13.1 Introduction

Spread spectrum signals typically refer to signals that use a bandwidth that is significantly higher than the minimum bandwidth required to transmit that information sequence. Although bandwidth is a precious commodity, increasing the bandwidth of the signal prior to transmission results in several advantages: (i) The resulting power spectral density (PSD) of the spread signal is lower than the PSD of the signal prior to spreading. (ii) The resulting signal is more resistant to narrow band interference/jamming. (iii) The signal has a low probability of being intercepted. (iv) The spread signal is readily suitable for multiple access. These advantages have resulted in several military and commercial applications for spread spectrum systems.

The early developments in spread spectrum systems were primarily in military applications, where the low probability of intercept and antijam features were especially useful. Subsequently, they found utility in the commercial area. For instance, commercial applications of spread spectrum in the cellular wide area networks are similar to the military predecessors in many aspects. However, a few specific innovations enable them to attain superior performance. For instance, in the cellular application, primary innovations include the use of universal frequency reuse with intelligent adaptive power control and the use of noise-like signal waveforms. Further, wireless personal area networks such as Zigbee [1,2] and Bluetooth networks [3,4] are based on spread spectrum technology.

Consider a simple multiuser spread spectrum system where the received signal is at power P and the interference from (M – 1) users equals (M – 1)P. Given a system bandwidth of W Hz, the received E_b/N₀ can be calculated as (P/R)/(N + (M – 1)P/W), where R is the data rate in bits per second and N is the noise power spectral density. Thus, the desired minimum value of E_b/N₀ can be satisfied in the presence of large number of users, if the bandwidth W is appropriately scaled. A similar analysis can be used to demonstrate the advantages of spread spectrum system, if the interference is from an intentional narrow band jammer.

This chapter is organized as follows. In Section 13.2, we discuss direct sequence spread spectrum systems. In Section 13.3, we focus on frequency hopping systems. Applications of spread spectrum are discussed in Section 13.4 followed by some brief concluding remarks in Section 13.5.

13.2 Direct Sequence Spread Spectrum

In a direct sequence (DS) code division multiple access (CDMA) system, the data sequence is multiplied by a pseudo-random signal before transmission. This pseudo-random signal is derived from a spreading code or spreading sequence. Each bit of the spreading sequence is also sometimes referred to as a chip. The time period, T_c, of a chip is usually much smaller than the symbol time period, T_s and the ratio L = T_s/T_c is referred to as the bandwidth expansion factor. In many systems this value L is selected to be an integer. The value L also represents the maximum number of orthogonal users that can be supported without any multiuser interference.

In a multiuser uplink scenario, the signals transmitted by the various users are not synchronous with each other. However, for simplicity of exposition, we first consider a synchronous DS–CDMA system. In a synchronous system, the signals of all the users are perfectly synchronized at the receiver. Such a model is appropriate for a downlink cellular system in which the signals of all the users are transmitted from a single base station. Let there be K users and let the bit sequence of the kth user be denoted by b_k(n). Let c_k(t) denote the spreading signal used to modulate the kth user's signal. Let the channel from the kth user to the central receiver be denoted by A_k. The received signal, r(t), is given by

where z(t) represents the additive noise at the receiver and is typically modeled as a Gaussian process. The spreading sequence, c_k(t) is usually nonzero only in the interval 0 ≤ t < T_s. Thus, there is no inter-symbol interference (ISI) in the system. This code sequence is generated based on the PN spreading sequence and a pulse shaping signal p(t) which is nonzero in the interval 0 ≤ t < T_c. An example of the time-domain and frequency domain representation of the DS-CDMA signal is shown in Figure 13.1a and b, respectively.

Depending on the length of the spreading codes, the sequences are also referred to as long codes or short codes. In a short code DS–CDMA system, the same spreading code is used for all symbols of a particular user. In other words, the spreading code length L_c equals L the bandwidth expansion factor. In this case, the spreading signal c_k(t) is given by

In a long code DS–CDMA system, the length of the code L_c is much greater than L the bandwidth expansion factor. In this case, the received signal r(t) is given by

where $\tilde{n}=n$ modulo L and the concatenation of the L_c/L segments of the spreading signal is based on the long spreading code. Many practical systems such as IS-95 and CDMA-2000 use a combination of both a short code and long code. Short orthogonal codes are used to separate the signals of the various users in a cell. Long codes are used for synchronization purposes and to differentiate the signals of users in different cells.

At the receiver, the signal is first typically passed through a set of matched filters (MFs) corresponding to each of the users. The output of the MF corresponding to the nth bit for the kth user is given by

where ${\rho }_{jk}={\displaystyle {\int }_{t=0}^T{c}_j(t)c_k(t)\text{d}t}$ represents the cross-correlation between the spreading waveforms of user j and user k.

The outputs y_k(n) from the MFs essentially represent a discrete time equivalent signal for the DS–CDMA signal. The collection of these outputs is succinctly represented in matrix form as follows:

where R and A represent, respectively, the code correlation and channel gain of the various users. The vector b = [b₁ b₂ … b_K]^T represents the information from all users.

In the special case of the cross-correlation between the various spreading waveforms being 0, the multiuser interference in Equation 13.6 is completely eliminated and the channel effectively behaves like K orthogonal single user channels. In general, for nonorthogonal spreading waveforms, the performance of the MF receiver is dependent on the amount of cross correlation and the relative amplitudes of the various users. For instance, the bit error rate (BER) P_e,_MF(k) of user k using a MF in the synchronous case is bounded as

If the spreading waveforms of the users are not orthogonal, then a multiuser detector is used to improve the performance [5] over that of a simple MF receiver. The optimum receiver that minimizes the probability of error is the maximum likelihood sequence detector. However, the complexity of such a receiver is high since it involves enumeration of a large set of possible sequences. Two alternate receivers with lower complexity are the linear minimum mean squared error (LMMSE) and the decorrelating receiver. In a decorrelating receiver, the signal in Equation 13.6 is multiplied by the inverse R⁻¹ of the cross-correlation matrix R. The advantage of such a decorrelating receiver is that it completely eliminates the multiuser interference. However, the decorrelating receiver can result in an increase in the effective noise variance. The LMMSE receiver on the other hand offers an optimal trade-off between reducing the multiuser interference and the effective noise. In the LMMSE receiver, the signal in Equation 13.6 is multiplied by M = (R + σ²A⁻2)⁻¹ and the resulting output is passed through a sign detector (in the case of antipodal signaling). The exact probability of error of the LMMSE receiver has been calculated in Reference 5, but as in the earlier case, its evaluation requires a computational complexity that is exponential in the number of users. Consequently, low complexity approximations for the error rate have been obtained using a Gaussian approximation for the multiuser interference. One such approximation for the error rate for user k is given by

where ${\mu }_k=\left(A_k{(\text{M}R)}_{kk}\text{}/\text{}\sigma \sqrt{{(\text{M}RM)}_{kk}}\right)$.

A plot of the BER of the various multiuser detectors is given in Figure 13.2a as a function of the SNR. In this case, each user is assigned a random binary code. It is clear from the figure that the performance of the MF is poor due to the presence of significant multiuser interference. As expected the LMMSE detector performs slightly better than the decorrelating receiver. The variation of the BER with the number of users is given in Figure 13.2b for the various multiuser detectors. Owing to the random nature of the spreading codes used to generate this figure, it can be clearly seen that the BER increases as the number of users increase. In addition to the various multiuser detectors discussed so far, there are several other linear and nonlinear multiuser detectors such as serial or parallel interference cancellation receivers [5].

Another advantage of the DS–CDMA systems when used for voice applications is that the voice users are not active for the entire duration of the call. The typical voice activity factor α is lesser than 50% and ensures that the effective interference is only a fraction α of the total interference. Thus, unlike in fixed TDMA systems, the actual interference experienced in a DS–CDMA system is smaller.

The analysis of an asynchronous CDMA system follows along similar lines with some additional modifications. In this case, the received signal is given by

The maximum likelihood sequence detector, LMMSE and decorrelating receivers are defined in a manner similar to the synchronous case. The performance evaluation can also be generalized to this case; however, the analysis is more involved since each bit for a particular user may be affected by portions of 2 bits from each of the other users. In other words, each bit sees interference from 2(K – 1) other bits.

The power spectral density (PSD) of a DS–CDMA system using a raised cosine transmit pulse is shown in Figure 13.1. For reference the PSD of the data sequence as well as a realization of the data sequence and spread sequence are shown in Figure 13.1. The increased bandwidth of DS–CDMA sequence is clearly evident from this figure.

A DS–CDMA signal has good performance even in the presence of a narrow band interference signal. Clearly, the process of spreading at the transmitter ensures that a narrow band signal is now spread over a wider bandwidth. The narrow band interference now additively interferes with the spread CDMA signal. The process of despreading gathers the energy from the spread signal into a narrow band. On the other hand, the despreading operation results in a spreading of the narrow band interference signal over a wider band. Hence, the effective power spectral density of the narrow band interference after despreading is reduced and thus its effect on the CDMA signal is also reduced. If ${\sigma }_I^2$ is the energy of the interfering signal, then the effective SNR of the DS–CDMA signal is given by $(P\text{/}{\sigma }_n^2+{\sigma }_I^2\text{/}N)$, where N is the length of the spreading code.

It has, however, been shown that the performance of the DS–CDMA signal can be significantly improved by actively suppressing the narrow band interference signal prior to despreading at the receiver [6,7]. These active noise suppression schemes have been widely investigated and can be classified into the following major schemes.

• Time domain estimation and interference subtraction. In these methods, an estimator is used to predict the narrow band signal and then subtract this estimated narrow band signal from the spread signal using a notch filter.

• Frequency domain filtering. A Fourier transform of the spread signal is first created and then the signal is filtered at a few selected frequencies. The resulting signal is passed through an inverse Fourier transform. The frequencies to filter can be calculated in an adaptive manner, for instance, by selecting spectral components that have energy higher than a certain threshold.

• Nonlinear techniques. Unlike the earlier two methods, there are also several nonlinear filtering methods that have been studied in the literature for narrow band interference suppression. One such method uses a decision feedback mechanism to predict the narrow band interference signal and then cancel its effect from the spread signal.

13.3 Frequency Hopping Spread Spectrum

Frequency hopping was first invented during World War II by Hedy Lamarr in a patent titled “Secret Communication System” [8]. In a frequency hopping spread spectrum system, the actual signal transmitted has narrow bandwidth. However, the center frequency of the transmission signal is varied over a wide band. The rate at which the center frequency of the signal is varied determines whether the system is a slow-hopping or fast-hopping system. The variation of the frequency, commonly referred to as hopping pattern, is usually based on a pseudorandom sequence. The advantage of a frequency hopping system is that its performance is not significantly affected by a narrow band interference source, since only the packets that are transmitted when the systems hops at the frequencies of the interfering signal are affected. The FH system also enables multiple point-to-point links in a given location if the hopping sequences used by the various links are effectively different. Note that the difference in the effective hopping pattern used by the various nodes could result from the use of different phases of the same hopping sequence for the various links. A block diagram of a typical FH spread spectrum signal is shown in Figure 13.3.

The modulation in an FH system is typically selected to be a form of Frequency Shift Keying (FSK) or Phase Shift Keying (PSK). For instance, in a binary FSK–FH system a binary “0” could be represented by a negative frequency deviation f_d from the carrier frequency f_c and a binary “1” could be represented by a positive frequency deviation f_d from f_c. The center frequency f_c is varied based on the PN sequence generated. In a MFSK system the frequency spacing between carriers is carefully selected to ensure optimum performance for a given receiver structure. In a noncoherent energy-based detector for the MFSK signal, it is desirable for the various tones representing the bits to be orthogonal to each other. This orthogonality can be achieved by spacing the carriers at integer multiples of 1/T_s. A common arrangement for a FH–MFSK system is to consider nonoverlapping blocks of spectrum of width M · 1/T_s. Each block of this spectrum is assigned to one hopping pattern of the PN code. Such an arrangement of tones is depicted in Figure 13.4. In Figure 13.4, there are N groups of M tones and each of these groups can be selected based on a PN code of length log(N).

As another example, consider a FH–PSK system. Using differential PSK in a FH system, the transmitted bandpass signal s(t) can be represented as

where m(t) is the low-pass signal. This differential PSK signal m(t) can be generated using a pulse shape p(t) as

where $m_k=m_{k-1}{e}^{\text{j}{\varphi }_k}$ is the differential PSK signal. For an 8-ary signal, φ_k ∈ {0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/8}. The pulse shape p(t) that is commonly used is the raised cosine pulse [9].

At the receiver, acquisition and tracking of the PN sequence used for frequency hopping is critical for successful decoding of the signal, as discussed in Section 13.4.4. Once the carrier frequency is obtained, the received signal is then converted into baseband in a process that is sometimes called frequency dehopping. The differential PSK signal can then be detected using a noncoherent detector.

In a slow frequency hop system multiple symbols are transmitted on each hop. In contrast, in a fast frequency hop system there are one or more hops for each symbol. A graphical representation of the slow and fast hopping FH systems is shown in Figure 13.5.

The BER of a slow FH system with orthogonal FSK can be derived as

This expression for the BER is derived for the signal transmission in the presence of AWGN. In a fast FH system with M hops per bit, the signal from each of the M subintervals corresponding to a bit are combined together before detection. In this scenario, it turns out that the probability of error is greater than Equation 13.12 and the difference is known as the noncoherent combining loss [9]. The performance of FH systems in fading channels is studied in Reference 10 and the performance in the presence of an interference signal is discussed in Reference 11.

13.4 Applications of Spread Spectrum

In this section, we highlight some of the important applications of spread spectrum signals. We first discuss cellular applications in Section 13.4.1, followed by applications in Bluetooth (Section 13.4.2) and ultrawideband systems (Section 13.4.3). Finally, we discuss synchronization issues in Section 13.4.4.

13.5 Conclusions

This chapter provides a basic introduction to spread spectrum signals. Additional details are available in several texts such as in References 17 through 20. A note of caution is appropriate when discussing the applicability of spread spectrum signals to very large bandwidths. It is well known that for the additive white Gaussian noise (AWGN) channel, as the bandwidth is increased the capacity equals P/N₀, where P is the power constraint and N₀/2 is the noise spectral density. It is also known that any orthogonal signaling (over time, frequency, or code space) can achieve this capacity for asymptotically large bandwidths. However, for time-varying channels, under certain conditions, it has been shown that uniformly spreading the signals in time and frequency as in DS–CDMA system does not achieve the asymptotic capacity for large bandwidths [21]. Instead, peaky signaling using for instance a combination of FH and DS–CDMA could be used to achieve capacity [21]. It is this author's expectation that spread spectrum signals will continue to play a vital role in the future wireless networks.

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