Category: 8-PSK

π/2 BPSK (pi/2 BPSK): 5G NR PHY modulation

The 5G New Radio (NR) supports quadrature phase shift keying (QPSK), 16- quadrature amplitude modulation (16-QAM), 64 QAM and 256 QAM modulation schemes for both uplink and downlink [1][2]. This is same as in LTE.

Additionally, 5G NR supports π/2-BPSK in uplink (to be combined with OFDM with CP or DFT-s OFDM with CP)[1][2]. Utilization of π/2-BPSK in the uplink is aimed at providing further reduction of peak-to-average power ratio (PAPR) and boosting RF amplifier power efficiency at lower data-rates.

π/2 BPSK

π/2 BPSK uses two sets of BPSK constellations that are shifted by 90°. The constellation sets are selected depending on the position of the bits in the input sequence. Figure (1) depicts the two constellation sets for π/2 BPSK that are defined as per equation (1)

\[d[i] = \frac{e^{j \frac{\pi}{2} \left( i \; mod \; 2\right) }}{ \sqrt{2}} \left[ \left(1 – 2b[i] \right) + j \left(1 – 2b[i] \right)\right] \quad \quad (1) \]

b[i] = input bits; i = position or index of input bits; d[i] = mapped bits (constellation points)

Ideal pi by 2 BPSK constellation as per 3GPP TS 38.211 5G specification odd even bits
Figure 1: Two rotated constellation sets for use in π/2 BPSK

Equation (2) is for conventional BPSK – given for comparison. Figure (2) and Figure (3) depicts the ideal constellations and waveforms for BPSK and π/2 BPSK, when a long sequence of random input bits are input to the BPSK and π/2 BPSK modulators respectively. From the waveform, you may note that π/2 BPSK has more phase transitions than BPSK. Therefore π/2 BPSK also helps in better synchronization, especially for cases with long runs of 1s and 0s in the input sequence.

\[d[i] = \frac{1}{ \sqrt{2}} \left[ \left(1 – 2b[i] \right) + j \left(1 – 2b[i] \right)\right] \quad \quad (2)\]
Figure 2: Ideal BPSK and π/2 BPSK constellations
Figure 3: Waveforms of BPSK and π/2 BPSK for same sequence of input bits

Figure 4, illustrates the constellations for BPSK and π/2 BPSK when the sequence of mapped bits are corrupted by noise.

Figure 4: BPSK and π/2 BPSK constellation for Eb/N0=50dB

Note: Though the π/2 BPSK constellation looks like a QPSK constellation, they are not the same. Give it a thought !!!

References

[1] 3GPP TS 38.201: Physical layer; General description (Release 16)
[2] 3GPP TS 38.211: Physical channels and modulation (Release 16)
[3] Gustav Gerald Vos, ‘Two-tone in-phase pi/2 binary phase-shift keying communication’, US patent number 10,931,492

Rician flat-fading channel – simulation

In wireless environments, transmitted signal may be subjected to multiple scatterings before arriving at the receiver. This gives rise to random fluctuations in the received signal and this phenomenon is called fading. The scattered version of the signal is designated as non line of sight (NLOS) component. If the number of NLOS components are sufficiently large, the fading process is approximated as the sum of large number of complex Gaussian process whose probability-density-function follows Rayleigh distribution.

Rayleigh distribution is well suited for the absence of a dominant line of sight (LOS) path between the transmitter and the receiver. If a line of sight path do exist, the envelope distribution is no longer Rayleigh, but Rician (or Ricean). If there exists a dominant LOS component, the fading process can be represented as the sum of complex exponential and a narrowband complex Gaussian process g(t). If the LOS component arrive at the receiver at an angle of arrival (AoA) θ, phase ɸ and with the maximum Doppler frequency fD, the fading process in baseband can be represented as (refer [1])

\[h(t)= \underbrace{\sqrt{\frac{K \Omega}{K +1}}}_\text{A:=} e^{\left( j2 \pi f_D cos(\theta)t+\phi \right)} + \underbrace{\sqrt{\frac{\Omega}{K+1}}}_\text{S:=}g(t)\]

where, K represents the Rician K factor given as the ratio of power of the LOS component A2 to the power of the scattered components (S2) marked in the equation above.

\[K=\frac{A^2}{S^2}\]

The received signal power Ω is the sum of power in LOS component and the power in scattered components, given as Ω=A2+S2. The above mentioned fading process is called Rician fading process. The best and worst-case Rician fading channels are associated with K=∞ and K=0 respectively. A Ricean fading channel with K=∞ is a Gaussian channel with a strong LOS path. Ricean channel with K=0 represents a Rayleigh channel with no LOS path.

The statistical model for generating flat-fading Rician samples is discussed in detail in chapter 11 section 11.3.1 in the book Wireless communication systems in Matlab (see the related article here). With respect to the simulation model shown in Figure 1(b), given a K factor, the samples for the Rician flat-fading samples are drawn from the following random variable

\[h= | X + jY |\]

where X,Y ~ N(μ,σ2) are Gaussian random variables with non-zero mean μ and standard deviation σ as given in references [2] and [3].

\[\mu = g_1 =\sqrt{\frac{K}{2\left(K+1\right)}} \quad \quad \sigma = g_2 = \sqrt{\frac{1}{2\left(K+1\right)}}\]

Kindly refer the book Wireless communication systems in Matlab for the script on generating channel samples for Ricean flat-fading.

Figure 1: Simulation model for modulation and detection over flat fading channel

Simulation and performance results

In chapter 5 of the book Wireless communication systems in Matlab, the code implementation for complex baseband models for various digital modulators and demodulator are given. The computation and generation of AWGN noise is also given in the book. Using these models, we can create a unified simulation for code for simulating the performance of various modulation techniques over Rician flat-fading channel the simulation model shown in Figure 1(b).

An unified approach is employed to simulate the performance of any of the given modulation technique – MPSK, MQAM or MPAM. The simulation code (given in the book) will automatically choose the selected modulation type, performs Monte Carlo simulation, computes symbol error rates and plots them against the theoretical symbol error rate curves. The simulated performance results obtained for various modulations are shown in the Figure 2.

Figure 2: Performance of various modulations over Ricean flat fading channel

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References

[1] C. Tepedelenlioglu, A. Abdi, and G. B. Giannakis, The Ricean K factor: Estimation and performance analysis, IEEE Trans. Wireless Communication ,vol. 2, no. 4, pp. 799–810, Jul. 2003.↗
[2] R. F. Lopes, I. Glover, M. P. Sousa, W. T. A. Lopes, and M. S. de Alencar, A simulation framework for spectrum sensing, 13th International Symposium on Wireless Personal Multimedia Communications (WPMC 2010), Out. 2010.
[3] M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems, Methodology, Modeling, and Techniques, second edition Kluwer Academic Publishers, 2000.↗

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