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Phase-shift keying (PSK) is a digital modulation scheme that conveys data by changing the phase of a carrier signal. The principle of PSK is that PSK uses some phases (2, 4 or 8) to represent digital data. After transmission, demodulator of PSK determines the phase of the received signal and then translates them back to the bits it presents. BPSK is the simplest form of phase shift keying, which use two phases separated by 180 degree. But it is not suitable for high data-rate application since it can only modulate 1 bit/symbol. However, compared with BPSK, QPSK use four phases which is able to double the data rate while keeping the same bandwidth of the signal. Additionally, the transmitter and receiver of QPSK is much complicated than BPSK, so the cost of QPSK is higher. Furthermore, 8PSK is the highest order PSK since if the order PSK is higher than 8, bit error rate (BER) becomes too high and there are better modulations available such as QAM. BPSK differs from BASK in an important respect: the envelop of the modulated signal is maintained constant at the value for all time t.
The equation of MPSK signal can be represented by
, i=1, 2…..M, where E is the signal energy per symbol, and Tis the symbol duration.
QPSK is a special case of M-ary PSK, which is commonly used in practice. This report will discuss QPSK as an example to explore some features of PSK.
In QPSK, as with binary PSK, information carried by the transmitted signal is contained in the phase. In particular, the phase of the carrier takes on one of four equally spaced values, such as , ,, and . The equation of QPSK is:
, where i=1, 2, 3, 4. Each possible value of the phase corresponds to a unique Gray-encoded dibit: 10, 00, 01 and 11.
Figure 1 shows a block diagram of a typical QPSK transmitter. The incoming binary data sequence is first transformed into polar form by a nonreturn-to-zero level encoder. Thus, symbols 1 and 0 are represented by and respectively. This binary wave is next divided by 'Series to parallel converter' into two separate binary waves consisting of the odd numbered inputs bits and even numbered input bits. These two binary waves are denoted by and . In addition, the amplitude of and equal to and , depending on the particular dibit that is being transmitted. After that, the two binary waves and are used to modulate a pair of quadrature basis functions: and. The following procedure is to perform pulse shaping to reduce intersample interference. The result is a pair of binary PSK signals; finally, the two binary PSK signals are added to produce the desired QPSK signal.
Transmitting a signal at high modulation rate through a band-limited channel can create intersymbol interference. The reason is that as the modulation rate increases, the signal's bandwidth increases. When the signal's bandwidth becomes larger than the channel bandwidth, the channel starts to introduce distortion to the signal. This distortion is usually seen as intersymbol interference. Then these data should be through pulse shaping to reduce intersymbol interference (ISI). This signal is then filtered with the pulse shaping filter, producing the transmitted signal. In addition, if the system transfer function H(f) is made rectangular, its impulse response, the inverse Fourier transform of H(f) is of the form h(t)=sinc (t/T), this h(t)=sinc (t/T)-shaped pulse is called the ideal Nyquist pulse, Nyquist established that if each pulse of a received sequence is the form sinc (t/T), the pulse can be detected without ISI. In signal transmission, Nyquist filter is often used to filtering signals to satisfy zero ISI at sampling points.
Principle Blocks of producing the MPSK signal is as bellows:
Figure Procedure of QPSK
The QPSK receiver: The received signal is first through pulse detection and down sample, and then produced by a locally generated pair of coherent reference signals and , as in Figure 2. The outputs and , produced in response to the received signal, are each compared with a threshold of zero. If >0, a decision is made in favor of symbol 1 for the 'I channel ' output, but if <0, a decision is made in favor of symbol 0. Similarly, if >0, a decision is made in favor of symbol 1 for 'Q channel', but if <0, a decision is made in favor of symbol 0. Finally, these two binary sequences at the ''I channel' and 'Q channel' outputs are combined in 'Parallel to series converter', producing the original binary sequences at the transmitter input with the minimum probability of symbol error in an AWGN channel. In the following figure, after transmitted signal add AWGN noise, signal should be filtered by raised-cosine filter, its function is to compensate for the distortion caused by both the transmitter and the channel.
Figure Receiver of the QPSK signal
The receiver gives the reverse process of the MPSK signal production. In the process of signal transmitting, errors such as channel noise would be brought into transmitted signal, so received signal may not be the same as the original ones. Therefore, it is necessary to evaluate the performance of MPSK by using the BER computation.
In this assignment, Matlab is used to develop a custom modulation schemes based on M-ary PSK.
Figure Matlab Process of MPSK Modulation and Demodulation
The lab procedure includes:
Set system parameters
Create signal source
Create signal source is a random binary data stream with size 30000.
%% Signal Source
% Create a binary data stream as a column vector.
x = randint(n*Rb,1);
% Plot first 40 bits in a stem plot.
title('Signal Source : Random Bits');
xlabel('Bit Index'); ylabel('Binary Value');
axis([0 40 -0.5 1.5]);
Figure First 40 random binary signal source
Map bits to symbols
In this step all the bit are mapped into symbols. The M is set as 4 which means each symbol can be represented by k=log2(M)=2 bits.
%% Bit-to-Symbol Mapping
% Convert the bits in x into k-bit symbols.
% Plot first 10 symbols in a stem plot.
figure; %Create new figure window.
title('Map bits to Symbols: Random Symbols');
xlabel('Symbol Index'); ylabel('Integer Value');
Figure First 10 random symbols after mapping
Create desired constellation
%% Create desired constellation
h=modem.pskmod('M',M,'SymbolOrder','Gray'); % Modulator object
mapping=h.SymbolMapping;%Symbol mapping vector
pt=h.Constellation; %Vector of all points in constellation
scatterplot(pt); % Plot the constellation.
% Inclide text annotations that number the points'
axis([-4 4 -4 4]);
Figure The desired constellation using Gray Modulation of QPSK
Modulate the signal
Perform the pulse shaping
%% Perform The Pulse Shaping
% Create a square root raised cosine filter.
rrcfilter = rcosine(1, nsamp,'fir/sqrt', rolloff, delay);
% Plot impulse response.
% Transmitted Signal
% Upsample and apply square root raised cosine filter.
Figure The impulse response of a square root raised cosine filter
Figure The scatter plot of the transmitted signal (QPSK )
Apply channel model
The applied channel is AWGN channel. The modulated signal is added white Gaussion noise. The ratio of bit energy to noise power spectral density is set as Eb/No(dB). The ratio of symbol energy to noise power spectral density Es/No can be got by addition of Eb/No and 10log10(k) . The factor oversampling rate is used to convert Es/No in symbol rate bandwidth to an SNR in the sampling bandwidth. Compared with Figure 8 and Figure 9, it can be seen that the PSK signal added the noise is hard to be recognized.
Figure The scatter plot of the received signal with AWGN
Perform pulse detection and down-sample the signal
%% Received Signal
% Perform the Pulse detection
% Filter received signal using square root raised cosine filter.
yrx=rcosflt(ynoisy,1, nsamp,'Fs/filter', rrcfilter);
% Downsample the signal
yrx = downsample(yrx,nsamp); % Downsample
yrx = yrx(2*delay+1:end-2*delay);
View constellation with a scatter plot
%% Scatter Plot
% Create scatter plot of receiver signal before and after filtering
h = scatterplot(sqrt(nsamp)*ynoisy, nsamp,0,'g.');
title('Recerived Signal, Before and After Filtering');
legend('Before Filtering','After Filtering');
axis([-5 5 -5 5]);
Figure The scatter plot of the received signal before and afetr filtering
It is clear that through filtering, the signal without noise can be detected.
% Demodulate signal using 16-QAM.
h = modem.pskdemod('M',M,'SymbolOrder','Gray');
zsym = demodulate(h,yrx);
% Plot first 10 demodulated symbols in a stem plot.
xlabel('Symbol Index'), ylabel('Integer Value');
Figure First 10 demodulated symbols
Map symbols to bits
%% Symbol-to-Bit Maping
% Undo the bit-to-symbol mapping performed earlier.
% Convert z from a matrix to a vector
z = reshape(z.',numel(z),1);
xlabel('Bit Index'); ylabel('Binary Value');
axis([0 40 -0.5 1.5]);
Figure First 40 demodulated bits
After the whole process of the baseband QPSK modulation and demodulation, the number of the error can be calculated. However, the bit error rate of QPSK is very small and this issue will be discussed in discussion.
The PSK signal in time domain
The transmitted QPSK signal is analog signal which contains real and image part. In the time domain, it is a continuous function.
Figure The real part of QPSK signal in time domain
Figure The image part of QPSK signal in time domain
The spectrum of the QPSK signal
The initial parameters for plotting the spectrum of the QPSK signal are shown below:
%% Plot the Spectrim
EbNo = 10;
filtorder = 40;
delay = filtorder/(SamplePerTb*2);
rolloff = 0.25;
Figure The PSD of QPSK
This figure shows power spectrum density (PSD) of transmitted signal without added AWGN noise.
Figure The PSD of received signal (Through AWGN channel)
This figure shows power spectrum density (PSD) of transmitted signal through AWGN channel.
Figure The PSD of received signal after filtering
This figure shows power spectrum density (PSD) of transmitted signal through AWGN channel and low pass filer.
Above three figures show the power spectrum density of the signals at three nodes. Comparing figure 15 and figure 17, it can be seen that two PSD have same shape with some differences. There are some characteristic values need to be discussed.
First, the flat part value of PSD of QPSK is 0 dB.
Consider QPSK in this case, there is 4 symbol levels which represented by 2 bits as shown in figure 6. The 2 bits in each constellation point can be considered as one bit each on independent QPSK modulation in I-axis and Q-axis respectively.
The average power is signal symbol is 0 dB.
Seen from figure 17, that value changes into 16 dB. The added 16 dB comes from added noise power. The ratio of signal and noise is SNR=Es/No -10*log10 (SamplePerBit). With fixed Es/No, the SNR only depends on the numbers of the samples per bit. Average power in single symbol through the channel would decrease if SamplePerBit decreases.
Second point is bandwidth. The theoretical transition bandwidth should be
, where is the roll off factor.
In this case roll off factor is set as 0.25 which means BW should be 0.624 as shown in figure 15 and 17.
The constellation of the QPSK signal
The constellation of the QPSK signal is shown in figure 6. The distribution of the points in the constellation is in rectangular form. The minimum distance between any two points is
Baseband digital communication architecture:
Number of the samples should be increased before passing through the square root raised cosine filter. Much more points are required to represent a single symbol. Oversampling rate should larger than twice the bandwidth of the signal.
Figure Comparison of the created filter with different oversampling rate
Seen from the above figure, higher oversampling rate gives much more sampled points which make the impulse response of the desired filter much more level and smooth.
Figure Comparison of the created filter with different roll off factor
Seen from the above figure, higher roll off factor makes the impulse response of the desired filter attenuate much more quickly. And also gives much wider bandwidth of the filtering signals. This is the raised-cosine filter characteristics.
The Pulse shape filtering
Figure Comparison of the scatter plot of transmitted signal with or without pulse shaping filter
Oversampling rate in the scatter plot of the pulse shaping filter is set as 4. It is much harder to see the original signals if the oversampling rate increases. Through the filter , the baseband signal can be got.
Pulse detection is the process of applying again the filter on the received signals. Figure 10 gives the results of the pulse detection. Through the pulse detection, noise can be filtered.
Figure Comparison of the original signals and demodulated signals in bits and symbols
From the previous results, the error bits in all transmitted 30000 bits are 0. The bit error rate is much small. So in Figure 21, which is for the first 40bits, there is no error. The displaying transmitted signals are the same as the demodulated signals.
The bit error rate for Gray coded QPSK in Additive White Gaussian Noise is
Furthermore, the average probability of symbol error for coherent MPSK as
, where it is assumed that M is larger than 2, is the signal energy per bit,
Figure Receiver Performance
The ratio of signal and noise is, so SER V.S. can also represent the receiver performance well. The Red line (-) shows the theoretical trend of SER with increased. The black ones (+) comes from the simulation. When or SNR increase, sample error rate decreases. Increase SNR means the ratio of the signal and noise increase so that the disturbances of the noise decrease. Obviously, now the performance of receiver should be much more effective which results in less error.
Figure Comparison of Receiver Performance with different M
Number of Error
Bit Error Rate
Table QPSK of 300000 Random bits signals
From Table 1, it can be seen that Bit Error Rate (BER) of BPSK and QPSK are very low, there is no error when transmitting 300000 random bits signals, and in 8PSK, BER is about 8.667*10^(-5), which is lower than M-QAM. From figure 23, we also can find that the spacing between red line and blue line in M=8 is larger than that in M=2 or 4, which shows the error rate is bigger in M=8.
Although more bits per symbol are transmitted in QPSK with bigger M, the constellation is close together, so probability of error is higher when the distance between two symbols in constellation is closer. In addition, figure 23 shows that theoretically, the BER of BPSK and QPSK (red line) are same, the reason is that: though QPSK can be viewed as a quaternary modulation, it is easier to see it as two independently modulated quadrature carriers. With this interpretation, the even (or odd) bits are used to modulate the in-phase component of the carrier, while the odd (or even) bits are used to modulate the quadrature-phase component of the carrier. BPSK is used on both carriers and they can be independently demodulated. As a result, the probability of BER for QPSK is the same as for BPSK: but if QPSK want to reach same BER as BPSK, QPSK would use twice the power (since two bits are transmitted simultaneously). Furthermore, figure 23 compares the bit-error rates of BPSK, QPSK and 8PSK,it is seen that higher-order modulation exhibit higher error-rates, in exchange however they deliver a higher raw data-rate.
A fundamental parameter for communication system is bandwidth efficiency, R/W, which represents a measure of data throughput per hertz of bandwidth and thus measure how efficiently any signaling technique utilizes the bandwidth resource. For QPSK, with the increase of the k, the bandwidth efficiency increases according to the following equation.
( bits /sec /Hz)
Comparison with other modulations
Although MASK, MPSK and MQAM have the same bandwidth efficiency, with the same Eb/No, BER of MPSK is smaller than that of MASK when M>2 and when M>8, BER of MQAM is smaller than that of MPSK.
When M is large, the constellation points on a circle become progressively less energy efficient and MPSK signaling schemes are no longer of practical interest.
To sum up, increasing bandwidth efficiency is an advantage of M-ary PSK, in other words, higher information transfer rate for a given symbol rate and channel bandwidth or reduce bandwidth requirement for a given bit rate. On the other hand, compared with binary PSK communications, noise/interference immunity is reduced in M-ary PSK. The process of baseband MPSK modulation and demodulation has been presented in this assignment. The receiver performance can be analyzed by calculated by the error bit rate. MPSK is a modulation with high bandwidth efficiency.