Chapter 3 Turbo Equalization
4.1 Transmitter
Adaptive Filter-Based Turbo Equalizer with Space-Time Decoder
In this chapter, we combine the techniques in the previous chapters and propose a turbo equalizer which utilizes the adaptive filtering technique and the space-time trellis coded system in wideband MIMO systems. We will describe how we transmit information bits to the antennas and how we recover the information bits from severely distorted received signals. The proposed receiver is composed of a low-complexity equalizer, but the good performance is still preserved due to the turbo process. The performance evaluation of the proposed receiver is carried out with different parameters, and the perfect-feedback case is used as reference in the figures.
These comparisons provide trade-off between complexity and performance. In addition, they reveal some interesting properties of a space-time coded system with wideband transmission.
4.1 Transmitter
A space-time trellis coded transmitter with
n transmit antennas is shown in
Tfigure 4.1. Assume the information bits in the sequence
a are independentChapter 4 Adaptive Filter-Based Turbo Equalizer with Space-Time Decoder
identically distributed (i.i.d.). The information bit sequence is encoded into
n
T coded bit sequencesc
i ,i
=1,…,n
T, and the coded bit sequences are then mapped into coded symbol sequencess
i ,i
=1,…,n
T. Interleaversπ
i ,i
= …1, ,n
T are used to shuffle coded symbol sequencess
i ,i
=1,…,n
T respectively to obtain new sequencesd
i ,i
=1,…,n
T . Before symbols are transmitted by antennas, pulse-shaping filters are applied to mitigate the ISI effect. A common choice of the pulse-shaping filter is the squared root raised cosine filter, which will be addressed later.a
nT
π π1
s1
nT
c
d1
nT
d nT
c1
nT
s
Figure 4.1 Transmitter of a space-time trellis coded system
4.1.1 Space-Time Trellis Encoder
We have introduced the space-time trellis codes in 2.3. Here we select some of these codes in our systems. In this thesis, the space-time trellis codes we choose are all 4-PSK modulated whose signal constellation is shown in figure 2.7. In 4.4.3, we will compare the performance between 2, 3, and 4 transmit antennas. In 4.4.5, we will compare the performance between 4, 8, 16, 32 and 64 states in the encoder.
Therefore, to carry out these simulations and comparisons, different codes are adopted for different simulations. Applying the notation in figure 2.8, we list the required codes described by generator sequences in table 4.1 and table 4.2. When the effect of transmit antenna number is concerned, table 4.1 provides codes of 2, 3 and 4
Chapter 4 Adaptive Filter-Based Turbo Equalizer with Space-Time Decoder
antennas with same state number. When the effect of state number is concerned,
table 4.2 provides codes with 4, 8, 16, 32 and 64 states with 2 transmit antennas. In
the coding theorem, a code with more states in the trellis results in better performance at the expense of extra computation complexity in the decoder. Moreover, more transmit antennas provide more diversity to the receiver, but the signal detection is more difficult. How to choose the number of transmit antennas and state number depends on the system designer. Simulations with respect to these parameters provide a powerful tool to determine these parameters.n
T v generator sequencesTable 4.1 Generator sequences of 32 states and 2, 3 and 4 transmit antennas
n
T v generator sequencesChapter 4 Adaptive Filter-Based Turbo Equalizer with Space-Time Decoder
Table 4.2 Generator sequences of different state numbers for 2 transmit antennas
Signal power normalization is applied after space-time trellis encoder. We constrain the total transmitted power of all transmit antennas to be 1. To meet the power constraint, each signal must be divided by sqrt n
(
T×2)
.4.1.2 Interleavers
All trellis codes are sensitive to burst errors. These burst errors are possibly caused by correlated noises and the sequential operations of an equalizer or a decoder.
Generally, the noise is assumed to be uncorrelated, i.e. white noise, at the receive antenna. However, the operation of the equalizer will produce a correlated noise, i.e.
the colored noise at the equalizer output. In addition, the residual ISI at the equalizer output is a form of correlated interference. These correlated interferences including the colored noise and the residual ISI have an impact on the coding gain and cause performance loss. To relieve the effect of burst errors, an interleaver is usually used.
In the receiver, the inverse operation, i.e. deinterleaver, is placed between equalizer and decoder. The deinterleaver will decorrelate the signals at the decoder input and thus avoid the performance loss cause by burst errors.
In a space-time coded system, the correlated interference not only appears in time domain but also in space domain. A space time equalizer outputs several sequences at the same time representing the estimated symbols for different antennas. The
Chapter 4 Adaptive Filter-Based Turbo Equalizer with Space-Time Decoder
cross filters in the equalizer introduce correlation between different output sequences and thus result in correlated interference in space domain. To handle this kind of correlation, different interleavers are applied for different antennas.
As mentioned in 3.4.3, interleavers plays a crucial role in a turbo system.
Because of the iterative process, it is important to remove the correlation at the output of each block. Otherwise, with the iteration increases, the correlation becomes bigger and results in larger performance loss. In the turbo equalizer receiver, the deinterleaver removes the correlation at the output of the equalizer while the interleaver removes the correlation at the output of the decoder.
In this thesis, we set the interleaver length to be 4096 and assign different permutation tables for different antennas. The i-th element in the permutation table is j , and we denote this as a pair
( )
i j, ,i=1,…, 4096, j 1,= …, 4096. The-th
i input of the interleaver will be the
j
-th output of the interleaver. On the other hand, the function of a deinterleaver according to a permutation table( )
i j, ,i=1,…, 4096, j 1,= …, 4096 is to put thej
-th input in the i-th output.4.1.3 Pulse Shaping Filter
The raised cosine filter is a simple spectrum shaping filter. The frequency response consists of a flat portion and a rolloff portion that has a sinusoidal form.
Define a parameter called the rolloff factor to be 1
f
rα
= −W
(4.1)where W is the Nyquist bandwidth of the signal and
f is the rolloff frequency.
r The rolloff factor also indicates the excess bandwidth over the Nyquist bandwidth W. Specifically, the transmission bandwidthB is defined by
TChapter 4 Adaptive Filter-Based Turbo Equalizer with Space-Time Decoder
( )
2 1
T r
B W f
W α
= −
= + (4.2)
Instead of using one raised cosine filter to meet the Nyquist criterion, we can split it into two filters each of which is a squared root raised cosine (SRRC) filter. By placing one in the transmitter before antenna, and one in the receiver after antenna, we can obtain a raised cosine response.
Define the symbol rate at the output of the interleavers to be
R . To design the
s SRRC filters, we first upsample the input by 2 to be 2R . Assigning the rolloff
s factor to be 0.5, and truncating the filter coefficients to be of length 13, we obtain the normalized SRRC filter of the impulse response shown in figure 4.2. The coefficients are normalized so that the signal power will not be amplified by this filter.Figure 4.2 Squared rooted raised cosine filter with rolloff factor 0.5 and truncated to be length 13
Chapter 4 Adaptive Filter-Based Turbo Equalizer with Space-Time Decoder
4.1.4 Training Symbols
We insert training symbols before information data is transmitted. The purpose of these training symbols is to train the equalizer coefficients. The packet format is shown in figure 4.3. The training mode will be described latter along with the equalization. We set the training length to be 4096 symbols in the following simulations.
Figure 4.3 Packet format