Chapter 3 B-Chase Detector
3.1 Introduce B-Chase Detector
3.1.5 The B-Chase Detector for Channel Estimation Errors
In previous sections, we always assumed that we have perfect the channel state information (CSI) at the receiver, which allows us to compare the performance. However, the channel information is typically not perfect. A channel estimator extracts from the received signal approximate channel coefficients during the transmission symbol. One method to accomplish this is to transmit the training signal prior to the transmission symbol. That are used as preamble at the start of each frame. Another way to estimate the channel fading coefficients is to embed the pilot bits, that is called pilot signal, inside the signal.
The impact from the channel estimation errors will degrade the performance of the system. To study the impact of the channel estimation errors on the B-CHASE detector algorithm, we introduce the error model at the receiver.
H = H + ΔH ′ (3.1.5.1)
where H represent the true channel matrix and ΔH denotes the channel estimation error. The elements ofΔH are assumed to be zero mean, variance is 0.01 and complex Gaussian. The B-CHASE*(16) is a measurement based on that we can accurately obtain the channel estimation. The B-CHASEer*(16) is a measurement based on that we can not accurately obtain the channel estimation. As shown in Figure, the channel estimation errors with The B-CHASEer*(16) given the B-CHASE decoding algorithm. It is clear from the figure, the B-CHASEer*(16) decoding algorithm starts to perform poorly. This poor performance is caused by inter-symbol interference (ISI).When we obtain the error channel matrix, find out the error outputs, ′F M etc., in the B-Chase preprocessing. From that obtain the error ′′ y = F r ′ produce the ISI. This cause a ISI problem since channel estimation error is the biggest contributor of the errors in the simulation at the high SNR region.
Table 3-6 System parameters
Modulation 16-QAM
Transmit antenna 4
Receive antenna 4
Channel is updated in T symbol periods 8
Rayleigh-fading Mean=0,Varance=1
Error of the Rayleigh-fading Mean=0,Varance=0.01
Channel order 0
Selection algorithm 1
List length l 16
5 10 15 20 25 30 35 10-5
10-4 10-3 10-2 10-1 100
SNR dB
BER
B-CHASE*(16) B-CHASEer*(16)
Figure 3-9 Bit error rate with channel estimation error and without channel estimation error
From figure we can know the channel estimation error demonstrate the error in the high SNR.
Chapter 4 B-Chase Detector of MIMO-OFDM Systems
The material in this Chapter is largely taken from [9], [10], [11], and [12].
4.1 OFDM System Models
We understand the single carrier (SC) that has the poor spectral efficiency in our communication system and when we have multipath so that have frequency selective fading and inter-symbol interference (ISI). So, we will employ the principle of multi carrier(MC) system that can combat them because only some subcarriers is fail to communication. We use orthogonal frequency division multiplexing (OFDM) that is to divide the available spectrum into several subchannels (subcarriers) and the frequency response of the subchannels are overlapping and orthogonal. That get the channel is flat fading per subcarrier and decrease ISI.
In the MC system the transmitter separate the data stream into several parallel ones and each modulated by a specific subcarrier that can use Inverse discrete Fourier Transform (IDFT) to implemt that in the baseband modulation. In the receive each demodulated by a specific subcarrier that can use discrete Fourier Transform (DFT) to implement that in the baseband demodulation.
When OFDM symbols pass through a time-dispersive channel, inter-symbol interference (ISI) and inter-carrier interference (ICI) usually occur in the receiver and cyclic prefix (CP) is introduced to combat ISI and ICI. Cyclic prefix, shown in Figure 4.1, is a copy of the tail part of a OFDM symbol is attached to its front. As long as the cyclic prefix length is longer than its experiencing time-dispersive channel length, ISI can be avoided. At the same time, the cyclic prefix along with its OFDM symbol makes a periodic OFDM signal and maintains the properties of circular convolution and subcarrier orthogonality that prevents the ICI effect.
For this system we employ the following assumptions:
z The channel impulse response is shorter than the cyclic prefix.
z Transmitter and receiver are perfectly synchronized.
z The fading is slow enough for the channel to be considered constant during one OFDM symbol interval.
z Channel noise is additive, white, and complex Gaussian.
Figure 4-1 Cyclic prefix of an OFDM symbol [10]
4.1.1 Continuous-Time Model
In this chapter, a continuous-time model is used to introduce the whole OFDM baseband system including the transmitter and receiver. In the transmitter, the transmitted data is split into multiple subchannels with overlapping frequency bands. The spectrum of OFDM signal is shown in Figure 3.2. It is clear that the spectrum of each subchannel is spreading to all the others, but is zero at all the other subcarrier frequencies, because of the sinc function property, which is the key feature of the orthogonality.
Assumeing an OFDM system with N subcariers, a bandwidth of WHz and a symbol length of T seconds, of which T seconds is the length of the cyclic prefix, the transmitter uses the g following waveforms
( )
common interpretation of OFDM is that it uses N subcarriers, each carrying a low bit-rate.The waveforms Φk
( )
t are used in the modulation and the transmitted baseband signal for OFDM symbol as Assume the given channel is quasi-static, i.e., constant during the transmission of an OFDM symbol, where the quasi-static impulse response is h( )
τ;t of the physical channel is restricted to the interval τ∈ ⎣⎡0,Tg⎤⎦ , i.e., to the length of the cyclic prefix. The received signalwhere n is additive, white, and complex Gaussian channel noise.
Calculating the sampled output at the k_ th matched filter.
( )( )
Figure 4.3 shows a typical continuous-time OFDM baseband modulator, in which the transmitted data is split into multiple parallel streams which are modulated by different subcarriers and then transmitted simultaneously. At the receiver, the received signal is demodulated simultaneously by multiple matched filters and then the data on each subchannel is obtained by sampling the outputs of matched filters, as shown in Figure 4.4.
Figure 4-2 Spectrum of an OFDM signal [10]
Figure 4-3 Continuous-time OFDM baseband modulator [10]
Figure 4-4 Continuous-time OFDM baseband demodulator [10]
4.1.2 Discrete-Time Model
To simultaneously transmit multiple data, the transmitter must modulate data with multiple subcarriers and the receiver must demodulate with multiple matched filters. In fact, the modulation and demodulation can be implemented efficiently by using digital IDFT/DFT operations, because they can be respectively represented as
1 2 1
0 0
( ) N ( ) jNki N ( ) k( )
k k
x i X k e X k i
− π −
= =
=
∑
=∑
Φ (4.1.2.1)1 2 1
0 0
( ) N ( ) jNki N ( ) ( )k
i i
Y k y i e y i i
π ψ
− − −
= =
=
∑
=∑
(4.1.2.2)which are the same as IDFT operation of the transmitted data X (k) and DFT operation of the received data y(i) , respectively.
Figure 4-5 shows the discrete-time baseband OFDM model. The IDFT transforms the frequency-domain data into time-domain data which is delivered over the air and passed through a multi-path channel, denoted as h(n,m) n is the time index and m is the channel path delay. At the receiver, to recover the signal in frequency domain, DFT is adopted in the demodulator as a matched filter. Then the frequency-domain signal of each subchannel is obtained from its DFT output.
Figure 4-5 Discrete-time OFDM system model [10]
4.1.3 Effect of Cyclic Prefix
Because of multipath channels, orthogonality as shown in Figure 4.2 will be destroyed by ISI and ICI. However, as long as the cyclic prefix length is longer than the channel order of h(n,m) , ISI effect can be avoided. It is known that circular convolution in time domain results in multiplication in frequency domain when the channel is stationary so that the received signal Y(k) in frequency domain is the product of transmitted data X(k) and channel response H(k) in the kth subcarrier . Thus, the orthogonality is maintained (if h(n,m) is fixed within the symbol length, then h(n,m)= h(m) is not about time) and data can be easily recovered by
one-tap channel equalizer, i.e., dividing Y(k) by the corresponding H(k).
( ) ( )* ( ) ( )
Y k =H k X k +N k (4.1.3.1)
Y(k), X(k), H(k),N(k) are the kth subcarrier after DFT according to y(i),x(i),h(i),n(i) .About that we can know the channel is flat fading at each subcarrier.
4.2 MIMO-OFDM Architecture
According to Section 4.1, OFDM technique turns frequency-selective fading channel into several flat-fading subchannels, and it solves the major problem in wideband transmission systems. We will employ V-BLAST technique to detect the transmitted signals on each subcarrier of a MIMO-OFDM systems. MIMO-OFDM transceiver and receiver architecturs are shown in Figures 4.5 and 4.6 respectively. Subchannels are orthogonal to each other in OFDM systems. Hence, in single-input-single-output (SISO) OFDM systems, the received signals are product of channel response and transmitted signal. In MIMO systems, signals transmitted from different antennas on a subcarrier simultaneously interfere each other, but signals at different subcarriers are independent. At each receiver antenna, a linear combination of the transmitted signal and channel response on each subcarrier is observed. That corresponds to assumptions of MIMO systems. On each subchannel, a space division multiplexing (SDM) is similar to V-BLAST is applied. That is, the task is to recover x from the received signal y and channel state information (CSI) H on each subcarrier.
S
Figure 4-6 Transmitter architecture of MIMO OFDM systems
RF
Figure 4-7 Receiver architecture of MIMO OFDM systems
4.3 B-Chase Detector in MIMO-OFDM Systems
We will employ MIMO-OFDM systems to extend the B-Chase detector in the MIMO systems. Due to OFDM systems can turns frequency-selective fading channel into several flat-fading subchannels and get high spectral efficiency. When the channel state is multipath
and the cyclic prefix length is longer than the path delay, we can use flat fading MIMO case to handle it in each subcarrier which do not interfere other subcarriers. That is robust for the frequency-selective fading channel when we detect the receiveed signal in the B-Chase detector. We can say that the frequency-selective fading channel can get time delay diversity when we can handle the frequency-selective fading channel as the flat fading channel. For that we can employ OFDM to get that. The subsequent MIMO signal processing takes place on each subcarrier identically. In order to describe the flat fading MIMO systems observed at each subcarrier in the frequency domain. We let [ ,1 ]
t
i i i T
a aN
=
a " denote the Nt × 1 transmit
signal vector of subcarrier i, then the corresponding Nr ×1 receive signal vector
[ ,1 ]
w " represents independent white Gaussian noise of variance ( )σni 2 observed at the Nr receive antennas while the average transmit Gaussian fading gains with unit variance. We assume that the channel matrix Hi is constant over a frame and changes independently between frames (block fading channel).That in the following the algorithms are given on the base of subcarrier i assuming an outer loop over all subcarrier i=1,…, NFFT. The index i is therefore omitted to simplify matters giving r = ri, the received signal on subcarrier i, H =Hi , the channel matrix, a = ai, the transmit symbols on
subcarrier i and w = wi , the noise vector respectively. From that we detect the transmitted signals on each subcarrier of MIMO-OFDM systems by B-Chase detector.
Table 4-1 System parameters
FFT length 16
Symbol period 16 samples
Cyclic prefix 4 samples
Modulation 16-QAM
Transmit antenna 4
Receive antenna 4
Channel is updated in T symbol periods 8
Rayleigh-fading Mean=0,Varance=1
Channel order 3
List length l 1 , 2, and 16
5 10 15 20 25 10-5
10-4 10-3 10-2 10-1 100
SNR dB
BER
B-CHASE*(1) B-CHASE*(2) B-CHASE*(16)
Figure 4-8 Bit error rate versus SNR in the B-Chase detector* ( l ) with l =1,2,16 for MIMO-OFDM Systems
Table 4-2 System parameters
FFT length 16
Symbol period 16 samples
Cyclic prefix 4 samples
Modulation BPSK
Transmit antenna 4
Receive antenna 4
Channel is updated in T symbol periods 8
Rayleigh-fading Mean=0,Varance=1
Channel order 3
List length l 1 , 2
0 2 4 6 8 10 12 10-5
10-4 10-3 10-2 10-1
SNR dB
BER
B-CHASE*(1) B-CHASE*(2) ML
Figure 4-9 Bit error rate versus SNR in the B-Chase detector* ( l ) with l =1,2 for MIMO-OFDM Systems
Chapter 5 Conclusion
The B-Chase of detection algorithm is a combination of a list detector and a parallel bank of subdetectors. The B-Chase detector that can trade performance for reduced complexity by modifying the list length. When applying the B-Chase of detection algorithm in the MIMO-OFDM, we can improve performance in the frequency-selective fading channel.
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自 傳
方自民, 西元 1981 年生於台南市。 西元 2004 畢業於台灣國立台北科技大 學電子系，之後進入交通大學電子研究所攻讀碩士學位，於 2008 年取得碩 士學位。研究方向為無線通訊訊號處理。