Modems for Wireless Communications

Abstract

Digital modulation and demodulation techniques together (modem) form the fundamental building block of data transmission in communication systems in general and wireless communication systems in particular (Lucky et al. 1968; Oetting 1979; Amoroso 1980; Feher 1995; Benedetto and Biglieri 1999; Proakis 2001; Simon and Alouini 2005; Schwartz 2005; Couch 2007). The modems can be classified in a multitude of ways (Simon et al. 1995; Sklar 1993, 2001; Haykin 2001). They can be identified in terms of the signal property that is modulated such as amplitude, phase, or frequency. They can also be classified in terms of the number of levels of values (binary, quaternary, or in general M-ary) attributable to the property. Detection methods such as coherent or noncoherent ones can also be used for the classification. We can, in addition, use terms such as “linear” and “nonlinear” modulation to classify the modulation types (Sundberg 1986; Anderson et al. 1986; Gagliardi 1988; Gallager 2008). The output of the frequency modulated system has a constant envelope while the output of the amplitude or phase modulated system has a constant frequency with time varying amplitudes. This makes the amplitude and phase modulation a form of “linear” modulation and the frequency modulation a form of “nonlinear” modulation. Even for a specific modulation type such as phase modulation (phase shift keying, for example), it is possible to have a coherent or a noncoherent detector or receiver. Modems cover a wide range of possibilities with some common themes such as the initial building blocks of the “signal space.” The concepts of signal space make it possible to analyze modems, design them so that they meet certain criteria, provide a uniform framework, and make it possible to compare and contrast the different modems.

Appendix

We will now explore a few topics of interest that were mentioned in passing earlier in this chapter. These include the difference between signal-to-noise ratio and energy-to-noise ratio, bandwidth concepts as they pertain to digital signals, synchronization, intersymbol interference, and so on. We will examine the effects of phase mismatch in coherence detection and also problems with timing.

3.1.1 Noise, Signal-to-Noise Ratio, Symbol Energy, Bit Energy

We had introduced the notion of noise at the beginning of this chapter by stating that the noise in wireless systems is assumed to be additive white Gaussian noise (Taub and Schilling 1986; Sklar 1993, 2001; Proakis 2001). Typically signal-to-noise ratio (SNR) is defined as the ratio of the signal power (Pr) to the noise power (Nr)

$$ \mathrm{SNR}=\frac{P_r}{N_r}. $$

(3.130)

The noise power is given by the product of 2B and the spectral density of noise is given by (N ₀/2) where B is the message bandwidth. Since we are dealing with signals of finite energy (bit or symbol), the SNR can be written in terms of the energy as

$$ \mathrm{SNR}=\frac{E_{\mathrm{s}}}{N_0{BT}_{\mathrm{s}}}=\frac{E_{\mathrm{b}}}{N_0{BT}_{\mathrm{b}}}. $$

(3.131)

However, in most of the analysis of wireless systems, the SNR is expressed as either (Es/N ₀) or (Eb/N ₀) for the SNR per symbol or SNR per bit. The performance is generally measured using the average bit error rate. Thus, if one uses MPSK or MFSK, the symbol error rate must be converted to bit error rate for the purpose of comparison.

3.1.2 Bandwidth of Digital Signals

While the notion of bandwidth is reasonably clear in analog systems, the bandwidth of digital signals has several definitions, often leading to problems in understanding the spectral content (Amoroso 1980; Aghvami 1993; Sklar 2001; Haykin 2001). There are several ways of defining the bandwidth, each with different meanings. Primarily this is because the basic shape of the message signal is a time limited one and, thus, ideally the bandwidth is infinite. We will now look at the multiple definitions of bandwidth and understand the relationships among them. We will choose the example of a binary ASK or PSK. The modulated output can be represented as

$$ p(t)={p}_T(t) \cos \left(2\pi {f}_0 t\right),\kern1em 0\le t\le T. $$

(3.132)

In (3.132), the pulse train is represented by

$$ {p}_T(t)=\sum_{n=-\infty}^{\infty } m\left( t- nT\right) $$

(3.133)

with

$$ m(t)=\pm 1,\kern0.5em NT\le t\le \left( N+ T\right) T. $$

(3.134)

The data rate R = (1/T). The spectrum of p(t) is given as

$$ P(f)=\frac{T}{4}{\left[\frac{ \sin \pi T\left( f-{f}_0\right)}{\pi T\left( f-{f}_0\right)}\right]}^2+\frac{T}{4}{\left[\frac{ \sin \pi T\left( f+{f}_0\right)}{\pi T\left( f+{f}_0\right)}\right]}^2. $$

(3.135)

The spectrum of the modulated signal is shown along with the spectrum of the basic pulse, as seen in Fig. 3.51.

Fig. 3.51

Spectra of signals

We will use Fig. 3.51 to define and compare the multiple definitions.

3.1.3 Absolute Bandwidth

Absolute bandwidth of a signal is defined as the frequency range outside (positive frequencies) of which the power is zero. This is infinite both for the bandpass signal p(t) and baseband signal m(t). Thus, this absolute bandwidth is not useful in comparing digital signals in terms of bandwidth since both baseband and bandpass signals have infinite bandwidth.

3.1.4 3 dB Bandwidth

The 3 dB bandwidth or half power bandwidth is the frequency range (positive frequencies, f ₁, f ₂ with f ₂ > f ₁), where the power drops to 50% of the peak value.

The bandwidth is given by (f ₂ − f ₁). It will have a different value for baseband and bandpass signals.

3.1.5 Equivalent Bandwidth

This is the extent of the positive frequencies occupied by a rectangular window such that it has a height equal to the peak value of the positive spectrum and a power equal to the power contained within the positive frequencies. This is shown by the rectangular window in Fig. 3.51 (top), and has a value of R.

3.1.6 Null-to-Null Bandwidth

This corresponds to the frequency band between two “nulls” on either side of the peak. For the bandpass spectrum, this corresponds to 2R.

3.1.7 Bounded Spectrum Bandwidth

This is the value of (f ₂ − f ₁) such that outside the band f ₁ < f < f ₂, the power spectrum is down by at least a significant amount (about 50 dB) below the peak value.

3.1.8 Power (99%) Bandwidth

This is the range of frequencies (f ₂ − f ₁) such that 99% of the power resides in that frequency band.

For the BPSK signal, these values are given in Tables 3.6 and 3.7.

Table 3.6 Definition of bandwidths of BPSK

Table 3.7 Bandwidths comparison

3.1.9 Carrier Regeneration and Synchronization

For the matched filter for both BPSK and QPSK, we require a carrier wave of matching frequency and phase at the receiver. There are two ways of accomplishing this (Gagliardi 1988; Haykin 2001; Anderson 2005). One is to transmit a pilot tone along with the modulated signal. This leads to the waste of transmit power since the pilot tone also carries some power. The second method involves the regeneration of the carrier wave from the BPSK or QPSK signal. Since the carrier recovery approaches are different for BPSK and QPSK (as we will see), we will start with the carrier recovery system for BPSK first.

A block diagram of the carrier recovery system for BPSK is shown in Fig. 3.52. It is based on the squaring operation on the received signal. If x(t) represents the BPSK signal we have

$$ x(t)= Am(t) \cos \left[2\pi {f}_0 t+\phi (t)\right]+ n(t). $$

(3.136)

Fig. 3.52

Block diagram of carrier recovery

In (3.136), m(t) represents the bipolar amplitude values of ±1. The noise is n(t) and the phase of the signal which must also be tracked along with the carrier frequency of f ₀. The output of the squarer is

$$ y(t)={x}^2(t)=\frac{A^2}{2}+\frac{A^2}{2} \sin \left[2\pi \left(2{f}_0\right) t+2\phi (t)\right]+{n}^2(t)+\cdots . $$

(3.137)

Note that the second term does not depend on m(t) at all and contains the double frequency term. A phase locked loop with a voltage controlled oscillator (VCO) at 2f ₀ tracks this double frequency term. The VCO output will give a signal at a frequency of 2f ₀ and phase of 2f(t). This is fed through a frequency divider providing a signal at frequency of f ₀ and an approximate phase of f(t), the required signal. We will later examine the problems arising out of phase mismatching which can pose serious problems.

For the case of QPSK modulation, a squarer will not suffice since there are four symbols and, therefore, four possible combinations of the bit pairs. Thus, instead of the squarer, one must use an mth law device where m = 4, i.e., one needs to take the fourth power of the incoming QPSK signal and use a frequency division by 4 to get the matching phase and frequency as shown in Fig. 3.53. Note that even in this case, the phase mismatch will create problems. We will explore those problems in the next section.

Fig. 3.53

QPSK carrier reference generator

For higher order phase shift keying, the concept of the QPSK carrier phase reference generation can be extended by replacing the mth law device with the Mth law device and using a (1/M) frequency divider.

The same approach needs to be used with the binary and M-ary FSK with the phase and carrier frequency circuits for each of the carrier frequencies of the incoming modulated signal at the receiver.

Digital reception also requires timing information so that the symbols and bit can be correctly recovered at the receiver. The output is sampled at regular intervals of T plus a small time delay that can account for the traverse of the data from the transmitter to the receiver. This requires the need for a clock signal at the receiver. The procedure of generating this clock signal is called symbol synchronization or timing recovery.

Symbol synchronization can be accomplished in several ways. Often, the transmitter and receiver clocks are synchronized to a master clock which takes care of the clock recovery issues. The receiver can estimate the transit time delay and generate the complete information necessary. The other option is to transmit a clock frequency (1/T). The receiver can use a very narrow bandpass signal and receive this information. However, such a step will apportion a part of the transmit power for the clock signal. This signal will also occupy a small fraction of a bandwidth as well. Thus, other techniques using the received data must be used for the timing recovery. One such technique uses the symmetry properties of the signal at the output of the matched filter. This gives a peak at t = T for a rectangular pulse. Similar characteristics can be used with other pulse shapes as well. As in the case of carrier phase recovery, there can be timing errors.

There are also techniques that allow the simultaneous estimation of carrier phase and symbol timing. But the purpose of this brief discussion is to indicate that while the carrier phase and symbol timing are critical in demodulation of digitally modulated signals, errors in the estimation of these two will impact the overall performance of the wireless system since such errors can lead to increased error rates. We will explore such effects in the following section.

3.1.10 Problems of Phase Mismatch: Deterministic and Random, Timing Error, etc.

In reference to phase and frequency, the coherent detector requires a perfect match of the local oscillator. A common problem is the phase mismatch (Prabhu 1969, 1976b; Gagliardi 1988; Shankar 2002). If the phase mismatch is ψ _m, the error rate for BPSK becomes

$$ p\left( e;{\psi}_{\mathrm{m}}\right)= Q\left[\sqrt{\frac{2 E}{N_0}} \cos \left({\psi}_{\mathrm{m}}\right)\right]. $$

(3.138)

Often, because of the instabilities in the local oscillator, the phase mismatch becomes a random variable. If the pdf of the phase mismatch is Gaussian,

$$ f\left(\psi \right)=\frac{1}{\sqrt{2{\pi \sigma}_{\psi}^2}} \exp \left[-\frac{{\left(\psi -{\psi}_0\right)}^2}{2{\sigma}_{\psi}^2}\right]. $$

(3.139)

The error rate becomes

$$ p(e)={\int}_{-\infty}^{\infty } Q\left[\sqrt{\frac{2 E}{N_0}} \cos \left({\psi}_{\mathrm{m}}\right)\right]\frac{1}{\sqrt{2{\pi \sigma}_{\psi}^2}} \exp \left[-\frac{{\left(\psi -{\psi}_0\right)}^2}{2{\sigma}_{\psi}^2}\right]\mathrm{d}\psi . $$

(3.140)

The error rate in (3.140) is plotted in Fig. 3.54 for two values of fixed phase mismatch.

Fig. 3.54

Error rate in BPSK as a function of phase mismatch

Figure 3.55 shows the effects of angular mismatch for a fixed SNR .

Fig. 3.55

Error rates as a function of angular mismatch for three values of the SNR

The error rate in the presence of random phase mismatch is shown in Fig. 3.56 for a few of the average phase mismatch (ψ ₀ = 0) and standard deviation (σ _ψ ). Note that as the standard deviation of the mismatch increases, the error curves flatten out and one sees that beyond a certain SNR, the error rate does not come down, signifying the existence of an error floor. It is also possible to see that even when the mismatch is deterministic, as the mismatch approaches π/2, the argument of the Q function in (3.138) approaches zero, suggesting a significant increase in the error rate.

Fig. 3.56

BPSK bit error rates for random phase match

The importance of maintaining phase matching is seen from these results. The performance of a BPSK system is ultimately determined by the integrity of the local oscillator and its ability to provide a perfect match.

Another problem in BPSK is assumed from the assumption of perfect timing and absence of any timing “jitter ” (Cooper and McGillem 1986; Gagliardi 1988). This means that there is some imprecision in the location of the timing instant. The signal correlation which produces the output of the matched filter does not result in the ideal value because the deviation from the ideal timing instant is t (timing offset) (Fig. 3.57). We can quantify the effect of timing errors on the error probability by considering the receiver structure of the matched filter discussed earlier.

Fig. 3.57

Timing offset

The output of the matched filter is

$$ x\left( T,\tau \right)=\pm {\int}_0^{T-\tau} s\left( t+\tau \right) s(t)\mathrm{d} t\pm {\int}_{T-\tau}^T s\left( t+\tau - T\right) s(t)\mathrm{d} t. $$

(3.141)

In (3.141), s(t) is the signal. The first integral arises from the symbol/bit being considered (0,T), and the next integral comes from the adjoining symbol/bit (T, 2T).

The ± accounts for the bipolar nature, with the first ± depending on the bit in (0,T) and the second ± depending on the adjoining window. In an ideal case of no timing errors, t = 0 and the second term vanishes and the first term provides the maximum overlap and will be a scaled version of the energy in the bit. The integrals are forms of correlation and the correlation function can be written as

$$ \rho \left(\alpha \right)=\frac{1}{E}{\int}_0^T s\left( t+\alpha \right) s(t)\mathrm{d} t. $$

(3.142)

For the BPSK case

$$ s(t)=\pm \sqrt{\frac{2 E}{T}} \cos \left(2\pi {f}_0 t\right). $$

(3.143)

Using (3.143), (3.142) becomes

$$ \rho \left(\alpha \right)= \cos \left(2\pi {f}_0\alpha \right). $$

(3.144)

Thus, if the bits are same, (3.141) yields

$$ x\left( T,\tau \right)=\pm E\rho \left(\tau \right)=\pm E \cos \left(2\pi {f}_0\tau \right). $$

(3.145)

If the bits are different, we have

$$ x\left( T,\tau \right)=\pm \left\{ E \cos \left(2\pi {f}_0\tau \right)-2\left|{\int}_{T-\tau}^T s\left( t+\tau - T\right) s(t)\mathrm{d} t\right|\right\}. $$

(3.146)

Simplifying,

$$ x\left( T,\tau \right)=\pm \left\{ E \cos \left(2\pi {f}_0\tau \right)-2 E\left|\frac{\tau}{T} \cos \left(2\pi {f}_0\tau \right)\right|\right\}. $$

(3.147)

Note that when t is equal to zero, both (3.145) and (3.147) yield the same value, namely E. The error rate in the presence of the timing error now becomes

$$ p\left( e;\tau \right)=\frac{1}{2} Q\left(\sqrt{\frac{2 E}{N_0}}\right)+\frac{1}{2} Q\left[\sqrt{\frac{2 E}{N_0}}\left(1-2\left|\varepsilon \right|\right)\right]. $$

(3.148)

In (3.148), the normalized timing offset is

$$ \varepsilon =\frac{\tau}{T}. $$

(3.149)

It has been assumed that the carrier frequency is high and that

$$ \cos \left(2\pi {f}_0\tau \right)\approx 1. $$

(3.150)

Just as in the case of the phase mismatch, if the timing offset is also random, the error rate will increase further. It is essential that timing offset be reduced to a small fraction of the bit duration. If there are problems of phase mismatch and timing offset simultaneously, the error rate can be written as

$$ p\left( e;\tau, \psi \right)=\frac{1}{2} Q\left[\sqrt{\frac{2 E}{N_0}} \cos \left(\psi \right)\right]+\frac{1}{2} Q\left[\sqrt{\frac{2 E}{N_0}} \cos \left(\psi \right)\left(1-\frac{2\left|\tau \right|}{T}\right)\right]. $$

(3.151)

In (3.148) and (3.151), E is the bit energy. The error rates in the presence of timing mismatch are plotted in Fig. 3.58.

Fig. 3.58

BER plotted for the normalized timing offset for BPSK

A similar analysis for phase mismatch in coherent QPSK can be carried out (Gagliardi 1988). Noting that the bit error rate for QPSK is half of the symbol error rate, the bit error rate with no phase mismatch from (3.63) is

$$ {p}_{\mathrm{b}}(e)= Q\left[\sqrt{\frac{2 E}{N_0}} \sin \left(\frac{\pi}{4}\right)\right]= Q\left[\sqrt{\frac{E}{N_0}}\right]. $$

(3.152)

The bit error rate for coherent QPSK when phase mismatch exists becomes

$$ {p}_{\mathrm{b}}(e)=\frac{1}{2} Q\left\{\sqrt{\frac{E}{N_0}}\left[ \cos \left(\psi \right)+ \sin \left(\psi \right)\right]\right\}+\frac{1}{2} Q\left\{\sqrt{\frac{E}{N_0}}\left[ \cos \left(\psi \right)- \sin \left(\psi \right)\right]\right\}. $$

(3.153)

Note that E is the symbol energy equal to twice the bit energy in (3.152) and (3.153).

For the case of OQPSK, a slightly different result is obtained as

$$ \begin{array}{c}{p}_{\mathrm{b}}(e)=\frac{1}{4} Q\left\{\sqrt{\frac{E}{N_0}}\left[ \cos \left(\psi \right)+ \sin \left(\psi \right)\right]\right\}+\frac{1}{4} Q\left\{\sqrt{\frac{E}{N_0}}\left[ \cos \left(\psi \right)- \sin \left(\psi \right)\right]\right\},\\ {}\kern-14.5em +\frac{1}{2} Q\left\{\sqrt{\frac{E}{N_0}}\left[ \cos \left(\psi \right)\right]\right\}\end{array} $$

(3.154)

3.1.11 Comparison of Digital Modems

We explored several linear and nonlinear modulation schemes, all with coherent detection. While MPSK and MQAM are examples of linear modulation formats, orthogonal MFSK is an example of a nonlinear modulation scheme. The other modulation schemes discussed such as OQPSK and π/4-QPSK are similar to QPSK while GMSK, a nonlinear scheme, uses a different pulse shape. MSK can be treated as a derived form of OQPSK, albeit with a different pulse shape and, therefore, the error rates and bandwidth capabilities of MSK and GMSK will depend on the pulse shape. Hence, we will exclude them from this discussion and devote our attention to MPSK, MQAM, and MFSK, all using the same pulse shape, e.g., a rectangular one. Let T be the pulse duration. Because of the rectangular shape of the pulse, the null-to-null bandwidth required for transmission will be 2(1/T). Since MPSK and MQAM are identical in their pulse shape/symbol shape characteristics, with MPSK having identical vector lengths and MQAM having non-identical vector lengths, the bandwidth for these two schemes can be written as

$$ W=2\left(\frac{1}{T_{\mathrm{b}}}\right)\frac{1}{{ \log}_2 M}=\frac{2{R}_b}{{ \log}_2 M}. $$

(3.155)

In (3.155) Rb is the data (bit) rate equal to (1/Tb). If we now define the spectral efficiency r as

$$ \rho =\frac{R_{\mathrm{b}}}{W}, $$

(3.156)

using (3.155), the spectral efficiency becomes

$$ \rho =\frac{{ \log}_2 M}{2}=\frac{k}{2}. $$

(3.157)

Thus, we see that the spectral or bandwidth efficiency of the linear modulation schemes such as MPSK and MQAM go up with M and since M = 2^k, spectral efficiency is directly proportional to k/2.

The bandwidth requirement of MFSK is different from MPSK since the M signals are created from carrier frequencies that are orthogonal. This requires that they be separated by 1/2T. Thus, the bandwidth required for the transmission of M orthogonal FSK signals is

$$ W= M\left(\frac{1}{2 T}\right)=\frac{M}{2\left(\frac{k}{R_{\mathrm{b}}}\right)}=\frac{M}{2{ \log}_2 M}{R}_b. $$

(3.158)

The spectral efficiency for MSFK is

$$ \frac{R_b}{W}=\frac{2{ \log}_2 M}{M}=\frac{2 k}{2^k}=\frac{k}{2^{k-1}}. $$

(3.159)

We see from (3.159) that the spectral efficiency decreases as M or k increases, clearly demonstrating that MFSK is inferior to MFSK and MQAM in terms of bandwidth utilization.

While spectral efficiency allows us to compare the modems, we also need to examine the power or energy efficiency of the modems so that the minimum SNR required to maintain a certain bit error rate can be used for comparison. We have discussed the fact that the minimum SNR required to have a certain error rate increases with M > 4 for MPSK and MQAM. However, since MQAM has dual encoding (amplitude and phase), the SNR required for MQAM will be less than that for MPSK. On the other hand, the SNR required to achieve an acceptable error rate in MFSK decreases as M increases. The values are tabulated in Table 3.8. Note that the spectral efficiency is redefined in terms of the two-sided spectra such that the newly defined efficiency η = ρ/2 where r is defined in (3.156).

Table 3.8 Comparison of spectral (η) and power efficiencies

A more general way of showing the power and spectral efficiencies of the modems is through the use of Shannon’s channel capacity theorem . The channel capacity C can be written as

$$ C= W\kern0.5em { \log}_2\left(1+\frac{P_{\mathrm{av}}}{WN_0}\right). $$

(3.160)

In (3.160), Pav is the average power. Converting to energy units, (3.160) becomes

$$ C= W{ \log}_2\left(1+\frac{RE_{\mathrm{b}}}{WN_0}\right). $$

(3.161)

In an ideal system, C = R and (3.161) becomes

$$ \frac{R}{W}={ \log}_2\left(1+\frac{{ R E}_{\mathrm{b}}}{{ W N}_0}\right). $$

(3.162)

Rewriting, we have

$$ \frac{E_{\mathrm{b}}}{N_0}=\frac{2^{R/ W}-1}{\left( R/ W\right)}. $$

(3.163)

The plot of the spectral efficiency vs. the average energy-to-noise ratio is shown in Fig. 3.59. The values for MPSK, MQAM, and MFSK are shown as individual points on the plot. These values are indicated in terms of the improvement over the respective binary modulation schemes (BPSK and BFSK). We can clearly see the trade-off between power and spectral efficiencies of the linear and nonlinear modulation schemes. The region above the X-axis is the bandwidth limited region; the region below the X-axis is the power limited region. If bandwidth is available and power is at a premium, one would use MFSK. If power constraints are absent and bandwidth is limited, one would use the linear modulation scheme such as MPSK and MQAM. Furthermore, in the bandwidth limited region, we can trade-off between the SNR complexity by choosing MQAM, which saves power but uses a complicated transmitter/receiver.

Fig. 3.59

Channel capacity comparison

A few additional comments are in order. Even though we had discussed several modems, the error rates were not compared as a group. For the case of noncoherent differential phase shift keying (DPSK), the error rate is given by

$$ {p}_e=\frac{1}{2} \exp \left(-\frac{E_{\mathrm{b}}}{N_0}\right). $$

(3.164)

For noncoherent binary FSK, the error rate is

$$ {p}_e=\frac{1}{2} \exp \left(-\frac{1}{2}\frac{E_{\mathrm{b}}}{N_0}\right) $$

(3.165)

For noncoherent MFSK, the (symbol) error rate (upper limit) is

$$ p{(s)}_e<\frac{M-1}{2} \exp \left(-\frac{1}{2}\frac{E_s}{N_0}\right). $$

(3.166)

3.1.12 Q Function, Complementary Error Function, and Gamma Function

We had seen that the error rate is conveniently expressed in terms of Q functions defined as (Helstrom 1968; Sklar 2001; Papoulis and Pillai 2002; Gradshteyn and Ryzhik 2007)

$$ Q(x)=\frac{1}{\sqrt{2\pi}}{\int}_x^{\infty } \exp \left(-\frac{z^2}{2}\right)\mathrm{d} z. $$

(3.167)

The error function erf(.) is defined as

$$ \mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}{\int}_0^x \exp \left(-{z}^2\right)\mathrm{d} z. $$

(3.168)

The complementary error erfc(.) function is defined as

$$ \mathrm{erfc}(x)=\frac{2}{\sqrt{\pi}}{\int}_x^{\infty } \exp \left(-{z}^2\right)\mathrm{d} z=1-\mathrm{erf}(x). $$

(3.169)

Thus, there is also a relationship between the Q function and the complementary error function

$$ \mathrm{erfc}(x)=2 Q\left( x\sqrt{2}\right). $$

(3.170)

Inversely,

$$ Q(x)=\frac{1}{2}\mathrm{erfc}\left(\frac{x}{\sqrt{2}}\right). $$

(3.171)

The three functions, erf(.), erfc(.), and Q(.), are plotted in Fig. 3.60.

Fig. 3.60

Erf(.), Erfc(.), and Q(.) function

The error rate can also be expressed in terms of gamma functions (Wojnar 1986). We had seen that the error rates are functions of the SNR (or energy-to-noise ratio) E/N ₀. If we represent the error rate that can be expressed as

$$ z=\frac{E}{N_0}, $$

(3.172)

where

$$ {p}_{\mathrm{e}}(z)=\frac{\Gamma \left( b, az\right)}{2\Gamma (b)}, $$

(3.173)

$$ \Gamma \left( b, az\right)={\int}_{az}^{\infty }{t}^{b-1} \exp \left(- t\right)\mathrm{d} t $$

(3.174)

$$ \Gamma (b)={\int}_0^{\infty }{t}^b \exp \left(- t\right)\mathrm{d} t $$

(3.175)

$$ b=\left\{\begin{array}{cc}\hfill \frac{1}{2},\hfill & \mathrm{coherent}\kern0.5em \mathrm{detection}\hfill \\ {}\hfill 1,\hfill & \hfill \mathrm{noncoherent}/\mathrm{differential}\kern0.5em \mathrm{detection}\hfill \end{array}\right. $$

(3.176)

$$ a=\left\{\begin{array}{cc}\hfill \frac{1}{2},\hfill & \mathrm{Orthoganal}\kern0.5em \mathrm{FSK}\hfill \\ {}\hfill 1,\hfill & \hfill \mathrm{bipolar}/\mathrm{bipodal}\kern0.5em \mathrm{PSK}\hfill \end{array}\right.. $$

(3.177)

Since

$$ \Gamma \left(\frac{1}{2}\right)=\sqrt{\pi} $$

(3.178)

$$ \Gamma \left(\frac{1}{2}, x\right)=\sqrt{\pi}\kern0.5em \mathrm{erfc}\left(\sqrt{x}\right), $$

(3.179)

we have for coherent BPSK,

$$ {p}_e(z)=\frac{\Gamma \left(\left(1/2\right), z\right)}{2\Gamma \left(1/2\right)}=\frac{1}{2}\mathrm{erfc}\left(\sqrt{z}\right)= Q\left(\sqrt{2 z}\right). $$

(3.180)

The error probability in (3.173) is plotted in Fig. 3.61 for the four sets of values of (a,b).

Fig. 3.61

Error probability for BPSK (coherent and noncoherent) and orthogonal BFSK (coherent and noncoherent)

In some of the analysis involving bit error rates in fading channels, we would also require the use of the derivative of the complimentary error function. We have

$$ \frac{\partial }{\partial z}\left[ erfc(z)\right]=-\frac{2}{\sqrt{\pi}} \exp \left(-{z}^2\right) $$

(3.181)

$$ \frac{\partial }{\partial z}\left[ Q(z)\right]=-\frac{1}{\sqrt{2\pi}} \exp \left(-\frac{z^2}{2}\right). $$

(3.182)

Specifically,

$$ \frac{\partial }{\partial z}\left[{p}_e(z)\right]=-\frac{1}{2\sqrt{\pi z}} \exp \left(- z\right). $$

(3.183)

There are several approximations to the Q function (Chiani et al. 2003; Karagiannidis and Lioumpas 2007; Yunfei and Beaulieu 2007; Isukapalli and Rao 2008). The two bounds of the Q functions are

$$ \left(1-\frac{1}{\alpha^2}\right)\frac{1}{\sqrt{2{\pi \alpha}^2}} \exp \left(-\frac{\alpha^2}{2}\right)\le Q\left(\alpha \right)\le \frac{1}{\sqrt{2{\pi \alpha}^2}} \exp \left(-\frac{\alpha^2}{2}\right). $$

(3.184)

An upper bound for large values of the argument for the Q function is

$$ Q\left(\alpha \right)\le \frac{1}{2} \exp \left(-\frac{\alpha^2}{2}\right). $$

(3.185)

3.1.13 Marcum Q Function

While the Q function defined in (3.167) is seen in error rate calculations involving additive white noise, another form of Q function is seen in equations involving integrals such as the ones associated with the Rician density function mentioned in Chap. 2. One such form is the Marcum Q function defined as (Nuttall 1975; Simon 1998, 2002; Helstrom 1998; Corazza and Ferrari 2002; Simon and Alouini 2003, 2005)

$$ Q\left( a, b\right)={\int}_b^{\infty } x\kern0.5em \exp \left(-\frac{x^2+{a}^2}{2}\right){I}_0\left( a x\right)\mathrm{d} x. $$

(3.186)

Equation (3.186) can also be expressed as an infinite summation,

$$ Q\left( a, b\right)= \exp \left(-\frac{x^2+{a}^2}{2}\right)\sum_{k=0}^{\infty }{\left(\frac{a}{b}\right)}^k{I}_k\left( a b\right). $$

(3.187)

The limiting cases of the Marcum Q function lead to

$$ Q\left( a, a\right)=\frac{1+ \exp \left(-{a}^2\right){I}_0\left({a}^2\right)}{2} $$

(3.188)

$$ Q\left(0, b\right)= \exp \left(-\frac{b^2}{2}\right) $$

(3.189)

$$ Q\left( a,0\right)=1. $$

(3.190)

The Marcum Q function is shown in Fig. 3.62 for three values of b. The generalized Marcum Q function is defined as

$$ {Q}_m\left( a, b\right)={\int}_b^{\infty }{x}^m \exp \left(-\frac{x^2+{a}^2}{2}\right){I}_{m-1}\left( a x\right)\mathrm{d} x. $$

(3.191)

Fig. 3.62

Marcum Q function

It is obvious that when m = 1, the generalized Marcum function reduces to the Q function in (3.186) as

$$ {Q}_1\left( a, b\right)= Q\left( a, b\right). $$

(3.192)

The generalized Marcum Q function is shown in Fig. 3.63.

Fig. 3.63

The generalized Marcum Q function

The recursion relation for the generalized Marcum Q function is

$$ {Q}_m\left( a, b\right)=\left(\frac{b}{a}\right) \exp \left(-\frac{a^2+{b}^2}{2}\right){I}_{m-1}\left( a b\right)+{Q}_{m-1}\left( a, b\right). $$

(3.193)

For the integer values of m the generalized Marcum Q function can be expressed as

$$ {Q}_m\left( a, b\right)= \exp \left(-\frac{a^2+{b}^2}{2}\right)\sum_{k=1- m}^{\infty }{\left(\frac{a}{b}\right)}^k{I}_k\left( a b\right). $$

(3.194)

For integer values of m, (3.194) becomes

$$ {Q}_m\left( a, a\right)=\frac{1}{2}+ \exp \left(- a\right)\left[\frac{I_0\left({a}^2\right)}{2}+\sum_{k=1}^{m-1}{I}_k\left({a}^2\right)\right]. $$

(3.195)

One limiting case of the generalized Marcum Q function is

$$ {Q}_m\left(0, b\right)=\frac{\Gamma \left( m,\left({b}^2/2\right)\right)}{\Gamma (m)}. $$

(3.196)

Equation (3.196) can be simplified for the case of integer values of m as

$$ {Q}_m\left(0, b\right)=\sum_{k=0}^{m-1}\frac{1}{k!}{\left(\frac{b^2}{2}\right)}^k \exp \left(-\frac{b^2}{2}\right). $$

(3.197)

A further generalization of the Marcum Q function (Nuttall 1975), which will be useful in integrals involving fading distributions such as the 'T − m or k − m, is

$$ {Q}_{m, n}\left( a, b\right)={\int}_b^{\infty }{x}^m \exp \left(-\frac{x^2+{a}^2}{2}\right){I}_{n-1}\left( a x\right)\mathrm{d} x. $$

(3.198)

Note that m and n can take any positive values including zero, and a and b lie between 0 and 1. For the special case of m = n + 1, (3.198) can be expressed in terms of the generalized Marcum Q function of (3.191) as

$$ {Q}_{n+1, n}\left( a, b\right)={a}^n{Q}_{n+1}\left( a, b\right), $$

(3.199)

with a further special case of

$$ {Q}_{1,0}\left( a, b\right)={Q}_1\left( a, b\right)= Q\left( a, b\right). $$

(3.200)

3.1.14 Intersymbol Interference

The issue of intersymbol interference was brought up earlier in connection with the choice of the appropriate pulse shapes (Sklar 2001; Haykin 2001). We will now briefly review the concepts behind intersymbol interference and the way to characterize problems arising from it. Consider an ideal low pass filter with a transfer function as shown in Fig. 3.64. It has a bandwidth of (1/2T), and if one looks at the corresponding time function in Fig. 3.65 we see that the impulse response is a sin c function .

Fig. 3.64

Ideal LPF transfer function

Fig. 3.65

Impulse response(s) of ideal LPF

Two other sin c functions are also plotted, one delayed by +T and the other delayed by −T. We can see that the pulse shapes are such that at every T s interval, the maximum of any given pulse has zero contributions from the other pulses since the sin c function goes through zero. In other words, for transmission of data occupying a duration of Ts, there will be no contribution from adjoining bits (or symbols) if we can choose the sin c pulse. Since the bandwidth of such a pulse is (1/2T), use of such pulses will lead to transmission with the best spectral efficiency because this BW is only equal to (R/2) where R is the data rate given by (1/T).

However, such a condition is not ideal for two major reasons. First, the filter response is far from ideal and real filters would have a gradual fall off beyond f = (1/2T). This would mean that the pulses after passage through filters will be distorted leading to overlaps at sampling instants resulting in intersymbol interference (ISI). Second, ideal sampling at exact multiples of T is also not practical since timing errors would be present leading to contributions from adjoining symbols and resulting in ISI which leads to increase in error rates.

One of the ways to mitigate ISI is to use a raised cosine pulse shape as discussed earlier. The raised cosine pulse shape can be expressed as

$$ h(t)= \sin c\left(2 Wt\right)\left[\frac{ \cos \left(2\pi \alpha Wt\right)}{1-16{\alpha}^2{W}^2{t}^2}\right]. $$

(3.201)

In (3.201), α is the roll-off factor and W is the bandwidth of the ideal low pass filter shown in Fig. 3.66 and corresponding to the ideal pulse for the highest data rate transmission, i.e.,

$$ W=\frac{1}{2 T}. $$

(3.202)

Fig. 3.66

Raised cosine pulses

Pulse shapes corresponding to three values of α = 0, 0.5, and 0.9 are shown in Fig. 3.66. The corresponding spectra are shown in Fig. 3.67. As can be seen, α = 0 corresponds to the case of a sin c pulse and an increase in values demonstrates the shrinking of the sidelobes which can reduce the chances of ISI. The value of α = 1 corresponds to the full roll-off raised cosine pulse. One can see that the bandwidths of the pulses increase with α. The BW of the WR raised cosine pulse can be written in terms of W as

$$ {W}_{\mathrm{R}}= W\left(1+\alpha \right). $$

(3.203)

Fig. 3.67

Spectra of the raised cosine pulses

Thus, for a full raised cosine pulse with α = 1 requires twice the BW of a sin c pulse. Defining

$$ {f}_1= W\left(1-\alpha \right), $$

(3.204)

the spectrum of the raised cosine pulse shapes are

$$ H(f)=\left\{\begin{array}{ll}\frac{1}{2W},\hfill & 0\le \left|f\right|<{f}_1\hfill \\ {}\frac{1}{4W}\left\{1- \sin \left[\frac{\pi \left(\left|f\right|-W\right)}{2W-2{f}_1}\right]\right\},\hfill & {f}_1\le \left|f\right|2W-{f}_1\hfill \\ {}0,\hfill & \left|f\right|\ge 2W-{f}_1\hfill \end{array}\right.. $$

(3.205)

For ɑ = 1, we have

$$ \left\{\begin{array}{cc}\hfill \frac{1}{4W}\left\{1- \sin \left[\frac{\pi \left(\left|f\right|-W\right)}{2W-2{f}_1}\right]\right\},\hfill & \hfill 0\le \left|f\right|<2W\hfill \\ {}0,\hfill & \hfill \left|f\right|\ge 2W\hfill \end{array}\right. $$

(3.206)

and, the corresponding pulse shape becomes

$$ h(t)=\frac{ \sin c\left(4 Wt\right)}{1-16{W}^2-{t}^2}. $$

(3.207)

At the receiver, pulse shaping is employed to make the pulse shapes to be of a raised cosine type so that ISI will be minimum. We will now look at eye patterns which show the effects of ISI.

Eye patterns allow us to study the effect of dispersion (pulse broadening) and noise in digital transmission. An eye pattern is a synchronized superposition of all possible realizations of the signal by the various bit patterns. We can easily create them using Matlab by considering the transmission of rectangular pulse (width T) passing through a low pass filter of bandwidth B. For a given pulse width, increasing values of BT means less and less distortion. Two eye patterns are shown in Figs. 3.68 and 3.69. The one for a low value of BT appears in Fig. 3.68. In both cases, additive white noise has also been added. The one with a higher value of BT appears in Fig. 3.69.

Fig. 3.68

Eye pattern (low value of BT)

Fig. 3.69

Eye pattern (high value of BT)

The width of the eye opening provides information on the time interval over which the received signal can be sampled without being impacted by ISI. One can see that a higher value of BT results in a wider eye, while a lower value of BT results in a narrow eye.