Figure 11.13 and Figure 11.14 show the BER as a function of SNR for the PUSC subcarrier mode in vehicular channel with speeds of 30kmph and 120kmph, respectively, for a realistic 7. Note that the accuracy of the channel-tracking algorithm can be improved by increasing the number
of pilot subcarriers in the standard, but that would reduce data rate.
Table 11.3 Band AMC Gaina over PUSC Subcarrier Permutation
a. A negative gain implies a link loss in going from band AMC to PUSC subcarrier permutation.
QPSK 16 QAM
10–2 BER (dB) 10–4 BER (dB) 10–2 BER (dB) 10–4 BER (dB) R 1/2 R 3/4 R 1/2 R 3/4 R 1/2 R 3/4 R 1/2 R 3/4
Ped A 5.0 6.5 6.5 11 .0 5.0 6.5 6.0 12.0
Ped B 2.5 5.0 2.5 5.0 2.5 5.0 2.5 5.0
Veh A 30 –1.0 < –.5 –2.0 –1.0 –3.0 –2.0 –4.5 –4.0
Veh A 120 –2.0 –1.0 –2.5 –2.0 –3.5 –2.5 –5.5 –4.5
Figure 11.13 Performance of PUSC in Veh A channel with 30kmph speed for real receivers and receivers with perfect CSI
Figure 11.14 Performance of PUSC in Veh A channel with 120kmph speed for real receivers and receivers with perfect CSI
1.E-04 1.E-03 1.E-02 1.E-01 1.E+00
0 5 10 15 20 25 30
SNR (dB)
Bit Error Rate
QPSK R1/2 Veh A30 QPSK R1/2 Veh A120 QPSK R3/4 Veh A30 QPSK R3/4 Veh A120 QPSK R1/2 (Perfect CSI) QPSK R3/4 (Perfect CSI)
QPSK R3/4 QPSK R1/2
1.E-04 1.E-03 1.E-02 1.E-01 1.E+00
0 5 10 15 20 25 30
SNR (dB)
Bit Error Rate
16 QAM R1/2 Veh A30 16 QAM R1/2 Veh A120 16 QAM R3/4 Veh A30 16 QAM R3/4 Veh A120 16 QAM R1/2 (Perfect CSI) 16 QAM R3/4 (Perfect CSI)
16 QAM R3/4 16 QAM R1/2
channel-estimation algorithm that can be used in WiMAX. These figures also shown the BER for a receiver with perfect CSI; that is, the channel response over all the subcarriers of all the OFDM symbols is known a priori at the receiver.
To arrive at an initial estimate of the channel response, we use the frequency-domain linear minimum mean square error (LMMSE) channel estimator with partial information about the channel covariance.8 The DL frame preamble or the MIMO midambles are used for this pur- pose. Over the subsequent OFDM symbols, the channel is tracked, using a Kalman filter–based estimator. Thus, at vehicular speeds, if reliable channel tracking is not performed, considerable degradation in the performance of the system occurs.
The channel response of a multipath tap when sampled at a given instance in time can be related to its previous samples as
, (11.9)
wherel is the tap index, αl is the temporal correlation between two consecutive samples of the tap,pl is the average power of the tap, and is a sequence of independent and identically dis- tributed complex Gaussian random variables with zero mean and unit power. If we assume each multipath tap to be an independent Rayleigh channel, the correlation αl between two consecutive samples of a tap is J0(2πfl∆t), where fl is the Doppler frequency of the tap, ∆t is the elapsed time between two samples, and J0 is the zero-order Bessel function of the first kind. In the frequency domain, the channel response over the kth subcarrier can thus be written as
. (11.10)
We assume that the Doppler frequency of each tap is the same. Although this assumption is not necessary, it is used here without any loss of generality. In Equation (11.10), if we substitute the sequence by its time-reversed version , the last term can be written as the lth term of the convolution between the power delay profile ( ) and the time-reversed random sequence . Since is a sequence of independent and identically distributed random variables, the time-reversed version has exactly the same statistical properties as the original sequence. In the frequency domain after the DFT, the convolution between the power-delay profile and the time-reversed sequence appears as a product. This allows Equation (11.11) to be written in a more compact matrix notation as the following:
8. An LMMSE receiver with full channel covariance in not considered here, since a real receiver does not possess a priori information about the channel covariance. However, an LMMSE receiver with partial channel covariance based on the knowledge of the RMS delay spread is expected to be re- alistic. A knowledge of the RMS delay spread can bound the channel covariance, which can im- prove the fidelity of the channel estimates.
h˜l(n+1) αlh˜
l( )n + (1–αl2)pl z˜l( )n
=
z˜l
hk(n+1) αhn( )n 1–α2 plz˜l( )n 2πjfkτl ---N
⎝– ⎠
⎛ ⎞
exp
l=1 L
∑
+
=
z˜l z
˜l z˜L l
= –
pl z˜l z˜l
, (11.11) whereR is the covariance matrix of the frequency-domain channel, and z is a vector of indepen- dent and identically distributed random variables, with the same statistics as . This recursive relationship between the channel samples can be tracked using a Kalman filter.
Figure 11.15 compares the actual channel and the channel estimated by this tracking algo- rithm in a Veh A 120 kmph channel for various SNR values. Since we use the DL preamble for an initial channel estimate that is then tracked in subsequent OFDM symbols, the results shown here are for the last symbol of the DL subframe, where the estimates are likely to be most erro- neous. At 0dB SNR, the reliability of the estimates is quite poor, as expected, particularly in sub- channels that are experiencing a fade. However, at 20dB SNR, the reliability of the channel estimate is quite good.
Although the reliability of the channel-estimation algorithm improves as SNR increases, the difference between ideal and real channel-estimation schemes seem to be more prominent in higher SNR values, (Figure 11.10). The reason is that at low SNR, the error occurrences are dominated by noise and interference, not by channel estimation error, which starts to play an important role in the occurrence of detection errors only at high SNR.
Figure 11.15 Channel estimation and channel tracking in OFDM systems h(n+1) = αh( )n + 1–α2Rz( )n
z
˜l
1.E-02 1.E-01 1.E+00 1.E+01
0 100 200 300 400 500 600 700 800 900
Subcarrier Index
Actual Channel Estimate 20dB SNR Estimate 0dB SNR
Channel/Response
From an information theoretic perspective, supported by the results in Figure 11.13 and Figure 11.14, it is clearly not possible to design a real receiver that will perform at par with a receiver that has exact channel state information. In a real system with limited time, frequency, and power dedicated to the pilot signal, it is not possible to eliminate channel-estimation error. Such errors can be reduced by increasing the resources given to the pilot signal, which, however, comes at the cost of reducing the resources for data and thus system capacity. In order to maximize the capac- ity of an OFDM system, such as WiMAX, one needs to carefully balance the division of resources in terms of time, frequency, and power between the pilot signals and the data signals [10].