Spectrograms of high-rate Global Positioning System (GPS) power data at the L1 frequency (1.57542 GHz) commonly show patterns of fluctuation frequency with respect to time that resemble quasiperiodic scintillation observations. These fluctuation patterns have probably not been noted in the past because of the low sampling frequencies and coarse quantization of power measurements in most GPS receivers. We demonstrate that the source of these fluctuations is usually not edge-type diffraction from steep ionospheric gradients, as is often hypothesized for quasiperiodic scintillation, despite some features consistent with ionospheric generation. Instead, in most cases these spectrogram patterns derive from intersatellite signal interference between pairs of coarse/acquisition (C/A) codes broadcast on L1. As evidence of this mechanism, we show that most spectrogram features appear on pairs of satellite tracking channels and that the patterns can be reproduced using a multisatellite GPS simulator. We also model the spectrogram patterns, both with regard to the basic GPS C/A code correlation process and the differential Doppler shift conditions that cause the fluctuations. In this latter respect, the GPS quasiperiodic scintillation observations bear resemblance to a mechanism discussed by L. A. Hajkowicz for traditional satellite beacons. Despite the complex nature of the intersatellite interference phenomenon, we can model when the spectrogram patterns are expected to appear. Thus one can screen out these time-varying interference patterns to find potential occurrences of actual ionospheric edge diffraction in spectrograms. The intersatellite interference phenomenon is also of general interest to GPS radio engineering.
 During spectrogram studies of 50-Hz Global Positioning System (GPS) amplitude scintillation data, Beach  noted occasional V-shaped patterns, “spectral Vs” for short, that persisted for minutes and had slopes with absolute values of 0.2–0.4 Hz/s. These spectral Vs were not an anomaly peculiar to a specific receiver since the patterns appeared on another receiver nearby. On the basis of this fact and other considerations, the preliminary conclusion was that the source of the patterns was a moving scatterer near the GPS line of sight. Because earlier observations of quasiperiodic scintillation on VHF beacons were attributed to edge-type diffraction or reflections from ionospheric gradients [Basu and Das Gupta, 1969; Slack, 1972; Kelleher and Martin, 1975; Franke et al., 1984], and because the slopes of the Vs were consistent with ionospheric distances and drifts, Beach  explored diffraction from steep ionospheric density gradients as a candidate mechanism for generating spectral Vs at L band frequencies. In other words, the moving scatterer would be the region of the ionospheric density gradient and the characteristic V signature on the spectrogram would correspond to a Fresnel-like diffraction pattern from the associated phase object.
 Unfortunately for the edge diffraction model, systematic observations soon revealed that most V patterns repeated from one day to the next at a given site with a 4-min advance (H. Kil, private communication, 2000). The GPS satellites have 12-sidereal-hour orbits; therefore patterns of GPS satellite visibility repeat from day to day with the approximately 4-min/day sidereal advance. Correspondingly, basic patterns of measurements, such as signal strength or pseudorange errors from a given satellite, tend to repeat daily for a stationary receiver, as is commonly noted in studies of multipath [e.g., Bishop et al., 1985]. The evidence to be presented here indicates that the source of the spectrogram patterns is not multipath in most cases, however, but interference between pairs of GPS satellites. That such interference occurs for code division multiple access (CDMA) GPS signals may initially appear somewhat surprising. Nevertheless, we shall demonstrate that it is possible for the coarse/acquisition (C/A) codes broadcast on the GPS L1 frequency (1.57542 GHz) to interfere with each other to produce the observed patterns. The C/A codes are short and periodic and therefore have much more significant cross-correlation peaks as a function of delay and Doppler shift than the precise (P) codes, or their encrypted “Y” code versions [Spilker, 1996]. Ultimately, then, these GPS observations of quasiperiodic scintillation trace their source to a similar mechanism as that proposed for some classes of VHF quasiperiodic scintillation of nonionospheric origin [Hajkowicz, 1974, 1997].
 Reporting these observations and their interpretation is important on several counts. It is critical to future high-resolution GPS amplitude scintillation studies that investigators be cognizant of such effects so that they can eliminate artifacts from their data. Also, the GPS research community at large needs to be aware of subtle intersatellite interference potential with the C/A code. For example, patterns akin to spectral Vs have been observed in spectrograms of GPS radio occultation data, an effect attributed to interference between the direct signal and the signal reflected off the Earth's oceans or other surfaces [Beyerle and Hocke, 2001; Hajj et al., 2004]. We do not suggest that intersatellite interference is necessarily responsible for these radio occultation signal patterns but rather that one must generally be alert for unexpected interference possibilities during GPS data analysis.
2. Observed Characteristics
 During the past decade researchers have conducted observations worldwide using the Cornell scintillation monitor [Beach, 1998; Beach and Kintner, 2001], which is based on an L1-only Plessey (now Mitel/Zarlink) GPS chipset and development kit. The original scintillation monitor was geared toward high data rates and campaign-style studies, although later variants of the scintillation monitor were designed for remote operation and on-board statistical processing [Ledvina et al., 2000]. The base model scintillation monitor records 50-Hz samples of received power derived from the C/A code signal, with provisions to synchronize data collection between spaced receivers for cross-correlation studies of ionospheric drift [Kil et al., 2000, 2002; Kintner et al., 2004; Ledvina et al., 2004].
 Since high-rate power data at L1 were newly available with this receiver, Beach  examined its spectrograms to investigate the temporal characteristics of intensity scintillation spectra. Almost immediately, the V-shaped patterns described above appeared in the spectrograms of some of the first scintillation monitor data collected at Ancón, Peru. Once the patterns proved to be real, and not receiver anomalies, their possible connection to quasiperiodic scintillation of ionospheric origin came to the fore. To that date, there had been very few observations of quasiperiodic scintillation at or near the GPS frequencies. In one of the rare cases in the literature, Karasawa et al.  present midlatitude observations from a 1.5-GHz geostationary beacon of what they term “Type S” scintillation, which has the same appearance as “Fresnel fading” patterns characteristic of quasiperiodic scintillation in time series plots. If the prospective GPS quasiperiodic scintillation had had its origin in steep ionospheric density gradients, as hypothesized for VHF observations, one could have inferred smaller gradient scale sizes from the shorter GPS wavelength.
 The clearest way to see the Vs is to use thresholded spectrograms of GPS signal power, here represented as vertical strips of color-coded power spectra plotted against time on the horizontal axis. “Thresholded” means that the color code is binary; we plot a dark pixel if the power spectral density falls above a threshold level, as defined below, and a blank pixel otherwise. For this article we compute spectrograms using the MATLAB signal processing function with nonoverlapping, 256-point Fourier transform segments and a Hamming window. Prior to computing the spectrograms we eliminate any repeated power measurements resulting from occasional timing issues in data buffering [Beach, 1998] and linearly interpolate through these and other infrequent data gaps in the time series. The interpolation may cause vertical lines to appear in the spectrogram, depending on the amount of discontinuity introduced, but otherwise does not appear to introduce excessive high-frequency artifacts into the spectrograms.
 The spectrogram threshold is based on power spectra values at frequencies greater than 5 Hz and for times when no regular ionospheric amplitude scintillation occurs. For the purpose of threshold calculations, amplitude scintillation occurs when the root-mean-square deviation of the signal power is greater than or equal to 0.4 times the average signal power over each 256-point interval. The subset of nonscintillated power spectral densities then forms the “background” power environment for that day's pass from which we compute a mean and standard deviation of power spectral density. Finally, we establish a threshold for that day equal to the mean background power plus three (3) of its standard deviations.
Figure 1 presents examples of thresholded spectrograms from Cachoeira Paulista, Brazil (22.7°S, 45.0°W, geographic) for the period 11–17 March 2001 and the satellite with pseudorandom noise (PRN) code 27. One V is quite evident centered on 2151 UT on 11 March and whose occurrence clearly advances by about 0.07 hours (4 min) each day. There is also a more subtle, probable V-shaped pattern centered on about 2125 UT on 11 March. Additionally, there is an intermittent pattern with a parabolic outline, part of which is visible shortly after 2200 UT on 11 March, but is more clearly visible on 13 and 17 March. Finally, other, less well defined structures in the spectrograms appear to repeat with the sidereal advance.
 The evidence that the patterns in the spectrograms were real, and not anomalies of a particular receiver or a visual illusion, includes the following. We and others have commonly deployed multiple receivers to a given site with antenna spacings of 100 m–1 km for cross-correlation studies; in these cases, Vs and the recently noted parabolic patterns appeared on all receivers, albeit sometimes with varying levels of clarity. The appearance on multiple nearby receivers eliminates unique antenna patterns or anomalies peculiar to a given receiver as a source. The thresholding process shows that the Vs and other patterns have some statistical significance relative to the background power fluctuations. The pattern values showed fluctuation power spectral density of at least 10–15 dB above adjacent background points in the earliest reported cases [Beach, 1998]. Furthermore, the patterns are not artifacts of the spectrogram processing technique; they remain when we adjust the number of points, windowing and overlap for the spectrogram. Sometimes there are not only spectrogram signatures but also clearly visible Fresnel-like fluctuations in received signal power for multiple receivers (Figure 2), although the large fluctuation levels suggest that their origin is probably not intersatellite interference. Finally, we should note that Beach and Kintner  successfully tested the scintillation monitor for sensitivity to power fluctuations up to its 25-Hz Nyquist frequency with a 0.25-Hz/s linear, frequency-modulated “chirp” pattern in signal power from a GPS signal generator.
 In many observations, we can reject multipath as a likely candidate to generate these spectrogram features on the basis of considerations related to receiver spacing. The observations from Cachoeira Paulista, Brazil came from antennas spaced about 1 km apart [Kinter et al., 2004, Figure 8]. Specifically, we examine stations U, W and Z. Station Z lies about 1 km to the geomagnetic north of station U while station W is situated almost due west magnetically of station U by 0.7 km. The W–Z distance in this approximate right triangle configuration is 1.2 km. Despite the separation, the thresholded PRN 7 spectrograms from all three stations contain essentially the same patterns (Figure 3). This observation has held in all multiple receiver cases examined for this and other locations [e.g., Beach, 1998] (also H. Kil, private communication, 2000).
 The argument against multipath as a source of spectrogram patterns common to widely spaced receivers goes as follows. It is unlikely that stationary antennas 1 km or more apart will have the same local scattering environment and therefore exhibit the same time-varying multipath patterns. Even at only 100 m apart, the local scattering environments are often different; for example, one antenna may be mounted on or near a building while another is in a relatively open area. While some possibility exists to have nearly common scattering from objects at low elevation angles (e.g., mountains or large structures around the horizon), we observe the multiple-receiver spectrogram patterns at all elevation angles. In the case of Figure 3, for example, PRN 7 is nearly at its highest elevation of about 75° when the parabolic structure starts at 2242 UT on 11 March. Likewise, PRN 7 approaches 60°elevation when the V at 2148 UT occurs.
 In order to generate common spectrogram features near zenith for the outlying receivers in the Cachoeira Paulista experiment in 2001, a multipath scatterer would have to be almost overhead at all stations over a ∼1 km extent. At first, it appeared that scattering from the edge of an equatorial depletion extended along the magnetic field lines [Franke et al., 1984] might fulfill this role. Then the day-to-day repetition of most patterns came to the fore. The ionosphere will not develop an identical equatorial depletion each night and its driving timescales are solar, not sidereal. Note that the day-to-day repetition was not immediately obvious when we first examined the Vs since any particular pattern, like the parabolic structure in Figure 1 may be sporadic from one day to the next. We will discuss some possible reasons for this sporadic behavior later. In general, investigators usually choose not to site all their receiving antennas under large, fixed scattering structures, although conceivably in some cases overhead power lines might provide a common scatterer. Still, the appearance of spectrogram structures near zenith at many sites, including ones for which overhead power lines are known to be absent, argues against this mechanism as being typical.
 Another possibility was that the source of daily spectrogram patterns lay with the satellites themselves. Initially, we thought that the patterns would have to come from each satellite alone given the CDMA nature of the GPS signals, but a plausible mechanism did not arise. Closer inspection of thresholded spectrogram data for multiple satellites revealed an important clue. The spectrogram patterns generally appear on pairs of satellites simultaneously. The characteristic is perhaps most obvious with the parabolic features since they can be uniquely identified. Figure 4 presents a comparison of thresholded spectrogram structures for PRNs 2, 7, 9 and 26 on 13–14 March 2001 for station W. Table 1 lists the more notable coincident spectrogram patterns in Figure 4, of which the most evident is the parabolic structure centered on 2248 UT for PRNs 7 and 26. The simultaneous appearance of unique spectrogram patterns on pairs of satellites strongly suggests interaction between the satellite signals as their source. GPS simulator tests described in the next section confirm this hypothesis and we model the interaction in a later section.
Table 1. List of Notable Paired Spectrogram Features in Figure 4
Time at Feature
 Finally, although discussion in this paper predominantly focuses on measurements from Cachoeira Paulista, Brazil recorded during the period 11–17 March 2001 using the Cornell scintillation monitor, the observed spectrogram patterns are by no means unique to these circumstances. Figure 5 compares spectrogram patterns from two satellites for an Ashtech Z-12 receiver at Ascension Island (8.0°S, 14.4°W, geographic) in 2002. The Ashtech receiver collects 20-Hz amplitude fluctuation data for the C/A code on the L1 frequency. In this case, the spectrogram patterns also appeared on a 10-Hz version of the Cornell scintillation monitor, located approximately 0.5 km to the geographic northwest of the Ashtech receiver, and repeated from day to day with the sidereal shift. The successful cross comparison between different receiver types lends further evidence that factors external to any particular receiver govern the observations.
3. GPS Simulator Tests
 Testing the hypothesis that intersatellite C/A code interference causes the observed spectrogram patterns is straightforward with multichannel GPS simulators. In February 2006 we conducted tests on Cornell University's Spirent model GSS 7700 simulator. The test plan consisted of running the simulator for 2100–0100 UT on 13–14 through 15–16 March 2001 for the Cachoeira Paulista location. We used RINEX (Receiver INdependent EXchange format) navigation files collected during the 2001 field experiment to provide the historical ephemeris data for the GPS constellation. The RINEX files contain the higher-precision orbital elements broadcast for each satellite observed, as well as the clock models (bias, drift and drift rate) broadcast for each satellite's atomic frequency standard. The RINEX files also include that day's ionospheric model coefficients for single-frequency GPS users. The simulator does not replicate any amplitude or phase fluctuations due to ionospheric scintillation or multipath. It does, however, simulate the time variation of satellite signal power and Doppler shift due to satellite motion. For the tests we used a hemispherical pattern as an approximation to the pattern of the receiving antenna.
Figure 6 shows the results from the simulation representing 13–14 March 2001. We collected these data on 14 February 2006 using a Cornell scintillation monitor that recorded 50-Hz power samples. Notably, all of the paired patterns listed in Table 1 appear, although some of the patterns, like the parabolic structure centered on 2342 UT for PRNs 2 and 9, are more pronounced in the simulated case than in the field data. Also, Figure 6 shows many more clear spectrogram patterns than Figure 4 in general. The wealth of additional patterns visible in the simulator test is not fully understood but their appearance is probably related to increased signal levels or decreased noise levels in the simulator environment as compared to the field conditions.
 We also conducted simulator runs for successive days of March 2001. In addition to the expected 4-min/day advance of pattern occurrence, we also observed that the vertex frequencies of the parabolic structures changed from day to day as they do in the field experiment. Thus the vertex frequency change depends only on simulator parameters, including satellite ephemeris, and probably relates to a slight mismatch in orbital period between pairs of satellites. Note that careful comparison of Figures 4 and 6 also shows a minor discrepancy between the vertex frequency of the station W parabola common to PRNs 7 and 26 and the parabola appearing in the simulator. This discrepancy probably stems from the use of slightly different receiver coordinates in the simulator. A similar small discrepancy can be observed in Figure 3 in the comparison among stations at different coordinates. So, the vertex frequency of the parabolic features is sensitive to receiver position as well as satellite ephemeris information. The mechanism for influence, differential Doppler shift, will be explored in upcoming analysis.
 For additional tests, we turned off the simulator's ionospheric propagation delay model, which did not affect the spectrogram patterns at all, and the signal for PRN 26. Figure 7 shows the results of these combined variations for 15–16 March 2001 and conclusively demonstrates that intersatellite interference generates the spectrogram patterns since the patterns paired with PRN 26 disappear from the other satellites' spectrograms. Unfortunately for presentation purposes, the one remaining parabolic structure in Figure 7 does not show a large variation in vertex frequency between 13 and 15 March 2001. Noticeable vertex frequency changes are present in the full set of parabolic patterns for the 15–16 March 2001 case when all simulated satellite signals are turned on (not shown), particularly for the parabolic feature centered on 0036 UT in the 13–14 March 2001 data.
4. Interference Modeling
 Modeling the interaction of GPS signals is important for several reasons. Certainly, modeling is important to demonstrate the basic mechanism of intersatellite C/A code interference. Conditions required for interference can also be used to screen spectrogram data, especially if the repetition of a pattern is visually inconclusive: e.g., the parabola in Figure 1 if data only from 11–16 March 2001 were available. If we can check observations against intersatellite interference criteria then we may search for patterns that fall outside this model and possibly, for example, demonstrate true ionospheric edge diffraction effects at the GPS L1 frequency.
4.1. GPS Signal Interactions
 The goal of modeling in this subsection is to establish the conditions under which two GPS C/A code signals can interfere in a receiver channel to produce spectrogram patterns in postcorrelation signal power. We examine the correlation processing that occurs inside a conventional C/A code GPS receiver, where Figure 8 schematically illustrates a single channel [e.g., Braasch and Van Dierendonck, 1999; Beach and Kintner, 2001]. The signal V(t), which is usually at intermediate frequency (IF) after several steps of down conversion from the L1 band, enters the channel and is mixed with the in-phase and quadrature outputs of a local oscillator. After this final step of down conversion, a local replica of the PRN code multiplies each signal and the results are accumulated, i.e., cumulatively summed, over the code period Tc of the local code generator. Additional circuitry (not shown) resets the accumulators at the completion of the code period and provides samples of their final output for further processing. Note that most receiver designs introduce somewhat arbitrary amplitude scaling into the processing so actual results will be proportional to those listed below. Overall, a GPS receiver adjusts the frequency, and possibly phase, of the local oscillator and the code period and code phase of the local PRN generator to track a received GPS satellite's signal and ultimately to take measurements for navigation processing.
 A composite GPS signal from two satellites, with C/A PRN codes denoted A and B, can be represented as
where Ax is the received signal amplitude of PRN x in volts, Cx (±1) is the C/A code bit (“chip”), t is reception time of the signal, τx is the offset of the received code from the locally generated copy of the code, Ωx is the frequency of the received signal after down conversion and Φx is the phase of the received signal at t = 0. As a first approximation we treat all code rates (“chipping rates”) as identical and 1.023 MHz, which is the nominal chipping rate at the GPS satellites. The extremes of Doppler shifts experienced with stationary receivers violate this approximation by about 2.5 parts per million and we discuss the qualitative implications of a slight frequency mismatch later. Note that Ωx is time varying in general and we presently exclude the broadcast navigation message data bits, which modulate each signal at a 50 bit/s rate with an additional factor of ±1, from consideration. Additionally, formula (1) omits the effects of noise.
 If we temporarily remove the second satellite signal by setting AB = 0, the observables obtained after correlation from a receiver channel configured for PRN A are the in-phase (IAn) and quadrature (QAn) samples [e.g., Hinedi and Statman, 1990]. Here the A subscript denotes the local code used in the correlation channel and the n denotes the number of correlation periods elapsed. The in-phase and quadrature signals resulting from discrete time accumulation are mathematically expressed in a convenient, but arbitrary, amplitude normalization as
where N is the number of accumulation steps per correlation period Tc, Ts = Tc/N, ΔωA = ΩA − ω0 (where ω0 is the local oscillator frequency), and RAA(τ) ≤ 1 is the normalized correlation function of PRN code A with itself. The summation limits should be interpreted to indicate that there are no terms summed when n = 1, one term summed when n = 2, etc. We have adopted suitable definitions of the correlation periods so that the additional phase accumulated because of a frequency error (ΔωA) over the first correlation period is zero, unlike the definitions of Hinedi and Statman [1990, p. 173]. We have also assumed that ΔωA varies slowly so that it is essentially constant over each correlation period.
 For two satellite signals, the observables become
where RBA(τ) is the normalized correlation of code B with code A. The results (3) can readily be generalized to more than two satellites; however, satellite pairs are the fundamental unit of cross coupling in signal power. In principle, RBA(τ) is also a function of ΔωB since if ΔωB ≠ 0 the code period of the Doppler-shifted received signal B no longer matches the correlation period of the locally generated code A. Numerically, the peak C/A code cross correlations go from −23.8 dB at zero Doppler to −21.6 dB if any relative Doppler shift is allowed [Spilker, 1996]. The cross correlation also slowly varies with time because of the accumulated code phase difference between the received and local codes. Similar comments can be made about the autocorrelation function RAA(τ), but such concerns are usually obviated when the receiver channel maintains code tracking. We shall retain the simplified notation and discuss the Doppler-related effects on RBA(τ) later in a qualitative fashion.
 Now we introduce the GPS signal tracking conditions. When the channel assigned to PRN A successfully tracks satellite signal A two types of tracking are implied: code tracking and carrier or carrier phase tracking. When the channel is code tracking it has adjusted its local replica of the C/A code A to match the incoming code in phase, τA, and code chipping rate; therefore RAA(τA) → RAA(0) = 1. Likewise, when the receiver is carrier tracking, it has adjusted its local oscillator so that ω0 = ΩA ⇒ ΔωA → 0; carrier phase tracking takes the additional step of adjusting the local oscillator phase so that ΦA → 0. In the case of carrier tracking, sin(ΔωATc/2)/[N sin(ΔωATs/2)] → 1 and the (IAn, QAn) phasor defined by (2) does not rotate between samples. We incur no loss of generality by assuming carrier phase tracking or, equivalently, assuming that ΦA = 0.
 We now investigate what happens to the composite signal samples (3) when the channel is tracking PRN A. Applying code and carrier phase tracking conditions yields
where ω0 = ΩA so that ΔωB = ΩBA ≡ ΩB − ΩA. Ultimately, we are interested in postcorrelation signal power, PAn = (IAn2 + QAn2)/2 since its fluctuations are the quantity examined in the spectrograms. Samples of signal power derived from (4) become
after applying cos2 θ + sin2 θ = 1. Since RBA(τB) ≤ 10−21.6/20 ≈ 8 × 10−2 (from the −21.6 dB maximum cross-correlation figure quoted above) we neglect the second-order term in RBA and the following approximation applies
The cross-correlation term in (6) is symmetric upon interchange of the roles of B and A. Hence the fluctuating power patterns due to cross correlation would appear in two receiver channels, A and B, as in the actual observations.
Equation (6) demonstrates the basic potential to develop spectrogram patterns of interference between coded signals A and B. The differences between the present formula and the conventional interference formula for monochromatic satellite signals [e.g., Hajkowicz, 1974], which consists essentially of the cosine part of the second term of (6), are the presence of the cross-correlation factor, RBA, and the ratio of sines factor. In order for there to be observable interference, these factors need to attain sufficient amplitude. Note that AA and AB are generally of the same order of magnitude and slowly vary. The GPS satellites' antenna patterns are designed to compensate approximately for the inverse square losses due to varying satellite distance between zenith and the horizon [Spilker, 1996]. Nevertheless, a receiver's antenna pattern will have low-gain regions, particularly near the horizon, which may reduce the amplitude of one or both satellite signals. It is also worth noting that even though the magnitude of the cross correlation is relatively small, the interference term gains an additional 3 dB relative to the AA2 term, due to a factor of two brought in by squaring, and the magnitude of RBA is not squared. Overall, the power from the cross-correlation term may be only 8 dB below the power from AA2/2, worst case, if AA = AB.
 The principal maximum of the ratio of sines is at ΩBA = 0 but this peak is relatively broad and of nearly the same shape as sin(ΩBATc/2)/(ΩBATc/2) ≡ sinc(ΩBATc/2) for large N. A sinc factor would stay above 0.5 (3 dB down from its peak) when ∣ΩBA∣ ≤ 1.2/Tc ≈ 1200 rad/s, implying that the two signals would need to be within about 190 Hz of each other in Doppler-shifted frequency to interfere most strongly. The ratio of sines factor has similarly shaped maxima at all points where sin(ΩBATs/2) = 0 or, alternatively, when fBATs is an integer (where 2πfBA = ΩBA). If the accumulation step interval Ts is less than or equal to the chip period (inverse of the 1.023 MHz code chipping rate), however, the first such maximum beyond fBA = 0 would be fBA > 1 MHz, which is not achievable without radial motion of hundreds of kilometers per second.
 Maximizing RBA(τB) requires particular lags, τB, in general; the aforementioned Doppler enhancement to its amplitude is a 2-dB effect. It should be noted that near-peak absolute values for C/A code cross correlations are not rare. Spilker  states that near-peak values occur about 25% of the time for the 1023-chip C/A codes at zero Doppler. Furthermore, we have examined many C/A code cross-correlation functions at zero Doppler and observe that this magnitude of occurrence frequency holds regardless of the particular PRN pairs involved. A difference in Doppler shift between the received signals leads to other effects as well. When the received code rate B is not identical to the code rate of the locally generated code A, the two codes will drift slowly in alignment at rate proportional to ΩBA, causing τB effectively to vary in time. Therefore the requirement for near-peak cross correlation is almost always satisfied in an intermittent fashion as code B drifts in alignment with respect to code A.
 We shall not discuss the averaging of power samples that occurs in most receivers in any detail in this section except to note two items. First, under appropriate assumptions of small ∣ΩBATc∣ one can derive a formula similar to (6) for the expected value of the averaged power, where the averaging period replaces the correlation period (Tc) in the cosine factor. Secondly, it is only in the averaging process that effects appear because of the previously ignored GPS navigation message data bits—a low rate (20-ms bit period) modulation of the basic GPS signal for each satellite by an additional +1 or −1 factor. Differences in data bit transitions between the two satellite signals can potentially reduce the average absolute value of the cross-correlation term, leading to some additional intermittency.
 Overall, intersatellite C/A code interference can occur for any pair of PRN signals and is governed by the second term of equation (6). Controlling factors include the relative amplitude levels of the signals as well as the Doppler difference between the two signals. Moreover, it is fluctuations in the second term of (6) due to time variation of the Doppler difference that ultimately generate spectrogram patterns. Finally, for power fluctuations due to intersatellite interference to appear on a pair of channels, both channels must be tracking their assigned satellites. If channel A loses lock on PRN A, for example, that channel will not show the interference pattern resulting from PRN A interaction with PRN B. Events like lost lock on one channel may cause sporadic spectrogram patterns when compared between days. Whether the fluctuation power due to intersatellite interference can ever become significant enough to perturb the tracking of the primary PRN remains an open question and probably depends on the presence of additional tracking loop stress.
4.2. Application to Data
 Now we examine how the intersatellite interference results of the preceding subsection apply to the observed spectrogram patterns. The instantaneous fluctuation frequency of the cosine factor in (6) is ΩBA, which should be the dominant fluctuation frequency in postcorrelation signal power. The amplitude factors (AA and AB) and the ratio of sines are slowly varying and we will address the cross-correlation factor (RBA) later. The starting hypothesis will thus be that the spectrogram patterns reflect the instantaneous Doppler difference frequency between the two satellite signals. Furthermore, we shall include the effects of aliasing appropriate to these 50-Hz power samples since there is clearly some “wraparound” in frequency mapping in the spectrogram patterns.
Figure 9 shows the aliased Doppler frequency difference for PRNs 10 and 27 throughout the GPS week 1105; this scenario corresponds to the partially formed, sporadically appearing parabola in Figure 1. We used historical GPS almanac data (modified Keplerian elements broadcast by the satellites used by receivers for initial Doppler estimation and available at www.navcen.uscg.gov/gps/almanacs.htm) to calculate the Doppler shifts, rather than more accurate historical GPS ephemeris data, to investigate the sensitivity of the recreated spectrogram patterns to orbital precision. Qualitatively, the central parabolic portion of the Doppler difference calculation corresponds well to the intermittent parabolic shape in the PRN 27 spectrograms. The position of the parabola's vertex in the aliased Doppler frequency difference advances by ∼4 min/day and the vertex frequency also decreases with time. Quantitatively, the vertex frequency in the almanac-based calculation does not decrease as much by 17 March as that of the corresponding spectrogram feature in Figure 1. The almanac-based calculation yields 5.4 Hz for the vertex frequency on 17 March 2001, whereas Figure 1 gives 2.9 Hz. Nevertheless, the general match in characteristics is striking.
Figure 9 also demonstrates additional features of the aliased Doppler difference model that bear a useful relationship to the observed spectrogram patterns. First, aliased Doppler differences can generate both parabolic and V-shaped structures. This observation has held true for all satellite pairs examined, indicating that higher-order variations in frequency with respect to time are typically small. Note that the V-shaped structure to the left of the parabola in Figure 1 represents the combined result of PRN 27's interaction with PRNs 11 and 26, according to the more detailed visibility criterion described below, rather than aliasing of difference frequencies with PRN 10 that are significantly beyond 25 Hz. Secondly, the model already partially accounts for when the differential Doppler shift becomes emphasized in the spectrogram patterns. In this case, PRNs 10 and 27 experience nearly identical Doppler shifts as observed from Cachoeira Paulista during the time of the parabolic shape so that their Doppler difference is close to zero. Fluctuations at a much higher rate than the Nyquist frequency of 25 Hz tend to become filtered out because of the 20-ms averaging used to obtain 50-Hz power samples from 1-kHz correlator output samples. The fact that the parabolic structure occurs for nearly zero Doppler difference probably also has relevance to the spectrogram pattern appearing as a dashed line, as will be discussed shortly.
 Very quickly, however, it became clear that near-zero Doppler difference does not account for all the observed cases of spectrogram patterns. For example, the parabolic pattern common to PRNs 7 and 26, centered on 2248 UT in Figures 4 and 6, occurs when their Doppler difference, fBA, is nearly 2000 Hz (Figure 10a). The correct analysis of this case lies in the fact that there are two sampling stages involved. Initially, the power samples emerge from the correlation process with 1000-Hz sampling. Fluctuations in the cosine factor of (6) with a frequency greater than the 500-Hz Nyquist frequency of this primary sampling stage become aliased. Note that the precorrelation bandwidth of the IF signal, V(t), must be several thousand Hz to accommodate the normal range of Doppler frequencies expected for all satellites in view from a stationary receiver. Next, the processing in the Cornell scintillation monitor averages 20 of these 1-ms samples together to form 50-Hz power samples. If the aliasing in the prior sampling process brings the power fluctuations into approximately the range of ∣fBA∣ < 25 Hz, then those fluctuations will pass through the averaging operation. In practice, it appears that fluctuations of ∣fBA∣ < 37.5 Hz (75% of the secondary, 50-Hz sampling frequency) or more may pass through the low-pass filtering achieved by averaging over 20-ms periods.
Figure 10b shows net results of the two-stage sampling process in a form that will be explained in this paragraph. Figure 10b plots the Doppler difference frequency between PRNs 7 and 26 as a function of time, directly aliased to the range 0 to 25 Hz. The upper limit of 25 Hz corresponds to the Nyquist frequency of the 50-Hz secondary sampling. We highlight with solid lines the portions of the aliased fBA curve where aliasing from the 1000-Hz primary sampling brings the fluctuations into the range of ∣fBA∣ < 37.5 Hz, as described above. Essentially, the aliasing condition for the primary sampling is met when ∣fBA∣ lies near an integer multiple of 1000 Hz. The highlighted portions of the aliased fBA curve should roughly correspond to visible patterns in the spectrogram data of Figures 4 or 6.
 The aliasing analysis predicts a parabolic pattern centered on 2248 UT that is well verified in the spectrogram data. Interestingly, the two-stage aliasing analysis also indicates that narrow V shapes should be visible at about 2124 and 0015 UT. While the spectrogram of field data in Figure 4 is somewhat unclear regarding the existence of a possible pattern at 2124 UT, a nearly vertical pattern at 0015 UT appears on both PRNs. In the simulator (Figure 6) the spectrogram plots show many more spectrogram features than in the field experiment, perhaps because of a “cleaner” radio signal environment. It is at least plausible from Figure 6 that the predicted 2124 and 0015 UT structures faintly appear on both PRNs 7 and 26, although they become merged with other structures. One should also keep in mind that Vs with steep slopes and durations shorter than the Fourier transform segments making up the spectrogram tend to become lost.
 In addition to aliasing and filtering associated with the two-stage sampling process, other factors can govern the prominence of particular spectrogram patterns. The amplitude terms, AA and AB, depend on the distances from the satellites to the receiving antenna and both antenna patterns. The ratio of sines, sin(ΩBATc/2)/[N sin(ΩBATs/2)], depends on the actual (unaliased) Doppler difference between the two satellite signals. Figure 11 illustrates these factors for all satellites that interact with PRN 07 in the 13–14 March 2001 scenario of Figures 4 and 6 according to the two-stage aliasing considerations. Here we use more accurate satellite ephemeris data, interpolated from the SP3-format precise orbit files available at www.ngs.noaa.gov/GPS/GPS.html, in the Doppler and distance calculations. We adapt the GPS satellite L1 antenna pattern of Spilker [1996, p. 86] for the transmitting antenna; our modeled, azimuthally symmetric receiving antenna pattern is based on the data sheet for the M/A-COM model ANP-C-114-5 antenna that typically accompanies a GPS scintillation monitor in the field. We have normalized the amplitude factor AAAB so that it is unity if both satellites are at zenith and 20,200 km overhead (typical GPS orbital height above the Earth's surface). In general, as Figure 11 shows, many potential intersatellite interactions appear and the threshold conditions that lead to the prominence of any particular pattern are not straightforward. Further investigation will be required.
 What about the effect of the code cross-correlation factor, RBA(τB)? Its influence depends on the rate of C/A code relative drift or “slippage” due to the difference in Doppler-shifted frequencies between the satellite signals. When the actual Doppler difference is near zero there is relatively little code slippage over the period of observation. Consequently, if the code starts out far from a peak in cross correlation it may remain there for a long time and temporarily suppress the appearance of the fluctuation pattern. The dashed appearance of the sporadic parabolic pattern in Figure 1 supports this interpretation since the Doppler difference between PRNs 10 and 27 is indeed close to zero.
 In more quantitative terms, suppose that the average Doppler difference is 5 Hz between satellites over an interval. The average difference in the received code chipping rate between the two PRNs is 5 Hz/1540 = 3.2 mHz, where 1540 is the ratio of the GPS L1 frequency to the 1.023 MHz C/A code chipping rate [Spilker, 1996]; correspondingly, the time required for the alignment of the two PRN codes to slip by one full code period relative to each other is 0.09 hours. Therefore the cross-correlation factor RBA will pass through its discrete peaks in slow succession leading to a spectrogram pattern with a dashed appearance. A dashed appearance is perhaps most clearly evident in the PRN 27 spectrogram on 17 March 2001 near the vertex of the parabola, an interval in which the average Doppler difference is less than 2 Hz. It is presently uncertain whether the near-zero Doppler difference contributes to the sporadic nature of the parabolic pattern between days. Further investigation would be required to examine the detailed variation of code phase, and other factors thus far excluded, as a function of time with respect to the cross-correlation maxima.
 On the other hand, if the Doppler difference between the satellite signals lies near 1000 Hz, the next “window of visibility” in the two-stage sampling process, the difference in code chipping rates becomes 0.65 Hz. In this case, the time required for the accumulated code phase difference to equal one code period is only 1.5 s. Note that the duration of each individual vertical Fourier transform strip in the spectrograms is 5.1 s. Thus the cross-correlation function reaches its peak several times during the Fourier transform interval, keeping the average fluctuation power at the aliased Doppler difference frequency fairly strong. It does not appear that it would be possible to resolve the discrete cross-correlation peaks for the 1000-Hz or higher windows of visibility with any reasonable tradeoff between frequency resolution and time resolution in the spectrograms. Overall, we conclude that fluctuations due to an effectively time varying τB in the cross-correlation factor RBA(τB) of equation (6) are so rapid as not to be manifest in spectrograms, except in the occasional case where the unaliased GPS Doppler difference lies near zero.
 Finally, there remains the question how strong the fluctuations in power due to intersatellite interference can become. We have experimented with quantizing the 50-Hz GPS power measurements used in Figure 4 to 1-dB and 0.1-dB steps (although the power data originally come from the scintillation monitor with linear scaling). The purpose of this investigation was to determine how significantly the decibel quantization in reported power levels present in many commercial receivers degrades the ability to detect spectrogram patterns from intersatellite interference. At 0.1-dB quantization, most of the Figure 4 spectrogram patterns appeared; at 1-dB quantization, nearly all of the patterns vanished. We can also estimate maximum fluctuation levels under the following assumptions: (1) AA ≈ AB, (2) sin(ΩBATc/2)/[N sin(ΩBATs/2)] ≈ 1 and (3) RBA(τB) ≈ 8 × 10−2, its peak value for any Doppler shift. Subject to these assumptions, the peak-to-peak variation of signal power is about 1.4 dB. One may imagine relative differences due to antenna pattern effects of, say, factors of 2–4 between AA and AB that could potentially lead to an increase or decrease in peak-to-peak fluctuation amplitude.
 With this estimated bound on intersatellite interference fluctuation levels in mind, we can examine the case of large amplitude fluctuations shown in Figure 2. Coincidentally, the two-stage sampling criterion predicts a pair of nearly overlapping satellite interactions with PRN 2, two Vs whose low-frequency portions are approximately centered on the Fresnel-type fluctuations. These Vs represent interactions with PRN 8 at near-zero Doppler difference and PRN 1 with approximately 3000-Hz difference. Nevertheless, as pointed out above, the amplitude fluctuations for worst-case conditions are only 1.4 dB peak to peak. Two simultaneous cases of the strongest possible intersatellite interference with fortuitous phasing would approximately double this peak-to-peak fluctuation level (according to a small-argument logarithm expansion). We conclude that the 15-dB peak-to-peak fluctuations are probably not due to intersatellite interference. Other evidence to support this position is the inability to reproduce the pattern with large fluctuation amplitude in a GPS simulator. Tests in this latter case were conducted at the Air Force Research Laboratory's Sensors Directorate in April 2006 using their standalone Spirent 2760 multichannel GPS simulator and an Ashtech ZXtreme GPS receiver, with 10-Hz power samples and 1-dB quantization, to record data (T. Malicki, private communication, 2006).
 We caution that analysis like the above does not guarantee an ionospheric origin for Fresnel-type power fluctuations. For example, time-varying interference of direct and scattered signals from a single PRN, i.e., multipath with relative motion, can lead to nearly linear sweeps in fluctuation frequency with time [Beach, 1998; Beyerle and Hocke, 2001]. It is important to consider several factors like repeatability from one day to the next with the sidereal advance for stationary receivers. Note that transient multipath conditions, such as aircraft passing near the GPS line of sight, may be difficult to identify. Consequently, one must rely on other characteristics to help form a determination: the presence or absence of total electron content (TEC) fluctuations, the basic relationship of the slope of spectral Vs to the distance and relative motion of the scatterer [Beach, 1998], etc.
 By introducing an analysis technique, the spectrogram and its thresholded variant, to the study of high-rate GPS power fluctuations we have observed patterns of intersatellite interference between C/A codes as a function of time-varying differential Doppler shift. These patterns appear on both associated channels if they maintain GPS signal tracking. Phenomenology, GPS simulator tests, modeling and analysis all lead to this solid conclusion. Furthermore, since multipath and ionospheric scintillation can also generate patterns of power fluctuation, it is important to be able to distinguish among the various mechanisms. We provide here the fundamental tools to help determine whether intersatellite interference is the source of observed fluctuations in received signal power. Unfortunately, it is beyond the scope of this article to provide detailed guidance on identifying multipath versus edge-type diffraction from a steep ionospheric density gradient in power fluctuations.
 The detailed analysis also indicates why this particular intersatellite interference phenomenon appears not to have been reported earlier. Most GPS receivers provide only coarse measures of signal power (e.g., quantized in 1-dB steps) at low data rates. It is only with the advent of GPS scintillation monitors suitable for cross-correlation studies of irregularity evolution and drift that researchers began examining rapidly sampled power data. Although much of the analysis here has focused on the Cornell scintillation monitor, the phenomenology is not unique, as demonstrated by the Ashtech Z-12 example (Figure 5). Indeed, such conditions as aliasing from multiples of the 1000-Hz code repetition frequency should be nearly universal in C/A code correlation-type receivers. The phenomenon may also appear with other Global Navigation Satellite System (GNSS) signals with relatively short codes, although one does not expect significant cross correlation between the longer GPS P or Y codes, for example.
 In addition to the original intent of distinguishing background GPS power fluctuations from ones of ionospheric origin, further engineering applications may arise from the intersatellite interference phenomenology. Sensitivity to the interference patterns may be a useful benchmark of receiver performance in some cases. As the comparison of less precise almanac-derived Doppler shifts with field experiment data demonstrates, the spectrogram patterns show some sensitivity to the accuracy of GPS orbit determination, particularly in the vertex frequencies of the parabolic shapes. New interferometric techniques to fit orbital parameters to observed spectrogram patterns may become possible, although the details of this hypothetical method are unclear at present. Spectrograms of high-rate postcorrelation power data may also provide some benefit to assessing C/A code interference and the multipath environment. In any spectrogram-based study, as in the original application to studying ionospheric scintillation, it will be important to recognize the signature of intersatellite interference.
 E. R. de Paula provided the GPS data from selected stations at Cachoeira Paulista, Brazil for March 2001. A. Cerruti and S. Powell supported the GPS simulator tests at Cornell University. T. Malicki performed additional GPS simulator experiments at AFRL. We thank H. Kil, P. M. Kintner, K. M. Groves, and M. J. Starks for many valuable suggestions and discussions. This work was partially supported by AFOSR Task 2311AS and has made use of the NASA Astrophysics Data System (ADS).