assumed and the variance must be estimated by some non data aided method, as in (3.72).
3.9
Interrelationships
In order to summarize the interrelationships between MI, fidelity and symbol error variance we show in Fig. 3.5 a decoder/encoder witha priori , AWGN and extrinsic channels. The two metric calculators,AMetric andE Metric, are used to calculate convergence analysis measures of the a priori and extrinsic channels respectively. The AWGN channel Y is modelled using the encoded systematic data which has noise of variance σ2
n added and is multiplied by 2/σ2
n to generate an LLR. The a priori channel A is modelled using the systematic data multiplied by σ2
A/2 and the addition of noise with variance σA2. The a
priori metric can then be calculated using A and the systematic data x. Note that the values of σn and σA can be selected arbitrarily, however if theJ function (3.42)–(3.43) is used to select the value ofσAfor a desired IA, the MI measured by theAMetric block in Fig. 3.5 will be approximately equal toIA. The decoder takesAand Y as inputs and the extrinsic outputE is passed to the E Metric calculator. The extrinsic metric can then be calculated using E and x. The metrics calculated in Fig. 3.5 are those discussed in this chapter,
• Mutual informationIΛ using (3.17)
• Symbol error varianceσx2 using (3.23)
• Fidelity MΛ using (3.31)
where Λ represents A or E for a priori or a posteriori information respectively. It is assumed that A and E have a Gaussian consistent distribution as in (2.18). The three metrics are then related by (3.61) and (3.65)–(3.66) which we repeat for convenience,
MΛ= 1−σ2x, (3.74) IΛ≈0.74MΛ+ 0.26MΛ2, (3.75) MΛ≈ −0.74 0.52 + sµ 0.74 0.52 ¶2 + IΛ 0.26, (3.76)
It is possible using the tools provided by (3.74)–(3.76) to easily translate between MI, fidelity and symbol error variance. Furthermore, these relationships together with Fig. 3.5 illustrate the similarities between the three metrics and enable us to select the most con- venient metric for each receiver component.
3.10
Summary
In this chapter the IMUD receiver was introduced. The IC and CE were described and illustrated mathematically. We discussed MI, symbol error variance and fidelity, which are used to model the receiver such that the BER performance can be predicted without the need to run simulations. We then introduced several methods of calculating these metrics without knowledge of the systematic data, which is important in online analysis where the
Decoder Encoder EMetric Y A E N ∼σ 2 A N ∼σ2 n x∈ {+1,−1} 2/σ2 n σ2 A/2 AMetric
Figure 3.5: Convergence analysis metrics using a priori and extrinsic channel models.
data is unknown. We derived the Js function which is similar to the J function but de- scribes the relationship between fidelity and variance. Finally, we showed the relationship between fidelity and symbol error variance and proposed a transfer function for translat- ing between MI and fidelity. We showed how these functions can be used to describe the interrelationships between all three convergence tracking metrics. These functions will be used extensively in the following chapters for convergence analysis of the IMUD receiver.
Chapter 4
Convergence Analysis of Iterative
Receivers
4.1
Introduction
MI, fidelity and variance techniques as defined in Chapter 3, can be used in transfer charts to analyze and predict the convergence behavior of iterative decoders and receivers. EXIT charts [25, 26] are a popular and powerful tool for analysis of iterative techniques. This chapter will give an introduction to EXIT analysis and further develop the tools for analysis of a range of receiver components and system configurations.
The design and optimization of a multi-user system present a number of challenges to the system engineers. The power level of each user and scheduling of the decoding process are two major issues addressed in this thesis. VT analysis, which has been successfully used to analyze an unequal power CDMA system, is convenient as a closed-form VT function exists for an IC. However, EXIT analysis is generally considered to be the superior method. Furthermore, EXIT functions for FEC codes and other receiver components (such as de- mappers) are widely available in the literature. It is therefore most convenient to use EXIT functions for convergence analysis of the IMUD receiver. However, no method exists for generating an EXIT function for an IC in the unequal power case. Since the system has multiple users and several receiver components, the EXIT chart of the receiver is multi-dimensional. In this chapter we investigate the properties of EXIT functions for the TD and IC, including the accuracy of the method and capacity considerations, and show how to derive a two-dimensional EXIT chart which can be used to accurately model the IMUD receiver in the unequal power case. We propose a closed-form expression for the EXIT function of an IC which we validate through simulations. We will also explore several approaches for improving the performance (BER) or efficiency (number of iterations required) of the IMUD receiver.