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)
b[i] = input bits; i = position or index of input bits; d[i] = mapped bits (constellation points)
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.
The phase transition properties of the different variants of QPSK schemes and MSK, are easily investigated using constellation diagram. Let’s demonstrate how to plot the signal space constellations, for the various modulations used in the transmitter.
Typically, in practical applications, the baseband modulated waveforms are passed through a pulse shaping filter for combating the phenomenon of intersymbol interference (ISI). The goal is to plot the constellation plots of various pulse-shaped baseband waveforms of the QPSK, O-QPSK and π/4-DQPSK schemes. A variety of pulse shaping filters are available and raised cosine filter is specifically chosen for this demo. The raised cosine (RC) pulse comes with an adjustable transition band roll-off parameter α, using which the decay of the transition band can be controlled.
The RC pulse shaping function is expressed in frequency domain as
Equivalently, in time domain, the impulse response corresponds to
A simple evaluation of the equation (2) produces singularities (undefined points) at p(t = 0) and p(t = ±Tsym/(2α)). The value of the raised cosine pulse at these singularities can be obtained by applying L’Hospital’s rule [1] and the values are
The function is then tested. It generates a raised cosine pulse for the given symbol duration Tsym = 1s and plots the time-domain view and the frequency response as shown in Figure 1. From the plot, it can be observed that the RC pulse falls off at the rate of 1/|t|3 as t→∞, which is a significant improvement when compared to the decay rate of a sinc pulse which is 1/|t|. It satisfies Nyquist criterion for zero ISI – the pulse hits zero crossings at desired sampling instants. The transition bands in the frequency domain can be made gradual (by controlling α) when compared to that of a sinc pulse.
Plotting constellation diagram
Now that we have constructed a function for raised cosine pulse shaping filter, the next step is to generate modulated waveforms (using QPSK, O-QPSK and π/4-DQPSK schemes), pass them through a raised cosine filter having a roll-off factor, say α = 0.3 and finally plot the constellation. The constellation for MSK modulated waveform is also plotted.
Conclusions
The resulting simulated plot is shown in the Figure 2. From the resulting constellation diagram, following conclusions can be reached.
Conventional QPSK has 180° phase transitions and hence it requires linear amplifiers with high Q factor
The phase transitions of Offset-QPSK are limited to 90° (the 180° phase transitions are eliminated)
The signaling points for π/4-DQPSK is toggled between two sets of QPSK constellations that are shifted by 45° with respect to each other. Both the 90° and 180° phase transitions are absent in this constellation. Therefore, this scheme produces the lower envelope variations than the rest of the two QPSK schemes.
MSK is a continuous phase modulation, therefore no abrupt phase transition occurs when a symbol changes. This is indicated by the smooth circle in the constellation plot. Hence, a band-limited MSK signal will not suffer any envelope variation, whereas, the rest of the QPSK schemes suffer varied levels of envelope variations, when they are band-limited.
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