that it is possible to easily derive the transfer chart without the use of Monte Carlo simulation. Note that the EXIT function can be derived using Monte Carlo simulation in similar fashion to the algorithm in Section 4.4.1. This will in fact be the method used in Section 7.4.1 for an IC in an IMUD with channel estimation as, to the authors knowledge, there is no closed-form solution toσ2
IC =fIC ³
Eb/N0,ˆh, σ2x
´
. The fidelity chart for the IC can be derived using (4.10) to obtain σE and theJs function to determine the corresponding value ofMIC E MEIC=Js Ãs 4 (1−MIC A )KN−1 +2REN0b ! . (4.12)
Derivation of the EXIT chart for the IC involves an extra step which is the translation of
σ2 x to MI, that is IEIC=J Ãs 4 ¡ 1−T−1¡IIC A ¢¢K−1 N +2REN0b ! , (4.13)
where T(·) is from (3.65)–(3.66). Furthermore, both (4.12) and (4.13) require the usual Gaussian assumptions on the input LLRs (2.18), as for EXIT/fidelity charts of component codes. In contrast to EXIT/fidelity charts for component codes generated through Monte Carlo simulations, (4.12) and (4.13) also require that the output LLRs are Gaussian, which is in general true if K and N are not too small [18]. The function (4.13) will be used throughout this thesis as a closed-form EXIT function for an IC.
EXIT functions for the IC are shown in Fig. 4.11 for various system loads, with spread- ing factorN = 20, at an SNR ofγb = 5dB. Note that in the SU case,K = 1, the MI of the extrinsic output is independent of thea priori input. The constantIEIC in the SU case is equal toJ(2/σn) as there is no MAI. Fig. 4.11 shows that increasing the number of users increases the steepness of the transfer characteristics. Note that at the point where full a priori knowledge is available, IAIC = 1, which corresponds to removal of all MAI, the MI of the extrinsic output is equal to the single user case.
4.6
EXIT Chart
An EXIT chart is a plot including the transfer functions of all the receiver components. The functions are arranged such that the output of one component shares an axis with the corresponding input of the component to which it is connected. The simple case as shown in Fig. 4.12(a), with an IC and rate 12 3GPP CC (K = 20,N = 20), is with two receiver components which results in a two-dimensional EXIT chart. Also shown, in Fig. 4.12(b), is the rate 1
3 3GPP CC with an IC in a fully loaded system (K = 20, N = 20). Note
that the SNR in Fig. 4.12(a) is 7dB and 3.5dB in Fig. 4.12(b), indicating that the more powerful rate 13 has a lower convergence threshold.
4.6.1 Capacity
Since EXIT functions show the MI transfer properties of receiver components it was ex- pected that EXIT charts could be used in capacity consideration. The relationship between
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IAIC I IC E K = 1 K = 2 K = 3 K = 5 K = 10 K = 20 K = 30
Figure 4.11: EXIT function of the IC under various system loads.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IAIC, IDecE I ICE , I D ec A CC Decoder IC (a) Rate 1 2 CC (561,753),γb= 7dB 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IICA, IEDec I ICE , I D ec A CC Decoder IC (b) Rate 1 3 CC (557,663,711),γb= 3.5dB Figure 4.12: EXIT charts with the 3GPP CC decoders and IC for KT = 20 users and
spreading factor N = 20.
EXIT curves, capacity and code rate in a binary erasure channel was proved in [27] and [29]. The area under an EXIT curve is given by A = R01IE(IA)dIA. Consider a serially concatenated code with outer code rate Rout and inner code of rate Rin = 1, which is
the case in the CDMA system of Fig. 4.2 where the inner “code” is the CDMA channel. The area under the (flipped) outer code EXIT curve, i.e. the area below the TD curve in Fig. 4.16, isAout =Rout and the area under the inner curve isAin=I(X;Y)/n/Rinwhere
Rin=k/n. For successful decoding the EXIT curves must not intersect andAin−Aout >0,
so
RoutRin< I(X;Y)/n≤C. (4.14)
Note that (4.14) shows that capacity can only be achieved if Rin = 1. Since the rate of
4.6 EXIT Chart 61
is Ain=C whereC is the capacity of the channel. Therefore
Rout ≤C. (4.15)
This is a form of Shannon’s capacity theorem [130], the outer code rate must be less than the capacity of the inner channel [43].
4.6.2 Multi-dimensional EXIT Charts
The turbo receiver of Fig. 4.2 can be thought of as a three-dimensional decoder as shown in Fig. 4.13. This allows the decoding trajectory to be visualized in three dimensions as in [131] and [83] which featured multiple concatenated codes. The transfer charts then become transfer surfaces and for the decoding to converge a tube must be open between the three surfaces. It is also possible to project this three-dimensional transfer chart onto a single two-dimensional chart as described in [83, 35]. Note that the CC decoders in Fig. 4.13 are for one group of users, k = 1, only. All other users, k 6= 1, are considered to be identical to group k = 1 so for simplicity the subscript k is dropped. It should also be noted that a parallel IC is used, so the receiver “waits” for all users to decode before activating the IC again. Whether the actual decoding is in practice carried out in serial or parallel is a hardware optimization problem beyond the scope of this work. The system shown in Fig. 4.13 should be considered to be the receiver as seen by a particular user. Thea priori and extrinsic information for the other users is passed to and from the decoders of each user following each activation of the IC. In this way the decoding for each user, or group of users (where groups of users activate their decoders in the same order), is independent. The only assumption made here is that each user finishes decoding before the IC is activated again. Decoding in the three-dimensional receiver can be visualized as a point moving from the center of Fig. 4.13 out to each receiver block, which corresponds to an activation of that receiver block, and back to the center again. Decoding proceeds as follows. The IC is always the first receiver block to be activated then the decoding point moves to the middle of the three spokes. We can then choose to activate the IC, decoder 1 or 2 and the decoding point again returns to the center, and so on. Note that it is possible to activate the same component multiple times in a row, although this will not improve soft estimates obtained from the first activation of that component.
The EXIT chart for the three-dimensional receiver is shown in Fig. 4.14 and is useful for visualizing the information exchange during decoding. The uncolored frame, multi- colored plane and grey plane are the EXIT functions of decoder 1, decoder 2 and the IC respectively. Decoding begins at the point (0,0,0) and activating the IC corresponds to moving up the y-axis to the IC EXIT curve. At this stage the decoding point lies in the center of the spokes in Fig. 4.13. At any time hereafter: activating the IC corresponds to moving the decoding trajectory parallel to the y-axis until it hits the IC EXIT function; activating decoder 1 corresponds to moving the trajectory parallel to the x-axis until it hits the decoder 1 EXIT function; and activating decoder 2 corresponds to moving the trajectory parallel to thez-axis to the EXIT function of decoder 2. Using the EXIT charts one can now see that activating a receiver block twice in a row achieves no further gain in the MI of the extrinsic output of a component. The question then arises as to which order,
IC DEC 1 DEC 2 Q−1 G QG Q−1 Q Ddec(2) D dec(2) s D dec(1) s Ddec(1) Adecs (2) A dec(1) s AIC1 E1IC Ydec(1) Ydec(2) EICk6=1 Y AICk6=1 − − − − D e M U X D eM UX DeM UX M U X QG
Figure 4.13: Turbo MUD system visualized as a multi-dimensional decoderfor one group of users.
or decoding schedule, is optimal, which will be addressed in Section 6. The difficulty with the EXIT chart in Fig. 4.14 lies in the fact that it is three dimensional. While it is useful for understanding the exchange of information it would be beneficial to include the same information in a two-dimensional EXIT chart.