Smrati Gupta, … Ricard Alegre-Godoy, in Cooperative and Cognitive Satellite Systems, 2015
12.2.3 Applications that could benefit from cognitive DSS
Satellite communication systems support a wide range of applications which need to satisfy a number and variety of requirements. As an illustration, a communication between an earth station and a lunar module through a satellite needs to be reliable, while transmissions from a control tower to airplanes must provide very low latencies. In this sense, the applications that could benefit more from CR in DSS are those demanding high reliable throughput and availability which are the main benefits provided by CR DSS. Typically these applications are related to the provision of fixed and mobile broadband multimedia services, for example, high definition (HD) broadcast [30] and HD interactive multimedia [31]. Other applications that need high throughput and availability which can benefit from CR in DSS are machine to machine communications [32], Internet of the things [33] and cloud computing.
P.L. Bargellini, B.I. Edelson, in Space and Energy, 1977
FUTURE TRENDS
Satellite communications systems will continue to expand because traffic growth forecasts indicate the need for greater communications capacity. For example, the yearly average growth rate in the INTELSAT system is expected to be at least 15 percent in the next decade. Satellites will become more competitive vis-a-vis other systems—in terms of magnitude of communications capacity, high reliability, flexibility, and progressively increasing cost effectiveness.
The two natural resources, the electromagnetic spectrum and the geosynchronous orbit, do, of course, constitute the bounds of satellite communications growth, and will be briefly discussed in the following sections.
Spectrum Utilization
As latecomers among the users of the electromagnetic spectrum, satellites have had difficulty in being assigned “optimum” spectral regions. The assignment to fixed satellite services of 500 MHz at around 4 and 6 GHz for the down- and up-links, respectively, is close, but not optimum since the overall noise power density exhibits its broad minimum between 1 and 2.5 GHz. In addition, power flux density limitations were imposed on satellite systems because the above-mentioned 4- and 6-GHz frequency bands had to be shared with existing terrestrial microwave systems. In spite of these constraints, satellite systems have been successful, the most serious limitation probably being that which derives from the constraints on the site selection for the earth terminals.
All commercial satellite systems at this time operate at 4 and 6 GHz. New frequency bands were assigned to satellite communications at the World Administrative Radio Conference of 1971. There are two new pairs of bands: one at 11 and 14 GHz, and another at 19 and 29 GHz. At 11 and 14 GHz, 500 MHz is available for the down- and up-links, respectively. These bands will have to be shared with terrestrial services and are therefore subject to power flux density limitations. There is 2,000 MHz of shared spectrum available between 17.7 and 19.7 for down-links and another 2,000 MHz between 27.5 and 29.5 GHz for up-links. Finally, bandwidths of 1,500 MHz will be exclusively available for down- and uplinks, respectively, between 19.7 and 21.2 GHz and 29.5 and 31.0 GHz.
Aside from frequency reuse techniques by separate antenna beams applicable in principle to any frequency band, the WARC 1971 frequency assignment results in bandwidth availability eight times greater than that at 4 and 6 GHz. The decrease in antenna size and the requirements of sharing, which are reduced or eliminated, make the new frequency bands extremely attractive. Future plans for INTELSAT and domestic or regional systems include the use of the 11/14-GHz bands as the RF hardware at these frequencies is in an advanced state of development and ready to be space-qualified. The 19/29-GHz bands will be used later as the art progresses and as the traffic needs require. Attenuation resulting from rain will be counteracted by diversity techniques and by adequate power margins when possible.
The use, and reuse, of all of the above-mentioned frequency bands will provide striking improvements. It has been pointed out (Pritchard, 1972; Knopow, 1974) that communications capacities of the order of 100,000 telephone circuits per satellite or greater can be attained without exceeding the mass and weight limits already encountered in satellites of the current generation (e.g., the INTELSAT IV or IV-A). The increase in capacity, which amounts to at least one order of magnitude, will be made possible by the expanded adoption of frequency reuse techniques via multibeam antennas and orthogonal polarization, and also by the introduction of the following technologies:
(a)
3-axis body stabilization,
(b)
higher efficiency solar cells,
(c)
higher efficiency energy storage devices,
(d)
electrical propulsion for stationkeeping and positioning,
(e)
onboard switching combined with time- and space-domain multiple-access techniques,
(f)
linearized transponders,
(g)
hybrid modulation techniques,
(h)
source encoding, and
(i)
intersatellite links.
The consequence of the above, plus the overall increased reliability and longer lifetime in orbit of the spacecraft, will be a progressive reduction of the space segment cost of satellite systems. Further overall cost reductions will derive from the application of new concepts in earth terminal design (Pollack, 1974). Finally, it is expected that the development of new space transportation systems (space shuttle and tug) will also be beneficial.
Orbit Utilization
On the basis of the following assumptions, it can be stated that the noise in each circuit is produced by radiation spillover:
(a)
ideal modulation-demodulation processes;
(b)
neglect of thermal noise, i.e., no power limitations;
(c)
finite available communications bandwidth;
(d)
sharing of this bandwidth by satellites uniformly spaced in equatorial synchronous orbit; and
(e)
earth station antennas as uniformly illuminated apertures.
Thus (Bradley, 1968), the noise power is constrained by an upper bound that is a function of a single geometric variable, that is, the satellite spacing. Neglecting differences in slant range and choosing an optimum spacing Δθ = λ/D results in a communications capacity per unit angle of
(6)C=2Dλ bps/Hz/rad
Consequently, the information rate (bps) which can be handled by a segment of synchronous orbit spanning θ rad with an available (common) bandwidth B is
(7)R=2DλBθ bps
A bandwidth of 500 MHz and antennas of 30-m diameter at 6 GHz would yield a theoretical global capacity of 3.77 × 1012 bps, or roughly 108 telephone channels.
In practice, these theoretical results must be corrected to take into account mechanical problems of tight orbital spacing, real modulation-demodulation processes, effects of thermal noise, and actual antenna configuration. While the first three items would lead to lower communications capacities, the last item leads to an increase in communications capacity.
Other possibilities of augmenting the communications capacity are intersatellite relaying, increase in the allowable interference ratio, channel interleaving, reversed use of frequencies, and pseudostationary satellites and 2-dimensional orbit space (Bargellini, 1969).
Since the two problems of spectrum and orbit utilization are inseparable, studies of all these possibilities are in progress.
Yasuo Hirata, Osamu Yamada, in Essentials of Error-Control Coding Techniques, 1990
6.2.1 Applicable FEC Codes and Their Performance
In satellite communication systems, FEC technique is often employed to effectively improve the transmission quality which is degraded due to the interference and the power limitation of the system. Furthermore, since there is a considerable amount of propagation delay (250–300 msec) in satellite communication systems, FEC technique tends to be more widely used than the ARQ technique, which requires retransmission of the data. One of the remarkable features of satellite communication systems is that the bit error occurs randomly in general due to Gaussian noise, which is considered to be of significant advantage when applying FEC codes.
A recent trend in digital satellite communication systems is to apply the powerful FEC code with large coding gain to use the limited satellite power as efficiently as possible. Since the usable frequency band is also becoming severely limited, it is more desirable to apply the FEC code of high coding rate with high coding gain.
Another important factor to be taken into account is the hardware complexity of the FEC codec. In particular, for high-speed transmission systems such as the time division multiple access (TDMA) in which many channels are accommodated by the single carrier, selection of the applicable FEC codes will be severely restricted by the factor of availability of the high-speed FEC codecs.
Table 6.1 gives a guideline of coding gain which is generally expected by applying typical random error-correcting codes to the additive white Gaussian noise (AWGN) channels employing coherent phase shift keying (PSK).
Table 6.1. Coding Gain of Typical Random-Error-Correcting Codes
| Coding gain (dB) | ||
|---|---|---|
| FEC | BER = 10−5 | BER = 10−8 |
| Concatenated coding (RS + Viterbi) | 6.5–7.5 | 8.5–9.5 |
| Sequential decoding (soft decision) | 6.0–7.0 | 8.0–9.0 |
| Block code (soft decision) | 5.0–6.0 | 6.5–7.5 |
| Viterbi decoding (hard decision) | 4.0–5.5 | 5.0–6.5 |
| Sequential decoding (hard decision) | 4.0–5.0 | 6.0–7.0 |
| Block code (hard decision) | 3.0–4.0 | 4.5–5.5 |
| Block code (threshold decoding) | 2.0–4.0 | 3.5–5.5 |
| Convolutional code (threshold decoding) | 1.5–3.0 | 2.5–4.0 |
Note: AWGN Channel, Coherent PSK.
Source: Clark and Cain (1981).
As indicated in the table, the concatenated coding of the convolutional coding–Viterbi decoding with the Reed–Solomon code and the convolutional coding–sequential decoding offer very high coding gain. These coding schemes have not yet been widely used in satellite communication because of the hardware complexity, except for the low-speed data-transmission systems. Block coding with soft decision decoding also provides high performance, i.e., coding gain of around 2 dB higher than that of block coding with hard decision decoding. However, this scheme is not widely applied due to the hardware complexity at present.
In view of the conditions previously mentioned, it can be said that the most viable FEC technique is the convolutional coding–Viterbi decoding (soft decision) which provides high coding gain with reasonable hardware complexity (Heller and Jacobs, 1971). In fact, the convolutional coding–Viterbi decoding is being widely employed in a variety of digital satellite communication systems. Research and development works are also being continued in various organizations to overcome the hardware complexity of the codec which can handle the high-speed data such as 100 Mbit/sec. In addition, the Viterbi decoder for the convolutional code of higher code rate such as 3/4 and 7/8 has been implemented (Yasuda et al.,1983b).
Figure 6.5 shows the Eb/No necessary to obtain the BER of 10−6 for typical FEC codes that have been already applied to satellite communication systems. In the figure, the performance of Viterbi decoding is for the convolutional code with the code rate of 1/2 and the constraint length of.,and for the punctured codes derived by this convolutional code (Cain et al.,1979; Yasuda et al.,1984). The abscissa in the figure is the bandwidth expansion ratio expressed in dB, which corresponds to the reciprocal of the code rate. Figure 6.5 also shows the Eb/No bound calculated from the channel cut-off rate which gives the upper bound of code rate R to provide reliable communication by the reasonable amount of decoding complexity, together with the Eb/No requirement for the uncoded coherent PSK system. Figure 6.5 demonstrates that Viterbi decoding provides higher coding gain than other FEC codes. The figure also shows that the concatenated coding of Viterbi decoding with the Reed-Solomon code can offer very high coding gain close to the theoretical bound, which is around 2 dB higher than in Viterbi decoding.
Fig. 6.5. Required Eb/No for typical FEC codes (BER = 10−6). K: Code constraint length; DEC: double error-correcting.
As is clear from the theoretical bound in the figure, the required Eb/No to get the specified BER tends to decrease according to the decrease of the code rate. In the region where the code rate is lower than 1/2, however, the degree of the Eb/No reduction becomes smaller. This implies that the remarkable increase of the coding gain cannot be expected even if the FEC code with a code rate lower than 1/2 is applied. Accordingly, there is no advantage to using the FEC code with lower code rate in satellite communications in which the usable frequency band is severely limited.
Reinaldo Perez, in Wireless Communications Design Handbook, 1998
1.1 Overview of Satellite Communications
A typical satellite communications system is shown in Figure 1.3. In the figure, the terrestrial network can correspond to the voice, data, or video signal from a public switching telephone network (PSTN) or from a cellular network. An earth transmitting station, also known as a gateway in cellular technology, is used for processing signals before they are transmitted to the satellite via an uplink protocol. Signals that arrive at an earth station through a terrestrial or cellular network include voice channels, analog channels, and data from computers and other digital data sources; these signals, which are usually called baseband signals, are deterministic or random. These signals are combined into a complex baseband signal. The combination of diverse signals coming from different sources is called multiplexing. The multiplexed signal is converted to a suitable form of signal for transmission; this is done by changing the amplitude, phase, or frequency of a sinusoidal signal. This process is known as modulation and can be analog or digital in form. Most preset modulation is done digitally. The main advantages of digital modulation are its error-correcting capability, insensitivity to noise, and nonlinearity.
Figure 1.3. Satellite communications network.
A digital satellite communications system is shown in Figure 1.4. The analog signals for the process of digital modulation are converted into binary data. These binary data are coded for error correction. The coded binary bits are modulated by several digital modulation schemes using RF modulators. These signals can be subjected to further spread-spectrum modulation for possible interference rejection. The signals are then up-converted and amplified and transmitted to the satellite via an antenna system. These signals are sent through space, where they are subject to several kinds of losses and arrive at the satellite’s antennas. Signals from other earth stations or gateways also arrive at the same time as the satellite antenna. This simultaneous access by several earth stations to the satellite transponder for the relay is achieved by a method of multiple access. The signals are down-converted at various transponders, then amplified and transmitted back to earth stations by multiple beams. At the earth stations, the signals go through low-noise amplifiers, are down-converted, despread, demodulated, and decoded at channel and source-to-baseband signals, and then demultiplexed and sent to the appropriate destination.
Figure 1.4. Components of a digital satellite communications system.
Rajat Acharya, in Satellite Signal Propagation, Impairments and Mitigation, 2017
7.2.1.7 Modelling of frequency scaling
In a practical satellite communication system, the uplink frequency is higher than the downlink frequency. Due to this difference, the attenuation value observed at the ground station over the downlink cannot be directly used for compensating the attenuation in the uplink. So to compensate the uplink fade in an open loop configuration, in a condition when the uplink fade information is not available to the ground terminal by any other mean, an efficient fade mitigation system needs precise estimation of the attenuation in uplink. This information is derived from the knowledge of the attenuation in downlink frequency. This is called frequency scaling of the attenuation. The scaling of the attenuation has an obvious dependence upon the frequencies in question. The knowledge of the relationship between attenuation values at different frequencies is needed, not only for carrying out the compensation activity with suitable FMT, but also for a broader use of evaluation of system performance based on measurements at other frequencies. The scaling relation is typically expressed by the Frequency Scaling Ratio, Rfs, which is the ratio between attenuation values, A1 and A2, expressed in dB, for two different frequencies, f1 and f2.
Two different frequency scaling approaches can be considered, viz. the long-term frequency scaling and instantaneous frequency scaling (IFS). The long-term frequency scaling gives the ratio of the attenuation values at two different frequencies for the same probability of occurrences. It is thus a statistically derived number which allows the calculation of the long-term cumulative distribution function of attenuation at one frequency, when the same for the other frequency is known or measured. Expressed mathematically, Rfs becomes,
(7.51)Rfsp=A2pA1p
The total long-term availability and performance of a system operating at one frequency can be predicted from attenuation measurements made at another frequency, using this ratio. Since attenuation is, in principle, the combination of contributions from different causes, like rain, gaseous absorption, clouds and scintillation, these contributions depend on frequency in different ways. So, any particular value of attenuation in one frequency may be arrived at by different contributions from the individual factors of the attenuation. This is the main cause why the same attenuation in one frequency does not always map to the same attenuation in the other. This leads to the variations in the long-term frequency scaling ratios. It is thus intuitive that the frequency scaling technique is most satisfactory when one cause predominates.
Long-term frequency scaling has been applied in the past to extrapolate rain attenuation mainly from Ku to Ka band. In Ku band, the nonrainy contributions to attenuation are generally negligible and the frequency scaling technique works particularly well. Further, the path independent nature of the ratio makes the frequency scaling more stable. One of the long-term frequency scaling models has been recommended by the ITU-R (ITU-R, 2015a) and is given below
(7.52)A2A1=gA2gA=φ2φ1q
Here
φf=f21+10−4f2andq=1−0.00112φ2φ1φ1A10.55
The term ‘A1’ represents the attenuation in frequency f1, while A2 represents equiprobable value of attenuation in the frequency f2. Both f1 and f2 are expressed in GHz. The term φ increases with frequency and therefore the ratio (φ2/φ1) > 1 when f2 > f1. (f1A)0.55 increases with A1 for a given f1. Thus, the frequency scaling ratio, Rfs = A2/A1, changes conditioned on the value of A1.
The IFS, on the other hand, relates simultaneous attenuation values at different frequencies. Thus, it provides a one to one mapping between the two attenuations at any instant of time. Modelling of the IFS is needed for real time adaptive FMT. That is, the timely activation of a fade countermeasure in the uplink relies on the attenuation measured on the downlink and then conversion to the higher uplink frequency. Such estimation is done at every instant and continued over the period of the attenuation. We shall learn about the particular fade mitigation techniques (FMT) later in this chapter.
The IFS ratio of rain is a stochastic quantity and has been empirically characterized for the variability around its mean values (Sweeny and Bostian, 1992; Laster and Stutzman, 1995). IFS ratio Rfs can be considered to be a log-normal variable about the mean, as long as rain dominates the attenuation phenomenon. The conditional mean, ‘μIFS‘ conditioned on the attenuation value experienced at the lower of the two frequencies, is very close to the long-term ratio, evaluated on an equiprobability basis. It also fits well with a power law expression (Matricciani and Paraboni, 1985)
(7.53)μIFS|A1=aA1−b
where a and b are the frequency-dependent regression parameters. The standard deviation σIFS, however, is found to be almost independent of the mentioned attenuation and its value is given by:
(7.54)σIFS=0.13
IFS fluctuations are essentially due to the natural variability of the various phenomena. As the natural variations of the individual contributors are slow, they do not change abruptly over shorter interval of time.
Reinaldo Perez, in Wireless Communications Design Handbook, 1998
6.0 Introduction
At the heart of a satellite communications system is the transponder. The transponder consists of input and output filters, up and down converters, phase-locked loops, and traveling wave tube amplifiers (TWTAs.) More modern transponders systems are using solid state power amplifiers (SSPAs). We now consider the nonlinear behavior of the transponder. A block diagram representation of a typical transponder was shown in Figure 5.36.
Let the input of the transponder be represented by
(6.1)Sit=Acosωct+fθ,
where ωc = 2πfc is the angular carrier frequency and phase of the input signal. The transponder output can be represented as
(6.2)Sout=gAcosωct+fθ+fA.
If g(A) and f(A) are independent of ωc, let Ak(t), ωc + ωk, and ƒ(θk(t)) denote the envelope, the angular carrier frequency, and the phase of the kth carrier. ωc is the midband frequency or center frequency, which can take any value within a transponder bandwidth. For m number of modulated carriers, access to a transponder input can be represented by
(6.3)Sit=∑k=1mAktcosωc+ωkt+fkθt=∑k=1mAktcosωkt+fkθtcosωct−∑k=1mAktsinωkt+fkθtsinωct=Xtcosωct−Ytsinωct=X2+Y2cosωct+tan−1YX=ReX2+Y2expjωct+jtan−1YX.
The corresponding transponder output is
(6.4)Soutt=RegX2+Y2.expjωct+tan−1YX+jfX2+Y2=RegX2+Y2.expjfX2+Y2X+jYX2+Y2expjωct.
Define the double Fourier transform
Luv=∫−∞∞∫−∞∞gX2+Y2X2+Y2expjfX2+Y2X+jYexp−juX−jvYdxdy,
which means
gX2+Y2expjfX2+Y2X+jYX2+Y2=12π∫−∞∞∫−∞∞LuvexpjuX+jvYdudv.
Therefore,
Soutt=12π2Reexpjωct∫−∞∞∫−∞∞LuvexpjuX+jvYdudv.
After further mathematical manipulations,
Soutt=Reexpjωct∑K1=−∞∞∑K2=−∞∞⋯∑Kn=−∞∞expjK1ω1t+fθ1+jK2ω2t+fθ2+…+jKnωnt+fθn.Nk,
where K1, K2, … ,Kn can be zero or any integers either positive or negative, and
(6.5)Nk=12π2∫−∞∞gX2+Y2X2+Y2expjfX2+Y2X+jY∏ℓ=1mJKℓAℓu2+v2×expj∑ℓ=1mKℓtan−1uvexp−jux−jvydxdydudv,
where JK is the Kth-order Bessel function. Using the polar coordinate transformation
X=ρcosξ,u=γsinηY=ρsinξ,v=γcosη
and performing the integration on ξ and η simplifies the expression to the following two cases. For K1 + K2 + … + Kn = 1,
Nk=∫0∞∫0∞γ∏ℓ=1mJKℓAℓγρgρexpjfρ⋅J1γρdγdρ
and for K1 + K2 + … + Kn ≠ 1, N(k) = 0.
Finally, the output of the transponder can be expressed as
(6.6)Soutt=ReNkexpjω¯t+fℓ¯θ,
where
fℓ¯θ=∑ℓ=1mKℓfℓθω¯=ωc+∑ℓ=1mKℓωℓ.
In the case of numerical computation, the factor g(ρ)exp[jf(ρ)] can be approximated by
gρexpjfρ=∑ℓ=1LbℓJ1αℓρ,
where L is the number of coefficients needed, J1 is the first-order Bessel function, and α = 2π/(period of the Fourier series). For a given transponder, the characteristics g(ρ) and f(ρ) are known. Since g(ρ) and f(ρ) are given, the coefficients bℓ can be obtained by an approximation, and N(k) reduces to
(6.7)Nk=∑ℓ=1Lbℓ∏ℓ=1mJKℓαℓAℓ,
which outlines the amplitude for each input signal. The bℓ are determined by best fit from the input data of g(ρ) and f(ρ) in terms of least-square error. Computer programs can calculate Sout(t) when Si(t), g(ρ) and f(ρ) are given.
Kinnosuke Kawakami, in Space and Energy, 1977
Publisher Summary
This chapter focuses on the Japanese Domestic Communications Satellite System. Japan’s space development program began in 1967 when the National Space Activity Council was instituted. Space development was later divided into two categories, namely, the scientific satellite program and the practical satellite program. Research since then has been carried out on rockets, satellites, missions, and ground facilities. The chapter also discusses Japan’s telecommunication community, radio wave broadcasting, and newly developed 12 GHz TV receiving system. In Japan, researches and developments are being conducted by NTT with the intention of coping with the increase in demands of public communications.
John R. Veastad, in Space and Energy, 1977
Publisher Summary
This chapter focuses on the Norwegian domestic communication satellite system. Radio communication for offshore oil rigs has for years been carried out by conventional medium frequency (MP) radiotelephony, usually on a one-channel simplex basis. This has been the method used in the Mexican Gulf and other well-known areas, and it was also the most likely means of communication to and from drilling platforms, barges, and supply vessels in the North Sea. The Norwegian Telecommunications Administration (NTA) started discussions on the possible use of satellites with the North Sea operators early in 1972. The reliability of the INTELSAT system today is beyond question. Many attempts have been made, within and outside Norway, at comparing costs between troposcatter systems and satellite systems in the North Sea. The need for additional drilling production facilities can be expected. The flexibility of the satellite system is of great importance for future development in the North Sea.
P A Bradley BSc MSc CEng MIEE, J A Lane DSc CEng FIEE FInstP, in Telecommunications Engineer’s Reference Book, 1993
9.10 Propagation and co-channel interference
The increasing need for different microwave terrestrial systems, satellite communications and broadcast systems to share frequencies has produced a correspondingly greater likelihood of mutual interference. The development of interference prediction methods leading to the efficient co-ordination of communication systems sharing a frequency band requires modelling the transmission loss due to all possible propagation mechanisms as a function of frequency, distance and time percentage. Until recently, the primary concern had been for situations occurring less than 1% of time, for which the main clear-air mechanisms are ducting and layer reflection. However the small antenna systems used in new services have generated a need for time percentages up to 20%, for which diffraction and tropospheric scatter are important.
In addition, scattering from different types of hydrometers can be important at frequencies above about 10GHz. This propagation mechanism may couple energy into the receiving antennas of unrelated systems, despite precautions such as site-shielding. Up to now there have been only very limited data available to develop prediction models.
A third group of studies in response to this general area has reflected that an accurate estimation of diffraction effects due to buildings and terrain is essential in the planning of radio services in the microwave bands. Significant data exist from diffraction over large-scale features of terrain, e.g. hills and mountains, which are of major concern at VHF and UHF, but little data are available on the diffraction losses for microwaves that can be produced by buildings or trees which could be used to shield antennas from interference signals.
All these aspects have been studied in recent years in a collaborative European Project ‘COST 210’(Influence of the Atmosphere on Interference Between Radiocommunication Systems at Frequencies Above 1 GHz). The full report should be consulted for details of the procedures developed.
K. Giridhar, … Ronald A. Iltis, in Control and Dynamic Systems, 1996
5.1.3 LMS Algorithm for Fading Channels
Channel fading is often encountered in wireless applications such as mobile radio and satellite communication systems. To recover the data stream, the equalizer must track the channel variations over time. The LMS algorithm in the previous section is inappropriate for such fading channels since the gain μ(k) exponentially decays to zero and prevents any further adaptation. A periodic restart of the LMS adaptation process is possible, as proposed for recursive least-squares (RLS) adaptive algorithms [6]. Instead, we propose a modified LMS adaptive scheme which can estimate and continually track the channel coefficients.
Consider the conditional measurement update in Eq. (45). By converting the covariance form update to information form, the Kalman gain term can be written as
(65)1σi2k|k−1Pik|k−1hiHk=Pik|khiHk1σn2
where Pi · (k|k) is the filtered error covariance matrix. This follows directly by post-multiplying both sides of Eq. (46) with the column vector hiH(k) to obtain
(66)Pik|khiHk=I−Pik|k−1hiHkhikhikPik|k−1hiHk+σn2Pik|k−1hiHk=Pik|k−1hiHkσn2σi2k|k−1
and then dividing both sides by σn2.
Next, assume that the one-step transition matrix is F = I, and that the process noise covariance matrix Q = 0 in the underlying AR model in Eq. (2). Let max be the index of the data subsequence with the largest metric. An approximate update for the KF covariance matrix is given by
(67)σn2Pmax−1k|k=∑l=1khmaxHlhmaxl+σn2P−10|0.
This update corresponds to the ordinary KF covariance equation (with known transmitted sequence) if the subsequence with the largest metric is assumed to be correct. For large values of k, it is reasonable to expect that the estimation errors become independent, and thus Pmaxk|k can be approximated by a diagonal matrix. We thus define
(68)μjk=σn−2Pmaxk|kjj≈1∑l=1k|hmaxlj|2+σn2(P0|0jj
where the subscript jj refers to the jth diagonal entry in the matrix and |(hmax (l))j|is the modulus of the jth element of hmax(l).9 The resulting diagonal matrix
(69)Λk=diagμ1k,μ2k,…,μNfk
contains a set of gains which resemble the normalized LMS algorithm [8]. These gains can be computed recursively according to
(70)1μjk=|(hmaxk)j|2+1μjk−1,j=1,…,Nf.
However, this update will cause the pseudo-Kalman gains to dacay to zero at the rate ≈ 1/k. To prevent this, a constant 0 < λ ≤ 1 is introduced to modify the update as follows:
(71)1μjk=|(hmaxk)j|2+λμjk−1.
The choice of λ decides the extent of influence the past gain values have on the present μj(k); it is usually referred to as a “forgetting” or finite-memory factor. For fading channels, λ is typically chosen in the range of 0.70-0.90, while for a slowly varying or time-invariant channel, the ideal choice would be λ ≈ 1.0.
Hence, the error-covariance updates are eliminated, and the entire MAP filter bank is updated with the single pseudo-Kalman diagonal gain matrix Λ(k). The only other modification involves the conditional innovations covariance update in Eq. (36). From Eqs. (65) and (68),
(72)Pik|k−1hiHk=σi2k|k−1ΛkhiHk≈σi2k|k−1Λk−1hiHk
where we have assumed that Λ(k – 1) ≪ I and λ is sufficiently close to unity. Using this expression for Pik|k−1hiHk in Eq. (36), we obtain
(73)σi2k|k−1=hikσi2k|k−1Λk−1hiHk+σn2=σn21−hikΛk−1hiHk.
Again, the notation from the previous section (for the time-invariant channel model) can be used to describe the various quantities. For example, the above innovations covariance is denoted by σi2k−1 and the measurement estimate becomes the conditional update
(74)f^ick=f^ik−1+ΛkhiHkrk−r^ik.