WO2020165211A1 - Method and device for adaptive in-orbit testing of a satellite transponder - Google Patents

Abstract

A new measurement procedure for satellite transponder parameters such as the amplitude response, the group delay, and the local oscillator stability of a wideband transponder in an in-orbit testing (IOT) context is proposed. The proposed procedure resorts to a direct sequence spread spectrum (DSSS) technique in order to reduce the interference caused by the measurement signals to the traffic to a negligible level. Further, an adaptive approach consisting in knowing the traffic power falling in the band of the measurement signals is proposed. Such knowledge, acquired either a priori by knowing the frequency plan of the transponder or by sensing the channel, allows to easily finetune the design parameters of the measurement signals in order to satisfy multiple performance requirements (i.e., accuracy, resolution, level of interference, and measurement duration) and limit at the same time the hardware complexity.

Description

METHOD AND DEVICE FOR ADAPTIVE IN-ORBIT TESTING OF A SATELLITE

TRANSPONDER Technical field

The invention lies in the field of satellite communication systems, and relates in particular to in-orbit testing of wide-band transponders of communication satellites. Background of the invention

Performing in-orbit testing, IOT, of a satellite payload is a fundamental operation that is required in several phases of the satellite’s life-time. In particular, during“begin of life” the satellite undergoes several tests in order to assess whether any damage occurred during the launch. Indeed, the launch remains the most critical part of a mission, in which the satellite undergoes strong mechanical vibrations that could damage the on-board hardware. Once the satellite reaches its final orbital slot and is activated for operation, it still requires regular test sessions to identify possible degradations caused by the aging of the on-board hardware. Monitoring the status of the satellite payload is hence necessary to guarantee the final end-customers’ quality of service. Known IOT methods require the interruption of the main customer signal (the payload signal) with consequences for both the operator and the customer, as outlined by Standing A.F. Measurement Techniques for In-Orbit Testing of Satellites. W. H. Freeman, Jan 1990. Disrupting the customer service for testing purposes, while normally being required, is a particularly costly and time consuming operation. Therefore, particular effort is normally put in the test planning, and in order to optimize the execution time and the regularity of the test. Further, in all the cases in which the satellite is moving toward a pre-determined orbital slot (such as during the initial“drift” phase or during relocation), the satellite cannot be normally tested until it reaches its final position due to the presence of nearby satellites, whose customer signal could be interfered by the testing signal. Finally, one of the most difficult issues to be resolved as part of a test campaign planning is the coordination of the test frequencies in presence of signal in the same or adjacent frequency bands used by own or other satellite operators’ payloads. The main known technology allowing for non-intrusive IOT systems is spread spectrum, SS, as outlined by Kleine-Ostmann et al. in“A new spread-spectrum measurement method for satellite transponder in-orbit testing,” Microwave J., vol.44, pp.66–70, Aug 2001. Spread Spectrum is based on the idea of using a testing signal whose power spectral density, PSD, is negligible with respect to the customer traffic signal, resulting in negligible interference, see Dixon, Spread Spectrum Systems with commercial applications. Wiley-Interscience, 3rd ed., Apr 1994. In general, the SS signal is generated starting from a standard narrow-band signal that is spread over a large bandwidth using one of the following two techniques:

- direct sequence, DS: the stream of information symbol is modulated by a pseudo-noise, PN, sequence of chips which have a much shorter duration than an information symbol;

- frequency hopping, FH: the carrier frequency changes in time according to a pseudo-random sequence. In both DS and FH methods, the pseudo-random sequence has to be known at the receiver in order to de-spread the signal. Since the main objective in the context of IOT measurements is to hide the measurement signal under the noise level in order to avoid interfering with commercial traffic or adjacent satellites, the DSSS scheme appears to be a more appropriate solution because it can decrease the power flux by spreading the measurement signal over a wide bandwidth. On the other hand, FH would create high peaks of interference which would hop from one frequency to another, deteriorating the quality of service perceived by the customers. Existing SS measurement methods include power spectrum estimation, PSE, delay locked loop, DLL, carrier suppression, CS, and small signal suppression, SSS. PSE is mainly used to measure the frequency response characteristic of a satellite transponder. The transponder is considered as a filter whose frequency response can be estimated as the ratio of the cross-correlation between the input and the output signals of the transponder, over the autocorrelation of the input signal, as developed in Stoica et al. Spectral Analysis of Signals. Prentice-Hall, 2005. Subsequently, amplitude and phase responses can be straightforwardly calculated from the frequency response, while the group delay is derived by differentiating the phase response. It should be noted that, in this method, the transmitted test signal must occupy the whole transponder bandwidth. On one hand, PSE is fast since it measures the whole bandwidth at once, but it is not very accurate because it has to adapt the power of the measurement signal to the traffic carrier with the worst signal-to-noise ratio (where“signal” here stands for the customer’s traffic). Also, in the case of wideband transponders, both the transmitter and the receiver units have to resort to a very high clock rate in order to generate and recover the measurement signal, and this requires very expensive hardware and a complicated design. As PSE, also DLL is used to measure the frequency response of the satellite transponder. The operation of the DLL method is based on the sweeping principle commonly used by commercial spectrum analyzers: a sounding signal sweeps the whole bandwidth under test. A SS signal centered at the sounding frequency is compared to a reference signal in order to determine amplitude response and group delay of the transponder. Indeed, in an IOT context only relative measurements of amplitude response and group delay are relevant. However, the measurement campaign takes longer compared to the PSE because of the sweeping process, which requires the iteration of the whole measurement procedure at each frequency under measurement. This may become a relevant issue especially for wideband transponders. In order to measure the amplitude response and the group delay of the transponder, CS requires detection, decoding, re-encoding, re-modulation, and subtraction of the traffic signal from the received signal, as outlined by Schwarzenbarth,“Carrier suppression for spread spectrum payload tests,” Master’s thesis, Fachhochschule Trier, Germany, Mar 2008. The traffic cancellation allows an important improvement of the measurement quality, but it also requires adding extra complexity and functionalities at the receiver terminal. Unfortunately, the measurements of wideband transponders should be performed at high speed in order to satisfy duration requirements, which makes the hardware implementation even more challenging and mostly impractical. SSS is used to measure the operating point of the satellite transponder (J. R. McCarthy, Spread Spectrum Communications over Nonlinear Satellite Channels. PhD thesis, University of South Australia, Feb 1999) and is based on the fact that, by injecting a small test signal in a loaded transponder, the main signal is not affected.“Small signal” typically refers to a 15 dB power difference compared to the main signal. The measurement of the operating point is based on the suppression of the test signal power. It should be noted that a calibration is needed beforehand in order to associate the SSS measurements with operating point values and generate the reference curves. Technical problem to be solved

It is an objective to present a method and device, which overcome at least some of the disadvantages of the prior art. Summary of the invention

In accordance with a first aspect of the invention, a method for in-orbit testing of a frequency band of a satellite transponder is proposed. The method comprises the steps of

a) at a ground station, using data transmission means, transmitting a set of input measurement signals using direct sequence spread spectrum modulation to the satellite on said transponder, the set of input measurement signals comprising a reference signal, which is centered on the central frequency of said frequency band, and at least one testing signal, which is centered on a first frequency of said frequency band , which is different to said central frequency ;

b) at the ground station, using data reception means, subsequently receiving a corresponding set of output measurement signals from the satellite;

c) at the ground station, using measuring means, measuring parameters that characterize the transponder at the frequencies on which said input measurement signals are centered, the parameters comprising the group delay and/or the amplitude response, wherein the measurement is based on the relationship between the set of input measurement signals and the set of corresponding output measurement signals ;

d) repeating steps a) to c) at least once by changing the frequency on which said at least one testing signal is centered, to a second frequency, which is different to said first frequency. The transmit power of each input measurement signal depends on the transmit power of at least one data traffic signal that concurrently occupies at least part of a frequency sub-band that comprises the frequency on which said input measurement signal is centered. Preferably, said input measurement signals comprise a plurality of testing signals, each being centered on a distinct frequency of said frequency band. Said second frequency of each testing signal may preferably be selected by the ground station to differ from the first frequency of said testing signal by a pre-provided offset value. Preferably, said second frequency may be selected by the ground station to lie in a frequency sub-band that is occupied by a data traffic signal having low transmit power. The transmit power of each testing signal may preferably be selected to be lower than the transmit power of a data traffic signal that concurrently occupies at least part of a frequency sub-band that comprises the frequency of which said input measurement is centered. The transmit power of each testing signal may preferably be lower than the transmit power of the corresponding data traffic signal by a predetermined threshold. Preferably, the measured parameters that characterize the transponder at the frequencies on which said testing signals are centered may be normalized with respect to the measured parameters that characterize the transponder at the central frequency, on which the reference signal is centered. The resulting measurements may preferably be stored in a memory element. Preferably, said frequency band may comprise the entire transponder bandwidth. The transponder may preferably be a wide-band transponder. Preferably, the transmit power of said data traffic signal may be provided at the ground station. The transmit power of said data traffic signal may preferably be estimated by said ground station. Preferably, for the direct-sequence spread spectrum modulation of each input measurement signal, a distinct pseudo-noise sequence may be generated at the ground station. In accordance with a further aspect of the invention, there is a device for testing of a frequency band of a satellite transponder is proposed. The device comprises data transmission means, data reception means and measuring means, the device being configured for:

a) using the data transmission means, transmitting a set of input measurement signals using direct sequence spread spectrum modulation to the satellite on said transponder, the set of input measurement signals comprising a reference signal, which is centered on the central frequency of said frequency band, and at least one testing signal, which is centered on a first frequency of said frequency band , which is different to said central frequency;

b) using the data reception means, subsequently receiving a corresponding set of output

measurement signals from the satellite;

c) using the measuring means, measuring parameters that characterize the transponder at the frequencies on which said input measurement signals are centered, the parameters comprising the group delay and/or the amplitude response, wherein the measurement is based on the relationship between the set of input measurement signals and the set of corresponding output measurement signals;

d) repeating steps a) to c) at least once by changing the frequency on which said at least one testing signal is centered, to a second frequency, which is different to said first frequency; wherein the transmit power of each input measurement signal depends on the transmit power of at least one data traffic signal that concurrently occupies at least part of a frequency sub-band that comprises the frequency on which said input measurement signal is centered. The data transmission and data reception means may preferably comprise a hardware interface for transmitting data symbols on the satellite communication link. The data transmission may preferably further comprise data and/or signal processing components, such as a data processing unit, for modulating and coding data symbols prior to transmission on said satellite communication link. Preferably the data processing unit may be programmed by computer software code to provide the desired functionality. Preferably, the measuring means may comprise a data processing unit programmed by computer software code to provide the desired functionality. The device may preferably further comprise pseudo-random number generation means, configured to generate distinct pseudo-noise sequences. The transmission means are configured to use direct- sequence spread spectrum modulation using one of said generated sequences for the transmission of one of said input measurement signals. The pseudo-random number generation means may preferably comprise a pseudo-random number generation unit, preferably functionally coupled to at least one memory element. The memory element may comprise a Random Access Memory, RAM, module or a Hard Disk Drive, HDD, Solid State Drive, SDD, or any other known volatile or persistent memory module. Preferably, the device may further be configured for carrying out any of the method steps in accordance with aspects of the invention. In accordance with yet another aspect of the invention, a computer program is proposed. The computer program comprises computer readable code means, which when run on a computer, cause the computer to carry out the method according to aspects of the invention. In accordance with a final aspect of the invention, a computer program product is provided. The computer program product comprises a computer-readable medium on which the computer program according aspects of the invention is stored. In accordance with aspects of the invention, a new measurement procedure for satellite transponder parameters such as the amplitude response, the group delay, and the local oscillator stability of a wideband transponder in an in-orbit testing, IOT, context is proposed. The proposed procedure resorts to a direct sequence spread spectrum, DSSS, technique in order to reduce the interference caused by the measurement signals to the traffic to a negligible level. Further, an adaptive approach consisting in knowing the traffic power falling in the band of the measurement signals is proposed. Such knowledge, acquired either a priori by knowing the frequency plan of the transponder or by sensing the channel, allows to easily finetune the design parameters of the measurement signals in order to satisfy multiple performance requirements (i.e., accuracy, resolution, level of interference, and measurement duration) and limit at the same time the hardware complexity. Brief description of the drawings

Several embodiments of the present invention are illustrated by way of figures, which do not limit the scope of the invention, wherein:

- figure 1 illustrates the main steps of a preferred embodiment of the method in accordance with the invention;

- figure 2 illustrates wideband satellite transponder and a set of input measurement signals as provided by preferred embodiment of the method in accordance with the invention; - figure 3 illustrates a system for implementing a preferred embodiment of the method in accordance with the invention, including a ground segment and a satellite;

- figure 4 provides a block scheme for modelling the system in which a preferred embodiment of the method in accordance with the invention is used;

- figure 5 provides a block scheme for a transmitter of input measurement signals as provided by a preferred embodiment of the invention;

- figure 6 provides a block scheme for the transponder channel as provided by a preferred

embodiment of the invention;

- figure 7 provides a block scheme for a receiver of the output reference measurement signal as provided by a preferred embodiment of the invention;

- figure 8 illustrates the amplitude of a transponder frequency response, and of the received power spectral densities for a traffic signal and for input measurement signals as provided by a preferred embodiment of the method in accordance with the invention;

- figure 9 illustrates the performance of amplitude estimators for measuring transponder

parameters;

- figure 10 illustrates the performance of a group delay estimator;

- figure 11 illustrates the power spectral densities of a traffic signal together with measured amplitude response and group delay, as provided by a preferred embodiment of the method in accordance with the invention. Detailed description

This section describes aspects of the invention in further detail based on preferred embodiments and on the figures. Figure 1 illustrates the main steps of a method in accordance with a preferred embodiment of the invention. The method aims at in-orbit testing of a frequency band of a satellite transponder. The satellite transponder 10 may preferably by a wide-band transponder. An illustration is provided in Figure 12, , wherein the frequencies of the transponder extend from left to right. The frequency band 10 that is tested using the proposed method may be the entire transponder, or only part thereof. At a first step a), a set of input measurement signals is transmitted from a ground station using direct sequence spread spectrum modulation to the satellite on said transponder. The set of input measurement signals comprises a reference signal 110, which is centered on the central frequency of said a frequency band 10, and at least one testing signal 120, which is centered on first frequency of said frequency band. The first frequency on which the testing signal 120 is centered is different to said central frequency. The corresponding infrastructure is illustrated in figure 3, wherein the ground station 200, comprising a satellite link interface, is shown communicating the input measurement signals 110, 120 to the satellite 210 at step a). The satellite receives the set of measurement signals and retransmits them to the ground station, so that the behaviours of the transponder on which the measurement signals are transmitted may be evaluated in a further method step. At step b) the ground station 210 acts as a receiver and subsequently receives a corresponding set of output measurement signals 110’, 120’ from the satellite 210. The output signals may be considered as filtered versions of the reference signal 110 and the testing signal 110’ respectively, wherein the filter is equivalent to the satellite channel response. At step c), parameters or features that characterize the transponder at the frequencies on which the measurement signals 110, 120 are centered, are computed using a data processing unit of the ground station 200. These parameters may for example comprise the group delay and/or the amplitude response of the transponder. The computation of these values depends on the correlation of the input measurement signals 110, 120 and the corresponding received output measurement signals 110’, 210’, as will be explained in details later. In accordance with step d), steps a) to c) are repeated at least once and preferably continuously at predetermined time intervals during a predetermined timespan. At each iteration, the frequency on which at least one testing signal 120 is centered is changed to a different frequency, as shown in figure 2, wherein the testing signal 120 is shown at a first time t1 corresponding to a first iteration of step a), and at a later time t1+X, corresponding to a later iteration of method step a). The frequencies on which the testing signal 120 is centered differ at the two instants, while the reference signal 110 remains at the center of the tested frequency band 10. During transmission of the input measurement signals 110, 120, the transmit power is selected by the transmission means so as to depend on the transmit power of at least one data traffic signal that concurrently occupies at least part of a frequency sub-band 112, 122, that comprises the frequency on which said input measurement signal 110, 120 is centered. By taking this adaptive approach, the transmit power of the measurement signal can be adapted for each measurement: if no or little traffic is present in a sub-band at the time of measurement, the transmit power of the input measurement signals 110, 120 may be increased, leading to more accurate measurements. On the other hand, for sub-bands having high traffic load, the transmit power of the input measurement signals 110, 120 may be reduced in order to limit interference with the data signals. In a preferred embodiment, the transmit power of the input measurement signals 110, 120 is selected to be lower by a predetermined threshold value to the transmit power of a data traffic signal that concurrently uses the corresponding frequency sub-band that includes the respective input measurement signal 110, 120. While a single testing signal 120 in the tested frequency band 100 is illustrated in figure 2, the invention extends to multiple testing signals that are simultaneously transmitted on different frequencies. This approach has the potential to reduce the time to test the bandwidth 100 considerably. All measurements are preferably normalized with respect to the measurement obtained from the reference input signal 110. Preferably, the testing signals 120 may continuously sweep a frequency sub-band. In that case the frequency offset for a given testing signal 120 between two iterations of method step a) is small. Alternatively, from one iteration of method step a) to the next, the testing signal 120 may be migrated to a different frequency sub-band, in which the transmit power of the data traffic signal is low at that instant in time. The knowledge of the data traffic signal allows to fine tune the sweeping algorithm in order to minimize the duration of the test. I. PREFERRED EMBODIMENT OF THE INVENTION In this embodiment, mainly the monitoring of the amplitude response and the group delay of a satellite ^{transponder is considered, under five design requirements:}

• _{level of interference: the interference caused by the measurement signals to the traffic must be} n^egligible;

• _{resolution: the measurement granularity must be fine;}

• ^{accuracy: the variance of the measurement errors must be below a chosen target;}

• _{duration: the whole measurement must take less than a chosen amount of time (which is particularly} r^{elevant for wideband transponders);}

• _{complexity: the hardware implementation of the measurement procedure must be feasible.} As measurement procedure, an enhanced adaptive version of the DLL approach is proposed, based on DSSS. For the measurement, a reference signal and one or multiple testing signals, each one characterized by a different pseudo noise, PN, sequence, can be used to sweep in parallel different portions of the bandwidth under measurement, in order to speed up the overall measurement procedure. All the measurements provided by the testing signals are normalized with respect to the measurement obtained with the reference signal. Both reference and testing signals shall respect a power constraint in order to make their interference to the traffic negligible. The transmitter shall use prior knowledge of the traffic power distribution (acquired either by knowing the traffic frequency plan or by channel sensing) and set the powers of reference and testing signals in order to satisfy such power constraint. Then, an analytic approach is proposed allowing the design of the measurement signals so as to satisfy the other requirements. By properly setting the symbol rates, the lengths of the PN sequences, and the number of possible sequence repetitions, it is usually possible to satisfy both the constraints coming from the measurement perspective (i.e., accuracy, resolution, measurement duration, and interference level) and the hardware perspective (i.e., implementation complexity). It is worth noting that the proposed method is particularly suitable for wideband transponders, since it allows parallel measurements in disjoint sub-bands. The proposed measurement procedure is therefore a useful design tool that can be used to check the feasibility of a specific design, the meeting of the system requirements, and the achievability of the measurement targets. In the following a particularly preferred embodiment of the invention will be described in detail, without limiting the invention thereto. The rest of the description is organized as follows: in Section II the main scenarios are described, the system model is provided, and the general measurement procedure is detailed. The description of the proposed algorithms is the focus of Section III, while Section IV and Section V provide some theoretical bounds and a numerical evaluation of the performance of the proposed algorithms, respectively. Finally, conclusions are drawn in Section VI. II. SCENARIOS, SYSTEM MODEL, AND MEASUREMENT PROCEDURE In this Section the scenarios of interest in an IOT context are highlighted, the measurement procedure is detailed, and the adopted system model is described. A. Scenarios Since the main purpose of the measurement procedure described in this embodiment is limiting the interference caused by the measurement signals to the traffic, the key scenario assumes that the satellite is located in its orbit and is already in use, i.e., the payload is loaded with customers’ signals (the traffic). Considering a geostationary, GEO, satellite, the Doppler effect is limited but not absent. In case of medium and low Earth orbit satellites, Doppler shifts and rates become more relevant. This would affect only the fine-tuning of the parameters of the synchronization chain (detailed in Section III-A) but not the whole measurement procedure detailed in Section II-C. A second scenario, relevant for IOT, assumes the satellite to be in relocation phase, i.e., traveling towards its orbit. In this scenario the payload is clearly not loaded, but during its movement the satellite may pass close to other loaded satellites. Therefore, the role of primary signal is now assumed by the traffic signals on the payloads nearby. The main difference between these two scenarios is mainly the amount of Doppler shift and rate affecting the measurement signals, which does not entail any modification to the proposed measurement procedure. Hence, without loss of generality, in the following the focus is on the first scenario only. B. System Model The considered system is composed of four functional entities (i.e. the transmission

means/reception/measuring means): the generator of the measurement signals , the generator of the traffic, the channel, and the receiver unit, as shown in figure 4. In a closed loop configuration, transmitter and receiver coexist in the same unit (e.g., a gateway). On the other hand, in an open loop configuration transmitter and receiver are separate units that can be located in different sites. The proposed procedure is based on two types of input measurement signals: the reference signal and the testing signal. The reference signal is unique and located at the central frequency with respect to the bandwidth under measurement, while the single or multiple testing signals sweep the desired bandwidth. The bandwidth under measurement can be the whole transponder bandwidth or a part of it; for the sake of generality, in the following the whole transponder bandwidth is considered. The advantage of using multiple testing signals, especially for wideband transponders, is that it is possible to perform measurements in parallel over disjoint sub-bands. This of course significantly reduces the total measurement duration, but also requires a higher hardware complexity caused by replicating the transmitter and receiver circuitry. Since the algorithms and the measurement procedure remain unchanged independently of the number of used testing signals, for the sake of simplicity only one testing signal is considered in the following description. As a consequence, the single testing signal sweeps the whole transponder bandwidth. 1) Measurement Signals: Each measurement signal is characterized by six degrees of freedom: the transmitted symbol sequence, the number of repetitions of the sequence, the symbol rate, the roll-off of the shaping pulse, the transmitted power, and the carrier frequency. The proposed design is inspired to multiuser code-division multiple access SS systems in the sense that a different PN sequence is used for each measurement signal (i.e., reference and testing). PN sequences are known for having good autocorrelation and cross-correlation properties (J. S. Lee et al., CDMA Systems Engineering

Handbook. Artech House, 1998), which are exploited by the measurement algorithms. Symbol sequences can be obtained by mapping the PN binary sequences on a tilted BPSK modulation, obtaining complex symbols. The baseband equivalent of the transmitted signals then read

where x_r(t) is the reference signal, f₀ is the central frequency of the bandwidth under measurement, x_t(t,f_k) is the testing signal measuring at frequency f_k, {s_q} is the sequence of binary complex symbols having length L_q and symbol period T_q, N_q is the number of repetitions of the sequence, p_q(t) is the shaping pulse, P_q is the transmitted power, and q = {r,t} identifies the reference and testing signals. If the bandwidth under measurement is the whole transponder bandwidth, then f₀ = 0. Figure 5 shows a functional block scheme of the transmitter of the measurement signals, in case K testing signals were used. Each PN sequence can preferably be repeated multiple times for each measurement, which is approximately equivalent to using a much longer sequence. The advantage of repeating multiple times the same medium-length sequence comes from the hardware perspective: shorter sequences require a shorter buffer for the computation of the correlation, and the full correlation is shorter, resulting in a smaller number of operations to be executed. Also, having multiple repetitions of the same sequence helps the synchronization procedure by reducing the time duration of the coherence window required to compute the correlation at the receiver. The symbol rate can preferably be different for reference and testing signals, according to the corresponding needs. For the reference signal a high symbol rate means that the bandwidth is large and captures a large portion of the transponder behaviour in its central band, making the measurement more stable. On the other hand, it makes a big portion of the traffic power fall inside the bandwidth of the reference signal, making the measurement noisier. Also, higher symbol rates may become challenging to implement in hardware. For the testing signal perspective, on the contrary, a smaller symbol rate allows a more precise and less noisy measurement. However, at the same time it makes the duration of the symbol longer, increasing the amount of time required to carry out the whole measurement. In the considered design, the powers of the measurement signals are always constrained to remain below the noise floor, limiting the interference to the traffic. Indeed, from the customer’s perspective, the signal-to-interference-and-noise ratio (SINR) degradation caused by the measurement signals (computed assuming the traffic as useful signal, and the measurement signals as interferers) must be negligible. A static approach (like the classic DLL) would simply dimension the power of the measurement signals in a conservative way, aiming at not degrading the customer’s SINR. On the contrary, the proposed adaptive approach allows the adaptation of the transmit power of the measurement signals to the traffic conditions. In particular, the power of a measurement signal is set to be X dB below the measured power of the traffic falling in the bandwidth of the measurement signal. In accordance with the present embodiment, the power difference X is the key design parameter that guarantees the preservation of the customer’s SINR. Such adaptability, stemming from the adaptive approach, allows either to improve the estimation accuracy for a given measurement duration, or to reduce the measurement duration for a given accuracy. Finally, the step-size of the frequency sweeping is determined by the desired resolution of the measurement. Of course, a finer granularity requires more measurements (one per each frequency value), which globally means that the total assessment of the whole transponder bandwidth will take more time. 2) Traffic and Satellite Models: From the measurement point of view, the channel is characterized by two main elements: the satellite payload and the traffic. The latter is a powerful and disruptive source of interference to the measurement signals, of which it is of course independent. Realistically, the traffic is composed by several single-carrier signals having different power, bandwidth, and time duration. Indeed during a measurement sweeping the whole bandwidth of a wideband transponder, it is possible that some single-carrier signals change their power, or disappear, or appear at a different carrier frequency, according to the customers’ needs. The payload, on the other hand, is the target of the measurement procedure. It is typically composed by local oscillators, input and output multiplexers (IMUX and OMUX, respectively), and a high power amplifier (typically a traveling-wave tube amplifier, TWTA), see also ETSI TR 102 376-2, V1.1.1 Digital Video Broadcasting (DVB), Implementation guidelines for the second generation systems for Broadcasting, Interactive Services, News Gathering and other broadband satellite applications; Part 2: S2 Extensions (DVB-S2X), Nov 2015. Available on ETSI web site (http://www.etsi.org). However, it can also be seen as a black box with a certain frequency response and introducing a (possibly time-varying) frequency error. For the design of the measurement algorithms, this second approach is chosen. On top of this, further impairments are introduced by the satellite and affect the received signal: frequency conversion errors and phase noise generated by the local oscillator, and Doppler shifts and rates (in both uplink and downlink) stemming from the satellite movements. A block scheme of the adopted channel model is shown in figure 6, where the traffic is omitted (consistently with figure 4). 3) Receiver Model: Since the focus of the description of the present embodiment is on measurements and satellite monitoring, a professional receiver is assumed. This means that high-quality hardware is used and negligible impairments are introduced by the components of the receive unit. Thus, the only additional noise source is the thermal noise. For the sake of simplicity, the analytic model for the received signals (both reference and testing) assumes perfect synchronization in frequency, phase, and timing (i.e., Doppler shifts and rates, phase noise, and frequency conversion errors are perfectly compensated, and the sampling occurs at the correct timing instants with the correct symbol rate). After match-filtering and sampling, the two perfectly synchronized received signals can be modeled as where s_q = [s_q(0),...,s_q((N_qL_q - 1)T_q)]^T is the vector containing the transmitted symbols, a_q is the received power of the signal, h_n is the transponder response at the measurement frequency f_n (assumed to be constant over the bandwidth of the received testing signal, which is narrow-band if compared to the whole transponder bandwidth), models the traffic, and w_q is the thermal noise. Since no prior assumption can be made on the traffic samples g_q = [g_q(0),...,g_q((N_qL_q -1)T_q)]^T, for simplicity they are approximated as independent and identically distributed, IID, samples following a complex circularly-symmetric standard Gaussian distribution (i.e., with zero mean and unit variance). The received power of the traffic is represented by the factor b_q. The samples of the additive white Gaussian noise, AWGN, w_q are IID and follow a complex Gaussian distribution with mean zero and variance s_w ² per component. A functional block scheme for the receiver of the reference signal is shown in figure 7 (the receiver structure for the testing signal is exactly the same). The figure also shows the synchronization blocks, that will be considered only in the numerical analysis. C. Measurement Procedure Monitoring all the features of a transponder is a challenging operation that cannot be accomplished in a single step. However, some relevant features can be measured by resorting to the same procedure, as will be highlighted in the following. In particular, the focus is on three key characteristics of the transponder: the amplitude response, the group delay, and the local oscillator stability. The procedure detailed in Algorithm 1 is applicable to both amplitude response and group delay; the measurement of the local oscillator stability is a by-product of the procedure itself. The proposed measurement procedure is a useful design tool because it links together all the five design requirements listed in the above introduction to this embodiment. In particular: • the level of interference limits the powers P_q of each measurement signal, forcing it to be X dB b^{elow the traffic power in the corresponding bandwidth B} _q ^;

• ^{the resolution requirement sets the number of desired measurements K (in the frequency domain);} • _{the accuracy requirement provides an upper bound s 2}

m_{eas,target to the measurement variance;}

• _{the duration requirement provides an upper bound to the total measurement time, that can be} evaluated as ^• the complexity requirement typically limits bandwidth B_q and the sequence length L_q of the measurement signals. Hence, it is assumed that some parameters of the measurement signals and some values for the target performance are chosen a priori in order to satisfy system or performance requirements. If it is possible to compute the measurement variance s_meas ² in a closed form for a given measurement algorithm, and link it to the other design parameters, then the designer can easily trade-off the various performance metrics and check the feasibility of a specific design. For example, the relationship between the measurement variance and the number of repetitions clarifies how the constraints on accuracy and measurement duration are connected one another. From the designer perspective, the knowledge of such connection helps the design process: by choosing the minimum number or repetitions that satisfies the required accuracy, the measurement duration is reduced to its minimum. Whereas the theoretical computation of s_meas ² depends on the chosen algorithm, the measurement procedure remains unchanged. Different measurement algorithms are presented in Section III and their theoretical accuracy is evaluated. Algorithm 1 Measurement Procedure

Given: transponder bandwidth, measurement resolution

• compute the number of desired measurements K

Given: measurement accuracy s_meas,target ² , bandwidths B_r and B_t, lengths L_r and L_t, power threshold X (in dB) for measurement signals

F^{or k = 1 to K}

• Acquire P_tr,r: traffic power in bandwidth B_r centered at the transponder central frequency f₀ • Acquire P_tr,r: traffic power in bandwidth B_t centered at the measurement frequency f_k

• Compute transmit powers of reference signal as P_r = P_tr,r - X dB

• Compute transmit powers of testing signal as P_t = P_tr,t-X dB

• Compute the minimum number of repetitions of the testing sequence N_t * that satisfies

• Compute the number of repetitions of the reference sequence as

• Transmit N_t * consecutive testing sequences and N_r c* consecutive reference sequences

• Synchronize (in frequency and time)

– Output: estimation of the local oscillator stability

• Correlate the received sequences with the transmitted ones ^{• Accumulate over the number of repetitions}

• ^{Measure (amplitude and/or group delay)}

• _{Normalize w.r.t. the measurement of the reference signal}

– Output: estimation of the amplitude response and the group delay

End Loop III. MEASUREMENT ALGORITHMS This section describes in details some estimation algorithms targeting the amplitude response, the group delay, and the oscillator stability of the transponder. In particular, the algorithms for the amplitude response and the group delay are based on the correlation function and follow the common de-spreading procedure used in SS systems. On the other hand, the estimation of the oscillator stability is a natural by-product of the synchronization procedure. The descriptions of the algorithms that follow are intended for a generic measurement frequency f_k belonging to the band under measurement. A. Local Oscillator Stability It is assumed that, at its frontend, the receiver of the satellite’s output measurement signals operates in the oversampled domain (i.e., it processes more than one sample per symbol period). The oversampling is a design parameter that depends on the hardware capability and is driven by the ^{complexity-accuracy trade-off. The synchronization procedure consists of multiple steps:}

• _{coarse frequency synchronization}

• _{timing synchronization}

• _{sequence synchronization}

• fine frequency synchronization. From the receiver perspective, the frequency error generated onboard the satellite and the Doppler shift introduced by the satellite movements are indistinguishable and are seen as a single quantity (i.e., a time-varying frequency offset). In the following it is assumed that the total frequency offset changes slowly with respect to the sequence duration. Such assumption allows to approximate it as a constant over multiple repetitions of the same sequence. Also, it is reasonable to assume that the ranges of variation of the onboard frequency conversion error and the Doppler shift are known a priori. For example, the range for the onboard frequency conversion error may be extrapolated from the data sheet of the onboard local oscillator, while the range for the Doppler shift can be computed from the ephemerides of the satellite and the location of the terminal. When the total range of variation is known, a simple way to perform a coarse estimation of the total frequency offset is the binning (or hypothesis testing). The range of variation of the total frequency offset is partitioned in N_F intervals he bins), and the value of the central frequency of each interval is used to compensate the offset in a trial-and-error fashion (i.e., N_F frequency hypotheses are assessed in parallel). Each of the resulting N_F compensated signals (obtained by compensating the received signal with a different frequency hypothesis) is then matched-filtered and down-sampled to one sample per symbol. The main advantage of this hypothesis-testing approach is that it does not require acquisition/tracking loops, which normally have to operate at a very high clock rate. Indeed, processing a SS oversampled signal may become quickly very challenging from the hardware point of view. The same hypothesis-testing approach can be used for timing synchronization: if N_T samples per symbol are extracted, N_T timing hypotheses are possible. Alternatively, if the oversampling factor is not high enough to ensure a negligible residual timing error, a classical algorithm (e.g., the early-late gate) can be used before downs-sampling (see for example U. Mengali et al., Synchronization Techniques for Digital Receivers (Applications of Communications Theory). Plenum Press, 1997). Sequence synchronization (i.e., the identification of the start of the transmitted sequence) is then performed by computing the full correlation between the transmitted and the received symbol sequences, and by finding its peak. This procedure has to be repeated for all the N_FN_T signals obtained by using the hypothesis-testing approach for coarse frequency and timing synchronization. Finally, the hypothesis with the highest correlation peak will be chosen as the correct one, and provide the coarse estimate of the frequency error f_coarse. Of course, the choice of N_F and N_T is driven by the trade-off between the affordable complexity of the receiver and the required accuracy. In order to refine the frequency estimation f_coarse, we exploit the fact that the transmitted sequence is repeated multiple times. By correlating the received N_q consecutive repetitions of the sequence with the transmitted sequence itself, N_q correlation peaks are identified. The phases of these peaks are then used to estimate the residual frequency error by means of a linear regression approach, see P. Broersen, Automatic Autocorrelation and Spectral Analysis. Springer Verlag, 2006. By denoting with f =

[f₀,...,f_Nq-1]^T the vector of the measured peak phases, by z = [J,f_res]^T the vector of unknown parameters (namely, a normalized phase offset J and the residual frequency error f_res), and by A the sampling matrix defined as

it is possible to write the peak phases as F = 2pL_qT_qAV Then, vector z can be computed by linear regression exploiting the Moore-Penrose pseudo-inverse of A as

The sum of the estimated coarse frequency error f_coarse and the estimated residual frequency offset f_res provides the required estimation of the total frequency error. By storing and processing such estimates, it is possible to extract statistics over time of the local oscillator stability. The described procedure is summarized in Algorithm 2. B. Amplitude Response Once the synchronization is achieved, the receiver can estimate the amplitude response n at the frequency f_k by processing the received symbols. In the following analysis, only the testing signal is considered because the measurement obtained from the reference signal (by using the same algorithm) is used for normalization purposes only. Hence, the subscript q is dropped for the ease of notation. Three algorithms are evaluated in the following, based on matching different moments of the received signal. ^{Algorithm 2 Synchronization Procedure}

• _{[Coarse Frequency Sync] Compute the frequency hypotheses {fi}}

• For i = 1 to N_F

– Compensate the frequency error by f_i

– Matched filter

– [Timing Sync] For j = 0 to N_T - 1 * Downsample at

* [Sequence Sync] Compute the full correlation with the transmitted sequence

* Compute the amplitude of the correlation peak

– ^{End Loop}

• ^{End Loop}

• _{Choose the frequency and timing hypotheses (fcoarse) giving the correlation peak with the highest} amplitude

• [Fine Frequency Sync] For n = 0 to N_q - 1

– ^{Compute the phase of the correlation peak}

• ^{End Loop}

• _{Linear regression on the peak phases to find fres}

• _{Compensate for fres}

• Output: 1) Mean-based Estimator: An L-sized vector

is considered to be obtained by taking only the entries of (1) corresponding to the n-th repetition of the transmitted sequence. Since

, one defines

which corresponds to the de-spreading processing in SS systems. In the following,

denotes a complex circularly-symmetric Gaussian distribution with mean µ and variance s² per component. Since

are linear combinations of the traffic samples and AWGN samples, respectively, y_n can be seen as a complex circularly-symmetric Gaussian variable with the following parameters

(3)

where the polar notation for the transponder response h = ne^jf has been introduced, and

(4)

From (3), the amplitude of the transponder response n can be estimated by taking the absolute value of an estimate of the mean of y_n. An unbiased estimator of the mean is the sampling mean, constructed by averaging the N realizations of y_n available at the receiver (see H. L. Van Trees, Detection, Estimation, and Modulation Theory - Part I. John Wiley & Sons, 1968). So, by averaging over the N repetitions, one obtains (5)

where e is the error term. Thus, the estimator for n is simply the absolute value of , i.e.

(6)

which is a Rice-distributed random variable because (J. Proakis and M. Salehi, Digital Communications.5th ed., 2008). In other words, and its first- and second- order statistics are

where is the Laguerre polynomial of order m. From (7), appears to be a

biased estimator; however, it can be shown that it is asymptotically unbiased for NL growing to infinity. Also, it is convergent in quadratic mean (i.e., its mean squared error vanishes asymptotically), which implies that it is also consistent (see A. van den Bos, Parameter Estimation for Scientists and Engineers. John Wiley & Sons, 2007). 2) Variance-based Estimator: An alternative approach to the use of the sample mean in (5) to estimate n from the moments in (3) is resorting to the sample variance

By inverting (4) with respect to n, the variance-based estimator can be obtained as

The computation of its statistics is now a bit more involved. Starting from (3), it is known that the sample variance of normal IID variables in (9) is c²-distributed when properly scaled (see A.

Papoulis, Probability, Random Variables and Stochastic Processes. New York, NY: McGraw-Hill, 1991 ). By removing the scaling, the c² distribution becomes a more general Gamma distribution. Finally, by applying the scaling, shifting, and rooting as in (10),

results following the Amoroso distribution, which is a four parameter extension of the generalized Gamma distribution

(see G. Crooks,“The amoroso distribution,” Jul 2015). In detail,

and the statistics of the estimator are

As the mean-based estimator, the estimator

is biased, but now a part of the bias (namely, the last additive term in (11)) can be removed because it is constant, yielding to the regularized estimator

The regularized estimator is asymptotically unbiased, convergent in quadratic mean (and hence

consistent), but only if the thermal noise vanishes (i.e., when tends to zero). Also, it is well defined only if the traffic power b is non-zero. 3) Kullback-Leibler Estimator: Since the parameter n appears in both mean and variance in (3), a more general approach requires taking both moments into account. To this purpose, one can define the measured distribution of y_n as and the desired distribution of y_n as

replacing (4) into (3). The estimate can be computed as the value that makes p_d(y_n) as close as possible to p_m(y_n). A common method to quantify the distance between two distributions p and q is the Kullback-Leibler divergence D(p||q) (see. M. Cover and J. A. Thomas, Elements of Information Theory. New York: John Wiley & Sons, 2nd ed., 2006). Hence, the estimation problem can be recast as a minimization problem It can be shown that the minimization of the Kullback-Leibler divergence is equivalent to the maximization of the likelihood function in a parametric estimation framework. Since p_m and p_d are both Gaussian distributions, the Kullback-Leibler divergence can be written as

By differentiating it with respect to nˆ, a third-order polynomial in is obtained. Namely, the polynomial reads where its coefficients are

The value of nˆ that minimizes (14) is a root of the polynomial in (15). Since must be a positive real quantity, only positive real roots are acceptable. Unfortunately, the analysis of the variance of the estimation error results to be intractable in this case. C. Group Delay It is clear from (3) that the phase of the frequency response of the transponder appears only in the mean of the distribution. Therefore, in order to derive an estimator for the group delay, the sample mean (5) is manipulated as follows: where

The phase estimate required for the computation of the group delay can be obtained by resorting to the following unbiased estimator:

where and denote the real and imaginary operators, respectively. If the residual noise is small when compared to the amplitude response of the transponder (i.e., when| , then the error e can be approximated as

(16)

which is the ratio of two real Gaussian-distributed variables (with same variance but different mean). Such ratio is known to have a Cauchy-like distribution, which does not allow the definition of moments. However, under mild conditions it can be shown that moments do exist (see G. Marsaglia, “Ratios of normal variables,” Journal of Statistical Software, vol.16, no.4, 2006), and in particular the variance of e in (16) can be approximated as (17) where The group delay is defined as (see D. Manolakis, V. Ingle, and S. Kogon, Statistical and Adaptive Signal Processing. Artech House, 2005 )

where in numerical systems the first-order derivative can be approximated as a central finite difference of the second order:

where∆f = |f_k - f_k-1|. Therefore, by using (17) and (18), the variance of the estimation error for the group delay can be evaluated as

IV. THEORETICAL BOUNDS It is known that the Cramér-Rao bound, CRB, provides a lower bound to the accuracy of an unbiased estimator. However, the CRB can be extended so as to take a bias into account (see A. van den Bos, Parameter Estimation for Scientists and Engineers. John Wiley & Sons, 2007). Therefore, in this Section the CRBs for the amplitude response and the group delay are derived, in both unbiased and biased cases. A. Cramér-Rao Bound for Unbiased Estimators As done in the previous section, in the following only the testing signal at the frequency f_k, is considered and the subscript q is dropped for the ease of notation. Under the hypothesis of perfect synchronization, the received sample of the testing signal at the sampling instant

where

(i.e., it is assumed that E{|s`|<sup>2</sup>} = 1 and the use of a BPSK modulation), and {h<sub>`</sub>} are known. The parameters to be estimated are the amplitude n and the phase of the transponder frequency response (which is assumed constant over the bandwidth of the testing signal). Consider the parameter vector

. Then, the likelihood function of the received sample given the parameter vector q becomes

The CRB for an unbiased estimator can be computed as

where the expectation is taken with respect to the received signal. Since AWGN and traffic samples are both assumed independent, the CRB obtained by considering NL samples (i.e., symbols) results to be the same obtained with one sample and scaled by the number of samples. After solving the expectation integral in (20), one can find that the two CRBs are

B. Cramér-Rao Bound for Biased Estimators

Since both

and

are biased (as can be seen from (7) and (11)), it is worth computing the CRB for biased estimators having the same type of bias. It can be shown that, if an estimator is such that

where B (q) is the bias, then its CRB can be computed as

This can be specialized to both nˆ_M and nˆ_V by using (21), yielding to

where G(x) is the Gamma function, is the generalized Laguerre polynomial of order m and parameter a, and G is a constant defined as

It is worth mentioning that these CRBs hold only for the estimators having a bias whose first derivative is the same as for nˆ_M and nˆ_V , respectively. C. Cramér-Rao Bound for the Group Delay From (18) it is possible to consider the group delay as a function of the phase response of the transponder, which can be treated as a vector of unknown parameters

. Then, by using the practical computation of the group delay in (18) and the CRB for the phase estimator in (22), it is possible to obtain the CRB for the group delay as [14]

V. NUMERICAL RESULTS In order to assess the performance of the proposed algorithms, as use case the measurement of amplitude response and group delay for a wideband transponder when the satellite is on station and loaded with traffic is considered . In such condition, a realistic load of the transponder is shown in figure 8: the useful payload bandwidth (delimited by the transponder amplitude response in black dashed line) is irregularly occupied by the traffic signal (in dark grey). Of course, the receiver is affected by thermal noise (in light grey). On top of this, the measurement signals, located at the central frequency of the transponder (for the reference signal, in black dotted line) and at a generic measurement frequency f_k (for the testing signal, in black solid line), are both below the noise level and may overlap with the traffic. A. Set-up Description For the simulation campaign a traffic signal composed by 26 single-carrier signals is considered, randomly located in adjacent and non-adjacent sub-bandwidths. Each carrier modulates a different signal, obtained by pulse-shaping an independent stream of QPSK random symbols. For simplicity, each single-carrier signal has a bandwidth equal to 4 MHz, which corresponds to a narrowband signal if compared to the whole transponder bandwidth. Since in an IOT perspective only relative measurements are relevant, for simulation purposes any amplitude in the simulated system has been normalized with respect to the central value of the transponder amplitude response. In this way, the transponder amplitude response at the central frequency is always 0 dB. Concerning the traffic, the normalized power of each single-carrier signal has been randomly extracted from the range [-7,0] dB. It is worth highlighting that the purpose of the simulation campaign is not assessing the reception of the traffic, but the measurement of the transponder frequency response. In this perspective, the traffic plays the role of a very strong interferer. Since the proposed design has many degrees of freedom (i.e., bandwidths, lengths, and repetitions, for both reference and testing signals), an exhaustive analysis of the ultimate performance of the proposed measurement procedure is out of the scope of this simulation. Also, these degrees of freedom highly depend on the affordable hardware capabilities and the required accuracy, which are clearly not the same for any application or satellite, and may vary according to contingent needs. Therefore, in the following the investigation is limited to one use case where most of these parameters are fixed to realistic values. In the chosen set-up, the reference signal has a bandwidth equal to 20 MHz and a sequence length of 2¹¹ symbols, whereas the testing signal has bandwidth equal to 4 MHz and a sequence length of 2¹³ symbols. This is a convenient choice from the simulation point of view because all the quantities are multiples one another, which simplifies the management of such a multi-rate set- up. The squared-root raised-cosine filters used for shaping the symbol sequences for traffic, reference, and testing signals have roll-off 0.2, which is one of the prescribed roll-offs in the DVB-S2 standard (ETSI EN 302307-1 Digital Video Broadcasting (DVB), Second generation framing structure, channel coding and modulation systems for Broadcasting, Interactive Services, News Gathering and other broadband satellite applications, Part I: DVB-S2, Nov 2014. Available on ETSI web site (http://www.etsi.org) ). All the shaping pulses have order 40 so as to guarantee a realistic numerical approximation of the analog waveform, and oversampling factor 8 (with respect to the corresponding symbol rate). In order to greatly limit the interference caused by reference and testing signals, their power is kept X = 30 dB below the traffic power measured inside their bandwidth.

TABLE I: TRANSMITTER PARAMETERS Such value guarantees a SINR degradation below 0.2 dB according to the worst case of the DVB-S2 standard (i.e., the use of the modulation and coding scheme with the highest spectral efficiency), and below 1 dB according to the DVBS2X standard (ETSI EN 302307-2 Digital Video Broadcasting (DVB), Second generation framing structure, channel coding and modulation systems for

Broadcasting, Interactive Services, News Gathering and other broadband satellite applications, Part II: S2-Extensions (DVB-S2X), Oct 2014. Available on ETSI web site (http://www.etsi.org) ). The transmitter parameters are reported in Table I, where N_c indicates the number of carriers and P_tr,q the traffic power falling in the bandwidth of reference or testing signal, and q = {r,t,tr} discriminates over reference, testing, and traffic signal. As an example, in the following the performance of the proposed measurement procedure is assessed at one specific frequency f_k. In particular, we assume that at f_k the power of the testing signal is P_t = -34 dB (i.e., the traffic power falling in a 4 MHz bandwidth centered at f_k is P_tr,t = -4 dB), while the power of the reference signal at the central frequency is P_r = -30 dB. The considered channel model includes several elements: the Doppler effects (in both uplink and downlink), the transponder, and the receiver thermal noise. In order to assess the performance in a worst-case scenario a Doppler shift equal to 180 Hz and 121 Hz were considered for uplink and downlink, respectively. These values are realistic for an uplink carrier at 30 GHz and a downlink carrier at 20.17 GHz, which are typical values for GEO communication satellites operating in the Ka- band. Also, it is worth mentioning that these values for the Doppler shift hold for the central frequency only. Since the considered transponder is wideband, the frequency components of the received signal that are the furthest from the central carrier (e.g., the testing signal when it is measuring the knees of the transponder response) will experience a warped Doppler shift. This also implies that the frequency synchronization has to be performed independently for both reference and testing signals. Since the receiver in an IOT application is assumed to be a high-quality professional equipment, the impairments generated by the onboard local oscillator are no longer negligible. To this purpose, a worst-case frequency conversion error equal to 12 kHz has been considered, as well as a realistic phase noise mask. Typical frequency responses for the IMUX and OMUX filters, as well as the AM/AM and AM/PM characteristics of the TWTA, can be found in ETSI TR 102376-2, V1.1.1 Digital Video Broadcasting (DVB), Implementation guidelines for the second generation systems for Broadcasting, Interactive Services, News Gathering and other broadband satellite applications; Part 2: S2 Extensions (DVB-S2X), Nov 2015. Available on ETSI web site (http://www.etsi.org). Since the targets are mainly wideband transponders, the filter characteristics are rescaled in order to have 480 MHz of useful bandwidth. Also, the IBO has been assumed equal to 9 dB in order to limit the nonlinear distortion caused by the linearized TWTA.

TABLE II CHANNEL AND RECEIVER PARAMETERS Finally, the thermal noise generated by the hardware components at the receiver side has been included. For simplicity, its PSD has been assumed constant over the transponder bandwidth, and in particular equal to -15 dB (normalized with respect to the amplitude of the transponder frequency response at the central frequency). As front-end, a matched filter having the same characteristics as the corresponding shaping pulse has been used for both reference and testing signals. The numerical values used for the parameters characterizing the satellite channel and the receiver are reported in Table II. In order to compute statistics on the measurements, 128000 partial measurements were taken for each frequency. With“partial measurements” the acquisition of samples {y_n} in (3) is denoted. Each partial measurement is obtained by using only one repetition of the testing sequence. The final measurements are obtained by accumulating and processing N_t consecutive partial measurements. In particular, Eq.(6), (10), and (15) are used to compute the different estimates. The performance of the proposed estimators are evaluated in terms of variance, assuming that one of the measurement requirements is keeping its variance below a certain threshold. Two types of variances are considered: the estimator theoretical variance, computed in closed form in the analysis in Section III, and the measured variance, computed by resorting to a Monte Carlo approach and multiple realizations of the measurements. Theoretical variances are functions of the number of repetitions of the testing sequence N_t and of the actual value of the transponder amplitude n, as shown in their closed- form expressions (8), (12), and (19). In order to make their evaluation possible at the receiver, the actual n has been replaced by 1, which is indeed its maximum possible value and maximizes (8), (12), and (19). No theoretical variance is available for the KL-based amplitude estimator. In the following each variance is converted for the amplitude response in the log-domain as

The performance of each estimator was assessed for N_t Î [1,256]. When N_t = 1 the measured variances are computed over 128000 realizations of the measurements, whereas when N_t = 256 they are computed over 500 realizations, which are still enough to provide stable values of the measured variances. Assuming N_t = 256, this leads to a total measurement duration equal to D_tot < 5 minutes for the whole transponder bandwidth (i.e., 480 MHz). It is worth highlighting that (2) does not take into account the processing time required at the receiver to output a measurement, and hence it can be considered as the acquisition time for the raw data. B. Amplitude Response The performance of the proposed estimators for the amplitude response is shown in figure 9. The dotted lines represent the biased CRBs in (23) and (24) for the mean-based and the variance-based estimators, respectively. No CRB is shown for the KL-based estimator because no closed-form for its mean is available and, therefore, no assumption can be done on its bias. From figure 9 it can be seen that the CRBs are very close one another, but not identical. This is related to the different biases of the two estimators. It is worth mentioning that all the other curves in figure 9 do not reach the bounds because the same SNR value is assumed for any plotted point. The theoretical variances, computed by assuming Gaussian traffic, no impairments, and ideal synchronization, are shown in figure 9 as continuous lines. It clearly appears that the mean-based estimator outperforms the variance-based one, even though both of them are quite far from the corresponding CRBs. Finally, the measured variances are shown in figure 9 in dashed lines. It can be seen that the measured variances for both the mean- based estimator and the variance-based estimator are very close to their corresponding theoretical variances. The difference stems from the fact that the measured variances are computed using realistic traffic (i.e., non-Gaussian), channel and hardware impairments (i.e., Doppler shifts, phase noise, frequency conversion error), and non-ideal synchronization. The KL-based estimator outperforms the other two, but its performance is still quite close to that of the mean-based estimator. The fact that the theoretical variances are slightly higher than the measured variances allows to safely use the theoretical ones as s_meas ² in Algorithm 1. Having a closed-form expression for the theoretical variance is of great help since it allows to easily compute the minimum number of repetitions satisfying the target accuracy (and consequently minimizing the measurement time). This is of course possible only for the mean-based and the variance-based estimators. Nevertheless, an adaptive procedure can be used for the KL-based estimator: at the beginning a high number of repetitions is set, multiple measurements are performed, the measured variance is calculated, and then if the measured variance is lower than the target accuracy, then the number of repetitions can be progressively reduced. The mean-based estimator is not only more accurate than the variance-based estimator, but it is also less complex than both variance-based and KL-based estimators. Complexity evaluations are reported in Table III taking into account the number of real additions and multiplications required to compute an estimate from the samples {y_n} in (3). Also, in Table III“best case” and“worst case” refer to the calculation of the roots of the polynomial in (15). In this perspective the mean-based estimator provides the best accuracy-complexity trade-off among the considered amplitude estimators.

TABLE III COMPLEXITY ANALYSIS C. Group Delay The performance of the proposed group delay estimator is shown in figure 10. It can be seen that the CRB in (25) (dotted curve) and the theoretical variance in (19) (continuous line) converge even with a low number of repetitions. The measured variance, obtained via Monte Carlo simulations, is shown as the dashed curve. The fact that the measured variance is constantly below the CRB is related to the assumptions made to simplify the theoretical analysis (i.e., the Gaussian distribution of the traffic, the absence of channel and hardware impairments, and the ideal synchronization). In other words, the CRB is a valid bound for the theoretical variance in (19) but not for the measured variance because they refer to two different systems (an ideal one for the CRB and the theoretical variance, and a realistic one for the measured variance). However, the fact that the theoretical variance is slightly higher than the measured variance turns out to be an advantage. By setting a priori the target accuracy, it is possible to compute analytically the minimum number of repetitions that provides a theoretical variance below the target variance. Then, by using such value, the measured variance will be lower and will satisfy the accuracy requirement. D. Duration-Accuracy Trade-off Since in Section V-B it has been shown that the mean-based estimator provides the best performance-complexity trade-off among the three investigated estimators for the amplitude response, in the following we consider only the mean-based estimator to assess the trade-off between duration and accuracy. Being the theoretical variance of the mean-based estimator a good approximation for its measured variance, for such exercise it is resorted to the theoretical variance only. As before, a wideband transponder is considered with useful bandwidth equal to 480 MHz, as marked in figure 11 by the two dashed vertical lines. The figure also shows the considered traffic realization that occupies the transponder only partially. The target accuracy for the amplitude response of the transponder s_ampl,target ² has been chosen equal to 0.05 dB, while all the other parameters (except the powers and the bandwidth of the testing signal) have been kept as in Table I and Table II. Three values for the bandwidth of the testing signal have been considered, namely 1 MHz, 2 MHz, and 4 MHz. Also, a resolution of 1 MHz has been considered for the sweeping of the 480 MHz bandwidth of the transponder. The proposed adaptive procedure in accordance with the present embodiment, as detailed in Section II-C, is here compared to a static procedure that dimensions the power of the testing signal by taking into account only the worst-case scenario. In other words, in the static procedure the power of the testing signal is set X = 30 dB below the lowest value of P_tr,t across the whole transponder bandwidth, and then it is kept unchanged during the whole measurement. This is a conservative approach that guarantees that the constraint on the interference level caused to the traffic signal is satisfied throughout the whole measurement. In the proposed adaptive procedure the power of the testing signal is instead adapted step by step (i.e., frequency by frequency) to the traffic power falling in the bandwidth of the testing signal. Also, when no traffic is present, the power of the testing signal is still set to P_t = -30 dB. Table IV compares the performance of the adaptive approach with respect to the static one, for different values of the bandwidth of the testing signal. In both approaches, the maximum and the minimum P_tr,t across the 480 MHz bandwidth of the transponder are of course the same, since they only depend on the bandwidth of the testing signal. In order to provide a practical threshold to the adaptive acquisition of P_tr,t, the traffic is estimated to be absent if P_tr,t < -30 dB. This also provides a lower bound equal to -60 dB to the power of the testing signal. The impact of the adaptive approach can be clearly seen in the maximum and minimum powers of the testing signal P_t: in the static approach it is always constant and determined by the worst-case scenario (i.e., when the traffic power falling in the bandwidth of the testing signal P_tr,t is low), while in the adaptive approach it follows the traffic power profile P_tr,t at a distance of 30 dB. In order to achieve the target accuracy ^s _ampl,target ² = 0.05 dB, the static approach requires many more repetitions with respect to the adaptive approach. Such number is indeed computed in the worst-case scenario, when the testing signal (whose power is dimensioned with respect to the weakest P_tr,t) is affected by the strongest P_tr,t. Since the power and the number of repetitions of the testing signal are kept constant during the whole measurement, the measurement accuracy is much improved with respect to the target accuracy when the testing signal is not in the worst-case condition (i.e., when P_tr,t is not at its maximum). On the other hand, the adaptive approach allows to increase the testing power according to the traffic power, and therefore to reduce the number of repetitions. This directly translates into a significant reduction of the total measurement duration. Also, the measurement accuracy does not depend any longer on the traffic power, but remains constant over the whole measurement. Testing signals with different bandwidths show an irregular trend in their performance, since the relationship among powers, bandwidth, accuracy, and number of repetitions is particularly complex (as can be seen in (8)).

TABLE IV: POWERS AND REPETITIONS FOR THE AMPLITUDE RESPONSE

TABLE V: POWERS AND REPETITIONS FOR THE GROUP DELAY The same exercise can be repeated for the group delay. Table V has been obtained by using the same assumptions used for the amplitude response. The target accuracy is now set to ^s _GD,target ² = 4 ns². Even though the numbers are clearly different from the assessment for the amplitude response, the same time-saving property of the proposed adaptive approach can be seen. It is important to highlight that the purpose of these two examples (one for the amplitude response and one for the group delay) is to exemplify a possible usage of the proposed measurement procedure (detailed in Section II-C) as a design tool. It is up to the system designer to choose the constraints and parameters that are more important and can be used as targets (in these examples, the interference level, the accuracy, and the hardware complexity given by the bandwidth), and the ones that can be set or tuned in a freer way in order to optimize the global system (in these examples, the number of repetitions and the duration). E. Out-of-Band Measurements

It is worth mentioning that the proposed adaptive procedure also allows to measure the knees of the transponder amplitude response and the horns of the group delay. However, since both the amplitude response and the group delay are typically very steep in those regions (as can be seen for example in the ranges [-270,-240] and [240,270] MHz in figure 11), they create significant distortion and attenuation of the testing signal. A way to partially counteract such behaviour is reducing the bandwidth B_t and increasing the power P_t. Indeed, by narrowing its bandwidth, the testing signal sees the amplitude response of the filter as more similar to a flat response, which reduces the linear distortion. It should be noted that features described for a specific embodiment may be combined with the features of other embodiments, unless the contrary is explicitly mentioned. Based on the description and figures that have been provided, a person with ordinary skills in the art will be able to construct a computer program for implementing the described method steps without undue burden. It should be understood that the detailed description of specific preferred embodiments is given by way of illustration only, since various changes and modifications within the scope of the invention will be apparent to the person skilled in the art. The scope of protection is defined by the following set of claims.

Claims

Claims 1. A method for in-orbit testing of a frequency band (100) of a satellite transponder (10),

comprising the steps of

a) at a ground station (200), using data transmission means, transmitting a set of input measurement signals (110, 120) using direct sequence spread spectrum modulation to the satellite (210) on said transponder, the set of input measurement signals comprising a reference signal (110), which is centered on the central frequency of said frequency band (100), and at least one testing signal (120), which is centered on a first frequency of said frequency band, which is different to said central frequency;

b) at the ground station (200), using data reception means, subsequently receiving a corresponding set of output measurement signals (110’, 120’) from the satellite (210); c) at the ground station, using measuring means, measuring parameters that characterize the transponder at the frequencies on which said input measurement signals are centered, the parameters comprising the group delay and/or the amplitude response, wherein the measurement is based on the relationship between the set of input measurement signals (110, 120) and the set of corresponding output measurement signals (110’, 120’);

d) repeating steps a) to c) at least once by changing the frequency on which said at least one testing signal (120) is centered, to a second frequency, which is different to said first frequency;

wherein the transmit power of each input measurement signal (110, 120) depends on the transmit power of at least one data traffic signal that concurrently occupies at least part of a frequency sub-band (112, 122) that comprises the frequency on which said input measurement signal (110, 120) is centered.

2. The method according to claim 1, wherein said input measurement signals comprise a

plurality of testing signals, each being centered on a distinct frequency of said frequency band.

3. The method according to any of claims 1 or 2, wherein said second frequency of each testing signal is selected by the ground station to differ from the first frequency of said testing signal by a pre-provided offset value.

4. The method according to any of claims 1 or 2, wherein said second frequency is selected by the ground station to lie in a frequency sub-band that is occupied by a data traffic signal having low transmit power.

5. The method according to any of claims 1 to 4, wherein the transmit power of each testing signal is selected to be lower than the transmit power of a data traffic signal that concurrently occupies at least part of a frequency sub-band that comprises the frequency of which said input measurement is centered.

6. The method according to any of claims 1 to 5, wherein the measured parameters that

characterize the transponder at the frequencies on which said testing signals are centered are normalized with respect to the measured parameters that characterize the transponder at the central frequency, on which the reference signal is centered.

7. The method according to any of claims 1 to 6, wherein said frequency band comprises the entire transponder bandwidth.

8. The method according to any of claims 1 to 7, wherein the transmit power of said data traffic signal is provided at the ground station.

9. The method according to any of claims 1 to 8, wherein for the direct-sequence spread

spectrum modulation of each input measurement signal, a distinct pseudo-noise sequence is generated at the ground station.

10. A device for testing of a frequency band of a satellite transponder, comprising data

transmission means, data reception means and measuring means, the device being configured for:

a) using the data transmission means, transmitting a set of input measurement signals using direct sequence spread spectrum modulation to the satellite on said transponder, the set of input measurement signals comprising a reference signal, which is centered on the central frequency of said frequency band, and at least one testing signal, which is centered on a first frequency of said frequency band, which is different to said central frequency;

b) using the data reception means, subsequently receiving a corresponding set of output measurement signals from the satellite;

c) using the measuring means, measuring parameters that characterize the transponder at the frequencies on which said input measurement signals are centered, the parameters comprising the group delay and/or the amplitude response, wherein the measurement is based on the relationship between the set of input measurement signals and the set of corresponding output measurement signals; d) repeating steps a) to c) at least once by changing the frequency on which said at least one testing signal is centered, to a second frequency, which is different to said first frequency;

wherein the transmit power of each input measurement signal depends on the transmit power of at least one data traffic signal that concurrently occupies at least part of aa frequency sub- band that comprises the frequency on which said input measurement signal is centered.

11. The device according to claim 10, further comprising pseudo-random number generation means, configured to generate distinct pseudo-noise sequences, and wherein the transmission means are configured to use direct-sequence spread spectrum modulation using one of said generated sequences for the transmission of one of said input measurement signals.

12. The device according to any of claims 10 or 11, wherein it is further configured for carrying out any of the method steps in accordance with claims 2 to 9.

13. A computer program comprising computer readable code means, which when run on a

computer, causes the computer to carry out the method according to any of claims 1 to 9.

14. A computer program product comprising a computer-readable medium on which the computer program according to claim 13 is stored.