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(a) QPSK modulator; (b) waveforms for (a).
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TABLE 10.1
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QPSK Modulator States pq(t) 1 1 1 1 QPSK
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Phase diagram for QPSK modulation.
low-pass filtering, the output of the upper BPSK demodulator is 0.5pi(t) and the output of the lower BPSK demodulator is 0.5pq(t). These two signals are combined in the parallel-to-serial converter to yield the desired output p(t). As with the BPSK signal, noise will create errors in the demodulated output of the QPSK signal.
10.6.3 Transmission rate and bandwidth for PSK modulation
Equation (10.14), which shows the baseband signal p(t) multiplied onto the carrier cos 0t, is equivalent to double-sideband, suppressed-carrier modulation. The digital modulator circuit of Fig. 10.12a is similar to the single-sideband modulator circuit shown in Fig. 9.2, the difference being that after the multiplier, the digital modulator requires a bandpass filter, while the analog modulator requires a single-sideband filter. As
Demodulator circuit for QPSK modulation.
Digital Signals
shown in Fig. 9.1, the DSBSC spectrum extends to twice the highest frequency in the baseband spectrum. For BPSK modulation the latter is given by Eq. (10.11) with Rsym replaced with Rb: BIF 2B (1 )Rb (10.15)
Thus, for BPSK with a rolloff factor of unity, the IF bandwidth in hertz is equal to twice the bit rate in bits per second. As shown in the previous section, QPSK is equivalent to the sum of two orthogonal BPSK carriers, each modulated at a rate Rb /2, and therefore, the symbol rate is Rsym Rb/2. The spectra of the two BPSK modulated waves overlap exactly, but interference is avoided at the receiver because of the coherent detection using quadrature carriers. Equation (10.15) is modified for QPSK to BIF (1 1 2 )Rsym (10.16) Rb
An important characteristic of any digital modulation scheme is the ratio of data bit rate to transmission bandwidth. The units for this ratio are usually quoted as bits per second per hertz (a dimensionless ratio in fact because it is equivalent to bits per cycle). Note that it is the data bit rate Rb and not the symbol rate Rsym which is used. For BPSK, Eq. (10.15) gives an Rb/BIF ratio of 1/(1 ), and for QPSK, Eq. (10.16) gives an Rb/BIF ratio of 2/(1 ). Thus QPSK is twice as efficient as BPSK in this respect. However, more complex equipment is required to generate and detect the QSPK modulated signal.
10.6.4 Bit error rate for PSK modulation
Referring back to Fig. 10.13, the noise at the input to the receiver can cause errors in the detected signal. The noise voltage, which adds to the signal, fluctuates randomly between positive and negative values, and thus the sampled value of signal plus noise may have the opposite polarity to that of the signal alone. This would constitute an error in the received pulse. The noise can be represented by a source at the front of the receiver, shown in Fig. 10.13 (this is discussed in detail in Chap. 12). It is seen that the noise is filtered by the receiver input filter. Thus the receive filter, in addition to contributing to minimizing the ISI, must minimize noise while maximizing the received signal. In short, it must maximize the received signal-to-noise ratio. In practice for satellite links (or radio links), this usually can be
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achieved by making the transmit and receive filters identical, each having a frequency response which is the square root of the raised-cosine response. Having identical filters is an advantage from the point of view of manufacturing. The most commonly encountered type of noise has a flat frequency spectrum, meaning that the noise power spectrum density, measured in joules (or W/Hz), is constant. The noise spectrum density will be denoted by N0. When the filtering is designed to maximize the received signalto-noise ratio, the maximum signal-to-noise voltage ratio is found to be equal to 22Eb >N0 , where Eb is the average bit energy. The average bit energy can be calculated knowing the average received power PR and the bit period Tb. Eb PRTb (10.17)
The probability of the detector making an error as a result of noise is given by Pe Eb 1 erfc a b N0 2 (10.18)
where erfc stands for complementary error function, a function whose value is available in tabular or graphic form in books of mathematical tables and as built-in functions in many computational packages. A related function, called the error function, denoted by erf( ) is sometimes used, where erfc(x) 1 erf(x) (10.19)
Equation (10.18) applies for polar NRZ baseband signals and for BPSK and QPSK modulation systems. The probability of bit error is also referred to as the bit error rate (BER). A Pe of 10 6 signifies a BER of 1 bit in a million, on average. The graph of Pe versus Eb/N0 in decibels is shown in Fig. 10.17. Note carefully that the energy ratio, not the decibel value, of Eb/N0 must be used in Eq. (10.18). This is illustrated in the following example.
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