8.3.1 OFDM Technique
In a conventional serial data system such as microwave digital radio data transmission and telephone lines, in which the symbols are transmitted sequentially, adaptive equal- ization techniques have been introduced to combat ISI. However, the system complexity precludes the equalization implementation if the data rate is as high as a few megabits per second.
A parallel data system can alleviate ISI even without equalization. In such a system the high-rate data stream is demultiplexed into a large number of sub-channels with the spectrum of an individual data element occupying only a small part of the total available bandwidth.
A parallel system employing conventional frequency division multiplexing (FDM) without sub-channel overlapping is bandwidth inefficient. A much more efficient use of bandwidth can be obtained with an OFDM system in which the spectra of the individual sub-channels are permitted to overlap and the carriers are orthogonal. A basic OFDM system is shown in Fig. 8.1 [32].
Let us assume that the serial data symbols after the encoder have a duration of Ts =
1
fs seconds each, where fs is the input symbol rate. Each OFDM frame consists of K coded symbols, denoted byd[0], d[1], . . . , d[K−1], where d[n]=a[n]+j b[n] anda[n]
and b[n] denote the real and imaginary parts of the sampling values at discrete time n, respectively. After the serial-to-parallel converter, the K parallel data modulate K sub- carrier frequencies,f0, f1, . . . , fK−1, which are then frequency division multiplexed. The sub-carrier frequencies are separated by multiples off = KT1s, making any two carrier frequencies orthogonal. Because the carriers are orthogonal, data can be detected on each of these closely spaced carriers without interference from the other carriers. In addition, after the serial-to-parallel converter, the signaling interval is increased from Ts toKTs, which makes the system less susceptible to delay spread impairments.
The OFDM transmitted signalD(t )can be expressed as
D(t )=
K−1 n=0
{a[n] cos(ωnt )−bnsin(ωnt )} (8.17)
X cosω0t
cosω0t
ωK1 ωK1 cos t ω0t sin
ωK1 ωK1 sin t
ωK1 ωK1 sin t
ωK1 ωK1 cos t ω0t sin
Encoder Converter
X
X X S/P
MULTIPLEX
Serial Data Stream D(t)
(a) Transmitter
(b) Receiver fs =
T
X
X
X
X
P/S
Converter Decoder
Integration
Integration
Integration
Integration
d[n]=a[n]+jb[n]
a[0]
b[0]
a[K1]
b[K1]
1 s
Figure 8.1 A basic OFDM system
where
ωn=2πfn
fn=f0+nf (8.18)
Substituting (8.18) into (8.17), the transmitted signal can be rewritten as D(t )=Re
e
K−1 n=0
{d[n]ej ωnt}
=Re K−1
n=0
{d[n]ej2π nf tej2πf0t}
=Re{ ˜D(t )ej2πf0t} (8.19)
STC in Wideband OFDM Systems 251
where
D(t )˜ =
K−1 n=0
{d[n]ej2π nf t} (8.20)
represents the complex envelope of the transmitted signalD(t ).
At the receiver, correlation demodulators (or matched filters) are employed to recover the symbol for each sub-channel. However, the complexity of the equipment, such as filters and modulators, makes the direct implementation of the OFDM system in Fig. 8.1 impractical, whenN is large.
Now consider that the complex envelope signalD(t )˜ in (8.19) is sampled at a sampling rate offs. Lett =mTs, wheremis the sampling instant. The samples ofD(t )˜ in an OFDM frame,D[0],˜ D[1], . . . ,˜ D[K˜ −1], are given by
D[m]˜ =
K−1
n=0
{d[n]ej2π nf mTs}
=
K−1 n=0
d[n]ej (2π/K)nm
=IDFT{d[n]}, (8.21)
Equation (8.21) indicates that the OFDM modulated signal is effectively the inverse discrete Fourier transform (IDFT) of the original data stream and, similarly, we may prove that a bank of coherent demodulators in Fig. 8.1 is equivalent to a discrete Fourier transform (DFT).
This makes the implementation of OFDM system completely digital and the equipment complexity is decreased to a large extent [30]. If the number of sub-channels K is large, fast Fourier transform (FFT) can be employed to bring in further reductions in complexity [31]. An OFDM system employing FFT algorithm is shown in Fig. 8.2. Note that FFT and IFFT can be exchanged between the transmitter and receiver, depending on the initial phase of the carriers.
8.3.2 STC-OFDM Systems
We consider a baseband STC-OFDM communication system withK OFDM sub-carriers, nT transmit andnRreceive antennas. The total available bandwidth of the system isW Hz.
It is divided intoK overlapping sub-bands. The system block diagram is shown in Fig. 8.3.
Data
Encoder Converter
Converter S/P
P/S
Channel Data
Decoder
IFFT
FFT
Figure 8.2 An OFDM system employing FFT
Figure 8.3 An STC-OFDM system block diagram
At each timet, a block of information bits is encoded to generate a space-time codeword which consists ofnTLmodulated symbols. The space-time codeword is given by
Xt =
xt,11 xt,21 ã ã ã xt,L1 xt,12 xt,22 ã ã ã xt,L2 ... ... . .. ... xt,1nT xt,2nT ã ã ã xt,LnT
(8.22)
where the i-th row xit = xt,1i , xit,2, . . . , xt,Li , i = 1,2, . . . , nT, is the data sequence for thei-th transmit antenna. For the sake of simplicity, we assume that the codeword length is equal to the number of OFDM sub-carriers, L = K. Signals xt,1i , xt,2i , . . . , xt,Li are OFDM modulated onKdifferent OFDM sub-carriers and transmitted from thei-th antenna simultaneously during one OFDM frame, wherext,ki is sent on thek-th OFDM sub-carrier.
In OFDM systems, in order to avoid ISI due to the delay spread of the channel, a cyclic prefix is appended to each OFDM frame during the guard time interval. The cyclic prefix is a copy of the lastLp samples of the OFDM frame, so that the overall OFDM frame length isL+Lp, whereLp is the number of multipaths in fading channels.
In the performance analysis, we assume ideal frame and symbol synchronization between the transmitter and the receiver. A sub-channel is modeled by quasi-static Rayleigh fading.
The fading process remains constant during each OFDM frame. It is also assumed that channels between different antennas are uncorrelated.
At the receiver, after matched filtering, the signal from each receive antenna is sampled at a rate ofW Hz and the cyclic prefix is discarded from each frame. Then these samples are applied to an OFDM demodulator. The output of the OFDM demodulator for the k-th
STC in Wideband OFDM Systems 253
OFDM sub-carrier,k=1,2, . . . , K, at receive antennaj,j =1,2, . . . , nR, is given by [5]
Rt,kj =
nT
i=1
Hj,it,kxt,ki +Nt,kj (8.23) whereHj,it,k is the channel frequency response for the path from thei-th transmit antenna to thej-th receive antenna on thek-th OFDM sub-channel, andNt,kj is the OFDM demodula- tion output for the noise sample at thej-th receive antenna and thek-th sub-channel with power spectral densityN0. Assuming that perfect channel state information is available at the receiver, the maximum likelihood decoding rule is given by
Xˆt =arg min Xˆ
nR
j=1
K k=1
Rt,kj −
nT
i=1
Hj,it,kxt,ki
2
(8.24) where the minimization is performed over all possible space-time codewords.
Recall that the channel impulse response in the time domain is modeled as a tapped-delay line. The channel impulse response between the i-th transmit antenna to the j-th receive antenna is given by
hj,i(t;τ )=
Lp
=1
ht,j,iδ(τ−τ) (8.25)
whereLp is the number of multipaths,τ is the time delay of the-th path andht,j,i is the complex amplitude of the-th path. Let us denote by Tf the time duration of each OFDM frame and byf the separation between the OFDM sub-carriers. We have
Tf =KTs Ts = 1
W = 1
Kf (8.26)
Now the delay of the-th path can be represented as τ=nTs = n
Kf (8.27)
wheren is an integer. Performing the Fourier transform of the channel impulse response, we can get the channel frequency response at timet as
Hj,it,k= Hj,i(t Tf, kf )
= +∞
−∞ hj,i(t Tf, τ )e−j2π kf τdτ
=
Lp
=1
hj,i(t Tf, nTs)e−j2π kn/K
=
Lp
=1
hj,i(t, n)e−j2π kn/K (8.28)
Let
htj,i=[ht,1j,i, ht,2j,i, . . . , ht,Lj,ip]H
wk=[e−j2π kn1/K, e−j2π kn2/K, . . . , e−j2π knLp/K]T (8.29) The equation (8.28) can be rewritten as
Hj,it,k=(htj,i)Hãwk (8.30) From (8.28), we can see that the channel frequency response Hj,it,k is the digital Fourier transform of the channel impulse responsehtj,i. The transform is specified by the vectorwk
for thek-th OFDM sub-carrier,k=1,2, . . . , K.