Local Oscillator Phase Noise Effects on GNSS Code Tracking © iStockphoto.com/Eugene Kazimiarovich iStockphoto.com/Eugene © A new perspective for GNSS designers quantifies the performance loss due to phase noise effects on the baseband signal. These performance bounds may then be used as the basis for local oscillator design. The authors develop bounds by identifying front- end local oscillator phase noise effects on the correlation loss, while tracking receiver performance. Until now, this effect has been rarely documented in research literature. Extensive simulations are used to validate results, while drawing conclusions regarding the relationship between phase noise, correlation time, and loss in the carrier-to-noise ratio. ERNEstO PÉREZ SERNA, correlation for a given delay is achieved. Furthermore, longer integration ACORDE TECHNOLOGIES S.A This provides an indirect measurement intervals ideally result in higher sensi- SARANG THOMBRE, MIKKO VALKAMA, SIMONA of the satellite-receiver range. tivity. However, because phase noise is LOHAN, VILLE SYRJÄLÄ One of the performance limiting fac- translated into a rotation of the constel- TAMPERE UNIVERSITY OF TECHNOLOGY tors of GNSS receivers is the imperfec- lation diagram of a modulated signal MARCO DETRATTI tion of the radio frequency (RF) oscil- that can make integration (correlation) ACORDE TECHNOLOGIES S.A lator. This imperfection translates into comparatively less effective as the inter- random deviations of instantaneous val increases. HEIKKI HURSKAINEN, JARI NURMI phase or frequency, typically modeled as Phase noise models have been pro- TAMPERE UNIVERSITY OF TECHNOLOGY a phase imperfection, and often referred posed for various wireless communica- to as phase noise. tion receivers. (See the sidebar, “Mod- NSS systems rely on direct The receiver oscillator phase noise eling Phase Noise Effects on Receivers: sequence spread spectrum narrows the carrier tracking loop band- A History” on page 54.) However, the (DSSS) transmissions to achieve width, while diminishing the achievable effects of phase noise on GNSS receiv- high receiver sensitivity. Typi- carrier-to-noise ratio (C/N ). The corre- ers’ performance have been rarely doc- G 0 cally, GNSS user equipment compares lation outputs in the code-tracking loop umented, leaving key design questions the signal received from a satellite with are also affected, creating correlation unanswered: What is the maximum an internally generated replica of its cor- noise and losses at the receiver that are acceptable phase noise level as required responding code until the maximum measured as reductions in C/N0. by an RF designer in order to achieve a 52 InsideGNSS NOVEMBER/DECEMBER 2010 www.insidegnss.com I I I I ADC Antenna Mixer LPF BPF LNA Carrier Tracking Wipe-off Loops Algorithms Navigation Correlators Accumulators Discriminators LPF Q Q Mixer Q Q ADC Reference carrier Replica Code Carrier Code Delay Correction NCO NCO Frequency and Phase Correction Direct Conversion Radio Front-End Digital Baseband Processing FIGURE 1 Block-diagram of a GNSS direct-conversion receiver minimum pre-defined C/N0? How do correlation losses relate to considered for this study. Let us first consider the operation of phase noise levels? the local oscillator (LO) in a receiver. In this article, we propose an analytical approach using a Any oscillator is defined by three parameters: phase φ( ), given phase noise model, validating it through simulations to amplitude (A), and frequency (f0). In a general case phase and quantify the effect of oscillator noise on the performance of amplitude noise exist, as well as distortion, which makes both GNSS receivers. From this we provide a first estimation of the A and φ functions of time, such that requirements of a radio front-end for a given baseband imple- (1) mentation, as well as an insight into the relationship between correlation time and performance degradation due to phase The nonlinear dynamics of an oscillator make the ampli- noise. The following section of our discussion provides a theo- tude A(t) relatively constant, so it becomes considerably less retical analysis, in both time-domain and frequency-domain, important than phase noise. Furthermore, if the modulation of the effect of this phase noise on various properties of code used to carry information does not affect the amplitude, the correlation. receiver can also ignore this component. Phase noise, however, We then perform numerical simulations of GPS L1 pseu- cannot be overlooked, and it actually becomes the defining dorandom noise (PRN) code correlation for various values of parameter of the quality of any oscillator. phase noise in order to validate the theoretical model. Next, Phase noise specifications are typically given in decibels a datastream simulation using a Galileo E1 receiver, complete relative to the carrier per hertz (dBc/Hz) at a certain frequency with carrier and code tracking loops, quantifies the effect of offset, fm, from the desired center frequency of the oscillator. oscillator noise on GNSS receiver performance. Here, the Phase noise Φ(t) for a free running oscillator is a non-station- simulated PRN code correlation in the presence of phase noise ary random variable, and is a stationary random variable demonstrates that the model (and GPS results) may be applied whose power spectral density (PSD), Px(f), is given in Equa- to Galileo signal receivers as well. tion 2. Finally, we compare results from all the simulations and recommend a practical limit for the maximum phase noise permissible from the front-end oscillator in order to maintain (2) the post-correlation signal-to-noise ratio (SNR) of a correlation peak beneath a given threshold. Our results provide a first estimate of the noise floor require- where is the unit time variance of the random phase noise ments for a receiver given a particular baseband implemen- process, so that total variance is measured by (t). In terms of tation. Also, this study provides insight into the relationship PSD, the phase noise is given by: between correlation time and performance degradation due L(f ) = 10log (P (f )) to phase noise. m 10 x m where L(fm) is the phase noise in units of dBc/Hz, and Px(fm) is Phase Noise Model given by simplifying Equation 2 as shown in Equation 3: Figure 1 shows the block diagram of a typical GNSS receiver www.insidegnss.com NOVEMBER/DECEMBER 2010 InsideGNSS 53 PHASE nOISE EFFECTS (3) ences in the results when other phase noise models are considered, this can still be considered a first approach to the problem. and further simplifying phase noise to: Phase Noise and Code Correlation (4) Let us turn now to the factors involved in the effects of phase noise on code correction. Post-Correlation Signal to Noise Ratio. where f0 is the oscillation frequency, fm is the frequency off- Letting x(t) be an ideal set, and σΦ is the constant that defines the noise level of the real signal, having no quadrature component, we initially jω0t oscillator. In practice all free-running oscillators show a noise modulate a carrier at ω0 as x(t)·e . Complex down-conver- roll-off of 20 decibels per decade when far enough from the sion, assuming coherent detection, is described by (6): fundamental frequency. (6) The defining assumption of this model is that the phase difference between time instants t1 and t2 follows a Gaussian where the down-converted signal is then correlated with an distribution with null mean and variance linearly increasing aligned replica over a period T. If the code is optimally aligned, with the length of the interval, also known as Wiener or ran- Equation 7 results: dom-walk processes, thus (7) (5) This model is also suitable for PLL-based synthesizers with narrow loop bandwidths where the voltage controlled oscillator Making several simplifications in the notation, we first (VCO) noise dominates. Even though there are significant differ- normalize the energy of x(t) over T (indicated as |Y|) to 1. Secondly, we assume that x(t) equals {-1,1}, which is true for simple modulations such as binary phase shift keying (BPSK) Modeling Phase Noise Effects on Receivers: A History or binary offset carrier (BOC(m,n)). Finally, we only consider In 1966 D.B. Leeson proposed an empirical linear model for the noise spec- pilot signals, excluding the factor of navigation data. Therefore, trum of an oscillator, which has been extensively cited in the literature since in the absence of phase noise, then. G. Sauvage generalized this model to other resonant circuits in 1977, (8) providing a deeper mathematical background. In 1998 A. Hajimiri and T.H. Lee proposed a linear time variant model (LTV) to explain the effect of each Let us now consider the effect of the phase noise of the local of the noise sources of an oscillator on its phase noise. oscillator where the carrier is down-converted with an addi- Phase noise models for various wireless communication receivers have tional random phase φ(t), thus: been proposed by T. Schenk (2006) (For the complete citations of this and other references, please see the Additional Resources section near the end (9) of the main article.) The article by D. Petrovic et alia uses orthogonal fre- quency division multiplexing (OFDM) systems for modeling noise, while A. Demir et alia (2000) developed extensive and generic theory for phase noise models. In a 2002 article, Demir extended this theory to jitter in opti- (10) cal and wireless communications, and K. Kundert also developed a jitter measurement for phase noise effects on phase locked loops. During the correlation process in a GNSS receiver, this A simulation-based study of phase noise introduced by the receiver and signal is multiplied by an ideal version of itself. Theoretically, the satellite payload was discussed by E. Rebeyrol et alia in 2006. Deriving when both are perfectly aligned the integral or area of the the phase noise power spectral densities, frequency comparisons are made, resulting function is maximized.