Spread-Spectrum Communication Systems

This tutorial is part of the National Instruments Signal Generator Tutorial series. Each tutorial in this series, will teach you a specific topic of common measurement applications, by explaining the theory and giving practical examples. This tutorial covers the history of communication systems. For additional signal generator concepts, refer to the Signal Generator Fundamentals main page.

Spread-Spectrum Communication Systems

In our treatment of signal design for digital communication over an AWGN channel, the major objective has been the efficient utilization of transmitter power and channel bandwidth. Channel coding allows us to reduce the transmitter power by increasing the transmitted signal bandwidth through code redundancy and, thus, to trade off transmitter power with channel bandwidth. This is the basic methodology for the design of digital communication systems for AWGN channels.

In practice, one encounters other factors that influence the design of an efficient digital communication system. For example, in multiple-access communication when two or more transmitters use the same common channel to transmit information, the interference created by the users of the channel limits the performance achieved by the system. The system designer must take into account the existence of such interference in the design of a reliable digital communication system.

Even in this complex design problem, the basic system design parameters are transmitter power and channel bandwidth. To overcome the problems of intentional or unintentional interference, we may further increase the bandwidth of the transmitted signal, as described below, so that the bandwidth expansion factor B_e = W/R is much greater than unity. This is one characteristic of a spread-spectrum signal. A second important characteristic is that the information signal at the modulator is spread in bandwidth by means of a code that is independent of the information sequence. This code has the property of being pseudorandom ; i.e., it appears to be random to receivers other than the intended receiver that uses the knowledge of the code to demodulate the signal. It is this second characteristic property that distinguishes a spread-spectrum communication system from the conventional communication system that expands the transmitted signal bandwidth by means of channel code redundancy. However, we should emphasize that channel coding is an important element in the design of an efficient spread-spectrum communication system.

Spread-spectrum signals for digital communications were originally developed and used for military communications either (1) to provide resistance to jamming (antijam protection), or (2) to hide the signal by transmitting it at low power and, thus, making it difficult for an unintended listener to detect its presence in noise (low probability of intercept). However, spread-spectrum signals are now used to provide reliable communications in a variety of civilian applications, including digital cellular communications and interoffice communications.

In this section we present the basic characteristics of spread-spectrum signals and assess their performance in terms of probability of error. We concentrate our discussion on two methods for spreading the signal bandwidth, namely, by direct sequence modulation and by frequency hopping. Both methods require the use of pseudorandom code sequences whose generation is also described. Several applications of spread-spectrum signals are presented.

Model of a Spread-Spectrum Digital Communication System

The basic elements of a spread spectrum digital communication system are illustrated in Figure 10.37. We observe that the channel encoder and decoder and the modulator and demodulator are the basic elements of a conventional digital communication system. In addition to these elements, a spread-spectrum system employs two identical pseudorandom sequence generators, one which interfaces with the modulator at the transmitting end and the second which interfaces with the demodulator at the receiving end. These two generators produce a pseudorandom or pseudonoise (PN) binary-valued sequence, which is used to spread the transmitted signal at the modulator and to despread the received signal at the demodulator.

Time synchronization of the PN sequence generated at the receiver with the PN sequence contained in the received signal is required in order to properly despread the received spread-spectrum signal. In a practical system, synchronization is established prior to the transmission of information by transmitting a fixed PN bit pattern which is designed so that the receiver will detect it with high probability in the presence of interference. After time synchronization of the PN sequence generators is established, the transmission of information commences. In the data mode, the communication system usually tracks the timing of the incoming received signal and keeps the PN sequence generator in synchronism. The synchronization of the PN sequence generators is treated in Section 10.3.7.

Interference is introduced in the transmission of the spread-spectrum signal through the channel. The characteristics of the interference depend to a large extent on its origin. The interference may be generally categorized as being either broadband or narrowband (partial band) relative to the bandwidth of the information-bearing signal, and either continuous in time or pulsed (discontinuous) in time. For example, an

interfering signal may consist of a high-power sinusoid in the bandwidth occupied by the information-bearing signal. Such a signal is narrowband. As a second example, the interference generated by other users in a multiple-access channel depends on the type of spread-spectrum signals that are employed by the various users to transmit their information. If all users employ broadband signals, the interference may be characterized as an equivalent broadband noise. If the users employ frequency hopping to generate spread-spectrum signals, the interference from other users may be characterized as narrowband. We shall consider these types of interference and some others in Section 10.3.6.

Our discussion will focus on the performance of spread-spectrum signals for digital communication in the presence of narrowband and broadband interference. Two types of digital modulation are considered, namely, PSK and FSK. PSK modulation is appropriate for applications where phase coherence between the transmitted signal and the received signal can be maintained over a time interval that spans several symbol (or bit) intervals. On the other hand, FSK modulation is appropriate in applications where phase coherence of the carrier cannot be maintained due to time variations in the transmission characteristics of the communications channel. For example, this may be the case in a communications link between two high-speed aircraft or between a high-speed aircraft and a ground-based terminal.

The PN sequence generated at the modulator is used in conjunction with the PSK modulation to shift the phase of the PSK signal pseudorandomly, as described below at a rate that is an integer multiple of the bit rate. The resulting modulated signal is called a direct-sequence (DS) spread-spectrum signal. When used in conjunction with binary or M-ary (M > 2) FSK, the PN sequence is used to select the frequency of the transmitted signal pseudorandomly. The resulting signal is called a frequency-hopped (FH) spread-spectrum signal. Although other types of spread-spectrum signals can be generated, our treatment will emphasize DS and FH spread-spectrum communication systems, which are the ones generally used in practice.

Direct-Sequence Spread-Spectrum Systems

Let us consider the transmission of a binary information sequence by means of binary PSK. The information rate is R bits/sec and the bit interval is T_b = 1/R sec. The available channel bandwidth is B_c Hz, where . At the modulator, the bandwidth of the information signal is expanded to W = B_c Hz by shifting the phase of the carrier pseudorandomly at a rate of W times/sec according to the pattern of the PN generator. The basic method for accomplishing the spreading is shown in Figure 10.38.

The information-bearing baseband signal is denoted as u(t) and is expressed as

where {a_n = ±1, –¥ < n < ¥} and g_T (t) is a rectangular pulse of duration T_b. This signal is multiplied by the signal from the PN sequence generator, which may be
expressed as

where {c_n} represents the binary PN code sequence of ±1's and p(t) is a rectangular pulse of duration T_c as illustrated in Figure 10.38. This multiplication operation
serves to spread the bandwidth of the information-bearing signal (whose bandwidth is R Hz, approximately) into the wider bandwidth occupied by PN generator signal c(t) (whose bandwidth is 1/T_c, approximately). The spectrum spreading is illustrated in Figure 10.39, which shows, in simple terms, using rectangular spectra, the convolution of the two spectra, the narrow spectrum corresponding to the information-bearing signal and the wide spectrum corresponding to the signal from the PN generator.

The product signal u(t)c(t), also illustrated in Figure 10.39, is used to amplitude modulate the carrier A_c cos 2pf_ct and, thus, to generate the DSB-SC signal

Since u(t)c(t) = ±1 for any t, it follows that the carrier-modulated transmitted signal may also be expressed as

where q(t) = 0 when u(t)c(t) = 1 and q(t) = p when u(t)c(t) = –1. Therefore, the transmitted signal is a binary PSK signal.

The rectangular pulse p(t) is usually called a chip and its time duration T_c is called the chip interval. The reciprocal 1/T_c is called the chip rate and corresponds (approximately) to the bandwidth W of the transmitted signal. The ratio of the bit interval T_b to the chip interval T_c is usually selected to be an integer in practical spread spectrum systems. We denote this ratio as

Hence, L_c is the number of chips of the PN code sequence/information bit. Another interpretation is that L_c represents the number of possible 180° phase transitions in the transmitted signal during the bit interval T_b.

The demodulation of the signal is performed as illustrated in Figure 10.40. The received signal is first multiplifed by a replica of the waveform c(t) generated by the PN code sequence generator at the receiver, which is synchronized to the PN code in the received signal. This operation is called (spectrum) despreading, since the effect of multiplication by c(t) at the receiver is to undo the spreading operation at the transmitter. Thus, we have

since c²(t) = 1 for all t. The resulting signal A_c u(t) cos 2pf_ct occupies a bandwidth (approximately) of R Hz, which is the bandwidth of the information-bearing signal. Therefore, the demodulator for the despread signal is simply the conventional cross correlator or matched filter that was previously described in Chapter 7. Since the demodulator has a bandwidth that is identical to the bandwidth of the despread signal, the only additive noise that corrupts the signal at the demodulator is the noise that falls within the information-bandwidth of the received signal.

Effect of Despreading on a Narrowband Interference. It is interesting to investigate the effect of an interfering signal on the demodulation of the desired

information-bearing signal. Suppose that the received signal is

where i(t) denotes the interference. The despreading operation at the receiver yields

The effect of multiplying the interference i(t) with c(t), is to spread the bandwidth of i(t) to W Hz.

As an example, let us consider a sinusoidal interfering signal of the form

where f_I is a frequency within the bandwidth of the transmitted signal. Its multiplication with c(t) results in a wideband interference with power-spectral density I₀ = P_I /W, where is the average power of the interference. Since the desired signal is demodulated by a matched filter (or correlator) that has a bandwidth R, the total power in the interference at the output of the demodulator is

Therefore, the power in the interfering signal is reduced by an amount equal to the bandwidth expansion factor W/R. The factor W/R = T_b /T_c = L_c is called the processing gain of the spread-spectrum system. The reduction in interference power is the basic reason for using spread-spectrum signals to transmit digital information over channels with interference.

In summary, the PN code sequence is used at the transmitter to spread the information-bearing signal into a wide bandwidth for transmission over the channel. By multiplying the received signal with a synchronized replica of the PN code signal, the desired signal is despread back to a narrow bandwidth while any interference signals are spread over a wide bandwidth. The net effect is a reduction in the interference power by the factor W/R, which is the processing gain of the spread-spectrum system.

The PN code sequence {c_n} is assumed to be known only to the intended receiver. Any other receiver that does not have knowledge of the PN code sequence cannot demodulate the signal. Consequently, the use of a PN code sequence provides a degree of privacy (or security) that is not possible to achieve with conventional modulation. The primary cost for this security and performance gain against interference is an increase in channel bandwidth utilization and in the complexity of the communication system.

Probability of Error. To derive the probability of error for a DS spreadspectrum system, we assume that the information is transmitted via binary PSK. Within the bit interval 0 £ t £ T_b, the transmitted signal is

where a_o = ±1 is the information symbol, the pulse g_T(t) is defined as

and c(t) is the output of the PN code generator which, over a bit interval, is expressed as

where L_c is the number of chips per bit, T_c is the chip interval, and {c_n} denotes the PN code sequence. The code chip sequence {c_n} is uncorrelated (white); i.e.,

and each chip is +1 or –1 with equal probability. These conditions imply that E(c_n) = 0 and .

The received signal is assumed to be corrupted by an additive interfering signal i(t). Hence,

where t_d represents the propagation delay through the channel and f represents the carrier-phase shift. Since the received signal r(t) is the output of an ideal bandpass filter in the front end of the receiver, the interference i(t) is also a bandpass signal, and may be represented as

where i_c(t) and i_s(t) are the two quadrature components.

Assuming that the receiver is perfectly synchronized to the received signal, we may set t_d = 0 for convenience. In addition, the carrier phase is assumed to be perfectly estimated by a PLL. Then, the signal r(t) is demodulated by first despreading through multiplication by c(t) and then crosscorrelation with g_T(t) cos(2pf_ct + f), as shown in Figure 10.41. At the sampling instant t = T_b, the output of the correlator is

where y_i(T_b) represents the interference component, which has the form
where, by definition,

The probability of error depends on the statistical characteristics of the interference component. Clearly, its mean value is

Its variance is

But E(c_nc_m) = d_mn. Therefore,

where n = n_n, as given by Equation (10.3.19). To determine the variance of n, we must postulate the form of the interference.

First, let us assume that the interference is sinusoidal. Specifically, we assume that the interference is at the carrier frequency, and has the form

where P_I is the average power and Q_I is the phase of the interference, which we assume to be random and uniformly distributed over the interval (0, 2p). If we substitute for i(t) in Equation (10.3.19), we obtain

Since Q_I is a random variable, n_n is also random. Its mean value is zero; i.e.,

Its mean-square value is

We may now substitute for E(n ²) into Equation (10.3.21). Thus, we obtain

The ratio of to is the SNR the detector. In this case we have

To see the effect of the spread-spectrum signal, we express the transmitted energy as

where P_S is the average signal power. Then, if we substitute for in Equation (10.3.27), we. obtain

where L_c = T_b /T_c is the processing gain. Therefore, the spread-spectrum signal has reduced the power of the interference by the factor L_c.

Another interpretation of the effect of the spread-spectrum signal on the sinusoidal interference is obtained if we express P_IT_c in Equation (10.3.29) as follows. Since , we have

where I₀ is the power-spectral density of an equivalent interference in a bandwidth W. Therefore, in effect, the spread-spectrum signal has spread the sinusoidal interference over the wide bandwidth W, creating an equivalent spectrally flat noise with power-spectral density I₀. Hence,

Example 10.3.1

The SNR required at the detector to achieve reliable communication in a DS spread-spectrum communication system is 13 dB. If the interference-to-signal power at the receiver is 20 dB, determine the processing gain required to achieve reliable communication.

Solution
We are given (P_I /P_S)_dB = 20 dB or, equivalently, P_I /P_S = 100. We are also given (SNR)_D = 13 dB, or equivalently, (SNR)_D = 20. The relation in Equation (10.2.29) may be used to solve for L_c. Thus,





Therefore, the processing gain required is 1000 or, equivalently, 30 dB.

As a second case, let us consider the effect of an interference i(t) that is a zero-mean broadband random process with a constant power-spectral density over the bandwidth W of the spread-spectrum signal, as illustrated in Figure 10.42. Note that the total interference power is

The variance of the interference component at the input to the detector is given by Equation (10.3.21). To evaluate the moment E(n ²), we substitute the bandpass representation of i(t), given by Equation (10.3.16), into Equation (10.3.19). By neglecting the double frequency terms (terms involving cos 4p f_ct) and making use of the statistical properties of the quadrature components, namely E [i_c(t)] = E [(i_s(t)] = 0, and

we obtain the result

This is an integral of the autocorrelation function over the square defined by the region 0 £ t₁ £ T_c and 0 £ t₂ £ T_c, as shown in Figure 10.43. If we let t = t₁ – t₂, E(n ²)
can be reduced to the single integral (see Figure 10.43)

Since T_c » a/W, where a > 0, the above integral may be expressed as

and it can be numerically evaluated for any value of a. Figure 10.44 illustrates J(a). Note that J(a) £ 1 for any value of a and that as .

By combining the results in Equations (10.3.21), (10.3.35) and (10.3.36), we conclude that the variance of the interference component y_i (T_b) for the broadband interference is

Therefore, the SNR at the detector is

If we compare Equation (10.3.31) with Equation (10.3.38) we observe that the SNR for the case of the broadband interference is larger due to the factor J(a). Hence, the
sinusoidal interference results in a somewhat larger degradation in the performance of the DS spread-spectrum system compared to that of a broadband interference.

The probability of error for a DS spread-spectrum system with binary PSK modulation is easily obtained from the SNR at the detector, if we make an assumption on the probability distribution of the sample y_i (T_b). From Equation (10.3.18) we note that y_i (T_b) consists of a sum L_c uncorrelated random variables {c_nn_n, 0 £ n £ L_c – 1}, all of which are identically distributed. Since the processing gain L_c is usually large in any practical system, we may use the Central Limit Theorem to justify a Gaussian probability distribution for y_i (T). Under this assumption, the probability of error for the sinusoidal interference is

where I₀ is the power-spectral density of an equivalent broadband interference. A similar expression holds for the case of a broadband interference, where the SNR at the detector is increased by the factor 1/J(a).

The Interference Margin. We may express in the Q-function in Equation (10.3.39) as

Also, suppose we specify a required to achieve a desired level of performance. Then, using a logarithmic scale, we may express Equation (10.3.40) as

The ratio (P_I /P_S)_dB is called the interference margin. This is the relative power advantage that an interference may have without disrupting the communication system.

Example 10.3.2

Suppose we require an to achieve reliable communication. What is the processing gain that is necessary to provide an interference margin of 20 dB?

Solution
Clearly, if W/R = 1000, then (W/R)_dB = 30 dB and the interference margin is (P_J /P_S)_dB = 20 dB. This means that the average interference power at the receiver may be 100 times the power P_S of the desired signal and we can still maintain reliable communication.

Performance of Coded Spread-Spectrum Signals. As shown in Chapter 9, when the transmitted information is coded by a binary linear (block or convolutional) code, the SNR at the output of a soft-decision decoder is increased by the coding gain, defined as

where R_c is the code rate and is the minimum Hamming distance of the code. Therefore, the effect of coding is to increase the interference margin by the coding gain. Thus, Equation (10.3.41) may be modified as

where (CG)_dB denotes the coding gain.

Some Applications of DS Spread-Spectrum Signals

In this subsection, we briefly describe the use of DS spread-spectrum signals in three applications. First, we consider an application in which the signal is transmitted at very low power, so that a listener trying to detect the presence of the signal would encounter great difficulty. A second application is multiple-access radio communications. A third application involves the use of a DS spread-spectrum signal to resolve the multipath in a time-dispersive radio channel.

Low-detectability Signal Transmission. In this application, the information-bearing signal is transmitted at a very low power level relative to the background channel noise and thermal noise that is generated in the front end of a receiver. If the DS spread spectrum signal occupies a bandwidth W and the power-spectral density of the additive noise is N₀ W/Hz, the average noise power in the bandwidth W is P_N = W N₀.

The average received signal power at the intended receiver is P_R . If we wish to hide the presence of the signal from receivers that are in the vicinity of the intended receiver, the signal is transmitted at a power level such that . The intended receiver can recover the weak information-bearing signal from the background noise with the aid of the processing gain and the coding gain. However, any other receiver which has no knowledge of the PN code sequence is unable to take advantage of the processing gain and the coding gain. Consequently, the presence of the information-bearing signal is difficult to detect. We say that the transmitted signal has a low probability of being intercepted (LPI) and it is called an LPI signal.

The probability of error given in Section 10.3.2 applies as well to the demodulation and decoding of LPI signals at the intended receiver.

Example 10.3.3

A DS spread-spectrum signal is designed so that the power ratio P_R /P_N at the intended receiver is 10^–2; (a) if the desired /N₀ = 10 for acceptable performance, determine the minimum value of the processing gain; and (b) suppose that the DS spread-spectrum signal is transmitted via radio to a receiver at a distance of 2000 Km. The transmitter antenna has a gain of 20 dB, while the receiver antenna is omnidirectional. The carrier frequency is 3 MHz, the available channel bandwidth W = 10⁵ Hz and the receiver has a noise temperature of 300 K. Determine the required transmitter power and the bit rate of the DS spread-spectrum system.

Solution
(a) We may write /N₀ as





Since /N₀ = 10 and P_R /P_N = 10^–2, it follows that the necessary processing gain is L_c = 1000.

(b) The expression for the received signal power is





where L_sdB is the free-space path loss and G_TdB is the antenna gain. The path loss is




where the wavelength l = 100 meters. Hence,




Therefore,




The received power level can be obtained from the condition P_R /P_N = 10^–2. First of all, P_N = WN₀, where N₀ = kT = 4 ´ 10^–21 W/Hz and W = 10⁵ Hz. Hence,




and




or, equivalently, P_RdB = –174 dBw. Therefore,




or, equivalently, P_T = 2.5 ´ 10^–9 W. The bit rate is R = W/L_c = 10⁵/10³ = 100 bps.

Code Division Multiple Access. The enhancement in performance obtained from a DS spread-spectrum signal through the processing gain and the coding gain can be used to enable many DS spread-spectrum signals to occupy the same channel bandwidth provided that each signal has its own pseudorandom (signature) sequence. Thus, it is possible to have several users transmit messages simultaneously over the same channel bandwidth. This type of digital communication in which each transmitter-receiver user pair has its own distinct signature code for transmitting over a common channel bandwidth is called code division multiple access (CDMA).

In the demodulation of each DS spread-spectrum signal, the signals from the other simultaneous users of the channel appear as additive interference. The level of interference varies as a function of the number of users of the channel at any given time. A major advantage of CDMA is that a large number of users can be accommodated if each user transmits messages for a short period of time. In such a multiple access system, it is relatively easy either to add new users or to decrease the number of users without reconfiguring the system.

Next, we determine the number of simultaneous signals that can be accommodated in a CDMA system. For simplicity, we assume that all signals have identical average powers. In many practical systems, the received signal power level from each user is monitored at a central station and power control is exercised over all simultaneous users by use of a control channel that instructs the users on whether to increase or decrease their power level. With such power control, if there are N_u simultaneous users, the desired signal-to-noise interference power ratio at a given receiver is

From this relation, we can determine the number of users that can be accommodated simultaneously. The following example illustrates the computation.

Example 10.3.4

Suppose that the desired level of performance for a user in a CDMA system is an error probability of 10^–6, which is achieved when . Determine the maximum number of simultaneous users that can be accommodated in a CDMA system if the bandwidth-to-bit-rate ratio is 1000 and the coding gain is .

Solution
From the relationships given in Equations (10.3.43) and (10.3.44), we have





If we solve for N_u we obtain




For W/R = 1000 and , we obtain the result that N_u = 201.

In determining the maximum number of simultaneous users of the channel, we implicitly assumed that the pseudorandom code sequences used by the various users are uncorrelated and that the interference from other users adds on a power basis only. However, orthogonality of the pseudorandom sequences among the N_u users is generally difficult to achieve, especially if N_u is large. In fact, the design of a large set of pseudorandom sequences with good correlation properties is an important problem that has received considerable attention in the technical literature. We shall briefly treat this problem in Section 10.3.5.

CDMA is a viable method for providing digital cellular telephone service to mobile users. In Section 10.4 we describe the basic characteristics of the North American digital cellular system that employs CDMA.

Communication Over Channels with Multipath. In Section 10.1, we described the characteristics of fading multipath channels and the design of signals for effective communication through such channels. Examples of fading multipath channels include ionospheric propagation in the HF frequency band (3–30 MHz) where the ionospheric layers serve as signal reflectors, and in mobile radio communication systems, where the multipath propagation is due to reflection from buildings, trees, and the other obstacles located between the transmitter and the receiver.

Our discussion on signal design in Section 10.1.5 focused on frequency selective channels, where the signal bandwidth W is larger than the coherence bandwidth B_cb of the channel. If W > B_cb, we considered two approaches to signal design. One approach is to subdivide the available bandwidth W into N subchannels such that the bandwidth per channel . In this way, each subchannel is frequency nonselective and the signals in each subchannel satisfy the condition that the symbol interval , where T_m is the multipath spread of the channel. Thus, intersymbol interference is avoided. A second approach is to design the signal to utilize the entire signal bandwidth W and transmit it on a single carrier. In this case, the channel is frequency selective and the multipath components with differential delays of or greater become resolvable.

DS spread spectrum is a particularly effective way to generate a wideband signal for resolving multipath signal components. By separating the multipath components, we may also reduce the effects of fading. For example, in LOS communication systems where there is a direct path and a secondary propagation path resulting from signal reflecting from buildings and surrounding terrain, the demodulator at the receiver may synchronize to the direct-signal component and ignore the existence of the multipath component. In such a case, the multipath component becomes a form of interference (ISI) on the demodulation of subsequent transmitted signals.

ISI can be avoided if we are willing to reduce the symbol rate such that . In this case, we employ a DS spread-spectrum signal with bandwidth W to resolve the multipath. Thus, the channel is frequency selective and the appropriate channel model is the tapped-delay-line model with time-varying coefficients as shown in Figure 10.4. The optimum demodulator for this channel is a filter matched to the tapped-delay channel model called the RAKE demodulator, as previously described in Section 10.1.5.

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