By   July 28, 2020

8.22.1. Probability of Error
In Table 8.7. we have summarized the expressions for the bit error rate (BER) for coherent binary PSK, conventional coherent binary FSK with one-bit decoding, DPSK, noncoherent binary FSK, coherent QPSK, and coherent MSK, when operating over an AWGN channel. In figure 8.46, we have used the expressions summarized in Table 8.10 to plot the BER as a function of the signal energy per bit-to-noise spectral density ratio, Eb/N0.
Table 8.10. Summary of formulas for the bit error rate of different digital modulation schemes

Signaling Scheme Bit Error Rate
(a) Coherent binary PSK
Coherent QPSK
Coherent MSK
(b) Coherent binary FSK
(c) DPSK
(d) Noncoherent binary FSK

Based on the performance curves shown in figure 8.44, the summary of formulas given in Table 8.7, and the defining equations for the pertinent modulation formats, we can make the following statements :
(i)         The bit error rates for all the systems decrease monotonically with increasing values of Eb/N0. The defining curves have a similar shape in the form of a waterfall.
(ii)        For any value of Eb/N0, coherent binary PSK, QPSK, and MSK produce a smaller bit error rate than any of the other modulation schemes.
(iii)       Coherent binary PSK and DPSK require an Eb/N0, that is 3 dB less than the corresponding values for conventional coherent binary FSK and noncoherent binary FSK, respectively, to realize the same bit error rate.
(iv)       At high values of Eb/N0, DPSK and noncoherent binary FSK perform almost as well (to within about 1 dB) as coherent binary PSK and conventional coherent binary FSK, respectively, for the same bit rate and signal energy per bit.
(v)        In coherent QPSK, two orthogonal carries  cos(2pfct) and  sin (2pfct) are used, where the carrier frequency fc is an integer multiple of the symbol rate 1/T, with the result that two independent bit streams can be transmitted simultaneously and subsequently detected in the receiver.
FIGURE 8. 44 Comparison of the noise performance of different PSK and FSK schemes.
(vi)       In the case of coherent MSK, there are two orthogonal carriers, namely,  cos(2pfc(t) and  sin(2pfct), which are modulated by the two antipodal symbol shaping pulses cos(pt/2Tb) and sin(pt/2Tb), respectively, over 2Tb intervals, where Tb is the bit duration. Correspondingly, the receiver uses a coherent phase decoding process over two successive bit intervals to recover the original bit stream.
(vii)      The MSK scheme differs from its counterpart, the QPSK, in that its receiver has memory. In particular, the MSK receiver makes decisions based on observations over two successive bit intervals. Thus, although the transmitted signal has a binary format represented by the transmission of two distinct frequencies, the presence of memory in the receiver makes it assume a two-dimensional signal space diagram. There are four message points, depending on which binary symbol (0 or 1) was sent and the past phase history of the FSK signal.
■          Modulation is defined as the process by which some characteristics of a carrier is varied in accordance with a modulating signal.
■          In digital communications, the modulating signal consists of binary data or an M-ary encoded version of it.
■          The channel may be a telephone channel, microwave radio link, satellite channel or an optical fiber. In digital communication, the modulation process involves switching or keying the amplitude, frequency or phase of the carrier in accordance with the input data.
■          There are three basic modulation techniques for the transmission of digital data. They are known as amplitude-shift keying (ASK), frequency shift keying (FSK) and phase-shift keying (PSK) which can be viewed as special cases of amplitude modulation frequency modulation and phase modulation respectively.
■          When we have to transmit a digital signal over a long distance, we need continuous-wave (CW) modulation. For this purpose, the transmission medium can be in form of radio, cable or other type of channel. Also, a carrier signal having some frequency fc is used for modulation. Then the modulating digital signal modulates some pameter like frequency, phase or amplitude of the carrier.
■          There is some deviation in carrier frequency fc. This deviation is known as the bandwidth of the channel. This means that the channel has to transmit some range or band of frequencies. Such type of transmission is known as bandpass transmission and the communication channel is known as bandpass channel.
■          When it is required to transmit digital signals on a bandpass channel, the amplitude, frequency or phase of the sinusoidal carrier is varied in accordance with the incoming digital data. Since the digital data is in discrete steps, the modulation of the bandpass sinusoidal carrier is also done in discrete steps. Due to this reason, this type of modulation (i.e., Digital modulation) is also known as switching or signaling.
■          Because of constant amplitude of FSK and PSK, the effect of non-linearities, noise interference is minimum on signal detection. However, these effects are more pronounced on ASK. Therefore, FSK and PSK are preferred over ASK.
■          In digital modulations, instead of transmitting one bit at a time, we transmit two or more bits simulteneously. This is known as M-ary transmission. This type of transmission results in reduced channel bandwidth. However, sometimes, we use two quadrature carriers for modulation. it’s process is known as Quadrature modulation.
■          There are a number of modulation schemes available to the designer of a digital communication system required for data transmission over a bandpass channel.
Every scheme offers system trade-offs of its own. However, the final choice made by the designer is determined by the way in which the available primary communication resources such as transmitted power and channel bandwidth.
■          Coherent digital modulation techniques are those techniques which employ coherent detection. In coherent detection, the local carrier generated at the receiver is phase locked with the carrier at the transmitter. Thus, the detection is done by correlating received noisy signal and locally generated carrier. The coherent detection is a synchronous detection.
■          Non-coherent digital modulation techniques are those techniques in which the detection process does not need receiver carrier to be phase locked with transmitter carrier.
■          The binary (i.e., digital) modulation has three basic forms amplitude-shift keying (ASK), phase-shift keying (PSK) and frequency-shift keying (FSK). In this section let us discuss different coherent binary modulation techniques.
■          ASK signal may be generated by simply applying the incoming binary data (represented in unipolar form) and the sinusoidal carrier to the two inputs of a product modulator (i.e., balanced modulator).
■          The demodulation of binary ASK waveform can be achieved with the help of coherent detector. It consists of a product modulator which is followed by an integrator and a decision making device.
■          In a binary phase shift keying (BPSK), binary symbol ‘1’ and ‘0’ modulate the phase of the carrier.
■          The BPSK signal may be generated by applying carrier signal to a balanced modulator. Here, the baseband signal b(t) is applied as a modulating signal to the balanced modulator.
■          It is not possible to determine whether the received signal is equal to b(t) or – b(t). Infact, this results in ambiguity in the output signal.
■          In binary frequency shift keying (BFSK), the frequency of the carrier is shifted according to the binary symbol. However, the phase of the carrier is unaffected. That is we have two different frequency signals according to binary symbols.
■          Orthogonal carriers are used for M-ary PSK and QASK.
■          The generation of BFSK is quite easy, even that it has many disadvantages compared to BPSK signal. Firstly, its bandwidth is greater than 4fb, which is almost double the bandwidth of BPSK. Also, the distance between the signal points is less in case of BFSK. Therefore, the error rate of BFSK is more as compared to BPSK.
■          Coherent detection exploits knowledge of the carrier wave’s phase reference, and thus providing the optimum error performance attainable with a digital modulation format of interest. However, when it is impractical to have knowledge of the carrier phase at the receiver, we make use of non-coherent detection.
■          In the case of phase-shift keying (PSK), we cannot have “non-coherent PSK” since non-coherent means doing without phase information. However, there is a ‘pseudo PSK’ technique known as differential phase-shift keying (DPSK) which can be viewed as the non-coherent form of PSK.
■          We can view differential phase-shift keying as the non-coherent version of the PSK. Differential phase shift keying (DPSK) is differentially coherent modulation method. DPSK does not need a synchronous (coherent) carrier at the demodulator.
■          DPSK does not need carrier at the receiver end. This means that the complicated circuitry for generation of local carrier is not required.
■          The bandwidth requirement of DPSK is reduced as compared to that of BPSK. The probability or error (i.e., bit error rate) of DPSK is higher than that of BPSK.
■          Because DPSK uses two successive bits for its reception, error in the first bit creates error in tho second bit. Therefore, error propagation in DPSK is more. On the other hand, in BPSK single hit can go in error since detection of each bit is independent.
■          In communication systems, we have two main resources. These are the transmission power and the channel bandwidth. The channel bandwidth depends upon the bit rate or signaling rate fb. In digital bandpass transmission, we use a carrier for transmission. This carrier is transmitted over a channel. If two or more bits are combined in some symbols, then the signaling rate will be reduced. Thus, the frequency of the carrier needed is also reduced. This reduces the transmission channel bandwidth. Hence, because of grouping of bits in symbols, the transmission channel bandwidth can be reduced. In quadrature phase shift keying (QPSK), two successive bits in the data sequence are grouped together. This reduces the bits rate or signaling rate (i.e., fb) and thus reduces the bandwidth of the channel.

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