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Low PAPR reference signal transceiver design for 3GPP 5G NR uplink
EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 182 (2020)
Abstract
Low peaktoaveragepower ratio (PAPR) transmissions significantly improve the cell coverage as they enable high power transmissions without saturating the power amplifier. To support the low PAPR transmissions, π/2BPSKbased data and DMRS are introduced in the latest version of 5G NR specifications. In addition to that, the spatial multiplexing support is also extended to π/2BPSK data transmissions. The DMRS sequences corresponding to these spatial streams (users) are frequency division multiplexed (FDM). However, the spectrum shaping process employed in the generation of π/2BPSK waveforms is frequency selective and hence results in asymmetric spectrum shaping effect on DMRS sequences, when they are frequency multiplexed. This subsequently results in a nonuniform block error (BLER) and PAPR performances across the spatial users, which in turn may result in inter user interference across the spatial users. In this paper, we propose two transmitter architectures, namely method 1 and method 2, to generate low PAPR π/2BPSKbased DMRS waveforms. The proposed architectures ensure that the spectrum shaping effect is uniform across all the spatial streams. We corroborate through simulations that the proposed architectures will result in identical block error and PAPR performances across all the spatial streams.
Introduction
For a cellular network, uplink transmissions define the coverage area. The transmission power in the uplink is limited to 23 dBm as opposed to 43 dBm in the downlink [1], owing to hardware limitations (such a battery size) and regulatory constraints. This limited transmission power in the uplink must, therefore, be judiciously used to enhance the cell coverage without increasing the CAPEX/OPEX costs of deploying more cell sites. Therefore the uplink design of a cellular standard is often crucial.
To enhance the cell coverage of the newly designed 3GPP 5G NR, a new modulation scheme, namely π/2BPSK, was introduced for the uplink data channel (physical uplink shared channel PUSCH) and control channel (physical uplink control channel  PUCCH) transmissions. The π/2BPSK modulation scheme, when transmitted using discrete Fourier transform spread orthogonal frequency division multiplexing (DFTsOFDM waveform), offers low PAPR when compared to higherorder modulation schemes including QPSK, 16QAM, and others. The complementary CDF (CCDF) of PAPR for various modulation schemes is shown in Fig. 1, which clearly shows the low PAPR behavior of the π/2BPSK modulation scheme. To further reduce the PAPR, spectrum shaping is applied to the π/2BPSK symbols. Spectrum shaping is a PAPR reduction technique which can be performed either in time domain or frequency domain [2, 3]. Specifically, the PAPR of the π/2BPSK modulation scheme with DFTsOFDM waveform and spectrum shaping is smaller than 2 dB. Moreover, it is shown in [2, 3] that the power amplifier can be driven to saturation, and yet the error rate performance of this modulation scheme is not compromised. Hence, this modulation scheme plays a crucial role in significantly enhancing the cell coverage for 3GPP 5G NRbased cellular networks.
The demodulation reference signals (DMRS) employed for coherent data demodulation in uplink of the current 5G NR systems are generated using ZadoffChu (ZC) sequences or QPSKbased computergenerated sequences(CGS) as specified in Section 5.2.2 in [4] and Section 6.2.2 in [5]. The CCDF of PAPR for a DFTsOFDM waveform with spectrumshaped π/2BPSK data symbols and the ZadoffChu based DMRS sequences is shown in Fig. 2. It can be seen that PAPR of π/2BPSK data is lower than that of the ZC sequences by over 2dB. Therefore, even though the data transmissions have low PAPR and potentially allow for larger coverage, the DMRS design still limits the cell size due to its high PAPR. For this reason, 3GPP introduced a new study item in Rel16 to design new reference sequences with lower PAPR [6]. In one of our associated contribution [7], a stepbystep procedure to generate low PAPR DMRS sequences for various lengths is provided. The sequences in [8, 10–13] were agreed to be used as lowPAPR reference sequences. In this paper, we employ them as input to the proposed transceiver architectures.
The specifications for 5G NR also support multiple stream transmissions using the DFTsOFDM waveform. In other words, a single user can be scheduled to transmit multiple streams, or multiple users can be configured simultaneously to transmit multiple streams depending on the channel conditions. In order to support these multiplestream transmissions, multiple DMRS sequences are required, one for each stream. This is achieved by introducing the concept of baseband DMRS port, where one single port is assigned for demodulation of each stream [4, Sec 6.3.1.3]. Where in [4, Sec 4.4.1], a DMRS port is defined such that the channel over which a symbol on the DMRS port is conveyed can be inferred from the channel over which another symbol on the same DMRS port is conveyed. Since the DMRS of each stream must be independently decoded to derive channel estimates, the DMRS sequences must be orthogonally separated to avoid any interference.
The orthogonality across the ports can be achieved by either frequency division multiplexing (FDM) or code division multiplexing (CDM) or timedivision multiplexing (TDM). However, the current 3GPP 5G NR [8] supports only FDM with a maximum of two streams when DFTsOFDM waveform is enabled. In the FDM method, the same DMRS sequence is employed by all the DMRS ports, but orthogonalized in frequency, as shown in Fig. 3, in FDM, the length of DMRS on each port will be \(\frac {M}{Q}\), corresponding to the data allocation of length M subcarriers, where Q indicates the number of DMRS ports, which is limited to 2 (port−0 and port−1) as per the current 3GPP specifications [9] and also for the scope of this paper. A typical data DMRS multiplexing in the current 5G NR systems is shown in Fig. 4. Here, a few of the DFTsOFDM symbols are dedicated for DMRS transmission while the other symbols carry the user data. Channel estimates derived from these DMRS symbols will be employed for coherent demodulation of data.
The existing ZC or QPSK based DMRS generation does not include the spectrum shaping; however, spectrum shaping for π/2BPSK data waveform is briefly outlined in [1]. To accurately derive the channel estimates from DMRS for coherent demodulation, the spectrum shaping has to align between data and DMRS transmissions. Since the exact spectrum shaping procedure for DMRS is not known, and if we continue to follow the same spectrum shaping principles for DMRS as well as data, then it results in asymmetric spectrum shaping effect on port0 and port1, which eventually result in portspecific channel estimation and PAPR performances, which will be shown later in the Section 4.
In this work, we address the problem of Low PAPR DMRS waveform generation in the context of spatial multiplexing. The key contributions in the paper are

1
Firstly, we present the generation of π/2BPSK based DMRS waveform and show that the PAPR of π/2BPSK based DMRS waveforms is almost 2 dB lesser than the conventional ZC based DMRS waveform.

2
For the case of spatial multiplexing, we propose two transmitter architectures, namely transmission method 1 and transmission method 2. The proposed methods ensure that the spectrum shaping is aligned across all the scheduled DMRS ports and thereby similar PAPR and BLER performances across the ports.

3
Transmission method 1 implements the spectrum shaping in frequency domain, where some fundamental DFT properties are invoked to ensure the similar spectrum shaping effect.

4
Transmission method 2 implements the spectrum shaping in the time domain through convolution procedure.

5
We establish the equivalence between method 1 and method 2 and show that the DMRS waveforms generated by both the methods are exactly identical.
At the receiver, we employ a DFTbased channel estimation procedure to estimate the joint impulse response of the wireless channel and spectrum shaping filter, which further is employed for data demodulation. We corroborate through simulations that both transmission method 1 and transmission method 2 result in identical channel estimation and PAPR performance on all the scheduled DMRS ports. Note that, as per the latest 5G specifications [9], when π/2BPSK data transmission is enabled, a maximum of 2 spatial users are supported. Hence, the analysis in this paper is confined to 2 DMRS ports; however, the proposed methods as such can be readily extended to any number of DMRS ports, which is explained later in Sections 2.4 and 2.5.
The rest of the paper is organized as follows. The π/2BPSK data and π/2BPSK DMRS generation using the proposed transmitter designs is discussed in Section 2. The receiver architecture, mainly the channel estimator employed to derive the channel estimates on both port0 and port1, is described in Section 3. The block error rate(BLER) and PAPR performances comparing the proposed and existing designs are presented and discussed under Section 4. Finally, we summarize and conclude the paper in Section 5.
Notation: The following notations are used in this paper. Bold uppercase letters X denote matrices, bold lowercase letters x denote vectors, nonboldface letters represent scalars, and x_{t},y_{f} indicates the time domain and frequency domain vectors x and y respectively. x^{T} and X^{†} represent the transpose and Hermitian operations on the vector x and matrix X respectively. We use the symbol x to denote the data symbols and r to denote reference signal symbols. x.y represents the Hadamard product of vectors x and y.
Transmitter architecture for π/2BPSK data and DMRS generation
In this section, we present transmitter designs to generate low PAPR data and DMRS waveforms. We first describe the system model, including the design of the DFTsOFDM data waveform as per the current 5G NR specifications, and then discuss the proposed transmitter designs.
π/2BPSK symbol generation
The current NR specifications [4, 5] specify DFTsOFDM for uplink transmissions in the coverage limited scenarios. Also, in addition to QAM modulation techniques, a new modulation scheme, namely, π/2BPSK, was introduced in 5G NR. In BPSK modulation; the input bit sequence b(m) is mapped to complexvalued modulation symbol x_{t}(m) as given below
whereas π/2BPSK is a special constellationrotated BPSK modulation, in which the evennumbered symbols are transmitted as like in BPSK, and the oddnumbered symbols are phase rotated by π/2 as given in (1)
Here, the subscript p in x_{p}(m) indicates a phase rotated sequence and the subscript t in x_{t}(m) indicates a time domain sequence. \(i=\sqrt {1}\) and M is the length of a BPSK sequence x_{t}(m).
Note: In order to maintain consistency with the other modulation mappings, for BPSK, the input bits are mapped to the rotated PSK [4], rather than from the set {−1,1},
The π/2phase rotation can be equivalently expressed in vector notation as given below,
where x_{t} is a M length BPSK vector and P is M×M diagonal matrix with diagonal entries \(p_{mm}=e^{i\hspace {0.5mm}\left (m \hspace {5pt} \mod 2 \right)\frac {\pi }{2}}\). Note that, although the constellation is similar to QPSK, we can only transmit 1bit on one π/2BPSK modulation symbol.
Spectrum shaping
The π/2BPSK symbols, when combined with an appropriate spectrum shaping, enables low PAPR transmissions without compromising the error rate performance [2–4]. Spectrum shaping can be performed either in time or frequency domain. In the case of frequencydomain processing, spectrum shaping can be performed by means of a spectrumshaping function w_{f}=D_{M}w_{t}, where w_{t} is zeropadded time domain impulse response of the Ltap spectrum shaping filter i.e., \(\mathbf {w}_{t}=\left [w(0),w(1),..w(L1),\underbrace {0,\ldots,0}_{ML}\right ]^{T}\). Where D_{M} is an M×M DFT matrix given by
Remark on the length of the spectrum shaping filter: The spectrum shaping is implementationspecific and is generally unknown at the receiver. In such a case, the receiver needs to estimate the impulse response of the spectrum shaping filter and wireless channel jointly. Let L_{DMRS},L_{CIR},L_{Fil} be the length of the DMRS sequence, channel impulse response (CIR) of the wireless channel, and length of spectrum shaping filter, respectively. According to [12], the DMRS sequence length should be at least equal to the length of the joint impulse response of the wireless channel and the spectrum shaping filter, i.e.,
As per [4, 5], the minimum allocation size for data transmissions is 12 subcarriers, for which the length of the DMRS sequence (L_{DMRS}) will be 6 subcarriers for the case of twostream transmission. Also, as per 3GPP channel models [14], for an allocation size of 12 subcarriers, the maximum length of channel impulse response (L_{CIR}) will be ≤3. Invoking the relation in (4), it can be inferred that the spectrum shaping filter can have a maximum of 3 taps. Hence, the spectrum shaping filters should be chosen such that their impulse response is limited to 3 taps or less. Commonly employed spectrum shaping filters with 2 and 3tap impulse response are w_{1}=[1,−1] and w_{2}=[−0.28,0,−0.28] respectively. The corresponding frequency spectrum of these filters is shown in Fig. 5.
DMRS signal structure
As discussed in Section 1, when multiple DMRS sequences are transmitted on frequency division multiplexed DMRS ports, spectrum shaping needs to be performed properly to ensure identical spectrum shaping effect on data and DMRS, which otherwise results in nonidentical channel estimation performance (and thereby equalization and demodulation) across the DMRS ports, which is not desirable.
Hence, the DMRS transmitter design, besides minimizing the PAPR of the waveform, should also ensure that the characteristics of the waveform (like autocorrelation and crosscorrelation) are similar across all the DMRS ports.
Transmission method 1
In this section, we present method 1 of data and DMRS waveform generation, wherein the spectrum shaping is performed in the frequency domain.
Data waveform design method 1
The transmitter architecture for data waveform generation is shown in Fig. 6. Let x_{p} denote an M×1 vector of π/2BPSK modulated data symbols generated as per (1). For transmission via DFTsOFDM, the π/2BPSK data symbols are first DFTprecoded as
The subscript f in x_{f}(k) indicates a frequency domain sequence. The DFT precoding shown in (5) can be equivalently represented in vector notation form as
where D_{M} is a M×M DFT matrix given by (3). The spectrum shaping is performed on the DFTprecoded data vector as \({\mathbf {x}_{f}^{s}}=\mathbf {w}_{f}\mathbf {\Huge {.}}\mathbf {x}_{f}\), where \({\mathbf {x}_{f}^{s}}\) indicates the spectrumshaped frequency domain data x_{f}. The spectrumshaped data vector \({\mathbf {x}_{f}^{s}}\) is then mapped to a set of subcarriers in frequency domain via an N×M mapping matrix M_{f} where M≤N. The block diagonal mapping matrix M_{f} is defined below (without loss of generality, the DFT precoded data is mapped to the initial M subcarriers)
Finally, the output of this mapping operation is converted to N×1 time domain vector s_{t} as
where \(\mathbf {D}_{N}^{\dagger }\) is an inverse DFT matrix and N is the total number subcarriers corresponding to the system bandwidth. An appropriate length cyclic prefix is added to s_{t} to generate s_{t}(t) as given by equation (5.3.1) in [4].
DMRS waveform design method 1
As mentioned in Section 1, the current 5G system supports only 2 MIMO streams in the uplink, and the corresponding DMRS is multiplexed in FDM manner [10]. Hence, \(\frac {M}{2}\)length DMRS sequences will be transmitted on port 0 and port 1 corresponding to M length data allocation. We will next present transmitter architectures for port 0 and port 1, respectively, such that the resultant DMRS waveforms from either of the ports have low PAPR and similar characteristics.
The transmitter architectures for port 0 and port 1 are shown in Figs. 7 and 8. In this architecture, the transmitter design is such that a given time domain DMRS sequence r_{t} will result in an identical spectrumshaped frequency domain sequence for both DMRS ports. This subsequently results in similar auto and crosscorrelation properties and hence produces an identical channel estimation performance at the receiver. The summary is tabulated in Table 1.
DMRS waveform generation for port 0:
Let r_{t} be a predetermined \(\frac {M}{2}\) length DMRS sequence with BPSK modulated symbols chosen as per the designs in [6–13]. This will be cyclically extended to give a M length vector \({\tilde {r}}_{t} (n)\) as follows
Using the diagonal matrix P defined in (2), a π/2phase rotation is applied on \({\tilde {r}}_{t} \) to give \({\tilde {r}}_{t}^{p}=\mathbf {P}{\tilde {r}}_{t}\). The resultant π/2BPSK sequence is DFT precoded as \(\mathbf {r}_{f}^{p_{0}}=\mathbf {D}_{M} {\tilde {r}}_{t}^{p}\). Invoking the DFT property that when a sequence is repeated twice in time domain, then it will have a comb like structure in frequency domain. Hence, the DFT output \(\mathbf {r}_{f}^{p_{0}}\) will have nonzero entries at the even locations as shown in Fig. 3 (and hence the notation \(\mathbf {r}_{f}^{p_{0}}\)). The DFTprecoded DMRS symbols are spectrumshaped using w_{f} defined in Section 2.2 to give the spectrumshaped port0 DMRS as
DMRS waveform generation for port 1:
When multiple DMRS ports are frequency multiplexed, the DMRS sequence should be identical on both ports [8] i.e., the input BPSK sequence r_{t} and the resulting \(\frac {\pi }{2}\) BPSK sequence \({\tilde {r}}_{t}^{p}\) has to be same for both port 0 and port 1. However, different from port 0, to generate the spectrumshaped frequency domainDMRS sequence on port 1, the following additional steps need to be performed

a precoder T is applied on \({\tilde {r}}_{t}^{p}\), where T is a M×Mdiagonal matrix with diagonal entries T_{mm}=e^{i2πm/M} followed by DFT precoding as shown below
$$ \mathbf{r}_{f}^{p_{1}}=\mathbf{D}_{M} \mathbf{T}{\tilde{r}}_{t}^{p}. $$Invoking the DFT property defined in the Section 2.4.2 and the frequency shift property, it can be seen that \(\mathbf {r}_{f}^{p_{1}}\) is a comblike structure with nonzero entries only at odd subcarriers equivalent to port1 mapping as given in Fig. 3.

Spectrum shaping of \(\mathbf {r}_{f}^{p_{1}}\) is done as follows,
$$ \mathbf{r}_{f}^{s_{1}}=(\mathbf Z \mathbf w_{f}) \mathbf{\Huge{.}} \mathbf{r}_{f}^{p_{1}} $$where, Z is a squarecirculant matrix of size M×M whose 1st row entries are \(\left [\underbrace {0,0,...,0}_{M1}, 1\right ]\).
Note: The precoder (Z) on the spectrum shaping filter (w_{f}) circularly rotate the frequency coefficients of spectrum shaping filter by 1 sample for port 1. This ensures that the same set of filter coefficients gets applied on π/2BPSK sequence \({\tilde {r}}_{t}^{p}\) for both port 0 and port 1 DMRS. Hence, the effect of spectrum shaping is identical on both port 0 and port 1 DMRS, whereas, without precoder Z, the nonzero entries of the spectrumshaped sequences \(\mathbf {r}_{f}^{s_{0}}, \mathbf {r}_{f}^{s_{1}}\) are not identical as shown in Fig. 9. This will be shown in asymmetric PAPR and channel estimation performance on port 0 and port 1, which is not acceptable in any MIMO system.
Using the proposed architecture, it can be shown that the output of the spectrum shaping filter is identical for both ports, i.e.,
where \(\mathbf {r}_{f}^{p_{0}}(k)\) is the M point DFT of π/2BPSK sequence \({\tilde {r}}_{t}^{p}\). Therefore, the same reference signal is transmitted on each baseband DMRS port, thereby satisfying the specifications provided in [10]. We further show in Section 3 that the estimated channel impulse response will also be the same on both ports, considering identical channel conditions.
The spectrumshaped DMRS vectors \( \mathbf {r}_{f}^{s_{0}}, \mathbf {r}_{f}^{s_{1}}\) are mapped to a set of subcarriers in frequency domain using the matrix M_{f} as discussed in Section 2.4.1. The resulting output is converted to time domain via inverseDFT operation similar to the method employed for data transmission as shown below
Using the above, the overall time domain baseband signals \(\mathbf {s}_{t}^{0}(t), \mathbf {s}_{t}^{1}(t)\) with an appropriate cyclic prefix are generated as given by equation (5.3.1) in [4].
Note: As mentioned in the introduction section, the latest 5G specifications [9] only support a maximum of 2 spatial users; hence, the transmitter design is discussed in the context of 2 ports. However, the proposed transmitter design can be easily extended to any number of ports. Let “Q” be the total number of spatial users scheduled on a given time frequency resource. The proposed method can be easily extended to generate the “Q” DMRS waveforms with the following minor adjustments.

The DMRS sequence corresponding to each of “Q” DMRS ports will be of length \(\frac {M}{Q}\).

The \(\frac {M}{Q}\) length DMRS is cyclically extended to give a M length vector \({\tilde {r}}_{t} (n)\) as follows
$$ {\tilde{r}}_{t} (n)={r}_{t}\left(n \mod\frac{M}{Q}\right), n=0,1,\ldots,M1. $$ 
A portspecific precoder T_{q} is applied on \({\tilde {r}}_{t}^{q}\), where T_{q} is a M×Mdiagonal matrix with diagonal entries T_{mm}=e^{i2πmq/M}, where q is the port number,
q={0,1,2,...Q−1}
$$ \mathbf{r}_{f}^{p_{q}}=\mathbf{D}_{M} \mathbf{T}_{q} {\tilde{r}}_{t}^{q}. $$ 
A portspecific precoder Z_{q} is applied on the respective spectrum shaping filters of each port to result in a spectrumshaped DMRS sequence as follows
$$ \mathbf{r}_{f}^{s_{q}}=\left(\mathbf Z_{q} \mathbf w_{f}\right) \mathbf{\Huge{.}} \mathbf{r}_{f}^{p_{q}} $$
Transmission method 2
In the method 1based transmitter design, the π/2BPSK data and DMRS sequences are spectrumshaped in the frequency domain. Further, the DFTprecoded DMRS sequences corresponding to each DMRS port are generated and spectrumshaped independently; however, some additional processing is required to design the spectrumshaped sequence for the port 1 compared to port 0. In the method 2based design, we propose a low complexity architecture, where spectrum shaping is performed in time domain for both data and DMRS via circular convolution operation. Specifically, a single DMRS sequence is spectrumshaped in time domain and mapped to both DMRS ports. In contrast to method 1, no additional processing is required to generate the reference sequence for port 1. The architecture for this transmitter design for the data and DMRS is shown in Figs. 10 and 11, respectively.
Data waveform design method 2
Let x_{t} be the M length data vector to be transmitted from the UE to base station that undergoes a π/2phase rotation through an M×M diagonal matrix P. Here, P is the same matrix used in method 1. This results in an M length data vector \(\mathbf {x}_{t}^{p}= \mathbf {P} \mathbf {x}_{t}\) with π/2BPSK symbols. The spectrum shaping of \(\frac {\pi }{2}\)BPSK data is performed in time domain through a circularconvolution procedure with zeropadded w_{t} to produce a spectrumshaped data as,
The spectrumshaped data sequence is DFT precoded by means of M point DFT matrix as \(\mathbf {x}_{f}^{s} =\mathbf {D}_{M}\mathbf {x}_{t}^{s}\). The DFT precoded spectrumshaped data vector is mapped to a set of subcarriers in frequency domain via a mapping matrix M_{f} (described in Section 2.4.1). Finally, this mapped sequence is converted to time domain via inverseDFT operation as.
Using the above, the overall time domain baseband signals s_{t}(t) with appropriate length cyclic prefix are generated as per equation (5.3.1) in [4].
DMRS waveform design method 2
Let r_{t} be the predetermined \(\frac {M}{2}\) length DMRS sequences (as mentioned earlier in method 1) with BPSK modulated symbols, which undergo π/2phase rotation through diagonal matrix P_{1} of sizes \(\frac {M}{2}\times \frac {M}{2}\). The diagonal entries of P_{1} are given by \(\left (e^{i\left (m \hspace {5pt} \mod 2\right)\frac {\pi }{2}}\right)\). This results in an \(\frac {M}{2}\) length DMRS vector \(\mathbf r_{t}^{p}= \mathbf P_{1} \mathbf r_{t}\) with π/2BPSK symbols. The spectrum shaping of the DMRS symbols is performed in time domain through a circularconvolution procedure with zeropadded w_{t} to produce a spectrumshaped DMRS sequences as
The spectrumshaped DMRS sequence is DFT precoded by means of \(\frac {M}{2}\) point DFT matrix as \(\mathbf {r}_{f}^{s} =\mathbf {D}_{\frac {M}{2}}\mathbf {r}_{t}^{s}\). The DFT output of DMRS sequence generated above is mapped to port 0 as
and to port 1 as
In the above equations, \(\mathbf {r}_{f}^{s_{0}}\) and \(\mathbf {r}_{f}^{s_{1}}\) indicate the frequency domain DMRS sequences on port 0 and port 1 respectively. It can be seen that with the proposed architecture the nonzero entries of DMRS sequence are exactly identical for both ports i.e.,
where \(\mathbf r_{f}^{p}(k), \mathbf w_{f}(k)\) are the \(\frac {M}{2}\)DFT outputs of π/2BPSK DMRS symbol \(\mathbf {r}_{t}^{p}\) and filter w_{t} respectively. The DFT precoded spectrumshaped data and DMRS vector of each port is mapped to a set of subcarriers in frequency domain via mapping matrix M_{f} (described in Section 2.4.1). Finally, this mapped sequence is converted to time domain via inverseDFT operation as
Using the above, the overall time domain baseband signals for DMRS transmission i.e., \(\mathbf {s}_{t}^{0}(t), \mathbf {s}_{t}^{1}(t)\) with appropriate length cyclic prefix are generated as per equation (5.3.1) in [4].
Note: Similar to the transmission method 1, method 2 can also be extended to arbitrary number of scheduled ports “Q” using the following steps:

The predefined \(\frac {M}{Q}\) length vector r_{t} with BPSK modulated symbols is \(\frac {\pi }{2}\) rotated using an \(\frac {M}{Q} \times \frac {M}{Q}\) diagonal matrix P_{q} as,
$$ \mathbf{r}_{t}^{p} = P_{q} \mathbf{r}_{t}. $$where the diagonal entries of P_{q} are given by \(e^{\left (i(m\hspace {2pt} mod \hspace {2pt} 2)\right)\frac {\pi }{2}}\).

The time domain spectrum shaping filter w_{t} is used to perform time domain spectrum shaping of \(\mathbf {r}_{t}^{p}\) as follows:
$$ \mathbf{r}_{t}^{s} = \sum_{n = 0}^{{\frac{M}{Q}}  1} {\mathbf{r}_{t}^{p} \mathbf{w}_{t}(n+m) \hspace{4pt} mod \hspace{4pt} \frac{M}{Q} }. $$ 
Matrix \(D_{\frac {M}{Q}}\) is used to perform DFT precoding as
$$ \mathbf{r}_{f}^{s} = \mathbf{D}_{\frac{M}{Q}} \mathbf{r}_{t}^{s} $$ 
Portspecific mapping for an arbitrary port q is performed as
$$\begin{array}{*{20}l} \mathbf{r}_{f}^{p} (k) = \mathbf{r}_{f}^{s} \left(\frac{kq}{2} \right),\\ & where, p = \left\{0,1,2,....Q1\right\}\\ & k = \left\{p,p+Q,p+2Q......\right\} \end{array} $$
Summary of the transmission methods
We presented two transmission methods for the data DMRS waveform generation. Specifically, in method 1, the processing happens in frequency domain, while in method 2, the processing happens in the time domain via the circularconvolution operation. In the frequency domain method, the spectrum shaping is performed on the M length sequence. Hence, the spectrum shaping filters must be defined for length M; an additional care must be taken to design this spectrumshaped sequence for the port 1 vs. port 0, whereas, in the time domain method, the spectrum shaping is performed in the time domain via the circular convolution method. Note that at this stage, the length of the sequence is M/2. Further, the length of the filter is a maximum of 3 taps. Therefore, circular convolution must be performed between a length3 and length M/2 sequences appropriately. In contrast to frequency domain method, no care needs to be taken to generate the reference sequence for port 0 and port 1 in time domain case as the same output sequence can be mapped to both ports.
The following DFT properties are invoked to establish equivalence between transmission method 1 and transmission method 2. For any arbitrary sequence with length less than or equal to M/2, the even coefficients of its M point DFT are identical to its M/2point DFT coefficients.
where \(\mathbf r_{f,M}^{p_{0}}, \mathbf r_{f,\frac {M}{2}}^{p_{0}}\) are M point and \(\frac {M}{2}\) point DFT outputs of \(\mathbf {r}_{t}^{p}\) respectively.
Using (17), we can rewrite (10) as
which is exactly identical to (15). Since input to IDFT is identical for both methods, we conclude that the inverse DFT outputs of port 0 and port 1, i.e., \(\mathbf {r}_{f}^{s_{0}}, \mathbf {r}_{f}^{s_{1}}\) and the subsequent baseband signals generated through method 1 will be identical to that of generated using method 2. Using an example, we show in the Appendix that the channel estimation performance when these different transmitter methods are used will remain the same.
Receiver design
In this section, the receiver architecture to decode the received π/2BPSK data symbols is discussed. The receiver procedure is common for both transmission methods (explained in Section 2). The receiver architecture is shown in Fig. 12.
The receiver front end operations such as sampling, synchronization, CP removal, and FFT are similar to a conventional DFTsOFDMbased system. Further, the ISI introduced by the propagation channel is assumed to be less than that of the CP length. Therefore, after CP removal and FFT, the data and DMRS signals on kth subcarrier can be represented as (without loss of generality, we consider only the initial M subcarriers of the DFT output, i.e., k∈[0,M−1])
Above, y_{d} corresponds to the received data vector with data symbols from both ports. yDMRS0,yDMRS1 corresponds to the received DMRS vectors on port 0 and port 1 respectively. \(\mathbf h_{f,\text {\texttt {DMRS}}}^{0}=\mathbf {D}_{M}\mathbf {h}_{t}^{0}\) and \(\mathbf h_{f,\text {\texttt {DMRS}}}^{1}=\mathbf {D}_{M}\mathbf {h}_{t}^{1}\) correspond to frequency response of CIR on port 0 (\(\mathbf {h}_{t}^{0}\)) and port 1 (\(\mathbf {h}_{t}^{1}\)) respectively, \(\mathbf x_{f}^{s_{0}}\) and \(\mathbf x_{f}^{s_{1}}\) are the transmitted data sequences on port 0 and port 1 respectively, and \(\mathbf r_{f}^{s_{0}}\) and \(\mathbf r_{f}^{s_{1}}\) are the transmitted DMRS sequences on port 0 and port 1 respectively. The noise vectors v,v_{0}, and v_{1} are i.i.d. complex Gaussian random variables with zeromean and covariance σ^{2}I where I is an identity matrix and σ^{2} is a constant indicating the variance of each noise sample.
In practice, for low to medium user speeds, the time variations of the multipath channel across consecutive OFDM symbols will be minimal and hence we consider that the channel on the data and DMRS symbol will be same, i.e.,
Channel estimation
As mentioned in the Section 2.2, the channel estimator needs to estimate the joint impulse response of the spectrum shaping filter and the wireless channel. In our work, we employ a DFTbased channel estimation technique to estimate the joint channel response for the M allocated subcarriers. A simple least squaresbased technique with tone averaging or linear interpolation will not be effective in this case due to the presence of the spectrum shaping filter. Tone averaging or linear interpolation is based on the assumption that the channel is constant across consecutive subcarriers, which does not hold in this case, because the spectrum shaping considerably changes channel across consecutive subcarriers as shown in Fig. 5.
The data vector of length M will be associated with \(\frac {M}{2}\) length DMRS vector; since the data is carried on M subcarriers, the channel on all of these M subcarriers must be estimated for coherent demodulation. We show that an M length frequency domain channel vector corresponding to M length data symbol can be perfectly constructed from \(\frac {M}{2}\)length DMRS sequence for both the ports.
Channel estimation on port 0
As mentioned in the Section 2.4.2, port 0 carries DMRS only on even numbered subcarriers. From the received DMRS symbol, the even numbered subcarriers are extracted and expressed in terms of transmitted π/2BPSK based DMRS sequence as follows
where (20) results from (18), and (21) results from (10). Invoking the equivalence between M point DFT and \(\frac {M}{2}\)point DFT (17), the above equation can be represented as
where ⊙ indicates the circular convolution operation; \(\mathbf {{r}}_{t}^{p}\) is defined in Section 2.5.2. As mentioned in the Section 2.2, the length of spectrum shaping filter can be a maximum of 3 taps and the reference sequence design [14, 15] ensures that the length of DMRS sequence is always greater than impulse response \(\mathbf {h}_{t}^{0}\) of any possible wireless channel. Hence, the following holds
\(\text {\texttt {length}}\left (\mathbf w_{t} \odot \mathbf h_{t}^{0}\right)=\max \left (\text {\texttt {length}}(\mathbf {w}_{t}), \text {\texttt {length}}(\mathbf h_{\text {\texttt {t}}}^{0})\right) \leq \frac {M}{2}.\)
We invoke the following DFT relation to derive the joint impulse response. Any arbitrary time domain sequence g_{t} with length \(\mathrm {L\leq \frac {M}{2}}\) can be reconstructed with only either even or odd coefficients of its M point DFT.
We perform channel estimation on \({\tilde {\mathbf {y}}^{0}_{\text {\texttt {DMRS}}}}\) as follows: we first perform a least squaresbased channel estimation followed with an \(\frac {M}{2}\) point IDFT. This gives the joint impulse response of filter and the wireless channel as
Invoking the above remarks on lengths of spectrum shaping filter and channel impulse response, it can be deduced that h_{eff} completely captures the joint impulse response of the spectrum shaping filter and the wireless channel.
A denoising time domain filter [16] is then applied to reduce noise in (23). This filter f(n) is defined as
where f_{c} is the “cutoff” point, which is commonly chosen as the length of the wireless channel length(ht0) if it is known a priori; otherwise, it is set to the cyclic prefix length. This filtering extracts only the useful samples of the CIR by excluding the rest of the possible noise samples. The effective impulse response after denoising is given as
Lastly, the time domain filtered samples are transformed via an M point DFT to recover the frequencydomain channel estimates on each subcarrier k ε {0,1,2,......M−1} as \(\hat {\mathbf {h}}^{f}_{\text {\texttt {eff}}}=\mathbf {D}_{M}\hat {\mathbf h}_{\text {\texttt {eff}}}\). These channel estimates can be further used for port0 data demodulation using wellknown techniques.
Channel estimation on port 1
As mentioned in the Section 2.4.2, port 1 carries DMRS only on odd numbered subcarriers. From the received DMRS symbol, the odd numbered subcarriers are extracted and expressed in terms of transmitted π/2BPSK based DMRS sequence as follows
Using (19), the above equation can be written as
Using (10), (24) can be expressed as
Further processing steps such as the least squaresbased channel estimation, denoising, and transforming the effective impulse response to frequency domain are identical to the procedure followed for channel estimation on port 0. For the case of AWGN channel, i.e.,
the estimated joint impulse response h_{eff} on port 0 and port 1 is shown in Fig. 13. It can be noticed that the estimated impulse response is identical for both ports.
Equalization and data demodulation
The estimated channel on port 0 and port 1 will be employed for channel equalization of data streams. Specifically, we construct an MMSE filter employing the channel estimates obtained previously. The MMSE filter is then applied to the received signal samples from all the receive antennas of the base station to result in equalized data symbols. The equalized data symbols are demodulated to generate soft loglikelihood ratio values, which are subsequently fed to the channel decoder module for subsequent bitlevel processing.
Numerical results
In this section, we present various numerical results that show

The PAPR comparison between the π/2BPSKbased DMRS sequences and the existing ZC or CGSbased DMRS sequences.

Link level block error rate (BLER) comparison for the data transmissions employing π/2BPSKbased DMRS sequences and existing ZC or CGSbased DMRS sequences for various sequence lengths and various bandwidth allocations.

Link throughput vs. SNR comparison for the data transmissions employing π/2BPSKbased DMRS sequences and existing ZC or CGSbased DMRS sequences for various sequence lengths.

BLER performance for the data transmissions on port 0 and port 1 in the case of MIMO twostream transmissions.
Unless otherwise mentioned, the simulation assumptions shown in Table 2 are used throughout this paper.
The CCDF of PAPR for ZC sequences and π/2BPSKbased DMRS sequences is shown in Fig. 14. The ZC sequences considered in this case are as defined in [4, Section 5.2.2] with length 96. The PAPR of ZC sequences with and without spectrum shaping is shown in the figure. As can be seen from the figure, the ZC sequences without spectrum shaping have a PAPR (at the 10^{−3} CDF point) of 2.8 dB more than the π/2BPSKbased DMRS sequences. When spectrum shaping is applied to the ZC sequences, the PAPR is slightly reduced from that of unfiltered ZC sequences. However, the PAPR of the filtered ZC sequence is still 2.0 dB larger than the PAPR of the π/2BPSKbased DMRS sequences with the same spectrum shaping. Moreover, as we increase the number of allocated subcarriers for data transmission, the PAPR gap between 3GPP ZC sequence and π/2BPSKbased DMRS sequence increases even further.
The CCDF of QPSK CGSbased DMRS and π/2BPSKbased DMRS for smaller lengths (M=12) is shown in Fig. 15. As discussed in Section 2, for smaller lengths (M<30), 3GPP employs computergenerated sequences (CGS) as DMRS. It can be seen from the figure that the PAPR of the spectrumshaped CGS sequences is almost 1.2 dB larger than the PAPR of the π/2BPSK sequences. Moreover, it can also be noticed that for CGS, the PAPR is further increased with filtering, because for shorter lengths, the frequency variations of the spectrumshaping filter across subcarriers are quite rapid. The IFFT size of the OFDM modulator (N) will be very much larger than sequence length (M); hence, there will be many unused subcarriers at both ends of the sequence and results in the spectrumshaped sequence effectively being oversampled in the time domain. This significantly affects the spectrum flatness of the CGS. The OFDM outputs of CGS sequences with and without spectrum shaping are shown in Fig. 16. The large amplitude variations in OFDM output of spectrumshaped CGS sequences can be seen from the figure. Hence, the results shown in Figs. 14, 15 conclude that the π/2BPSK sequences designed in [13] are far superior compared to the existing sequences in improving the cell coverage.
The block error rate performance for a single stream PUSCH transmission is shown in Fig. 17. Here, DMRS is transmitted on port 0. Note that ZC sequences are used for comparing the BLER performance because these sequences have a flat frequency spectrum. The frequency flatness ensures unbiased channel estimation across all the allocated subcarriers, which results in the best channel estimation performance. Hence, the goal for the newly designed sequences is to ensure that they match the performance of these ZC sequences. In this figure, the results are shown for the cases when the base station receiver employs 2 and 4 receive antennas. From Fig. 18, it can be seen that irrespective of the number of receive antennas, the link level performance of π/2BPSK DMRS sequence is equivalent to that of existing CGS based DMRS sequences, although the newly designed sequences are not frequency flat.
The throughput vs. SNR comparisons are shown in Figs. 19 and 20 for the DMRS lengths 96 and 24 respectively. The link throughput is computed by invoking the HybridARQ protocol (HARQ) with a maximum of 4 retransmissions. The throughput shown is the percentage of transport blocks that got decoded correctly, i.e., throughput \(= \frac {B_{D}}{B}\times 100\). Here ‘ `B_{D}” is the number of transport blocks that got decoded correctly, and “B” is the total number of transport blocks transmitted. It can be seen from the Figs. 19 and 20 that the throughput performance of π/2BPSK DMRS is equivalent to that of existing ZC and CGS based sequences. From the BLER and throughput analysis, it can be deduced that the π/2BPSK DMRS sequences do not cause any performance degradation but minimizes the PAPR.
We next consider the performance of the proposed transmitter designs for the case of two MIMO streams transmission where DMRS is transmitted on both port 0 and port 1. Firstly, we show the drawbacks of the existing design in 3GPP in Figs. 21 and 22. It can be seen that when the 3GPP transceiver is used, there is a clear difference in the performance both in terms of PAPR and BLER across port 0 and port 1. This is highly undesirable as the data on two different ports will behave differently. This problem is addressed using the proposed transceiver design, as claimed earlier. We next show that it is indeed the case.
In Figs. 21 and 23, we show the PAPR and BLER performance for the two MIMO streams transmission setting where DMRS is transmitted on both port 0 and port 1 using our proposed method 1 transceiver design. It can be seen that both PAPR, as well as BLER, are identical for both ports confirming that the proposed transmitter design produces identical DMRS sequences on both ports.
In Fig. 24, PAPR of the DMRS sequences on port 0 and port 1 generated by method 1 and method 2 is shown. It can be seen that PAPR is the same for both port 0 and port 1 in both methods confirming that the proposed transmitter designs are equivalent. The same is the case with BLER performance as well. Therefore, the proposed methods 1 and 2 have shown to be equivalent to both analytically and numerically.
Conclusion
In this paper, a low PAPR reference signal transceiver design for 3GPP 5G NR π/2BPSKbased uplink transmissions is proposed. The PAPR of the reference signal is significantly minimized compared to the current design of Rel15 5G NR systems using the proposed design. Such a design considerably helps to improve the coverage of the 5G systems. Specifically, we have shown a frequency domain and a time domain transceiver design, both of which are equivalent and result in the same system performance in terms of PAPR and also BLER. We have shown how the proposed design can be extended to the case of a MIMO transmission without causing any discrepancy on different MIMO streams, which is not the case for the current Rel15 3GPP 5G NR uplink design.
Methods/experimental
The transmit power for the uplink transmissions is significantly lesser than the downlink transmissions. Hence, to improve the power efficiency of the uplink transmissions and subsequently the cell coverage, it is required that the uplink waveform to have a low peak to average power ratio (PAPR). To support the low PAPR transmissions, a new modulation scheme, namely π/2BPSK, was introduced in the Rel15 3GPP 5G NR specifications. However, the reference signals employed for data demodulation have higher PAPR than the data signals, which potentially limits the cell coverage. Also, in the case of spatial multiplexing, the reference signal design should ensure that each data stream experiences a similar channel estimation performance. In this contribution, we present two transmitter designs, namely method 1 and method 2, to generate low PAPR data and reference signals. The proposed transmitters generate π/2BPSKbased reference signals with appropriate spectrum shaping and DFT precoding such that the reference signals on each DMRS port have similar PAPR and results in similar channel estimation performance. In method 1, the processing happens in the frequency domain, where an M/2 length binary sequence is cyclically extended and phasemodulated to result in M length π/2BPSK sequence, which is further DFT precoded and spectrum shaped. In method 2, the processing happens in the time domain via the circularconvolution operation. To decode the received π/2BPSK data symbols, we propose a receiver architecture which is common for both the transmission methods. We justify our claims using computer simulations.
Appendix
In this section, we present the spectrumshaped DFT outputs of port 0 and port 1 generated using the proposed transmitter design. In Table 4, we present the effective CIR estimated from both ports in a noiseless scenario. We consider a 3tap spectrum shaping filter with impulse response w_{t}=[−0.28 1 −0.28]. For convenience, we consider a flat and identical wireless channel for both DMRS ports, i.e., hf,DMRS0(k)=hf,DMRS1(k)=1 ∀ k∈[0,M−1]. Let r_{t}=[111011] be a 6length low PAPR DMRS sequence corresponding to length12 data allocation. This sequence is passed through the transmitter design as shown in Figs. 7 and 8 or 11 corresponding to the method 1 or method 2 transmitter designs. The resulting output is shown in Table 3. As proved earlier, it can be seen from this table that the nonzero entries of the DMRS sequences on both DMRS ports are the same.
For this setting, the channel can be estimated perfectly on both baseband antenna ports, as shown in Table 4, thereby allowing for correct data demodulation.
Abbreviations
 PAPR:

Peaktoaverage power ratio
 3GPP:

3rd Generation Partnership Project
 NR:

New radio
 BPSK:

Binary phaseshift keying
 QAM:

Quadrature amplitude modulation
 DFTsOFDM:

Discrete Fourier transform spread orthogonal frequency division multiplexing
 DMRS:

Demodulation reference signals
 TDL:

Tapped delay line
 LDPC:

Lowdensity paritycheck
 MMSE:

Minimum mean squared error
 FFT:

Fast Fourier transform
 IDFT:

Inverse discrete Fourier transform
 DFT:

Discrete Fourier transform
 BLER:

Block error rate
 CCDF:

Complementary cumulative distribution function
 PRB:

Physical resource block
 MIMO:

Multiple inputmultiple output
 CIR:

Channel impulse response.
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Funding
This work is supported by the Department of Telecommunications(DOT), grant funded by the Ministry of Communications, Government of India (No. 423/5G Test Bed /2017NT dt. 22.03.18 Indigenous 5G Test Bed (Building an end to end 5G Test Bed)).
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The transceiver designs proposed in the paper are mainly developed and analyzed by M Sibgath Ali Khan and Koteswara Rao, where M Sibgath Ali Khan is also responsible for generating all the required performance plots and writing of the manuscript till the final submission. Koteswara Rao and Saidhiraj Amuru have extensively reviewed and edited the manuscript to improve the quality of the manuscript. Kiran Kuchi being the supervisor of M Sibgath Ali Khan monitored the whole work from transceiver design till the final submission. All authors read and approved the final manuscript.
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Khan, M., Rao, K., Amuru, S. et al. Low PAPR reference signal transceiver design for 3GPP 5G NR uplink. J Wireless Com Network 2020, 182 (2020). https://doi.org/10.1186/s13638020017871
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Keywords
 PAPR
 Spectrum shaping filter
 Impulse response
 BPSK