8.2.1 Frequency-Selective Fading Channels
Frequency-selective fading channels can be modeled by a tapped-delay line. For a multipath fading channel withLp different paths, the time-variant impulse response at time t to an impulse applied at timet−τ is expressed as [1]
h(t;τ )=
Lp
=1
ht,δ(τ−τ) (8.1)
Space-Time Coding Branka Vucetic and Jinhong Yuan c2003 John Wiley & Sons, Ltd ISBN: 0-470-84757-3
whereτrepresents the time delay of the-th path andht,represents the complex amplitude of the-th path.
Without loss of generality, we assume thath(t;τ )is wide-sense stationary, which means that the mean value of the channel random process is independent of time and the autocor- relation of the random process depends only on the time difference [1]. Then,ht, can be modeled by narrowband complex Gaussian processes, which are independent for different paths. The autocorrelation function ofh(t;τ )is given by [1]
φh(t;τi, τj)= 12E[h∗(t, τi)h(t+t, τj)] (8.2) wheret denotes the observation time difference. If we lett =0, the resulting autocor- relation function, denoted byφh(τi, τj), is a function of the time delaysτi andτj. Due to the fact that scattering at two different paths is uncorrelated in most radio transmissions, we have
φh(τi, τj)=φh(τi)δ(τi−τj) (8.3) whereφh(τi)represents the average channel output power as a function of the time delayτi. We can further assume that theLpdifferent paths have the same normalized autocorrelation function, but different average powers. Let us denote the average power for the -th path byP (τ). Then we have
P (τ)=φh(τ)=12E[h∗(t, τ)h(t, τ)] (8.4) Here,P (τ),=1,2, . . . , Lp, represent thepower delay profile of the channel.
The root mean square (rms)delay spread of the channel is defined as [2]
τd =
Lp
=1
P (τ)τ2
Lp
=1
P (τ)
−
Lp
=1
P (τ)τ
Lp
=1
P (τ)
2
(8.5)
In wireless communication environments, the channel power delay profile can be Gaus- sian, exponential or two-ray equal-gain [8]. For example, the two-ray equal-gain profile can be represented by
P (τ )=12δτ +δ(τ−2τd) (8.6)
where 2τdis the delay difference between the two paths andτdis the rms delay spread. We can further denote the delay spread normalized by the symbol durationTs byτd= τTds.
8.2.2 Performance Analysis
In this section, we consider the performance analysis of space-time coding in multipath and frequency-selective fading channels. In the analysis, we assume that the delay spreadτd is relatively small compared with the symbol duration. In order to investigate the effect of
Performance of Space-Time Coding on Frequency-Selective Fading Channels 247
frequency-selective fading on the code performance, we assume that no equalization is used at the receiver.
Consider a system withnT transmit andnR receiver antennas. Lethj,i(t, τ )denote the channel impulse response between the i-th transmit antenna and j-th receive antenna. At timet, the received signal at antennaj after matched filtering is given by [8]
rtj = 1 Ts
(t+1)Ts
tTs
n T
i=1
∞
0
ui(t−τi)hj,i(t, τi)dτi
dt+njt (8.7) whereTs is the symbol period,njt is an independent sample of a zero-mean complex Gaus- sian random process with the single-sided power spectrum densityN0 andui(t )represents the transmitted signal from antennai, given by
ui(t )= ∞
k=−∞
xkig(t−kTs) (8.8)
where xki is the message for the i-th antenna at the k-th symbol period and g(t ) is the pulse shaping function. The received signal can be decomposed into the following three terms [7][8]
rtj =α
nT
i=1 Lp
=1
ht,j,ixti+Itj+njt (8.9) where Itj is a term representing the intersymbol interference (ISI), and α is a constant dependent on the channel power delay profile, which can be computed as
α= 1 Ts
Ts
−Ts
P (τ )(Ts − |τ|)dτ (8.10)
For different power delay profiles, the values ofαare given by [8]
α=
1−τd Exponential or two-ray equal-gain profile 1−√
2/π τd Gaussian profile (8.11)
The mean value of the ISI termItj is zero and the variance is given by [8]
σI2=
3nTτdEs Exponential or two-ray equal-gain profile
2nT(1−1/π )τ2dEs Gaussian profile (8.12)
where Es is the energy per symbol. For simplicity, the ISI term is approximated by a Gaussian random variable with a zero-mean and single-sided power spectral densityNI = σI2Ts. Let us denote the sum of the additive noise and the ISI bynjt.
njt =Itj+njt (8.13)
The received signal can be rewritten as rtj =α
nT
i=1 Lp
=1
ht,j,ixti +njt (8.14)
wherenjt is a complex Gaussian random variable with a zero mean and the single-sided power spectral densityNI +N0. Note that the additive noise and the ISI are uncorrelated with the signal term. The pairwise error probability under this approximation is given by [8]
P (X,X)ˆ ≤ n
T i=1
1+λi α2Es 4(N0+NI)
−nR
≤ r
i=1
λi
α2 NI/N0+1i
−nR Es 4N0
−rnR
(8.15) whereris the rank of the codeword distance matrix, andλi,i=1,2, . . . , r, are the nonzero eigenvalues of the matrix. From the above upper bound, we can observe that the diversity gain achieved by the space-time code on multipath and frequency-selective fading channels isrnR, which is the same as that on frequency-nonselective fading channels. The coding gain is
Gcoding= r
i=1
λi 1/r
α2 NI/N0+1
du2 (8.16)
The coding gain is reduced by a factor of α2
NI/N0+1
compared to the one on frequency flat fading channels. Furthermore, it is reported that at high SNRs, there exists an irreducible error rate floor [7] [8].
Note that the above performance analysis is performed under the assumptions that the time delay spread is small and no equalizer is used at the receiver. When the delay spread becomes relatively high, the coding gain will decrease considerably due to ISI, and cause a high performance degradation. In order to improve the code performance over frequency- selective fading channels, additional processing is required to remove or prevent ISI.
It is shown in [4] that a space-time code on frequency-selective fading channels can achieve at least the same diversity gain as that on frequency-nonselective fading channels provided that maximum likelihood decoding is performed at the receiver. In other words, an optimal space-time code on frequency-selective fading channels may achieve a higher diversity gain than on frequency-nonselective fading channels. As the maximum likelihood decoding on frequency-selective channels is prohibitively complex, a reasonable solution to improve the performance of space-time codes on frequency-selective fading channels is to mitigate ISI. By mitigating ISI, one can convert frequency-selective channels into frequency-nonselective channels. Then, good space-time codes for frequency-nonselective fading channels can be applied [9].
A conventional approach to mitigate ISI is to use an adaptive equalizer at the receiver. An optimum space-time equalizer can suppress ISI, and therefore, the frequency-selective fading channels become intersymbol interference free. The main drawback of this approach is a high receiver complexity because a multiple-input/multiple-output equalizer (MIMO-EQ) has to be used at the receiver [17] [18] [19].
An alternative approach is to use orthogonal frequency division multiplexing (OFDM) techniques [5] [6]. In OFDM, the entire channel is divided into many narrow parallel sub- channels, thereby increasing the symbol duration and reducing or eliminating the ISI caused
STC in Wideband OFDM Systems 249
by the multipath environments [15]. Since MIMO-EQ is not required in OFDM systems, this approach is less complex.
An OFDM technique transforms a frequency-selective fading channel into parallel cor- related frequency-nonselective fading channels. OFDM has been chosen as a standard for various wireless communication systems, including European digital audio broadcasting (DAB) and digital video broadcasting (DVB), IEEE broadband wireless local area networks (WLAN) IEEE802.11 and European HIPERLAN [26] [27]. In OFDM systems, there is a high error probability for those sub-channels in deep fades and therefore, error control coding is combined with OFDM to mitigate the deep fading effects. For a MIMO frequency- selective fading channel, the combination of space-time coding with wideband OFDM has the potential to exploit multipath fading and to achieve very high data rate robust trans- missions [5][10][11][14][15][16]. In the next section, we will discuss space-time coding in wideband OFDM systems, which is called STC-OFDM.