In coherent detection, the receiver derives its demodulation frequency and phase references using a carrier synchronization loop. Such synchronization circuits may introduce phase ambiguity in the detected phase, which could lead to erroneous decisions in the demodulated bits. For example, Costas loop exhibits phase ambiguity of integral multiples of radians at the lock-in points. As a consequence, the carrier recovery may lock in radians out-of-phase thereby leading to a situation where all the detected bits are completely inverted when compared to the bits during perfect carrier synchronization. Phase ambiguity can be efficiently combated by applying differential encoding at the BPSK modulator input (Figure 1) and by performing differential decoding at the output of the coherent demodulator at the receiver side (Figure 2).
In ordinary BPSK transmission, the information is encoded as absolute phases: for binary 1 and for binary 0. With differential encoding, the information is encoded as the phase difference between two successive samples. Assuming is the message bit intended for transmission, the differential encoded output is given as
The differentially encoded bits are then BPSK modulated and transmitted. On the receiver side, the BPSK sequence is coherently detected and then decoded using a differential decoder. The differential decoding is mathematically represented as
This method can deal with the phase ambiguity introduced by synchronization circuits. However, it suffers from performance penalty due to the fact that the differential decoding produces wrong bits when: a) the preceding bit is in error and the present bit is not in error , or b) when the preceding bit is not in error and the present bit is in error. After differential decoding, the average bit error rate of coherently detected BPSK over AWGN channel is given by
Following is the Matlab implementation of the waveform simulation model for the method discussed above. Both the differential encoding and differential decoding blocks, illustrated in Figures 1 and 2, are linear time-invariant filters. The differential encoder is realized using IIR type digital filter and the differential decoder is realized as FIR filter.
File 1: dbpsk_coherent_detection.m: Coherent detection of D-BPSK over AWGN channel
Figure 3 shows the simulated BER points together with the theoretical BER curves for differentially encoded BPSK and the conventional coherently detected BPSK system over AWGN channel.
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For a given modulation technique, there are two ways to implement the simulation model: passband model and equivalent baseband model. The passband model is also called waveform level simulation model. The waveform level simulation techniques, described in this chapter, are used to represent the physical interactions of the transmitted signal with the channel. In the waveform level simulations, the transmitted signal, the noise and received signal are all represented by samples of waveforms.
Typically, a waveform level simulation uses many samples per symbol. For the computation of error rate performance of various digital modulation techniques, the value of the symbol at the symbol-sampling time instant is all the more important than the look of the entire waveform. In such a case, the detailed waveform level simulation is not required, instead equivalent baseband discrete-time model, described in chapter 3 can be used. Discrete-time equivalent channel model requires only one sample per symbol, hence it consumes less memory and yields results in a very short span of time.
In any communication system, the transmitter operates by modulating the information bearing baseband waveform on to a sinusoidal RF carrier resulting in a passband signal. The carrier frequency, chosen for transmission, varies for different applications. For example, FM radio uses carrier frequency range, whereas for indoor wireless networks the center frequency of transmission is . Hence, the carrier frequency is not the component that contains the information, rather it is the baseband signal that contains the information that is being conveyed.
Actual RF transmission begins by converting the baseband signals to passband signals by the process of up-conversion. Similarly, the passband signals are down-converted to baseband at the receiver, before actual demodulation could begin. Based on this context, two basic types of behavioral models exist for simulation of communication systems – passband models and its baseband equivalent. In the passband model, every cycle of the RF carrier is simulated in detail and the power spectrum will be concentrated near the carrier frequency . Hence, passband models consume more memory, as every point in the RF carrier needs to be stored in computer memory for simulation.
On the other hand, the signals in baseband models are centered near zero frequency. In baseband equivalent models, the RF carrier is suppressed and therefore the number of samples required for simulation is greatly reduced. Furthermore, if the behavior of the system is well understood, the baseband model can be further simplified and the system can be implemented entirely based on the samples at symbol-sampling time instants.
BPSK stands for Binary Phase Shift Keying. It is a type of modulation used in digital communication systems to transmit binary data over a communication channel.
In BPSK, the carrier signal is modulated by changing its phase by 180 degrees for each binary symbol. Specifically, a binary 0 is represented by a phase shift of 180 degrees, while a binary 1 is represented by no phase shift.
BPSK is a straightforward and effective modulation method and is frequently utilized in applications where the communication channel is susceptible to noise and interference. It is also utilized in different wireless communication systems like Wi-Fi, Bluetooth, and satellite communication.
Implementation details
Binary Phase Shift Keying (BPSK) is a two phase modulation scheme, where the 0’s and 1’s in a binary message are represented by two different phase states in the carrier signal: \(\theta=0^{\circ}\) for binary 1 and \(\theta=180^{\circ}\) for binary 0.
In digital modulation techniques, a set of basis functions are chosen for a particular modulation scheme. Generally, the basis functions are orthogonal to each other. Basis functions can be derived using Gram Schmidt orthogonalizationprocedure [1]. Once the basis functions are chosen, any vector in the signal space can be represented as a linear combination of them. In BPSK, only one sinusoid is taken as the basis function. Modulation is achieved by varying the phase of the sinusoid depending on the message bits. Therefore, within a bit duration \(T_b\), the two different phase states of the carrier signal are represented as,
\begin{align*}
s_1(t) &= A_c\; cos\left(2 \pi f_c t \right), & 0 \leq t \leq T_b \quad \text{for binary 1}\\
s_0(t) &= A_c\; cos\left(2 \pi f_c t + \pi \right), & 0 \leq t \leq T_b \quad \text{for binary 0}
\end{align*}
where, \(A_c\) is the amplitude of the sinusoidal signal, \(f_c\) is the carrier frequency \(Hz\), \(t\) being the instantaneous time in seconds, \(T_b\) is the bit period in seconds. The signal \(s_0(t)\) stands for the carrier signal when information bit \(a_k=0\) was transmitted and the signal \(s_1(t)\) denotes the carrier signal when information bit \(a_k=1\) was transmitted.
The constellation diagram for BPSK (Figure 3 below) will show two constellation points, lying entirely on the x axis (inphase). It has no projection on the y axis (quadrature). This means that the BPSK modulated signal will have an in-phase component but no quadrature component. This is because it has only one basis function. It can be noted that the carrier phases are \(180^{\circ}\) apart and it has constant envelope. The carrier’s phase contains all the information that is being transmitted.
BPSK transmitter
A BPSK transmitter, shown in Figure 1, is implemented by coding the message bits using NRZ coding (\(1\) represented by positive voltage and \(0\) represented by negative voltage) and multiplying the output by a reference oscillator running at carrier frequency \(f_c\).
The following function (bpsk_mod) implements a baseband BPSK transmitter according to Figure 1. The output of the function is in baseband and it can optionally be multiplied with the carrier frequency outside the function. In order to get nice continuous curves, the oversampling factor (\(L\)) in the simulation should be appropriately chosen. If a carrier signal is used, it is convenient to choose the oversampling factor as the ratio of sampling frequency (\(f_s\)) and the carrier frequency (\(f_c\)). The chosen sampling frequency must satisfy the Nyquist sampling theorem with respect to carrier frequency. For baseband waveform simulation, the oversampling factor can simply be chosen as the ratio of bit period (\(T_b\)) to the chosen sampling period (\(T_s\)), where the sampling period is sufficiently smaller than the bit period.
function [s_bb,t] = bpsk_mod(ak,L)
%Function to modulate an incoming binary stream using BPSK(baseband)
%ak - input binary data stream (0's and 1's) to modulate
%L - oversampling factor (Tb/Ts)
%s_bb - BPSK modulated signal(baseband)
%t - generated time base for the modulated signal
N = length(ak); %number of symbols
a = 2*ak-1; %BPSK modulation
ai=repmat(a,1,L).'; %bit stream at Tb baud with rect pulse shape
ai = ai(:).';%serialize
t=0:N*L-1; %time base
s_bb = ai;%BPSK modulated baseband signal
BPSK receiver
A correlation type coherent detector, shown in Figure 2, is used for receiver implementation. In coherent detection technique, the knowledge of the carrier frequency and phase must be known to the receiver. This can be achieved by using a Costas loop or a Phase Lock Loop (PLL) at the receiver. For simulation purposes, we simply assume that the carrier phase recovery was done and therefore we directly use the generated reference frequency at the receiver – \(cos( 2 \pi f_c t)\).
In the coherent receiver, the received signal is multiplied by a reference frequency signal from the carrier recovery blocks like PLL or Costas loop. Here, it is assumed that the PLL/Costas loop is present and the output is completely synchronized. The multiplied output is integrated over one bit period using an integrator. A threshold detector makes a decision on each integrated bit based on a threshold. Since, NRZ signaling format was used in the transmitter, the threshold for the detector would be set to \(0\). The function bpsk_demod, implements a baseband BPSK receiver according to Figure 2. To use this function in waveform simulation, first, the received waveform has to be downconverted to baseband, and then the function may be called.
function [ak_cap] = bpsk_demod(r_bb,L)
%Function to demodulate an BPSK(baseband) signal
%r_bb - received signal at the receiver front end (baseband)
%N - number of symbols transmitted
%L - oversampling factor (Tsym/Ts)
%ak_cap - detected binary stream
x=real(r_bb); %I arm
x = conv(x,ones(1,L));%integrate for L (Tb) duration
x = x(L:L:end);%I arm - sample at every L
ak_cap = (x > 0).'; %threshold detector
End-to-end simulation
The complete waveform simulation for the end-to-end transmission of information using BPSK modulation is given next. The simulation involves: generating random message bits, modulating them using BPSK modulation, addition of AWGN noise according to the chosen signal-to-noise ratio and demodulating the noisy signal using a coherent receiver. The topic of adding AWGN noise according to the chosen signal-to-noise ratio is discussed in section 4.1 in chapter 4. The resulting waveform plots are shown in the Figure 2.3. The performance simulation for the BPSK transmitter/receiver combination is also coded in the program shown next (see chapter 4 for more details on theoretical error rates).
The resulting performance curves will be same as the ones obtained using the complex baseband equivalent simulation technique in Figure 4.4 of chapter 4.
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