27-12-2012, 02:22 PM
Space-Time Block Codes versus Space-Time Trellis Codes
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Abstract
Transmit diversity schemes for the coherent multiple-antenna flat-fading channel range from space-
time block codes (STBC) to space-time trellis codes (STTC). In this letter we compare the performance
of STBC and STTC by means of frame error rate. We discover that a simple concatenation of STBC
with traditional AWGN (additive white Gaussian noise) trellis codes outperforms some of the best known
STTCs at SNRs (signal to noise ratios) of interest. Our result holds for small numbers of receive antennas
and trellis states, and may extend to greater numbers of antennas and states with improved block codes.
Introduction
The challenge of transmit diversity design for the multiple-antenna fading channel has been
met with several novel modulation and error correction techniques in the recent past. Prominent
among these space-time block codes ([1], [2]) and space-time trellis codes ([3], [4], [5], [6]).
Space-time block codes operate on a block of input symbols producing a matrix output over
antennas and time. Unlike traditional single-antenna AWGN block codes, full rate space-time
block codes do not provide coding gain. Their key feature is the provision of full diversity with
extremely low encoder/decoder complexity. The best known orthogonal block codes are provided
in [2].
Space-time trellis codes operate on one input symbol at a time producing a sequence of spatial
vector outputs. Like traditional TCM (trellis coded modulation) for the single-antenna channel,
space-time trellis codes provide coding gain. Since they also provide full diversity gain, their
key advantage over space-time block codes is the provision of coding gain. However, they are
extremely difficult to design and require an expensive encoder and decoder.
This paper proposes a compromise between STBC (known designs, simple ML (maximum
likelihood) decoding, no coding gain) and STTC (difficult to design, expensive ML decoding,
coding gain). We concatenate AWGN trellis codes with STBC in order to obtain coding gain.
Then we compare the performance of concatenated STBC with STTC by keeping transmit power
and spectral efficiency fixed. The results are somewhat surprising - with the same number of
trellis states (4 and 8), concatenated STBC outperforms STTC for 1 and 2 receive antennas.
Space-Time Block Codes
The input to the encoder is a stream of real or complex modulated symbols. The encoder
operates on a block of K symbols producing an Mt×T codeword XnT whose rows correspond
to transmit antennas and columns correspond to symbol times. At the receiver, ML decoding is
simplified by the orthogonal structure of the codewords.
The effective channel induced by space-time block coding of input symbols (before ML de-
tection) is an AWGN channel with receive SNR equal to |H|2
FEs
MtN0
[10], [11]. This motivates the
concatenation of traditional single-antenna TCM with STBC. The system used here consists
of an outer TCM encoder/decoder concatenated with the STBC encoder/decoder. The overall
coding gain is only due to the outer TCM encoder since we consider full rate STBCs.
Recently, new block codes were designed in [12] by maximizing average capacity, some of which
outperform the orthogonal codes in [2]. Here we will only consider the codes in [2], which are the
best orthogonal linear codes with respect to the union bound on symbol error probability [13].
Space-Time Trellis Codes
Space-time trellis codes encode the input scalar symbol stream into an output vector symbol
stream. Unlike space-time block codes, space-time trellis codes map one input symbol at a time
to an Mt×1 vector output. Decoding is performed via ML sequence estimation.
Code performance is quantified by the diversity advantage and the coding advantage [4]. For
a given number of transmit antennas, the design objective is to construct the largest possible
codebook with full diversity advantage (= Mt) and the maximum possible coding advantage.
A number of hand-crafted codes with full diversity advantage were provided in [4]. Codes with
greater coding advantage than those in [4] were reported in [5] and [14] after exhaustive computer
searches over a feedforward convolutional coding (FFC) generator. A structured method of code
construction that ensures full diversity was provided in [15] along with new designs. Recently
new codes with better distance spectrum properties and performance were reported in [6].
Performance Analysis
Note that any space-time code can be analyzed in terms of the criteria presented for space-
time trellis codes, namely diversity advantage and coding advantage. These criteria affect the
performance curve in different ways. Diversity advantage affects the asymptotic slope of the
FER (frame error rate) versus SNR graph - greater the diversity, the more negative the slope.
Coding advantage affects the horizontal shift of the graph - greater the coding advantage, the
greater the shift to the left.
Coding Advantage
We will only consider full diversity codes for 2 transmit antennas at a bit rate of 2 bits per
symbol time. We will focus on a small number of trellis states, equal to 4 and 8, both for STTCs
and concatenated STBCs. In order to predict the performance of different codes, we computed
their coding advantages by means of a computer search and listed them in Table I. These values
agree with those found in published literature [16], [5], [6]. Note that the values in Table I are
normalized by Mt=2.
The block code labeled STBC stands for the complex Mt = 2 Alamouti code [1] and has a
4-PSK constellation. The codes labeled STBC-TCM represent the Alamouti code concatenated
with outer AWGN trellis codes and have 8-PSK constellations. The outer codes are both rate 2/3
and are the best 4-state (parallel transition) and 8-state trellis codes in [16]. The codes labeled
STTC denote space-time trellis codes, of which the 4 and 8 state codes in [4] (STTC-Tarokh),
[5] (STTC-Grimm), and [6] (STTC-Yan) are shown, all of which have 4-PSK constellations. The
4-state Grimm and Yan codes have the best possible coding advantage in the class of FFC codes
[5].
Conclusions
We have shown that for a small number of receive antennas and trellis states, a simple concate-
nation of space-time block codes with traditional AWGN trellis codes can significantly outperform
space-time trellis codes. The concatenated scheme separates spatial modulation from temporal
error correction and is therefore attractive from the aspect of computational complexity as well.