3.0. FDMA LINK

As with CDMA, we first derive the FDMA channel capacity for a fixed-beam system and then modify the resulting expressions for the FDMA scan-beam system. The main differences between the two multiple-access systems is that in FDMA there is a higher sensitivity to fading, no self-interference, and more bandwidth is required to accommodate Doppler, oscillator frequency drift and the bandwidth expansion due to FEC coding.

FDMA link fading

In contrast to CDMA, where fading impairs transmission only when reflections have time delays shorter than a code chip period, the FDMA signals suffer from both phase and amplitude fading.

In the system under discussion the bit-rate of the vocoded voice is 2.4 to 4.8 kbps. The coded voice undergoes FEC, using a rate 1/2 or 3/4 convolutional code. The FEC coded voice signal is then QPSK modulated. The signal fade must be compensated by increasing the transmitted power or by using a more powerful code. Lower rate codes require more signal bandwidth, but then there is the danger of reaching the bandwidth limitation sooner than the power limitation. Therefore one must be careful not to overburden the system with too much error protection.

For a postulated error rate of 10^-3, and with no fading, the required Eb/No ratio with rate 1/2 FEC is about 5 dB (including equipment losses). Under Ricean fading conditions, with fading factor k = 10, the required Eb/No ratio increases. The additional delta in Eb/No, required to achieve the same bit error rate, on the loss in detectability due to fading, is denoted by X. It enhances the required bit energy/noise density ratio for the normal path condition.

3.1. FDMA Single Beam

The total noise/desired carrier ratio is
(N / C)_r = ( N / C )_u + (N / C )_d + ( N / C )_IM + ( N / C )_ADJ       (22)

For arbitrary modulation, using (4) and solving for the number of channels m gives

(23)

where B_ch is the predetection channel bandwidth.

For digital modulation with information bit period T and channel bandwidth B_ch

(N / C ) _r = (N_o / E_b )B_ch T/X (24)

where X is the loss of detectability due to fading. The C-band uplink noise to carrier ratio in bandwidth B_ch is :
(25)

IM noise is generated in both the multicarrier uplink and downlink

(C / N) _IM  = (C / N)1 B^v_o  dB
where (C/N)₁ = 10 dB, v = 2.0 for a non linearized SSPA and v = 1.4 for a TWT. For VOX operation
 
(26)

For linearized TWTAs or SSPAs, other empirical (C/N)_IM laws may be found. Combining equations (22)-(26), one obtains for digital modulation and power limited operation
m (FDMA single beam)
 

(27)
3.2. FDMA Multiple Spot Beam System

In the FDMA spot beam system the spot beam gain increases by the ratio of the total coverage area to the scan-beam area. The signals get distributed among the beams such that on average there are M b/B signals per beam. The IM noise gets also distributed among the B/b beams. In addition there is processor quantizing noise and self interference due to sidelobe of other beams having the same frequency as that beam in which the mobile user is located : the signal power radiated through antenna sidelobes interferes with the desired signals. Therefore, for the FDMA forward link of a clustered spot beam antenna system, the FDMA channel capacity is

M (FDMA forward, clustered)

(28)
The amount of sidelobe interference depends on the type of beam cluster, i.e. the distance between beams of equal frequency. For the seven beam cluster of Figure 4b the distance to a beam of equal frequency is about three beamwidth, and the amount of sidelobe power is 0.023 relative to desired carrier power.

The reason for the above implied improvement of L-band IM noise within the spot beams by the factor d is the following : the carriers are superimposed due to frequency reuse FR times before L-band power amplification, after which they are distributed through beam forming among the spot beams. The IM noise follows the carriers. Therefore the NPR is reduced by FR and improved by IM distribution among B/b beams.

The effects of IM dispersion improve the NPR by d = B/(b FR) = n which is identical to the number of beams in a cluster. For a 7 beam cluster the improvement due to IM dispersion is about 8.5 dB.

For the FDMA return link the faded L-band uplink signal reduces all elemental CNRs. The up-link fade reduces the up-link CNR, reduces the down-link carrier share of the available down-link power (thus reducing the down-link CNR) and reduces IM CNR and sidelobe CNR. Therefore the channel capacity for the FDMA return link is :

M (FDMA return clustered)

(29)

TDMA/FDMA

M (TDMA/FDMA forward, clustered) =

(30)
where N/P is the total received power in the L-band received carrier bandwidth. The channel capacity of the return link with Z channels per carrier is

M (TDMA/FDMA return, clustered) =
(31)

3.4. FDMA Carrier Bandwidth and Bandwidth limitation

The channel bandwidth is related to the information and coding rate as well as Doppler and oscillator uncertainties in QPSK by

 B_ch = 

+ 2 (Doppler effects + oscillator drift) + overhead

The frequency uncertainty of the incoming carrier is due to Doppler shift and oscillator drift. The Doppler shift is given by

D f_osc = ± f_c (D f/f)_osc

The carrier frequency uncertainty

Therefore with a rate 1/2 code, an information rate of 4.8 kb/s a frequency uncertainty of ± 287 Hz, the required bitrate per channel is 9.6 kbps, and using QPSK the required bandwidth is 0.6 x 9.6 + 2 x 0.287 = 6.4 KHz. Using a 1.4 spacing factor (ratio of spacing to symbol rate of 4.8 kbps) the required carrier separation is about 6.8 KHz. With a 4 beam cluster, the frequency reuse is 35 and the frequency limited capacity is 35 x 4 x 10⁶ / 6.8 x 10³ = 20,587 circuits.

With an allocation of 4 MHz L-band spectrum, 140 beams, 7 beams/clusters, frequency reuse FR = 20 and carrier spacing of 6.8 KHz the bandwidth limited capacity is

M_FDMA = 

Table II. Link Parameters (Forward Link)

Parameter	4 beam/cluster  CDMA	7 beam/cluster  FDMA	7 beam/cluster TDMA/FDMA
EIRP at back off, EIRPb,d, dBW, NPR = 13 dB	72.0	72.0	72.0
Nr beams, B/b, (140) dB	-21.5	-21.5	-21.5
Power control advantage, p, dB	2.5	2.5	2.5
EIRP into area coverage antenna, dBw	53.0	53.0	53.0
Lfd (fading), dB	-10.0	-10.0	-10.0
Ld (path loss) and atmospheric loss, dB	-187.6	-187.6	-187.6
Gd, dB	0.0	0.0	0.0
Received total power, P, dBw	-144.6	-144.6	-144.6
Boltzmann Constant KB1, dBw / Hz K	228.6	228.6	228.6
T, dBk	-25.0	-25.0	-25.0
W or Bch, dBHz	59.5	-38.0	-52.0
Nd, dBw	-144.1	-165.6	-151.6
(N/P)d, dB (Noise to total power ratio)	0.5	-21	-7.0
Chiprate, bps	0.9 x 106	9.6 x 103	270 x 103
Bitrate, bps	4.8 x 103	4.8 x 103	4.8
Bandwidth, KHz	900	6.4	162
WT	187.5	1.332	33.75
Eb/No required, dB	3.2	5	6.0
(No/Eb) WT	89.5	0.4373	8.43
(N/C)u, dB	1.4	0.01	0.01
Asb	1.08	1.023	1.023
(N/C)fade, dB	0.1	0.10	0.1
NPR, dB	13	13	13
(N/C)sl	0.08	0.023	0.08

3.5. TDMA/FDMA Carrier bandwidth and Bandwidth limitation

In order to avoid narrowband operation with associated oscillator drift and acquisition problems, mobile satellite systems use TDMA/FDMA, e.g. 32 channel TDMA for the forward link and 8 channel TDMA for the return link.

The bandwidth and separation requirements for the TDMA/FDMA system including frame efficiencies (75%) would be as follows :

TDMA/FDMA Forward link, 32 channels, Rate 3/4, channel rate = 6.4 kbps

Bitrate = 32 x 4.8 x (4/3) / 0.75 = 270 kbps
Bandwidth = 270 x 0.6 = 162 kHz
Carrier Separation(s) = 270 x 0.5 x 1.41 » 200 kHz

Return link, 8 channels
Bitrate = 8 x 6.4 /0.75 = 68 kbps
Bandwidth (Bw) = 8 x 0.6 x 6.4 /0.875 = 41k kHz

Carrier Separataion(s) = 1.42 x 68/2 @ 50 kHz

With a 1° beamwidth for the spot beam, the pattern null can be expected at 1.35° and the first sidelobe at 1.75° . With a scan width of 7° one should be able to generate about 140 to 150 spots. With a 7 beam cluster and frequency reuse of 140/7 = 20 times, usable L-band frequency band of 4 MHz, the bandwidth-limited channel capacity is

M (Bandwidth limited, TDMA) = 

4.0. CDMA vs. FDMA vs. TDMA/FDMA

Link capacities have been calculated for a system where the L-band spot beam antenna has a gain of 42 dB and the total coverage area to spot beam area ratio is about 140; the total coverage area gain is therefore 20.5 dB. The spot beam EIRP is 72 dBw.

Next we compare CDMA, FDMA and TDMA/FDMA for identical information rate and antenna beam size. Link parameters are listed in Table II for the forward link and in Table III/IV a link analysis is performed for both forward and return links. The CDMA forward link capacity is calculated according to (20) and is plotted vs. EIRP for fixed antenna gain in Figure 5. The results show that the capacity is very much dependent on the Eb/No ratio and that the clustered beams give slightly better results.

The link capacities for the inbound link of the CDMA spot beam system are calculated according to (21) and are plotted in Figure 6 against the C-band EIRP x G/T. The capacities are almost constant over the chosen range of EIRP x G/T product.

The channel capacities vs spot beam EIRP for FDMA and TDMA/FDMA forward links (Figure 7) are calculated according to (28) and (30) respectively. One notices that for high EIRP values, FDMA runs sooner into bandwidth limitation than TDMA/FDMA. The return link capacities vs C-band EIRP & G/T are calculated according to (29) and (31) and shown in Figure 8.

The dependence of channel capacity of all three multiple access systems on EIRP is shown in Figure 9 and the dependence on beam size is shown in Figure 10. The CDMA capacity lies between the four beam cluster FDMA and TDMA/FDMA and surpasses the 7 beam cluster FDMA capacity because of FDMA bandwidth limitation (4 MHz allocated L-band bandwidth). For the design point AEIRP of 72 dBw and 140 beams, FDMA provides the highest capacity for the 4 beams/cluster antenna configuration.

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5.0. CONCLUSION