Binary Offset Carrier Modulations for Radionavigation

Transcription

1 Binary Offset Carrier Modulations for Radionavigation JOHN W. BETZ The MITRE Corporation, Bedford, Massachusetts Received September 2001; Revised March 2002 ABSTRACT: Current signaling for GPS employs phase shift keying (PSK) modulation using conventional rectangular (non return to zero) spreading symbols. Attention has been focused primarily on the design of the spreading code and selection of the keying rates. But better modulation designs are available for next-generation radionavigation systems, offering improved performance and the opportunity for spectrum sharing while retaining implementation simplicity. This paper describes a class of particularly attractive modulations called binary offset carrier (BOC). It presents important characteristics of modulations for radionavigation, introduces several specific BOC designs that satisfy different applications in evolving radionavigation systems, describes receiver processing for these modulations, and provides analytical and numerical results that describe the modulations performance and demonstrate advantages over comparable conventional PSK modulations with rectangular spreading symbols. INTRODUCTION The significant success of first-generation signals in GPS has fostered continuing expectations for improved accuracy. Current modulation designs have been restricted to phase shift keying with rectangular spreading symbols (referred to here as PSK-R), duplicating early modulation designs for digital communications. As other sources of error diminish, contributions from noise and multipath start to dominate. But bandwidth limitations preclude further improvements that might be obtained using PSK-R modulations with faster keying rates. Also, increasing transmitted power to improve accuracy is expensive and has a limited effect on multipath performance. Receiver design strategies such as very wide-bandwidth receiver front ends and very small early late spacing in code-tracking discriminators provide diminishing returns at increasing cost. In contrast, modulations designed specifically for radionavigation can outperform existing modulation designs while using the same or even less bandwidth and enabling simple transmitter and receiver designs. Further, more advanced modulations can better share existing frequency allocations with each other and with heritage signals, thus preserving spectrum. It remains important that modulations provide binary phase values, maintaining ease of signal generation (including multiplexing multiple signals onto a single carrier) and receiver processing. These binary PSK modulations can readily be extended to higher-order alphabets. NAVIGATION: Journal of The Institute of Navigation Vol. 48, No. 4, Winter Printed in the U.S.A. A significant study of modulations other than PSK-R for GPS was instigated by the need to modernize the military GPS service, adding a new military signal within the radio frequency (RF) bands already being used [1]. The binary offset carrier (BOC) modulation was developed for this purpose [2]. It has been found to provide the best overall performance [3], and is being implemented on Block IIR-M and Block IIF satellites to be launched as early as While BOC modulations were developed to provide spectral isolation from heritage signals modulating the same carrier frequency, it was quickly determined that they offer performance advantages as well. More recent work suggests that BOC modulations may allow yet another signal, in addition to heritage signals and the new military signal, to be added to currently allocated GPS bands [4]. While there has been previous consideration of subcarrier modulations for civilian signals (e.g., [5] mentions a C/A-code signal variant equivalent to a BOC(1,1)), this paper provides the first comprehensive discussion of BOC modulations in general, also demonstrating their enhanced performance and assessing their characteristics. The next section summarizes essential characteristics of BOC modulations. It defines the modulations, outlines the approach for generating them, and presents expressions for their second-order statistics. The third section introduces some example BOC modulations of particular interest, defines measures for evaluating characteristics of modulations for radionavigation, and evaluates these measures for the example BOC modulations. Next, the paper outlines approaches for receiver processing of BOC modulations. It then presents results on effects of noise and of multipath on code-tracking accuracy, 227

2 compared with equivalent results for PSK-R modulations. The final section summarizes the findings of this work. BINARY OFFSET CARRIER MODULATIONS This section presents the properties of BOC modulations. Definition of Binary Offset Carrier Modulation The complex envelope of an offset carrier signal is given by [2] (note that all the analysis here uses analytic signal representations and lowpass-equivalent representations of linear time-invariant systems): s(t) e i a k (t knt nt (1) s s t 0 )c (t t T s 0) k where {a k } is the data-modulated spreading code unit magnitude with phase values chosen randomly from an alphabet (restricted to two symbols for the binary modulation considered here, but readily expanded to higher order); c Ts (t) is the subcarrier a periodic function with period 2T s ; nts (t) is a spreading symbol, here a rectangular pulse with time support equal to nt s ; and n (restricted to be a positive integer) is the number of half-periods of the subcarrier during which the spreading code value remains the same. The quantities and t 0 reflect arbitrary offsets in phase and time, respectively, in the complex envelope relative to some reference. This representation shows that, when the spreading symbol is rectangular, offset carrier signals are conventional PSK-R signals modulating a periodic signal. Initial work on offset carrier modulations focused on a sinusoidal subcarrier and a lowpass-filtered spreading symbol. The resulting complex envelope was not constant modulus. This class of offset carrier modulation has been termed linear offset carrier, since its modulus takes on a continuum of values. Implementation considerations motivated consideration of a square-wave subcarrier c T s (t) in equation (1) and a rectangular (unfiltered) spreading symbol nts (t), producing a constant-modulus complex envelope. The resulting binary-valued modulation, referred to as binary offset carrier, is the subject of this paper. There is a very close relationship between these variations of offset carrier modulations; in fact, a linear offset carrier modulation can be processed by a receiver using a BOC reference signal (having the same subcarrier frequency and code rate) with very little effect on performance. A BOC modulation is denoted BOC(f s,f c ), where f s is the subcarrier frequency, and f c is the code rate, defined by f s 1 2T s (2) f c is the code rate f c 1 2 (3) nt s n f s Consistent with the convention introduced in [2] and followed subsequently in work on GPS and elsewhere, the designation BOC(, ) used in this paper is an abbreviation. The subcarrier frequency is actually MHz, while the spreading code rate is actually MHz. For example, BOC(10,5) means that the subcarrier frequency is MHz, and the spreading code rate is MHz. Generation of data, spreading code, subcarrier, and RF uses a common clock so that zero crossings are aligned. Figure 1 provides a block diagram of BOC signal generation. All of the baseband signals in Figure 1 are binary-valued, and thus can be implemented using binary logic. An equivalent representation to equation (1) describes a biphase BOC modulation as a PSK modulation with unconventional spreading symbol shape, so that its complex envelope can be written as s BOC(fs,f c )(t) e i a k q nts (t knt s t 0 ) k for n even, and s BOC(fs,f c )(t) e i (1) k a k q nts (t knt s t 0 ) k Fig. 1 Conceptual Generation of BOC Modulation (4) (5) 228 Navigation Winter

3 for n odd, with spreading symbol n1 q nts (t) (1) m Ts (t mt s ) (6) m0 so that q nts (t) consists of n half-cycles of a square wave, that is, n alternating values of 1 and 1. Whenever n is even, q nts (t) is a balanced (average value of zero) symbol. Considered in this sense, the BOC modulation generalizes a Manchester modulation to more than one zero crossing per spreading symbol (i.e., n 2 yields a BOC(f c,f c ) modulation, which is a Manchester modulation). Also, n 1 yields a BOC(f c /2,f c ) modulation, which uses a rectangular (non return to zero) spreading symbol, and thus has the same second-order statistics (autocorrelation function and power spectral density) as a conventional PSK-R modulation. While the second-order statistics are the same, the waveforms of a BOC(f c /2,f c ) modulation and conventional PSK-R modulation are not identical because of the 1 in equation (5). Spectra and Correlation Functions of Binary Offset Carrier Modulations Assume that the binary values of the BOC spreading sequence values are equally likely, independent, and identically distributed. The appendix shows that for n even, the normalized (unit area) baseband power spectral density of a BOC modulation is G BOC(fs,f e )(f) 1 nt s sin(ft s)sin(nft s ) f cos(ft s ) 2 f csin f 2f s sin f f c f cos f 2f s f c tan f 2f s sin f f (7) For n odd, the appendix shows that the normalized BOC power spectral density is G BOC(fs, f c )(f) 1 nt s sin(ft s)cos(nft s ) f cos(ft s ) 2 f csin f 2f s cos f f c f cos f 2f s f c tan f 2f s cos f f f c f c 2 2, 2f s f c 2 2, 2f s f c n even n odd (8) The expressions for n odd in equation (8) and n even in equation (7) are very similar; the only difference is whether a sine or cosine appears in the numerator. When n 1, equation (8) becomes G BOC(fc, /2,f c )(f) T s sin(ft s) f (9) ft s 2 1 f c 2 f csin f f c which is the conventional expression for the power spectral density of a PSK-R modulation. In the power spectral density, the sum of the number of mainlobes and sidelobes between the mainlobes is equal to n, twice the ratio of the subcarrier frequency to the code rate. As in conventional PSK, the zero crossings of each mainlobe are spaced by twice the code rate, while the zero crossings of each sidelobe are spaced at the code rate. For example, n 4 for both BOC(10,5) and BOC(8,4) modulations, and their spectra have two sidelobes between the two mainlobes. In contrast, n 5 for a BOC(5,2) modulation, and its spectrum has three sidelobes between the two mainlobes, while n 10 for a BOC(5,1) modulation, and its spectrum has eight sidelobes between the two mainlobes. Maxima of the mainlobes occur at frequencies somewhat below the subcarrier frequency because of the coherent interactions between the upper and lower sidebands. Examples of this point are provided in the next section. For an ideal BOC modulation with infinite bandwidth, the autocorrelation function consists of a set of connected line segments with (in general) multiple zero crossings and multiple lobes. The number of negative and positive peaks in the magnitude autocorrelation function is 2n 1, and the peaks are separated in delay by T s. Adopting the notation that is the index of the (normalized) correlation function peak (with 0 indicating the main peak, 1 being the index of the first peak to the right of the main peak, etc.), the value of the -th peak is ( 1) (n )/n, for 0,1,...,n 1. The zero crossings nearest the main peak occur at delay 1/[ (4f s f c )], and the support of the correlation function is 2/(f c ). Figure 2 shows correlation functions for BOC(5,1) and BOC(5,2) modulations computed over very wide bandwidths. For BOC(5,1), n 10, and the correlation function has 19 peaks, as predicted. The peaks are separated by 97.8 ns, and the first peaks away from the main peak have a value of 0.9. The zero crossings nearest the main peak occur at approximately 51 ns. For BOC(5,2), n 5, and the correlation function has 9 peaks, as predicted. The peaks are separated by 97.8 ns, and the first peaks away from the main peak have a value of 0.8. The zero crossings nearest the main peak occur at approximately 54 ns. Vol. 48, No. 4 Betz: Binary Offset Carrier Modulations for Radionavigation 229

4 Fig. 2 Correlation Functions for BOC(5,1) (upper) and BOC(5,2) (lower) Computed over 1 GHz Bandwidth Many useful properties of modulations for radionavigation have been identified and are provided below in terms of an arbitrary power spectral density G s (f), which is normalized to unit area over infinite bandwidth. When the signal is ideally bandlimited to complex bandwidth r, the autocorrelation function is given by r/2 R s () G s (f )e i2f d (10) r /2 The fraction of power remaining after bandlimiting to r is r /2 G s (f)df (11) r /2 so that G s (f ) 1 G s (f ) has unit area over complex bandwidth r. The corresponding normalized correlation function for the bandlimited signal is r /2 R s () 1 G s (f )e i2f df (12) r /2 The root mean square (RMS) bandwidth (in hertz) of a bandlimited signal having unit power is denoted rms, defined by 2 rms r /2 f 2 G s (f)df r /2 (13) Ideal BOC modulations have infinite RMS bandwidth evaluated over infinite bandwidth, like other modulations that use rectangular spreading symbols. When the RMS bandwidth is computed over finite bandwidth, two separate effects occur: loss of power and shaping of the spectrum. To distinguish between the two, equation (13) defines the RMS bandwidth using the normalized bandlimited power spectral density, and the loss of power is accounted for separately through equation (11). Lower Bound on Code-Tracking Accuracy A lower bound on code-tracking accuracy in white noise can be obtained using the RMS bandwidth [6], as a specific case of the more general lower bound in Gaussian noise and interference having arbitrary spectral shape [7, 8]. This lower bound is based on the performance of a maximum-likelihood estimator of time of arrival using T seconds of data, driving a tracking loop. The modulation has a normalized power spectral density G s (f) over receiver front-end bandwidth r, received in white noise at a given carrier-power-to-noise-density ratio C/N 0 (where the carrier power is over infinite bandwidth, consistent with convention), and a code-tracking loop with a one-sided equivalent rectangular bandwidth of B L. 230 Navigation Winter

5 For this case, the smallest RMS code-tracking error that can be achieved in white noise using any discriminator design is 1 LB 2 rms B L C N 0 where the units of LB are seconds. (14) Spectral Separation Coefficients Spectral separation coefficients quantify the degree to which interference having a given power spectral density degrades the effective C/N 0 [9, 4] of a receiver channel. Define the spectral separation coefficient so that the composite interference has normalized power spectral density G (f) G t /2 (f)/ G (f )df, f t /2 t /2 (15) 0, elsewhere over some transmitting bandwidth t, the reference signal in the receiver has infinite bandwidth and power spectral density G s (f), and the receiver s front end has complex bandwidth r. The spectral separation coefficient (having dimensions of seconds) is then (16) When the interfering signals have a composite received power of C within bandwidth r, the desired signal has a received power of C over an infinite bandwidth, and the thermal noise density is N 0, the resulting effective C/N 0 [9] is C N 0 eff r /2 s G (f )G s (f )df r /2 r /2 N 0 r /2 r /2 C G s(f )df r /2 G s (f )df C s (17) The effective rectangular bandwidth of a power spectral density is given by the bandwidth of a rectangular spectrum having both the same maximum power spectral density and the same area: rect r /2 G s (f )df r /2 /G s (f max ) (18) where f max is the frequency at which G s (f ) attains a maximum value. Observe that when the spectral separation coefficient in equation (16) is computed for an interfering signal that is a unit-power sinusoid at the frequency where G s (f) attains a maximum value, G s (f max ) (sinsuoid at fmax )s and, using equation (11), rect / (sinsuoid at fmax )s (19) APPLICATIONS This section illustrates the theory presented in the preceding section by examining three representative BOC modulations: The BOC(10,5) modulation used for the M-code signal, the new military signal on GPS carriers L1 and L2. A BOC(5,2) modulation that has been suggested as a candidate for a new civil signal on GPS carriers L1 and L2 in addition to the C/A-code or L2C signal, Y-code signal, and M-code signal, and that fits within the existing 24 MHz allocation on these carriers. A BOC(8,4) modulation that has very similar spectral properties to a PSK-R modulation with a spreading code rate of MHz. Two conventional GPS modulations, both using PSK-R, are also considered. A MHz PSK-R modulation represents both the heritage military signal, the Y-code signal, and the modulation selected for the new civilian signal on L5. A MHz PSK-R modulation represents both the C/A-code signal and the L2C signal that may be used on L2. Since the emphasis in this paper is on modulation design, long spreading codes are assumed, leading to smooth spectra without spectral lines. Signal characteristics provided below are computed for signals strictly bandlimited to 30 MHz bandwidth and normalized to unit power over that bandwidth. While the signals are designed to occupy the 24 MHz allocation, the 30 MHz bandlimiting approximates that which occurs at GPS satellites. The BOC(10,5) modulation was designed to allow the new military signal to be transmitted at much higher power while minimizing interference with existing signals and providing many other benefits, ranging from implementation ease to performance. Before being selected, it was subjected to rigorous scrutiny, ranging from analysis to hardware demonstrations, as were other candidate modulations. Figure 3 portrays the spectra of MHz PSK-R and BOC(10,5). The normalized (scaled so that main peaks have unit height) autocorrelation functions are portrayed in Figure 4, showing the narrower peak and multimode structure of the BOC(10,5) correlation function. BOC(5,2) is one of a family of BOC(5,x) modulations recently suggested [4] as suitable modulations for a fourth radionavigation signal that could coexist with the modernized signal architecture consisting of the Y-code signal, M-code signal, and C/A- (or L2C)- code signal within the existing 24 MHz allocations on L1 and L2. While BOC(5,1) and BOC(5,2.5) are other modulations within the BOC(5,x) family that might also be suitable, BOC(5,2) offers an attractive mix of performance and RF compatibility. Since it would serve as a new civil signal within L1 and L2, Vol. 48, No. 4 Betz: Binary Offset Carrier Modulations for Radionavigation 231

6 Fig. 3 Normalized Power Spectral Densities of Y-Code Signal Having MHz PSK-R Modulation and M-Code Signal Having BOC(10,5) Modulation Fig. 4 Normalized Autocorrelation Functions of Y-Code Signal Having MHz PSK-R Modulation and M-Code Signal Having BOC(10,5) Modulation, Computed Using 24 MHz Bandwidth its characteristics are compared with those of the current MHz PSK-R signals in those bands. Figure 5 portrays the spectra of the modulations, showing that BOC(5,2) s spectrum lies primarily within a bandwidth less than 20 MHz. Since n 5 for BOC(5,2), there is a spectral sidelobe at the band center. The normalized autocorrelation functions are portrayed in Figure 6, showing the narrower peak and multimode structure of the BOC correlation function. The support of the BOC(5,2) autocorrelation function is half that of the autocorrelation function for MHz PSK-R. Although a MHz PSK-R modulation was selected for the new civil signal on L5, a BOC(8,4) modulation is comparable in many respects. Figure 7 portrays the spectra of BOC(8,4) and MHz PSK-R. The normalized autocorrelation functions are portrayed in Figure 8, showing the narrower peak and multimode structure of the BOC correlation function. Figure 9 shows the cumulative spectra of the five specific modulations being considered. All have most or all of their power contained within 24 MHz, but the different locations of large slopes show that their power is located in different portions of the band. Table 1 provides the characteristics of these modulations. The locations of spectral peaks and their maximum power spectral densities are selfexplanatory. The 90 percent power bandwidth indicates the complex bandwidth needed to pass 90 percent of the signal power. (Recall, as noted above, that these calculations use signals strictly bandlimited to 30 MHz, representing bandlimiting at the satellite, even though the signals are designed to use the 232 Navigation Winter

7 Fig. 5 Normalized Power Spectral Densities of MHz PSK-R and BOC(5,2) Modulations Fig. 6 Normalized Autocorrelation Functions of MHz PSK-R and BOC(5,2) Modulations, Computed Using 24 MHz Bandwidth Fig. 7 Normalized Power Spectral Densities of MHz PSK-R and BOC(8,4) Modulations Vol. 48, No. 4 Betz: Binary Offset Carrier Modulations for Radionavigation 233

8 Fig. 8 Normalized Autocorrelation Functions of MHz PSK-R and BOC(8,4) Modulations, Computed Using 24 MHz Bandwidth Fig. 9 Cumulative Normalized Power Spectral Densities of Different Modulations allocated 24 MHz bandwidth.) The smaller is the out-of-band power loss outside 24 MHz, the greater is the in-band power available to a receiver. The greater is the RMS bandwidth, defined in equation (13), the better is the inherent ability to yield small code-tracking errors, under the assumption that the errors are small enough for linearized error analysis to apply. The greater is the equivalent rectangular bandwidth, defined in equation (18), the greater is the resistance to narrowband interference at the worst-case frequency. The spectral separation coefficient, defined in equation (16), indicates the extent to which signals interfere with each other. The smaller is the spectral separation coefficient of a modulation with itself, the greater is the modulation s resistance to multipleaccess interference with other signals having the same modulation, and thus the better a modulation can be used for many signals at different received power levels. In addition, a smaller spectral separation coefficient indicates the ability of a modulation to provide enough processing gain against multipleaccess interference from similar signals so that it can support higher data rates. The smaller is the spectral separation coefficient of the modulation with MHz PSK-R, the less it interferes with C/A- (and L2C)-code signals, and thus the more signals that can be accommodated at higher power without interfering with reception of existing civilian signals. The spectral separation coefficient of BOC(10,5) indicates the degree to which the modulation is isolated from the new military signal, so that high-power M-code signals do not interfere with its reception. The time delay of the first autocorrelation function sidelobe, combined with the magnitude of the first correlation function peak, indicates the degree to which receivers may have difficulty maintaining 234 Navigation Winter

9 Table 1 Characteristics of Different Radionavigation Modulations Characteristic MHz PSK-R MHz PSK-R BOC(5,2) BOC(8,4) BOC(10,5) Frequency Offsets of Main Spectral Peak from Band Center (MHz) Maximum Power Spectral Density (dbw/hz) % Power Bandwidth (MHz) Out-of-Band Loss* (db) RMS Bandwidth* (MHz) Equivalent Rectangular Bandwidth* (MHz) Spectral Separation Coefficient with Itself* (db/hz) Spectral Separation Coefficient with MHz PSK-R* (db/hz) Spectral Separation Coefficient of BOC(10,5)* (db/hz) Time Delay of First Autocorrelation Function Sidelobe* (ns) None None Ratio of Squared First Sidelobe Magnitude to Magnitude of Squared Main Peak* None None * Computed with receive bandwidth of 24 MHz. track of the true peak of the correlation function. Sidelobe peaks that are close both in delay and in magnitude can make it challenging for code tracking to maintain track on the main lobe of the correlation function. As long as the subcarrier frequency is not excessive (e.g., less than 12 MHz) the modulation provides adequate separation of the correlation function peaks, so that tracking strategies such as those documented in [10] provide reliable tracking of the main peak. Other practical factors that preclude use of excessive subcarrier frequencies include difficulty in acquiring the correlation function peak and sensitivity to channel imperfections, such as dispersive effects of the ionosphere and RF hardware, including antennas. However, studies and hardware testing have demonstrated that the BOC modulation designs described in this section provide robust and practical performance. The columns for BOC(10,5) modulation and MHz PSK-R modulation in Table 1 show the contrast between the characteristics of the new and the heritage military signals. BOC(10,5) places spectral peaks near edges of the 24 MHz band allocated to GPS, rather than in the center. The maxima of the power spectral densities for these two modulations are almost identical. While the 90 percent power bandwidth of BOC(10,5) is much wider than that of MHz PSK-R, reflecting the shift of power near the band edge, it remains within 24 MHz. Consequently, the out-of-band loss is only 0.4 db. The RMS bandwidth of BOC(10,5) is much greater than that of MHz PSK-R, indicating the potential for reducing RMS code-tracking error in white noise by a factor of 2.6 under equivalent conditions. Since the equivalent rectangular bandwidths of these two modulations are almost identical, they have almost the same processing gain against narrowband interference. BOC(10,5) provides slightly better spectral separation with itself, allowing more signals with greater power variation. More significant, BOC(10,5) offers 17 db better spectral separation from civilian signals that use MHz PSK-R modulations than does MHz binary PSK-R, allowing the M-code signal to be transmitted at much higher power than Y-code signals without interfering with reception of C/A-code or L2C signals. The peak closest to the main peak of the BOC(10,5) modulation s autocorrelation function is well separated from the main peak, both in amplitude and in delay. The columns for BOC(5,2) modulation and MHz PSK-R modulation in Table 1 show the contrast between the characteristics of a candidate new civilian signal that could fit within the current GPS allocation on L1 and L2 and the heritage civilian signal in those bands. BOC(5,2) places spectral peaks in spectral nulls of the heritage civilian signals and the new military signal [4]. The maximum of the power spectral density for the BOC(5,2) modulation is more than 6 db lower than that for PSK-R modulations, which is better for spectral compatibility. The 90 percent power bandwidth of BOC(5,2) is less than half the 24 MHz registered band, and even smaller than that of a MHz PSK-R modulation. The RMS bandwidth of BOC(5,2) is much greater than that of MHz PSK-R, indicating the potential for reducing RMS code-tracking Vol. 48, No. 4 Betz: Binary Offset Carrier Modulations for Radionavigation 235

10 error in white noise by a factor of 4 under equivalent conditions. Since the equivalent rectangular bandwidth of BOC(5,2) is 4 times greater than that of MHz PSK-R, BOC(5,2) offers 6 db greater resistance to narrowband interference. By providing almost 7 db better spectral separation with itself than is provided by MHz PSK-R, BOC(5,2) allows use of more signals having greater power variation, and also supports higher data rates while tolerating multiple-access interference. BOC(5,2) s better spectral separation from MHz PSK-R indicates that new signals using this modulation would interfere less with reception of C/A-code or L2C-code signals, providing backward compatibility with more new signals at higher power levels. Even with these advantages, the M-code signal is almost as separated from this new modulation as from heritage signals using MHz PSK-R modulation, allowing the M-code signal to be transmitted at higher power. Again, the first peak of the BOC(5,2) modulation s autocorrelation function is also well separated from the main peak. Comparing the columns for the BOC(8,4) modulation and MHz PSK-R modulation yields interesting contrasts. In many respects, including maximum power spectral density, out-of-band loss, and spectral separation with itself, the two modulations are very similar. Both fit readily within the 24 MHz allocated bandwidth. Yet the RMS bandwidth of BOC(8,4) is more than twice that of MHz PSK-R, allowing the RMS code-tracking error to be less than half under equivalent conditions. Viewed another way, obtaining the same codetracking error with MHz PSK-R in white noise would require more than a 6 db increase in C/N 0 compared with that for BOC(8,4). Conversely, since MHz PSK-R has greater equivalent rectangular bandwidth, it provides slightly (0.8 db) better carrier-tracking performance with a narrowband interferer at the spectral maximum. Yet the BOC(8,4) modulation provides slightly (0.1 db) better carrier tracking than MHz PSK-R in white noise because of BOC(8,4) s lower out-of-band loss. RECEIVER PROCESSING OF BOC MODULATIONS In most ways, receiver processing of BOC modulations is similar to receiver processing of PSK-R modulations. However, BOC modulations offer some unique opportunities for variations in receiver processing that can provide advantages in implementation and performance. The best receiver performance is obtained by processing both sidebands coherently as a single signal, not only because all the signal energy is combined coherently, but also because the unique spectral shape of the dual-sideband modulation leads to better ranging performance. However, since each of the two spectral sidebands redundantly contains all the information needed for ranging and data demodulation, the distinct sidebands can be processed separately if desired. Sideband processing is straightforward to implement and has been demonstrated in hardware [11]. The receiver treats the BOC modulation like a PSK-R modulation centered at one of the two subcarrier frequencies, and having the code rate of the BOC modulation. If desired, filtering can be used to select only the desired sideband and prevent aliasing at lower sampling rates. Thus, processing one sideband is identical to processing a conventional GPS signal with a PSK-R modulation. One degradation of singlesideband processing occurs from loss of approximately 3 db of C/N 0. Another degradation is due to the change in spectral shape and the resulting reduction in RMS bandwidth. In some applications, however, the implementation simplicity afforded by the opportunity to process narrower bandwidths may offset these disadvantages. One example in which single-sideband processing might be useful is in a portable wireless device, such as a cellular telephone, that includes a radio navigation satellite service (RNSS) receiver. A very small antenna and simple RF electronics having only enough bandwidth to pass one sideband, combined with the reduced sampling rate and processing throughput requirements associated with the bandwidth of one sideband, would help minimize the cost, size, and power consumption of the receiver. Another motivation for sideband processing of BOC modulations is interference avoidance. Simple circuitry in a receiver can detect when partial-band interference is encountered that obscures only one sideband of the BOC modulation, and can reconfigure the receiver to process only the sideband not being interfered with. This approach can provide substantial immunity to some types of interference without the complexity of interference mitigation circuitry. Acquisition of BOC modulations can also take advantage of the sideband structure. Acquisition architectures have been developed that process each sideband separately [12]. The correlation function from sidelobe processing is a bandlimited version of the correlation function for a PSK-R modulation at the spreading code rate of the BOC modulation a much broader correlation function than that of the dual-sideband BOC modulation. This wider correlation function allows the signal to be sampled at a much lower rate while still avoiding large scalloping losses that occur when sampling epochs fall far from the peak of the correlation function. Since (for a fixed integration time) acquisition processing complexity (measured in arithmetic operations) is proportional to the square of the sampling rate, and storage for noncoherent integration is proportional to the sampling rate, this simplification can be substantial even 236 Navigation Winter

11 when both sidebands are processed in parallel, and then the results are combined noncoherently. While the effect on acquisition performance is small (on the order of 1 db degradation that typically has a negligible effect on the acquisition time), this approach can provide more than an order of magnitude reduction in computational load and storage required for acquisition as compared with acquisition processing using the dual-sideband signal. Code tracking of the dual-sideband signal is based on well-known techniques developed for PSK-R modulations. A conventional code-tracking loop can be used with a discriminator based on early late processing or an analogous approach. The detailed design and performance of noncoherent early late processing (NELP) are presented in the following section. The sidelobes of BOC modulations allow the use of additional processing in the discriminator to maintain code track even under stressed conditions. One approach, described in [10], augments early and late processing with very early and very late taps positioned to coincide with the location of the first sidelobes of the BOC autocorrelation function. Simple logic indicates whether the prompt tap is no longer tracking the main peak and directs how to maintain track of the main peak. This very-early/very-late processing has been demonstrated in real-time hardware. A similar approach could employ a discriminator that would operate on the correlation function for one sideband (or the noncoherently combined correlation functions from each sideband) to maintain the coarse track of the main peak, cross-coupled with a fine tracking discriminator operating on the wideband signal, for best performance. The prompt tap from the discriminator provides estimates of the despread residual carrier, which can be tracked in a frequency-locked loop or phaselocked loop. This carrier processing is identical to that for PSK-R modulations, and also supports conventional approaches for data demodulation. In addition, it is easy to extract the separate upper and lower subcarriers from the received signal and to track these as well. While this extension has not yet been explored, it could provide redundancy, and might even offer opportunities for disambiguating range measurements from carrier-phase estimates using a narrowlaning approach. PERFORMANCE USING NONCOHERENT EARLY LATE PROCESSING As discussed in the previous section, code tracking of a signal with BOC modulation can use conventional early late processing, with early late spacing selected to take advantage of the modulation s characteristics. This section outlines a procedure for selecting early late spacing for NELP based on the theory presented in [7] and [8], and illustrates the process for the example BOC modulations introduced previously. The effect of specular multipath environment on bias errors in code tracking is then examined. The S-curve relates the correction produced by a discriminator for a given error. When NELP with early-to-late spacing of seconds is used, the S-curve is given by [6] S() R s (/2) 2 R s (/2) 2 (20) where is the error in the signal s time of arrival, and the correlations in equation (3) are assumed normalized to unit value at zero delay: R s () R s ()/R s (0). The discriminator gain is defined as the slope of the S curve at 0: g ds() & (21) d 0 S curves for NELP of the BOC(10,5) modulation, bandlimited to 24 MHz, are shown in Figure 10. The curves are well-behaved for early late spacings of less than 50 ns, but the slope reverses for early late spacings of ns. As long as an appropriate form of extended range correlation (see [10], for example) is used to ensure that the main peak of the correlation function is tracked, the shape of the S curve away from the zero crossing point does not describe how the code-tracking loop behaves. While the discriminator gain becomes smaller for early late spacing greater than 30 ns, the effect of early late spacing on code-tracking accuracy cannot be inferred from this behavior, as seen later in this section. S curves for NELP of the BOC(8,4) modulation, bandlimited to 24 MHz, are shown in Figure 11. The curves are well-behaved for early late spacings of less than 60 ns, but the slope reverses for early late spacings of ns. Again, while the discriminator gain becomes smaller for early late spacings of greater than 40 ns, the effect of early late spacing on code-tracking accuracy cannot be inferred from this behavior. S curves for NELP of the BOC(5,2) modulation, bandlimited to 24 MHz, are shown in Figure 12. The curves are well-behaved for early late spacings of less than 100 ns, but the slope reverses for early late spacings of ns. While the discriminator gain becomes smaller for early late spacings of greater than 60 ns, the effect of early late spacing on code-tracking accuracy cannot be inferred from this behavior. For first-order performance analysis of codetracking error in white noise, let the correlation integration time within the discriminator be T (units of seconds), and let B L (units of hertz) be the one-sided noise-equivalent rectangular bandwidth of the code-tracking loop. When code-tracking errors are small so that a linearized analysis applies, the variance of the code-tracking error for NELP (in Vol. 48, No. 4 Betz: Binary Offset Carrier Modulations for Radionavigation 237

12 Fig. 10 S-Curves for BOC(10,5) Modulation Bandlimited to 24 MHz Fig. 11 S-Curves for BOC(8,4) Modulation Bandlimited to 24 MHz units of seconds squared) is given for a particular power spectral density G s (f), normalized over infinite bandwidth, by [7] and [8]: r /2 2 NELP B G L(1 0.25B L T) s (f) sin 2 (f)df r /2 C r /2 2 2 fg s (f)sin(f)df N 0 r /2 r 2 /2 G s(f)cos 2 (f)df 1 r /2 (22) T C r /2 N 0 G s (f)cos(f)df r /2 The term in square brackets reflects the squaring loss due to the use of noncoherent processing. To assess the code-tracking accuracy of different discriminator designs for the BOC modulations, use a C/N 0 of 30 db-hz, and consider integration times of 20 ms (corresponding to the current channel bit rate of 50 bps) and 5 ms (corresponding to a postulated higher bit rate of 200 bps), a one-sided equivalent rectangular bandwidth of the code-tracking loop of 1 Hz, and a front-end bandwidth of 24 MHz. Figure 13 shows results for BOC(10,5) modulation, computed using equation (22). The error becomes very large for early late spacing of ns, corresponding to the slope reversal in the S-curves 238 Navigation Winter

13 Fig. 12 S-Curves for BOC(5,2) Modulation Bandlimited to 24 MHz observed in Figure 10. For early late spacing of less than 45 ns, the code-tracking accuracy closely approaches the information-theoretic lower bound computed using equation (14), indicating little benefit from using smaller early late spacings. Figure 14 shows results for BOC(8,4) modulation, computed using equation (22). The error becomes very large for early late spacing near 70 ns, corresponding to the slope reversal in the S-curves observed in Figure 11. For early late spacing of less than 50 ns, the code-tracking accuracy closely approaches the information-theoretic lower bound computed using equation (14), indicating little benefit from using smaller early late spacings. Figure 15 shows results for BOC(5,2) modulation, computed using equation (22). The error becomes very large for early late spacing near 100 ns, corresponding to the slope reversal in the S-curves observed in Figure 12. For early late spacing of less than 80 ns, the code-tracking accuracy closely approaches the information-theoretic lower bound computed using equation (14), indicating little benefit from using smaller early late spacings. Figure 16 shows results for BOC(5,2) modulation with the receiver front end limited to 12 MHz. While this bandwidth restriction would cause codetracking error for a MHz PSK-R modulation to increase by almost 50 percent, the effect on code Fig. 13 Code-Tracking Error for BOC(10,5) Modulation Bandlimited to 24 MHz Vol. 48, No. 4 Betz: Binary Offset Carrier Modulations for Radionavigation 239

14 Fig. 14 Code-Tracking Error for BOC(8,4) Modulation Bandlimited to 24 MHz Fig. 15 Code-Tracking Error for BOC(5,2) Modulation Bandlimited to 24 MHz tracking for BOC(5,2) is negligible, indicating that high performance can still be obtained from this modulation even with the hardware simplicity offered by narrower-bandwidth processing. Comparing the code-tracking errors in the preceding figures with corresponding discriminator gains plotted earlier reveals an important insight. Conventional thinking has indicated that codetracking accuracy can be inferred from discriminator gain, with larger discriminator gain indicating smaller code-tracking errors. The results presented here demonstrate that this relationship does not apply once the early late spacing is less than the reciprocal of the receiver front-end bandwidth. Similar results have been observed for BPSK-R modulations [13] and for BOC(10,5) specifically [6]. For example, while Figure 10 shows that discriminator gain increases as early late spacing decreases from 40 ns, Figure 13 shows that there is no further reduction in code-tracking error. To understand why this phenomenon occurs, note that two different and counteracting effects occur with early late processing of bandlimited signals as the early late spacing becomes small while the bandwidth of the codetracking loop is held constant. One effect is that the discriminator gain is reduced, tending to increase code-tracking error. The other is that the errors at the early and the late tap become more positively correlated so that they tend to cancel more in the difference, causing the variance of the difference signal to diminish. These two effects counteract each other for vanishing early late spacing, producing the asymptotic behavior that is observed. Yet another way to verify this asymptotic behavior is to recognize that, with vanishing early late spacing, early late processing approaches an ideal differentiator, which is the maximum-likelihood estimator of time of arrival in white Gaussian noise. 240 Navigation Winter

15 Fig. 16 Code-Tracking Error for BOC(5,2) Modulation Bandlimited to 12 MHz Consequently, the performance of early late processing with vanishing spacing should approach that of the lower bound, which relates the performance of maximum-likelihood estimation in white noise. Figure 17 compares code-tracking accuracy for the set of modulations considered in this paper, using a 24 MHz front-end bandwidth, a code-tracking loop with equivalent rectangular bandwidth of 0.1 Hz, and NELP with appropriate early late spacing for each modulation 0.05 chip period for MHz PSK-R, 0.5 chip for MHz PSK-R, 80 ns for BOC(5,2), 50 ns for BOC(8,4), and 40 ns for BOC(10,5). All the BOC modulations provide much better code-tracking accuracy than the PSK-R modulations under equivalent conditions (same front-end bandwidth, codetracking loop bandwidth, and C/N 0 ). Even BOC(5,2) has better code-tracking accuracy than MHz PSK-R, providing the same code-tracking accuracy as MHz PSK-R at 11 db lower C/N 0. The BOC(8,4) and BOC(10,5) modulations provide the same codetracking accuracy as MHz PSK-R at approximately 7 db lower C/N 0. To examine the effect of multipath on the code tracking of different modulations using NELP, consider a simple model of multipath as a specular reflection having some amplitude relative to the direct path, arriving at some phase and delay, with all values time-invariant over the time period of interest. The channel thus has the transfer function H(f ) 1 e i(2fd) (23) Fig. 17 Code-Tracking Errors for Different Modulations Bandlimited to 24 MHz Vol. 48, No. 4 Betz: Binary Offset Carrier Modulations for Radionavigation 241

16 The effect of this multipath channel on the S-curve can be computed analytically. For a given relative amplitude of the specular reflection, the bias error introduced into code tracking can be computed for a given multipath delay d. For simplicity, only the minimum and maximum bias errors (computed over channel phase ) are plotted for each delay value. Figure 18 shows the worst-case multipath bias errors for specular reflection received 6 db weaker than the direct path for MHz PSK-R and BOC(10,5). The calculation assumes the receiver has a 24 MHz front-end bandwidth, and that NELP uses good choices of early late spacing 0.5 chip for MHz PSK-R and 40 ns for BOC(10,5). BOC(10,5) reduces the worst-case bias error from 5.4 to 2.9 m, and provides smaller bias errors over most of the range of multipath delays. Figure 19 shows the worst-case multipath bias errors for specular reflection received 6 db weaker than the direct path for MHz PSK-R and BOC(5,2). The calculation assumes the receiver has a 24 MHz front-end bandwidth, and that NELP uses good choices of early late spacing 0.05 chip for MHz PSK-R and 80 ns for BOC(5,2). BOC(5,2) has substantially less multipath-induced bias at most multipath delays. However, the worst-case bias error is slightly greater (5.6 m) for BOC (5,2) than for MHz PSK-R (4.9 m). Figure 20 shows similar results for BOC(5,2) and MHz PSK-R with the front-end bandwidth Fig. 18 Bias Errors Caused by 6 db Specular Multipath for MHz PSK-R and BOC(10,5) Modulations Bandlimited to 24 MHz Fig. 19 Bias Errors Caused by 6 db Specular Multipath for MHz PSK-R and BOC(5,2) Modulations Bandlimited to 24 MHz 242 Navigation Winter

17 Fig. 20 Bias Errors Caused by 6 db Specular Multipath for MHz PSK-R and BOC(5,2) Modulations Bandlimited to 12 MHz limited to 12 MHz. While this reduction in front-end bandwidth causes the worst-case multipath error for MHz PSK-R to increase by almost 80 percent to 8.8 m, the worst-case multipath error for BOC(5,2) increases by only 22 percent to 6.0 m. Based on similar results for code-tracking accuracy in Figures 15 and 16, it is clear that BOC(5,2) allows reduced receiver front-end bandwidths with less receiver complexity, with very little effect on performance. In contrast, high-performance reception of PSK-R modulations relies on wide front-end bandwidths. Figure 21 shows the worst-case multipath bias errors for specular reflection received 6 db weaker than the direct path for MHz PSK-R and BOC(8,4). The calculation assumes the receiver has a 24 MHz front-end bandwidth, and that NELP uses good choices of early late spacing 0.5 chip for MHz PSK-R and 50 ns for BOC(8,4). BOC(8,4) has substantially less multipath-induced bias at most delay values and has a worst-case multipath error of 3.5 m, compared with 5.4 m for MHz PSK-R. CONCLUSIONS Modulation design offers an important dimension for obtaining better performance from new signals for radionavigation, complementing the more mature design dimensions of spreading code and data message. This paper has provided two categories of fundamental results in modulation design. First, it has introduced a set of quantitative measures that can be used to assess the capability of different modulations. Second, it has introduced binary offset carrier (BOC) modulations, and shown that they offer significant benefits over comparable phase shift key modulations using rectangular keying (PSK-R). Fig. 21 Bias Errors Caused by 6 db Specular Multipath for MHz PSK-R and BOC(8,4) Modulations Bandlimited to 24 MHz Vol. 48, No. 4 Betz: Binary Offset Carrier Modulations for Radionavigation 243

18 Much of the work on modulation measures was formalized as part of the recent design process for the new military signal the M-code signal. Some measures, such as those of spectral occupancy, have been used extensively for many years in assessment of modulations for radionavigation. Others, such as equivalent rectangular bandwidth and RMS bandwidth, were introduced to this design space as part of the effort for military signal design. Still others, such as the spectral separation coefficient, were formalized more recently. While the set of measures does not completely define the design space, it does provide important insights into the suitability of a modulation for radionavigation. BOC modulations have many good attributes. Since they are constant-modulus and have binary phase, they are straightforward to implement. By moving signal power away from the band center, they offer the potential for better code-tracking accuracy and multipath rejection. Since BOC modulations offer two independent design parameters subcarrier frequency and spreading code rate they provide more freedom for a designer to concentrate signal power within specific parts of the allocated band to reduce interference with the reception of other signals. Furthermore, the redundancy in the upper and lower sidebands of BOC modulations offers practical advantages in receiver processing for signal acquisition, code tracking, carrier tracking, and data demodulation. Engineering judgment is still needed in selecting practical BOC modulations. For example, when the subcarrier frequency is much greater (perhaps by more than a factor of 10) than the spreading code rate, the correlation function has many closely spaced (in amplitude and delay) peaks that can introduce anomalous effects in code tracking. Also, wide spacing of subcarriers can limit the spectral coherence of the received signal as a result of the dispersive effect of hardware imperfections and other channel conditions, such as ionospheric distortion. Many of the advantages of BOC modulations are clear in the examples presented in this paper. The BOC(10,5) modulation, selected for the M-code signal, offers comparable or improved performance relative to the heritage military signal, the Y-code signal, while offering reduced spectral overlap with C/A-code and L2C-code signals, allowing transmission of the M-code signal at higher power levels with tolerable interference. Further, the BOC(10,5) modulation of the M-code signal offers significant advantages in terms of code-tracking accuracy and mitigation of multipath effects. The BOC(5,1) and BOC(5,2) modulations offer an attractive opportunity to add a fourth signal within the current 24 MHz spectral allocation for GPS, augmenting the Y-, M-, and C/A-code signals with an additional civil signal that provides better performance and new capabilities. These modulations offer spectral isolation from the other signals (both so that they do not interfere and so that they are not interfered with). They also provide capacity for more signals at disparate power levels without selfinterference while offering advantages in performance, such as code-tracking accuracy and multipath resistance, and the opportunity for flexible receiver design as compared with the modulation used for the C/A-code signal. In fact, use of a BOC(5,1) or BOC(5,2) modulation allows high-performance receivers to employ front-end bandwidths as narrow as 12 MHz with almost no performance degradation as compared with wider bandwidths, offering the opportunity for exceptional performance with compact antennas and lower-power electronics. Simpler receivers can process only one sideband of the signal, using front-end bandwidths as low as 2 MHz. The BOC(8,4) modulation offers performance comparable to or better than that of a MHz PSK-R modulation for civilian use in a frequency band where high-power M-code signals are not employed. While BOC(8,4) fits comfortably within a bandwidth of less than 24 MHz, its superior code-tracking accuracy and multipath resistance would be desirable for a high-performance civilian signal. Through the continuing development of innovative modulations and prudent use of modulation attributes such as those employed in this paper, there will be ongoing opportunities to design signals that can be readily implemented, efficiently share spectrum with other signals used for radionavigation, and provide opportunities for enhanced performance. ACKNOWLEDGMENTS This work was sponsored by the U.S. Air Force under contract number F C The author thanks the anonymous reviewers for many useful suggestions that improved the presentation of the paper, and especially one reviewer for suggestions that simplified the derivation in the appendix. APPENDIX DERIVATION OF BOC POWER SPECTRAL DENSITY The power spectral density (considering only the stationary and not the cyclostationary part of the statistics) of a BOC modulation is found most readily from the representations of equations (4), (5), and (6). This is the case since the power spectral density of a PSK modulation with independent, identically distributed spreading code and spreading symbol p(t) with spreading symbol period is given by (see equations (11 49) in [14]) G(f ) P(f ) 2 / (A-1) 244 Navigation Winter