Simulating a wireless network that consists of a very large number of cell sites is often computa- tionaly prohibitive and inefficient. Therefore, the system simulator used to generate the results presented in this chapter consists of only two tiers of cell sites that are present in a hexagonal grid, as shown in Figure 12.1.
Owing to the finite size of the simulated network, cell sites that lie toward the edge of the simulated network (outer tier) have missing neighbor cell sites, which causes the other-cell cochannel interference to be modeled inaccurately in these cells. This edge effect mandates that statistics related to network performance indicators, such as data rate and throughput, should be sampled only in the center cell, where the modeling of the other-cell interference is accurate enough. One solution to edge effect is the wraparound approach, which allows the simulator to model the interference and collect statistics even in the cells at the edge of the network. The sys- tem simulator used to generate the results presented in this chapter implements such a wrap- around to mitigate the edge effect.
Figure 12.1 Two-tier layout of cells for the system simulator
Missing Neighbor Cells
The system simulator takes multiple Monte Carlo snapshots of the network to observe ergodic samples of the network and to determine how it behaves over long time scales [2]. Each Monte Carlo snapshot samples the behavior of the network over a 5msec frame. During each snapshot, the simulator randomly distributes various MSs in each sector and analyzes the expected instantaneous behavior of the network in terms of data rate, cell throughput, and outage probability. Information related to the location of each MS and the state of its traffic buffer are purged at the beginning of the next Monte Carlo snapshot of the network. This way, each Monte Carlo sample is random and completely independent of the previous sample. In each Monte Carlo snapshot, the simulator takes several steps to calculate the cell throughput and user data rates.
12.2.1.1 Computation of Time-Domain MIMO Channel
The instantaneous channel as observed by each MS from all the BSs in the network is calcu- lated as
, (12.4)
where is the median pathloss between the nth MS and the kth BS; Hn,k(τ) is the instan- taneous fast-fading component of the MIMO channel; D is the total number of paths—depends on the path-delay profile—τi is the delay of the ith multipath relative to the first path; dn,k is the distance between the MS and the BS, sn,k is the instantaneous shadow fading between the MS and the BS, gb and gm are the gain patterns of BS and MS antennas, respectively; and and are the angle of departure at the BS and angle of arrival at the MS with respect to the bore- sight directions of the BS and MS antennas, respectively. The fast-fading MIMO component Hn,k(τ-τi) is calculated using the methodology explained in Section 11.1.1, and is a complex matrix with dimension Nr × Nt for each path index i, where Nr is number of receive antennas, andNt is the number of transmit antennas. The instantaneous shadow-fading components sn,k for each Monte Carlo snapshot are generated as i.i.d. random variables with a lognormal distribu- tion. Appropriate correlation between the shadow fading observed by a given MS from various BSs, is also modeled,2 using a coloring matrix.
12.2.1.2 Computation of Frequency-Domain MIMO Channel
The time-domain MIMO channel is then converted to the frequency-domain MIMO channel, using a Fourier transformation, as given by the following:
2. Since the shadow fading between a BS and an MS is partially dependent on the local neighborhood of the MS, it is expected that the shadow fading at a given MS from a different BS is correlated to a certain degree.
H'n k, ( )τ PL d( n k, )sn k, gb(θn k, )gm θ˜
n k,
( )n Hn k, (τ τ– i)
i=1
∑D
= PL d( n k, )
θn k, θ˜
n k,
, (12.5) wherelis the subcarrier index,∆f is the frequency separation between two adjacent subcarriers, andNsub is the total number of subcarriers. If transmit precoding is used at the transmitter, as is done in the case of closed-loop MIMO or beamforming, the net channel in the frequency domain can be written as
, (12.6)
wherePn(l) is the precoding matrix (or vector) of the nth BS for the lth subcarrier.3 12.2.1.3 Computation of per Subcarrier SINR
Once the instantaneous channel response from each BS for the given MS is known in the fre- quency domain, the system simulator calculates the SINR per subcarrier. This calculation is per- formed for both matrix A and matrix B usage, since the system simulator does not possess a prior knowledge of the spacetime coding matrix used (see Chapter 8). In the case of matrix A, the simulator assumes a linear ML receiver for the 2 × 2 Alamouti block code; and in the case of matrix B, the simulator assumes a linear MMSE receiver. The post-detection SINR per sub- carrier for the nth MS is thus given by
. (12.7)
In Equation (12.7), Norm indicates the Forbenius norm of the matrix, and γA(n,l) is the post-detection SINR for the lth subcarrier of the nth MS if matrix A space/time encoding is used.
Similarly, γB(n,l,1) and γB(n,l,2) are the post-detection SINR of the first and the second streams, respectively, for the lth subcarrier of the nth MS and second streams, respectively, if matrix B space/time encoding is used. In the equation, β is the noise figure of the receiver, N0 is the noise 3. The precoder used for a given subcarrier and the instantaneous MIMO channel depends on the
closed-loop MIMO mode used and the optimization criteria used by the precoder, such as maxi- mum capacity, minimum MSE.
H'˜
n k, ( )l 1 Nsub
--- H'n k, ( )τi exp(–2π∆f l N( – sub⁄ 2)τi
i=1 D
∑
=
H'˜
n k, ( )l 1 Nsub
---Pn( )l H'n k, ( )τi exp(–2π∆f l N( – sub⁄ 2)τi
i=1 D
∑
=
γA(n l, ) Norm ΘA n, ( )H'l ˜
n m, ( )l
( )
βN0W Norm ΘA n, ( )H'l ˜
n k, ( )l
( )
k=1,k m≠ K
∑
+
---
γB(n l ss, , ) Norm D( )Θss B n, ( )l H'˜
n m, ( )l
( )
βN0W Norm D( )Θss B n, ( )H'l ˜
n k, ( )l
( )
k=1,k m≠ K
∑
+
---
=
=
power spectral density, and W is the noise bandwidth, which is the same as the channel bandwidth, assuming that a matched Nyquist filter is used at the receiver. The variables are the linear estimation matrices for the lth subcarrier of the nth user, assuming matrix A and matrix B usage respectively. The matrices D(ss) in are given by
. (12.8)
Thus, when used inside a Norm operator, D(1) provides the SINR for the first stream. When used inside the Norm operator, D(2) provides the SINR for the second stream.
12.2.1.4 Computation of per Subchannel Effective SINR
Next, the simulator calculates the effective SINR for each subchannel, based on the postprocess- ing SINR per subcarrier. The effective SINR is an AWGN-equivalent SNR of the instantaneous channel realization and can be calculated in different ways. Owing to the frequency selectivity of broadband channels, the average SINR over all the subcarriers that constitute a given subchannel is not a good indicator of the effective SINR, since averaging fails to capture the variation of the SINR over all the subcarriers. Several metrics, such as EESM (exponentially effective SINR map), ECRM (effective code rate map), and MIC (mean instantaneous capacity), are widely accepted as better representatives of the effective SINR and capture the variation of SINR in the subcarrier domain. In this chapter, the MIC metric is used, since it is considerably simpler to implement than the EESM and ECRM methods. In the MIC method, the SINR of each subcarrier is first used to calculate the instantaneous Shannon capacity of the subcarrier. Then the instanta- neous Shannon capacity of all the subcarriers are added to calculate an effective instantaneous Shannon capacity of the subchannel. The effective instantaneous Shannon capacities of the sub- channel is then converted back to an effective SINR for the subchannel of interest. Thus, the effective SINR for subchannel s of the nth user for matrix A and matrix B is given by
(12.9)
12.2.1.5 Link Adaptation and Scheduling
The effective per subchannel SINR is then used by the scheduler to allocated the slots4 in the DL and UL subframes among all the MSs that have traffic in the buffer. The effective SINR is also 4. A slot in WiMAX is the smallest quanta of PHY resources that can be allocated to an MS. A slot
usually consists of one subchannel by one, two, or three OFDM symbols, depending on the subcar- rier permutation scheme. For a more detailed definition of a slot, refer to Section 8.7.
ΘA n, ( ) and Θl B n, ( )l
D( )1 1 0
0 0 D( )2 0 0
= = 0 1
γ˜
A n, ( )s ( )2
1
48--- log2(1+γA(n l, i s, ))
i=1 48
⎝ ∑ ⎠
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎛ ⎞
1
γ˜
B n, (s ss, )
–
( )2
1
48--- log2(1+γB(n l, i s, , ss))
i=1 48
⎝ ∑ ⎠
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎛ ⎞
1 –
=
=
used to determine the optimum modulation, code rate, FEC code block size, and space/time encoding matrix (matrix A or matrix B) for each scheduled MS.
As discussed in Chapter 11, the link-level simulation results are used to determine the SINR thresholds for the selection of optimum modulation and code rate (see Figure 12.2). The slots allocated to each MS are then divided into one or more FEC code blocks, based on the code block segmentation (see Section 8.1). The AWGN link-simulation results are used to calculate the block error probability of each FEC code block. Based on the block error probability, the system simulator performs a Bernoulli toss to determine which of the FEC blocks are received erroneously and need to be retransmitted. When H-ARQ, is used the retransmissions of an FEC code block are combined with the previous transmissions to determine the block error rate.
12.2.1.6 Computation of per Sector Throughput and User Data Rate
The per sector throughput and per user data rate are then calculated, based on the size and the average number of transmissions needed for each of the FEC code blocks over multiple Monte Carlo snapshots. Table 12.2 shows the various parameters and assumptions used for the system-level simulation results. In the DL subframe, the system simulator assumes that Figure 12.2 FEC block error rate for turbo codes in AWGN channel
1.E-04 1.E-03 1.E-02 1.E-01 1.E+00
–5 0 5 10 15 20
SNR (dB)
FEC Block Error Rate
QPSK R1/2 (2× rep) QPSK R1/2 QPSK R3/4 16 QAM R1/2 16 QAM R3/4 64 QAM R1/2 64 QAM R2/3 64 QAM R3/4
16 QAM 2× Repetition Code
64 QAM QPSK
11 OFDM symbols are used for the various control messages, such as DL frame preamble, FCH, DL-MAP, UL-MAP, DCD, and UCD, and the MSs are not scheduled over the first 11 DLs. Thus, the calculated average throughput and data rate do not include any MAP over- heads and represent the effective layer 1 capacity available for data plane traffic. However, overheads due to the MAC header, convergence sublayer, mobility management, QoS man- agement, and so on, are not explicitly modeled. As a result, the net layer 3 capacity available for IP traffic is expected to be less that what is indicated here.