Calculation of the maximum power density (averaged over 4 khz) of an angle modulated carrier

Transcription

1 Re. ITU-R SF RECOMMENDATION ITU-R SF * CALCULATION OF THE MAXIMUM POWER DENSITY (AVERAGED OVER 4 khz) OF AN ANGLE-MODULATED CARRIER Re. ITU-R SF ( ) The ITU Radioommuniation Assembly, onsidering a) that administrations are requested to prepare the inmation listed in Appendix S4 to the Radio Regulations (RR) oordination and notifiation purposes; b) that one item of the inmation listed in RR Appendix S4 is the maximum power density per Hz at the input of the antenna averaged over the worst 4 khz band; ) that generalized methods are neessary the alulation of the maximum power density of an angle-modulated arrier, reommends 1. that the methods desribed in Annex 1 be used the alulation of the maximum power density, averaged over 4 khz, of an angle-modulated arrier; 2. that the determination of the maximum spetral power density, with agreement of the other administrations onerned, administrations should make use of the most preise alulation method available to them. ANNEX 1 Calulation of the maximum power density (averaged over 4 khz) of an angle modulated arrier Given below is the method of alulating the power level in the worst 4 khz (W/4 khz). The power density per Hz required by the RR is obtained by dividing this value by FM arrier 1.1 FM arrier modulated by a multi-hannel telephony signal The maximum power spetral density at full baseband loading is determined either by the residual arrier or by the peaks of the ontinuous spetrum, depending upon the nature of the modulation. The power of the residual arrier is given by the expression: P t e ψ mmmmmmw (1) ψ = m2 ε C + C 2 ε + C 4 3 (ε + ε2 + ε 3 ) (2) * Radioommuniation Study Group 4 and 9 made editorial amendments to this Reommendation in 2 in aordane with Resolution ITU-R 44.

2 2 Re. ITU-R SF In equation (2), m is the multi-hannel r.m.s. modulation index and the onstants C, C 2 and C 4 desribe the prevailing pre-emphasis harateristi in the general expression the pre-emphasis: p(ƒ/ƒ h ) = C + C 2 (ƒ/ƒ h ) 2 + C 4 (ƒ/ƒ h ) 4 (3) where ƒ is the speifi baseband frequeny under onsideration, to be given in the same units as ƒ h. In the range ε ƒ/ƒ h 1, the ITU pre-emphasis harateristi (Reommendation ITU-R F.275) is well approximated by: Thus, a system with ITU pre-emphasis: p(ƒ/ƒ h ) = (ƒ/ƒ h ) (ƒ/ƒ h ) 4 (4) ψ 2 m ε 2 3 ( ε +.25 ε +.25 ε ) (5) where m = ƒ /ƒ h. Note 1 Administrations should make available details of the spetrum shape and the value of the oeffiients used in equations (2) and (3), detailed oordination purposes. The maximum power of the spetral density in the ontinuous part of the spetrum an be obtained approximately from Figs. 1 and 2. FIGURE 1 Maximum spetral density of signal (distributed omponent) frequeny modulated by Gaussian noise (Pre-emphasis of Reommendation ITU-R F.275) 11 Normalized spetral density (power per unit bandwidth/total arrier power), Vmax (db) A B C D G J 3 H E F Normalized frequeny deviation, m = f /fh h 1 Curves A: ψ =.1 Curves B: ψ =.2 Curves C: ψ =.4 Curves D: ψ = 1. Curves E: ψ = 2. Curves F: ψ = 4. Curves G: ψ = Curves H: small-deviation approximation Curves J: large-deviation approximation Values standard radio-relay systems (as labelled) are the following base band limits: 1 12 hannels khz 1 96 hannels khz 1 26 hannels khz D1 FIGURE 1...[D1] = 15 CM

3 Re. ITU-R SF FIGURE 2 Equal ontour of Vmax (maximum spetral density) of the FDM-FM signal max (Pre-emphasis of Reommendation ITU-R F.275) Ratio of minimum to maximum baseband frequenies, ε Multihannel modulation (= f /fh), m h 8 D2 FIGURE 2...[D2] = 11 CM In the equations and the figures, the symbols have the following meanings: ƒ : total power of arrier (W) multi-hannel r.m.s. deviation (Hz) f f f f d d d ( 15+1 log N )/2 ( 1+4 log N )/2 (2.6+2 log N )/2 N 24 > N 6 > N ( 15+1 log N )/2 f 1 or d ( 1+4 log N )/2 fd 1 12 N 24 N < 24 ƒ d : N : ƒ h : ƒ 1 : r.m.s. test tone deviation (Hz) number of hannels top frequeny of the baseband (Hz) bottom frequeny of the baseband (Hz) m : multi-hannel modulation index (= ƒ /ƒ h ) ε = V max = W max : ƒ 1 / ƒ h W max ƒ h / P t maximum spetral power per unit bandwidth (W/Hz).

4 4 Re. ITU-R SF For arriers whih 1 < N 12, the maximum power density per 4 khz is approximated by the expression: P t os 2 m b 1.5 mmmmmmw/4 khzmmmmmm m b < 1 (6) m b : total power of the arrier (W) peak modulation index (rad) due to a dbm test tone in the highest frequeny baseband hannel. 1.2 FM arrier modulated by a multi-hannel telephony signal and an energy dispersal signal of a triangular wavem with fixed amplitude Triangular wave dispersal systems are normally designed to ensure that the maximum power spetral density per 4 khz entred on the arrier frequeny is maintained within 3 db of the fully loaded value. The power spetral density entred on the arrier frequeny is given by: P t F 4 mmmmmmw/4 khz (7) F : total power of the arrier (W) peak-to-peak frequeny deviation due to the energy dispersal signal (Hz). Note 1 Equation (7) assumes the use of a perfetly linear triangular dispersal wavem. 1.3 FM arrier modulated by a television video signal For the ase where an energy dispersal signal of a triangular wavem is superimposed on the video signal, the maximum power density per 4 khz in the worst ase is given by: P t F 4 mmmmmmw/4 khz (8) F : total power of the arrier (W) peak-to-peak frequeny deviation due to the energy dispersal signal (Hz). Note 1 Equation (8) assumes the use of a perfetly linear triangular dispersal wavem. Negligible error results from this assumption urrent FM-TV transmissions. For the ase where there is no modulation by video and energy dispersal signals, the maximum power density per 4 khz in the worst-ase is given by: P t mmmmmmw/4 khz 2. Phase modulated (PM) arrier modulated by a multi-hannel telephony signal When a PM arrier is modulated by a multi-hannel telephony signal, the maximum power density is found at the entre frequeny of the arrier. This is true if the top baseband frequeny is muh larger than the bottom baseband frequeny. An expression the maximum power density assuming this ondition is given as follows: βσ a 2: P t (βσ a ) ƒ h 3 2π 4 mmmmmmw/4 khz (9)

5 Re. ITU-R SF ßσ a < 2, the maximum power density per 4 khz is the sum of following two terms: ontinuous spetrum: P t S() 4 mmmmmmw/4 khz (1) S() an be found from Fig. 3 whih gives values the ratio of the total arrier power-to-power density in a bandwidth of ƒ h (Hz). residual arrier: P t exp { (βσ a ) 2 }mmmmmmw (11) βσ a : β : σ a : total power of the arrier (W) multi-hannel phase deviation (rad) r.m.s. test tone phase deviation (rad) loading fator of the multi-hannel telephony signal. σ a = ( 15+1 log N )/2 1 ( 1+4 log N)/2 1 N 24 N < 24 N : ƒ h : hannel number top baseband frequeny (Hz). FIGURE 3 Power density at the entre frequeny of ontinuous power spetrum of PM arrier in a bandwidth of f h 2 Ratio of total arrier power-to-power density in a bandwidth of fh, h S() fh h(db) Multi-hannel phase derivation, βσ a (rad) FIGURE 3...[D3] = 12 CM

6 6 Re. ITU-R SF PSK arrier The maximum power density per 4 khz of a PSK arrier modulated by a digital energy dispersal signal of a PN (pseudo-noise) sequene is given by (see Note 1): when the repetition yle of the PN sequene is longer than 25 µs; and by: P t L + 1 P t (4 / B)mmmmmmW/4 khz (12) mmmmmmw/4 khz (13) 1 / L t L 2 when the repetition yle of the PN sequene is equal to or less than 25 µs, B : L : t : total power of the arrier (W) symbol rate (symbol/s) length of the PN sequene (symbol) symbol duration (s) 4 1 / L t = integer part 4 1 / L t The expressions given above apply to the ase of PN sequene modulation of a PSK arrier and would be appliable to ases where the PSK message signal is ontinuously overlaid by a PN srambling sequene. Large errors ould result from the appliation of these expressions to systems, suh as TDMA, in whih the preamble portion of the signal, added parity bits, and the like, are not srambled. Furthermore, in multi-phase systems the spetral unimity expeted from PN energy dispersal may be destroyed by the ommon operation of differential enoding. It should be noted that the above treatment does not give any guidane on the assumptions whih should be made to over the ase of PSK systems without energy dispersal, under onditions in whih bit patterns ould repeat in suh a manner as to onentrate the power in a relatively small number of spetral lines. For the remaining radio-relay systems shown in Fig. 1, the maximum power spetral densities are valid any of the baseband limits speified in Reommendation ITU-R F.38, sine their β values are relatively onstant and math those used in Fig. 1. Note 1 The equations (12) and (13) an be used in alulating the maximum power density per 4 khz of a PSK arrier modulated by a digital energy dispersal signal of a PN sequene, by simply replaing the number 4 by the number 4. Equations (12) and (13) would then be, respetively, appliable to a PN sequene longer than 25 µs and to a PN sequene equal to or less than 25 µs. If the symbol rate, B, is less than 4, the maximum power density per 4 khz is the total power of the PSK arrier.