some cases, the data vector d[b] is composed of D data symbols drawn from a quadrature amplitude modulation (QAM) constellation and Z zeros, S = D + Z. Let D ⊆ ZS be the set of indices indicating the locations of D data symbols in d[b],∀b. All data symbols are assumed to be zero-mean, independent and identically distributed (i.i.d.) with symbol power Es, i.e.,
E {[d[b]]D} = 0D×1, ∀b, (2.9)
E{
[d[b]]D[d[b′]]HD }
= EsIDδb,b′, ∀b, b′, (2.10)
and jointly wide sense stationary and uncorrelated with noise. Let ¯P = [P]D. The matrix Q in (2.8) can be chosen as [59]
QZF = [
(H¯P)H(H¯P) ]−1
(H¯P)H (2.11)
or
in the sense of ZF-FDE or MMSE-FDE, respectively.
2.1.1 Quantifying Spectral Sidelobe Leakage Using OSBEP
Spectral sidelobe leakage of a user, referred to as out-of-subband emission (OSBE), is evaluated by calculating the power spectral density (PSD). Under the assumptions of (2.9) and (2.10), the PSD of (2.3) is given by [59] be concerned. To quantify the OSBE, a commonly used approach is to compute its total power, namely, OSBE power (OSBEP) [40–42], [59, Ch. 9]
γx =
It is worthy to note that an actual baseband PSD expression involves an interpolation filter used in a digital-to-analog converter (DAC). Specifically, the analog transmitted signal x(t) is obtained by passing x[n] through a DAC with a sampling period Ts and an interpolation filter G(f ). The PSD of x(t) is Sx˜(f ) = T1
sSx( ejf Ts)
|G(f)|2 [59, Eq.
(6.32)]. Since G(f ) mainly affects the spectral attenuation outside the sampling bandwidth [−π/Ts, π/Ts), for simplicity we assume |G(f)| = 1 for |f| < π/Ts and |G(f)| = 0 otherwise, and address the amount of OSBE by (2.14) in this study.
2.1.2 Quantifying Envelope Fluctuation Using VIP
A measure closely related to the nonlinear distortion caused by a power amplifier (PA), namely, variance of instantaneous power (VIP), is known as a more practical metric than peak-to-average power ratio (PAPR) [92]. The main reason is that keeping power effi-ciency sufficiently high is usually much more important to a user equipment (UE) than taking large input backoff (IBO) [61]. The VIP averaged over the bth block is defined as [28, 29, 37, 41] over the bth block given by
¯ easily found that (2.15) and (2.16) are independent of the block index b.
In the subsequent contents, the block index “[b]” is omitted for notational brevity, since the design of P taking (2.13)-(2.16) into account is invariant of incoming data and there is no IBI (2.6).
2.1.3 Subband Precoder Design Problem
This chapter studies the subband precoder design problem for the precoded OFDM system described in Section 2.1. Specifically, we intend to design the S× S precoding matrix P such that the OSBEP (2.14) and the VIP (2.15) can be simultaneously reduced as compared to OFDMA. The detailed problem statement will be formulated in Sections 2.2.2, 2.2.3, and 2.3.
In addition, it is also desirable for the precoding matrix P to possess a low-complexity implementation and at the same time not to cause significant receiver performance
degra-dation. For the complexity concern, we notice that in the most general form, there are S2 complex-valued coefficients in P to be specified, resulting in an undesired quadratic order complexity (i.e., O(S2)). We thus seek some constraints on the precoder structure to make the implementation efficient in linearithmic order. On the other hand, for the concern of performance degradation, it will be helpful when the precoder P is chosen to be unitary (i.e., PHP = IS), so that the noise enhancement penalty (NEP) at the receiver will be avoided. The above two issues will be addressed in more details in Sections 2.2.1 and 2.2.4, respectively.