2.6 Complexity Analysis and Numerical Results
2.6.2 Numerical Results
Selected simulated performance of the proposed RA algorithms are presented in this subsection. First we evaluate the performance of the proposed RA algorithms for the orthogonal precoding scheme in Fig. 2.2 ∼ 2.4. For Fig. 2.5 ∼ 2.10, we compare the performance of the proposed RA algorithm for the orthogonal precoding scheme and BD [16].
The performance of the proposed algorithms for uplink and downlink transmissions is shown in Fig. 2.2 and Fig. 2.3 respectively. In Fig. 2.4 we compare the performance of the proposed algorithms with that of the optimal solution. The average power is normalized by that of the single-user case, i.e., when a single user has access to all eigenchannels and all subcarriers. We define the average power ratio at BER=B as :
PB = 10 log10
Pavg,B Pavg,10−5,single
(2.23)
where Pavg,B represents the average transmit power for a given modulation scheme at BER=B and Pavg,10−5,singlerepresents the average transmit power for the single user case at BER=10−5.
We assume each Hmk is a 2× 4 (4 antennas at the BS and 2 antennas at each MS) matrix with i.i.d. zero-mean, unit-variance complex Gaussian entries. The system has six different modulation modes, BPSK, QPSK, 8QAM, 16QAM, 32QAM, and 64 QAM, respectively. For simplicity, we assume that the required data rate and BER are the same for all users.
In Fig. 2.2, we compare the performance of the proposed algorithms with that of the adaptive zero-forcing MIMO-OFDM receiver proposed in [23] and its non-adaptive
counterpart. Algorithm I (the efficient space/frequency resource allocation algorithm) is superior to the adaptive MIMO-OFDM ZF approach by a 2.5 dB margin and Algorithm II (the constraint-relaxation based greedy search algorithm) offers additional 0.5 dB performance gain. Both algorithms achieve more than 12 dB performance gain against the non-adaptive MIMO-OFDM ZF receiver.
−10 −5 0 5 10 15 20
10−5 10−4 10−3 10−2 10−1
Average Power Ratio (dB)
BER
non−adaptive MIMO−ZF adaptvie MIMO−ZF algorithm I algorithm II
Figure 2.3: Average power ratio per user for a MIMO-OFDM uplink; 32 subcarriers, 64 bits per OFDM symbol, 4 users.
In Fig. 2.3, we consider downlink transmission and compare the performance of the FDMA scheme and our algorithms. For the FDMA scheme (without bit-loading), the subcarriers are allocated to the users like the classical FDMA scheme according to their data rate requirements. Each user has access to all the eigenchannels on the corresponding subcarrier subset. Each required data rate (bits/transmission) is equally distributed among all eigenchannels available to a user. We also consider the FDMA scheme with bit-loading. Similar to the uplink case, Algorithm I outperforms the FDMA scheme with bit-loading by 2.5 dB and Algorithm II provides additional 0.5 dB gain.
Both algorithms have more than 10 dB advantage over the FDMA scheme without bit-loading.
−10 −5 0 5 10 15
10−5 10−4 10−3 10−2 10−1
Average Power Ratio (dB)
BER
FDMA with bit−loading FDMA without bit−loading algorithm I
algorithm II
Figure 2.4: Average power ratio per user for a MIMO-OFDM downlink; 32 subcarriers, 64 bits per OFDM symbol, 4 users.
Finally, Fig. 2.4 plots the performance of both the optimal solution and the proposed algorithms. It is found that Algorithm II yields performance almost identical to that of the optimal solution while Algorithm II suffers only minor degradation.
In Figs. 2.5–2.9, we compare the proposed orthogonal precoding scheme (using Algo-rithm I) and BD in downlink transmission. BD is a popular linear precoding technique for the multiple antenna multicast channel that involves transmission of multiple data streams to each receiver such that no multiuser interference is experienced at any of the receivers. In BD, each user data vector is multiplied by a precoding matrix to project the transmitted signal to the null space of the space spanned by all other users’ spatial channels.
We examine the performance for the 2-user and 4-user cases. In the 2-user case, we
−12 −10 −8 −6 −4 −2 0 2 10−5
10−4 10−3 10−2 10−1
Average Power Ratio (dB)
BER
optimal solution algorithm I algorithm II
Figure 2.5: Average power ratio per user for a MIMO-OFDM downlink; 8 subcarriers, 16 bits per OFDM symbol, 2 users.
assume that Tx = 4 and Rx = 2 and for the 4-user case, Tx = 8 and Rx = 2. As for the required user data rate, we consider three cases: 64 bits per OFDM symbol, 128 bits per OFDM symbol, and 256 bits per OFDM symbol, respectively. The system offers eight different modulation options ranging from from BPSK to 256 QAM.
First we examine the performance of the 2-user case in Figs. 2.5 and 2.6. It is found that the proposed precoding scheme outperforms the BD approach when desired data rate is 64 or 128 bits per OFDM symbol. However, Fig. 2.7 indicates that BD outperforms the proposed precoding scheme by almost 0.5 dB. The same behavior can be found in the 4-user case. In Figs. 2.8 and 2.9, the performance of the proposed precoding scheme is better than that of the BD precoder when desired data rate is 64 bits or 128 bits per OFDM symbol. However, when the data rate increases to 256 bits per OFDM symbol, BD is superior to our precoder by a 1.5 dB margin.
Such a performance trend can be explained by (2.21) which indicates that the
trans-mit power of an eigenchannel is a function of both GNR (GN Rmr(k)) and the num-ber of bits loaded (brmk). When the eigenchannel’s GNR is large (strong eigenmode) or the number of loaded bits is small, the corresponding required transmit power is low. For the BD precoder, the number of eigenchannels per subcarrier for each user is Tx − (Rx∗ (K − 1)) while that for the proposed precoder is R = min(Tx, Rx). For the 2-user case, the numbers of eigenchannel per subcarrier for the BD and GS precoders are 4 and 2. For the 4-user case, the number of eigenchannel per subcarrier for the BD precoder are 8 while that for the GS precoder remains to be 2. Moreover, both simu-lation and theoretical analysis show that the eigenmode magnitude suffer degradation by using BD (this will be proved in the last of this section). Define the channel loading coefficient η as
η =
K i=1Rk
RM bmax. (2.24)
When η is more close to 1, this means that each eigenchannel must be loaded more bits since the user’s data rate is close to the system limit such that the total consuming power is higher. When η is more close to 0, this means that the bits on each eigenchannel is less and the total consuming power is also less relatively. When the channel loading is light (η is close to 1), the bit number on each eigenchannel (brmk) is small and the eigenmode magnitude (GN Rmr(k)) will dominate the system performance. However, when we in-crease the user’s data rate such that the channel loading is heavy, the exponential term brmk in (2.21) becomes an important factor that will affect the system performance.
The channel loading coefficient for BD and the proposed precoding scheme is listed as follows:
Table 2.4: η for the 2 users case
Precoding scheme/Users’ data rate 64 bits 128 bits 256 bits
BD 0.125 0.25 0.5
Orthogoal precoding scheme 0.25 0.5 1
Table 2.5: η for the 4 users case
Precoding scheme/Users’ data rate 64 bits 128 bits 256 bits
BD 0.0625 0.125 0.25
Orthogoal precoding scheme 0.25 0.5 1
Numerical results given in the above two tables indicate that the BD precoder has lower channel loading coefficients (≤ 0.5) as it offers more eigenchannels. Therefore, when the data rate requirement is low, the performance of BD precoder is inferior to that of GS precoder due to the fact that BD precoding results in weaker eigenmodes; see Lemma 2.6.2 below. But for high data rate requirements, the per eigenchannel loading for the BD precoder remains relatively low which then leads to better performance.
−14 −12 −10 −8 −6 −4 −2 0 2 4
10−5 10−4 10−3 10−2 10−1
BER
Average Power Ratio (dB) BD
Proposed scheme
Figure 2.6: Average power ratio per user for a MIMO-OFDM downlink; 32 subcarriers, 64 bits per OFDM symbol, 2 users.
To prove the eigenmode degradation suffered in the BD scheme, we need
Lemma 2.6.1. (Weak Majorization Lemma) Let x1, . . . , xn, y1, . . . , yn be 2n given real numbers such that x1 ≥ . . . ≥ xn, y1 ≥ . . . ≥ yn and
k
i=1yi ≥ k
i=1xi, k = 1, . . . , n.
−12 −10 −8 −6 −4 −2 0 2 4
Figure 2.7: Average power ratio per user for a MIMO-OFDM downlink; 32 subcarriers, 128 bits per OFDM symbol, 2 users.
Then for any real-valued function f (·) which is increasing and convex on the interval [min{xn, yn}, y1],f (x1)≥ . . . ≥ f(xn), f (y1)≥ . . . ≥ f(yn), and
Using the above lemma we can prove that the sum magnitude of the eigenmodes associated with the augmented matrix XY is always less than the sum of magnitude product for the component matrixes X and Y.
Lemma 2.6.2. Consider the m× p and p × n matrices, X and Y and let σi(X) be the ith singular value of X. Then
k where Uk and Vk both contain the first k columns of U and V, respectively. SVD
−8 −6 −4 −2 0 2 4 6
Figure 2.8: Average power ratio per user for a MIMO-OFDM downlink; 32 subcarriers, 256 bits per OFDM symbol, 2 users.
of YVk yields YVk = U Σ V† = [ Uk U×] Σk
−10 −8 −6 −4 −2 0 2 4 6 10−5
10−4 10−3 10−2 10−1
BER
Average Power Ratio (dB) BD
Proposed scheme
Figure 2.9: Average power ratio per user for a MIMO-OFDM downlink; 64 subcarriers, 64 bits per OFDM symbol, 4 users.
It is noted that the inequality becomes equality when m = p = n and k = n. If we take log for both sides, we get
k i=1
ln(σi(X)σi(Y)) ≥ k
i=1
ln(σi(XY)). Let xi = ln(σi(XY)), yi = ln(σi(X)σi(Y)) and f (·) is to take the exponential of the argument matrix. The Weak Majorization Lemma then implies
k
i=1[σi(X)σi(Y)]≥ k
i=1[σi(XY)] for k = 1,· · · , q, where q = min{m, p, n}.
Let X = H and Y be the precoder matrix. For the BD precoder, σi(Y) = 1, ∀ i.
Hence, the above lemma tells us that BD precoding decreases the sum strength of all eigenchannels. Moreover, our simulation results show that not only the sum of the eigenmode magnitudes but also the individual eigenmode magnitude degrades after BD precoding.
−8 −6 −4 −2 0 2 4 6 10−5
10−4 10−3 10−2 10−1
BER
Average Power Ratio (dB) BD
Proposed scheme
Figure 2.10: Average power ratio per user for a MIMO-OFDM downlink; 64 subcarriers, 128 bits per OFDM symbol, 4 users.
−6 −4 −2 0 2 4 6 8 10 12
10−5 10−4 10−3 10−2 10−1
BER
Average Power Ratio (dB) BD
Proposed scheme
Figure 2.11: Average power ratio per user for a MIMO-OFDM downlink; 64 subcarriers, 256 bits per OFDM symbol, 4 users.
Chapter 3
Resource Allocation for MIMO Systems with Non-Orthogonal Precoding
3.1 System and Transceiver Models
In the previous chapter, we consider MIMO systems that use a orthogonal precoding scheme so that system users can transmit through distinct eigenchannels on the same subcarrier without causing interference to each other. For such a scheme, however, the maximum eigenchannel number is bounded by the rank of the MIMO channel matrix (R) and thus the spectrum efficiency may be constrained. To increase the spectrum efficiency, we allow more than R users to transmit over the eigenchannels on the same subcarrier. In this situation, the co-channel interference among users is no longer avoidable. Therefore, the associated optimization problem becomes more complicated due to the constraints on the tolerable inter-channel interference (ICI).
Similar to the previous system setup, we consider a MIMO-OFDMA system with a single base station (BS) equipped with Tx antennas and K mobile station (MS) users, each equipped with Rxantennas. The frequency band used contains M subcarriers which are to be allocated to the K MS’. Based on the GS precoder design, we provide R− 1
orthogonal eigenchannels for users with no interference and additional Q (R∼ R+Q−1) eigenchannels with various tolerable interference levels.
For the R− 1 orthogonal eigenchannels, the way to choose the pre-processing and post-processing vectors is the same as that described in Chapter 2. That is, for the user to whom the rth eigenchannel is given, the pre-processing and post-processing vectors are the linear combinations of first r left and right singular vectors, respectively. In order that the Q non-orthogonal eigenchannels do not induce interference to the R− 1 orthogonal eigenchannels, we require that the users who are allocated non-orthogonal eigenchannels to transmit over an eigenchannel which lies in the null space spanned by all R− 1 orthogonal eigenchannels. More specifically, they use linear combinations of R singular vectors as the processing vectors to project the transmitting signal to the null space of the R− 1 dimensional space spanned by orthogonal eigenchannels.
Although the non-orthoganal eigenchannels will not interfere with the R− 1 orthog-onal eigenchannels, the co-channel interference among the non-orthogorthog-onal eigenchannels is unavoidable. Here we define Bmk = 1 if user k is to transmit on the mth subcarrier’s non-orthogonal eigenchannel and Bmk = 0, otherwise. The GINR (gain to interference and noise ratio) for users k who is allowed to transmit data on the non-orthogonal eigen-channels can be expressed as:
GIN Rmk = correlation between user k and user i) as ρmki, then (1) can be simplified as:
GIN Rmk = |gmk|2 σ2+ K
i=1,i=k|ρmki|2pmi
(3.2)
It is noted that if a user is assigned more than one non-orthogonal eigen-channels,
the interference will become too large since the correlation is unity. Therefore, we will assign one non-orthogonal eigenchannel to the same user at most.
3.2 Problem Formulation
Let bmkand pmkbe the number of bits and the corresponding power transmitted over the mth subcarrier using the non-orthogonal eigenchannel by user k. The resource allocation problem can then be reformulated as:
min
subject to the following constraints:
R−1
The above optimization problem is a mixed-integer problem which is NP-hard. In this case, we have to assign eigenchannels to users more carefully.