Category: Constellations

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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QAM modulation: simulate in Matlab & Python

A generic complex baseband simulation technique, to simulate all M-ary QAM modulation techniques is given here. The given simulation code is very generic, and it plots both simulated and theoretical symbol error rates for all M-QAM modulation techniques.

Rectangular QAM from PAM constellation

There exist other constellation shapes (like circular, triangular constellations) that are more efficient (in terms of energy required to achieve same the error probability) than the standard rectangular constellation. Rectangular (symmetric or square) constellations are the preferred choice of implementation due to its simplicity in implementing modulation and demodulation.

In one of the earlier articles, I have discussed the method of constructing constellation for rectangular QAM modulation using Karnaugh-map walks, where the inherent property of Karnaugh-maps is exploited to construct Gray coded QAM symbols.

Any rectangular QAM constellation is equivalent to superimposing two Amplitude Shift Keying (ASK) signals (also called Pulse Amplitude Modulation – PAM) on quadrature carriers. For example, 16-QAM constellation points can be generated from two 4-PAM signals, similarly the 64-QAM constellation points can be generated from two 8-PAM signals.

Figure 1: Signal space constellations for 16-QAM and 64-QAM

The generic equation to generate PAM signals of dimension D is

For generating 16-QAM, the dimension D of PAM is set to . Thus for constructing a M-QAM constellation, the PAM dimension is set as . Matlab code for dynamically generating M-QAM constellation points based on Karnaugh map Gray code walk is given below. The resulting ideal constellations for Gray coded 16-QAM and 64-QAM are shown in Figure 1.

Matlab code

Full Matlab code available in the book Digital Modulations using Matlab – build simulation models from scratch

function [s,ref]=mqam_modulator(M,d)
%Function to MQAM modulate the vector of data symbols - d
%[s,ref]=mqam_modulator(M,d) modulates the symbols defined by the vector d
% using MQAM modulation, where M specifies order of M-QAM modulation and
% vector d contains symbols whose values range 1:M. The output s is modulated
% output and ref represents reference constellation that can be used in demod
if(((M˜=1) && ˜mod(floor(log2(M)),2))==0), %M not a even power of 2
  error('Only Square MQAM supported. M must be even power of 2');
end
  ref=constructQAM(M); %construct reference constellation
  s=ref(d); %map information symbols to modulated symbols
end

Python code

Full Matlab code available in the book Digital Modulations using Python

class QAMModem(Modem):
    # Derived class: QAMModem
    
    def __init__(self,M):
        
        if (M==1) or (np.mod(np.log2(M),2)!=0): # M not a even power of 2
            raise ValueError('Only square MQAM supported. M must be even power of 2')
        
        n = np.arange(0,M) # Sequential address from 0 to M-1 (1xM dimension)
        a = np.asarray([xˆ(x>>1) for x in n]) #convert linear addresses to Gray code
        D = np.sqrt(M).astype(int) #Dimension of K-Map - N x N matrix
        a = np.reshape(a,(D,D)) # NxN gray coded matrix
        oddRows=np.arange(start = 1, stop = D ,step=2) # identify alternate rows
        
        nGray=np.reshape(a,(M)) # reshape to 1xM - Gray code walk on KMap
        #Construction of ideal M-QAM constellation from sqrt(M)-PAM
        (x,y)=np.divmod(nGray,D) #element-wise quotient and remainder
        Ax=2*x+1-D # PAM Amplitudes 2d+1-D - real axis
        Ay=2*y+1-D # PAM Amplitudes 2d+1-D - imag axis
        constellation = Ax+1j*Ay
        Modem.__init__(self, M, constellation, name='QAM') #set the modem attributes

This article is part of the following books
● Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885
● Digital Modulations using Python ISBN: 978-1712321638
All books available in ebook (PDF) and Paperback formats

M-QAM demodulation (coherent detection)

Generally the two main categories of detection techniques, commonly applied for detecting the digitally modulated data are coherent detection and non-coherent detection.

In the vector simulation model for the coherent detection, the transmitter and receiver agree on the same
reference constellation for modulating and demodulating the information. The modulators generate the reference constellation for the selected modulation type. The same reference constellation should be used if coherent detection is selected as the method of demodulating the received data vector.

On the other hand, in the non-coherent detection, the receiver is oblivious to the reference constellation used at the transmitter. The receiver uses methods like envelope detection to demodulate the data.

The IQ detection technique is an example of coherent detection. In the IQ detection technique, the first step is to compute the pair-wise Euclidean distance between the given two vectors – reference array and the received symbols corrupted with noise. Each symbol in the received symbol vector (represented on a p-dimensional plane) should be compared with every symbol in the reference array. Next, the symbols, from the reference array, that provide the minimum Euclidean distance are returned.

Let x=(x1,x2,…,xp) and y=(y1,y2,…,yp) be two points in p-dimensional space. The Euclidean distance between them is given by

The pair-wise Euclidean distance between two sets of vectors, say x and y, on a p-dimensional space, can be computed using the vectorized code. The vectorized code returns the ideal signaling points from matrix y that provides the minimum Euclidean distance. Since the vectorized implementation is devoid of nested for-loops, the program executes significantly faster for larger input matrices. The given code is very generic in the sense that it can be easily reused to implement optimum coherent receivers for any N-dimensional digital modulation technique (Please refer the books Digital Modulations using Matlab and Digital Modulations using Python for complete simulation code) .

Matlab code

Full Matlab code available in the book Digital Modulations using Matlab

function [dCap]= mqam_detector(M,r)
%Function to detect MQAM modulated symbols
%[dCap]= mqam_detector(M,r) detects the received MQAM signal points
%points - 'r'. M is the modulation level of MQAM
   if(((M˜=1) && ˜mod(floor(log2(M)),2))==0), %M not a even power of 2
      error('Only Square MQAM supported. M must be even power of 2');
   end
   ref=constructQAM(M); %reference constellation for MQAM
   [˜,dCap]= iqOptDetector(r,ref); %IQ detection
end

Python code

Full Matlab code available in the book Digital Modulations using Python

Performance simulation results

The simulation results for error rate performance of M-QAM modulations over AWGN channel and Rician flat-fading channel is given in the following figures.

Figure 2: Error rate performance of M-QAM modulations in AWGN channel
Figure 3: Error rate performance of 64-QAM modulation in Rician flat-fading channels having different values for Rician K-factors.

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Reference

[1] John G. Proakis, “Digital Communciations”, McGraw-Hill; 5th edition.↗

Related Topics

Digital Modulators and Demodulators - Complex Baseband Equivalent Models
● Introduction
● Complex baseband representation of modulated signal
● Complex baseband representation of channel response
● Modulators for amplitude and phase modulations
 □ Pulse Amplitude Modulation (M-PAM)
 □ Phase Shift Keying Modulation (M-PSK)
 □ Quadrature Amplitude Modulation (M-QAM)
● Demodulators for amplitude and phase modulations
 □ M-PAM detection
 □ M-PSK detection
 □ M-QAM detection
 □ Optimum detector on IQ plane using minimum Euclidean distance
● M-ary FSK modulation and detection
 □ Modulator for M orthogonal signals
 □ M-FSK detection

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