Tag: Raised Cosine Filter

Constellation diagram – investigate phase transitions

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.

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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

Using the equations above, the raised cosine filter is implemented as a function (refer the books Digital Modulations using Python and Digital Modulations using Matlab for the code).

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.

Figure 1: Raised-cosine pulse and its manifestation in frequency domain

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.

Figure 2: Constellations plots for: (a) a = 0.3 RC-filtered QPSK, (b) α = 0.3 RC-filtered O-QPSK, (c) α = 0.3 RC-filtered π/4-DQPSK and (d) MSK

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.

References

[1] Clay S. Turner, Raised Cosine and Root Raised Cosine Formulae, Wireless Systems Engineering, Inc, (May 29, 2007) V1.2↗

In this chapter

Digital Modulators and Demodulators - Passband Simulation Models
● Introduction
● Binary Phase Shift Keying (BPSK)
 □ BPSK transmitter
 □ BPSK receiver
 □ End-to-end simulation
● Coherent detection of Differentially Encoded BPSK (DEBPSK)
● Differential BPSK (D-BPSK)
 □ Sub-optimum receiver for DBPSK
 □ Optimum noncoherent receiver for DBPSK
● Quadrature Phase Shift Keying (QPSK)
 □ QPSK transmitter
 □ QPSK receiver
 □ Performance simulation over AWGN
● Offset QPSK (O-QPSK)
● π/p=4-DQPSK
● Continuous Phase Modulation (CPM)
 □ Motivation behind CPM
 □ Continuous Phase Frequency Shift Keying (CPFSK) modulation
 □ Minimum Shift Keying (MSK)
● Investigating phase transition properties
● Power Spectral Density (PSD) plots
● Gaussian Minimum Shift Keying (GMSK)
 □ Pre-modulation Gaussian Low Pass Filter
 □ Quadrature implementation of GMSK modulator
 □ GMSK spectra
 □ GMSK demodulator
 □ Performance
● Frequency Shift Keying (FSK)
 □ Binary-FSK (BFSK)
 □ Orthogonality condition for non-coherent BFSK detection
 □ Orthogonality condition for coherent BFSK
 □ Modulator
 □ Coherent Demodulator
 □ Non-coherent Demodulator
 □ Performance simulation
 □ Power spectral density

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Raised cosine pulse shaping

As mentioned earlier, the shortcomings of the sinc pulse can be addressed by making the transition band in the frequency domain less abrupt. The raised-cosine (RC) pulse comes with an adjustable transition band roll-off parameter , using which the transition band’s rate of decay can be controlled. The RC pulse shaping function is expressed in frequency domain as

Correspondingly, in time domain, the impulse response is given by

This article is part of the book Wireless Communication Systems in Matlab, ISBN: 978-1720114352 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here).

A simple evaluation of the equation (2) produces singularities (undefined points) at and . The value of the raised-cosine pulse at these singularities can be obtained by applying L’Hospital’s rule [1] and the values are

Figure 1: Raised-cosine pulse and its manifestation in frequency domain

The following Matlab codes generate a raised cosine pulse for the given symbol duration and plot the time-domain view and the frequency response (shown in Figure 1). The RC pulse falls off at the rate of as , which is a significant improvement when compared to the decay rate of sinc pulse which is . It satisfies Nyquist criterion for zero ISI – the pulse hits zero crossings at desired sampling instants. By controlling , the transition band roll-off in the frequency domain can be made gradual.

Program 1: raisedCosineFunction.m: Function for generating raised-cosine pulse(click here)

Matlab code for Program 1 is available is available in the book Wireless Communication Systems in Matlab (click here).

Program 2: test_RCPulse.m: Raised-cosine pulses and their manifestation in frequency domain

Tsym=1; %Symbol duration in seconds
L=10; % oversampling rate, each symbol contains L samples
Nsym = 80; %filter span in symbol durations
alphas=[0 0.3 0.5 1];%RC roll-off factors - valid range 0 to 1
Fs=L/Tsym;%sampling frequency
lineColors=['b','r','g','k','c']; i=1;legendString=cell(1,4);

for alpha=alphas %loop for various alpha values
	[rcPulse,t]=raisedCosineFunction(alpha,L,Nsym); %RC Pulse
	
	subplot(1,2,1); t=Tsym*t; %translate time base for given duration
	plot(t,rcPulse,lineColors(i));hold on; %plot time domain view
	[vals,f]=freqDomainView(rcPulse,Fs,'double');%See Chapter 1
	
	subplot(1,2,2);
	plot(f,abs(vals)/abs(vals(length(vals)/2+1)),lineColors(i));
	hold on;legendString{i}=strcat('\alpha =',num2str(alpha) );i=i+1;
end
subplot(1,2,1);title('Raised Cosine pulse'); legend(legendString);
subplot(1,2,2);title('Frequency response');legend(legendString);

References

[1] Clay S. Turner, Raised cosine and root raised cosine formulae, Wireless Systems Engineering, Inc, May 29, 2007.

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Digital Modulations using Python
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Topics in this chapter

Pulse Shaping, Matched Filtering and Partial Response Signaling
● Introduction
● Nyquist Criterion for zero ISI
● Discrete-time model for a system with pulse shaping and matched filtering
 □ Rectangular pulse shaping
 □ Sinc pulse shaping
 □ Raised-cosine pulse shaping
 □ Square-root raised-cosine pulse shaping
● Eye Diagram
● Implementing a Matched Filter system with SRRC filtering
 □ Plotting the eye diagram
 □ Performance simulation
● Partial Response Signaling Models
 □ Impulse response and frequency response of PR signaling schemes
● Precoding
 □ Implementing a modulo-M precoder
 □ Simulation and results