Summary

In this chapter, the limited feedback MIMO system is presented. We introduce the 4-bit codebook in LTE Release 8 for full vector quantization in our system. The user feeds back optimum codeword (i.e., quantized CSI) to the transmitter and transmitter performs Zero-Forcing precoding based on these received quantized CSI from different users. Moreover, Jakes’ channel simulator is given as our temporal correlated channel model and we will further exploit this temporal correlation of CSI to reduce the number of feedback bits by proposed differential quantization scheme in the next chapter.

Chapter 3

Reduced CSI Feedback Based on Differential Quantization

In limited feedback systems, perfect CSI feedback is impossible due to finite-rate feedback link. How to reduce CSI feedback overhead in limited feedback systems while maintaining performance has became a crucial issue. Differential pulse code modulation (DPCM) gives us the motivation to apply the concept of differential quantization in limited feedback systems to reduce or, equivalently, to compress the number of feedback bits. We further extend DPCM to “Predictive Vector Quantization” (PVQ) [15] to implement differential quantization and thereby reduce the number of feedback bits required. The architecture of PVQ and its components, including a vector quantizer and a predictor, are presented in this chapter.

First, we use differential quantization to feedback CSI by exploiting its temporal correlation. Then, extend to feedback the subcarrier information of MIMO OFDM systems by exploiting the temporal correlation of subcarriers. Simulation results have shown that limited feedback systems with differential quantization scheme can achieve higher SINR under highly correlated channel environment.

3.1 Motivation

Differential quantization is frequently used in data compression or source coding to reduce the number of quantization bits. The most commonly used technique for differential quantization is “differential pulse code modulation＂ (DPCM) [26].

Intuitively, we can apply DPCM in limited feedback systems to reduce the number of feedback overhead. In Fig. 3-1, 1 bit per element, real and imaginary part separately, is assumed (i.e., 8 bits for each channel vector

h

∈

C

^{4 1×} ,

M = ) for both DPCM

4 and full scalar quantization. Fig. 3-1 shows higher SINR can be achieved by DPCM compared to the conventional full scalar quantization. However, DPCM is an

“element-wise” quantization scheme which requires a large number of bits to quantize a complex channel vector. Therefore, we propose the idea of using differential vector quantization by “Predictive Vector Qautization” (PVQ) in limited feedback systems to further reduce the feedback overhead.

2 3 4

-5 0 5 10 15 20 25

SI NR

Number of users DPCM

Full Scalar Quantization

Fig. 3-1 Comparison of DPCM and Full Scalar Quantization

3.2 Proposed Differential Quantization Method

The system model of limited feedback systems with proposed differential VQ scheme is presented in Fig. 3-2. We introduce PVQ to implement differential VQ. The block diagrams of PVQ at the transmitter and receiver are shown in Fig. 3-5 and Fig.

3-3 respectively. A vector quantizer and a predictor are incorporated at the receiver end

while the transmitter end only includes a predictor. The corresponding mathematical expressions of Fig. 3-3 and Fig. 3-5 are illustrated in Fig. 3-4 and Fig. 3-6 respectively. Instead of quantizing

h (full CSI), the difference of channel vector and

_n predicted channel vector,

e

_n =

h

_n −

h (differential CSI), is quantized. Note that

_n

n is denoted as the time index and user index i is ignored here for simplicity. The

quantization criterion is

ˆ_n arg min _n 2 j

= − _j

e e c

(3.1)

where

c ,

_j

j =

1,..., 2^Rd is the codeword from codebook C (

R is the number of

_d quantization bits for differential CSI). The received quantized

e is added by the

_n predicted channel vector

h to recover the quantized channel vector

_n

h

_n

Fig. 3-2 Limited feedback system with proposed differential VQ scheme

Fig. 3-3 Block diagram of PVQ system at receiver

Feedback Link

n 1

n ₂

Transmitter

ZF-Precoding

User 1

User K H

h h

Differential VQ system (Fig. 3-3) Differential

VQ system (Fig. 3-5)

⊕ VQ

⊕ p th

order Predictor

h n ^e _n ^e _n

h n

1 ,..., _{n p}

n − h −

h

Fig. 3-4 Mathematical expression of PVQ system at receiver

Fig. 3-5 Block diagram of PVQ system at transmitter

Fig. 3-6 Mathematical expression of PVQ system at transmitter

1

3.2.1 Incorporation of Predictor

The LMMSE Predictor [27] is incorporated in the proposed differential VQ system. The optimal predictor of LMMSE can be obtained by the “orthogonality principle.” Suppose the order of the predictor is p , we can express the orthogonality principle as US is given by zeroth-order Bessel function of the first kind

( )

( ) 0 2 _{d s} , 0,1,2,...,

r τ

=

J π τ f T τ

= .

the correlation matrix would simply be

R

_ij

= E [ h

_{n i−}

h

^H_{n j−}

] = r i ( − j ) I

_M.

3.2.2 Design of Differential VQ Codebook

For the design of the vector quantization codebook, Generalized Lloyd Algorithm (GLA), which is a widely used codebook generation technique [15, 26], is adopted in our system. The distortion function of GLA is the overall Mean Squared Quantization Error (MSQE) between vectors and quantized vectors.

2

Fig. 3-7. At the beginning, the initialization of codebook is randomly chosen from the

sample vectors. A threshold ε is selected to stop iterations when

D

^{( )k} −

D

^(k−1) <

ε

(

D represents the distortion (MSQE) at k th iteration). GLA training process is

^{( )k} mainly based on the two rules:

1. Nearest Neighbor Condition: The encoding partition

S should consist of all

_n vectors that are closer to codeword

c than any other codewords.

_n

{ ^:

^{2 ' 2}

^' }

n n n

S = x x − c ≤ x − c ∀ n ≠ n

(3.6)

2. Centroid Condition: The codeword

c should be average of all the vectors that

_n are in encoding partition

S .

_n

From

Fig. 3-7, we learn that GLA requires a large number of sample vectors for

training the codebook. If we take a closer look at Fig. 3-3, there is one problem:

quantized CSI

h is required to generate

_n

e . (i.e., codebook is needed before

_n

e

_n generation). The solution for this problem can be solved by two-step codebook training as shown in Fig. 3-8.

Fig. 3-7 GLA training process [15, 26]

Step3:

Update the codebook based on

Centroid Condition.

Step1: Start with

random codebook

( represents

codeword) and select a threshold ε

Step2: Classify the sample

vectors according to Nearest

Neighbor Condition.

Training samples

Final codebook

Step4: Go back to

step2 to classify the vectors again.

Iterations (step2 and step3)

Stop when

D

^{( )k} −

D

^(k−1) <

ε

Step 1 is to train the initial codebook by feeding real channel vectors

1,..., _{n p}

− −

h

n

h

into the predictor instead of quantized channel vectors

h

_n−1

,..., h

_n−p. Then the initial codebook can be used to generate another bunch of

e samples for

_n

Step 2 training. Because the initial codebook is generated by feeding the real channel

vectors, the magnitudes of training samples

e are smaller than the practical

_n implementation and hence the magnitudes of codewords of initial codebook are also smaller. By Step 2 training, the codebook would be closer to the practical optimum codebook.

Fig. 3-8 Two-step codebook training

Train initial codebook

⊕

3.2.2 Initial Full CSI and Error Accumulation

Initial Full CSI Quantization

Before differential CSI (

e ) feedback, the first p channel vectors ( p is the

_n order of predictor) have to be fed back by full CSI (

h ) quantization. For differential

_n quantization, the accuracy of initial full CSI is a dominant factor of the system performance. Therefore, we quantize initial full CSI “element-wise” by uniform scalar quantization. Separate the real and imaginary parts of each entry in

h

∈

C

^M×1. Then each channel vector h has 2M real elements, and these 2M elements are quantized by uniform scalar quantization. Intuitively, the more quantization bits are, the better performance is. Fig. 3-9 (Two users, the rest of parameter settings are in

Table 3-1) shows SINR goes up with the number of quantization bits of initial full

CSI,

R , and tends to saturate when

_f

R is larger than 56 bits (i.e., 7 bits per

_f element).

Error Accumulation Due to Differential Quantization

One major problem for differential quantization is error accumulation. Because the recovered quantized channel vector

h is based on the past quantized differential

_n CSI vectors (

e ), the quantization errors are accumulated over time as shown in

_n

Fig. 3-10 (Two users, the parameter settings are in Table 3-1 except

64 bits, 4 bits

f d

R = R =

). Our solution is to feedback full CSI periodically to maintain CSI quality and correct the past accumulated error. To sum up, we use large number of bits to quantize periodically fed-back full CSIs and few bits to quantize remaining differential CSIs. The average number of feedback overhead is roughly the same as that in Full VQ.

Fig. 3-9 SINR under different number of initial full scalar quantization bits

Fig. 3-10 Error Accumulation

0 200 400 600 800 1000

9 10 11 12 13 14 15 16 17 18

Time Index

SI NR

Diff. VQ Full VQ

10 20 30 40 50 60 70 80

6 8 10 12 14 16

R f

SI NR

( ) N

3.3 Computer Simulations

In this section, we will give some simulations to demonstrate the advantage of the proposed differential vector quantization scheme. This section is separated into two main simulation environments. The environment in first subsection is the time domain CSI feedback and the environment in second subsection is the frequency domain CSI, i.e., subcarrier information, feedback. Before the demonstration of simulation results, some parameters are defined below.

Definition of Parameters

y Differential CSI feeds back every

T sec.

_d y Full CSI feedback every

T

_f =

N T

× _d sec.

y

R : Number of quantization bits per channel vector for full CSI

_f

y

R : Number of quantization bits per channel vector for differential CSI

_d y

R : Average number of quantization bits per channel vector

_avg

y

p : Predictor order

Fig. 3-11 Definition of Parameters

… … …

3.3.1 Simulations in Time Domain CSI Feedback

The time domain CSI feedback is considered in this section. Table 3-1 lists all parameters used in our simulation. The simulation is based on a 4× wireless 1 system. Perfect channel knowledge known at the receiver and the error-free feedback channel are also assumed in the simulation. We compare the proposed differential vector quantization to the full vector quantization scheme. The codebook in LTE Release 8 is applied in full VQ. As for the proposed differential VQ, the system uses different self-trained codebooks for different velocities since the users with same velocity should have similar magnitude of codewords. The corresponding codebooks are listed in Table 3-1.

Table 3-1 Simulation Parameters

Parameter Value

Channel

Rayleigh fading channel

Number of Transmit antennas (M)

4

Predictor Order p

2

N

100

T

d 5ms

R

d 3 bits

R

f 56 bits

(

^{( )}

)

^/

avg f d

R

=

p R

× +

N

−

p

×

R N

4.06 bits

Carrier frequency f

_c 2.5 G Hz

Codebooks (Full VQ)

LTE Release 8 (Table 2-1)

Codebooks (Differential VQ)

Self-Trained Codebooks are listed in

Table 3-2 to Table 3-6

Table 3-2 codebook for v = 3 km/hr

index codewords 1 0.01861- 0.00444i -0.00301 - 0.00463i 0.00527+ 0.03416i -0.03592 - 0.01361i

2 -0.00973 - 0.01458i -0.02546 + 0.05598i 0.00356 - 0.00238i 0.00522 - 0.01194i 3 -0.01701 + 0.05485i 0.01718 + 0.01455i -0.02505- 0.00429i -0.00938 - 0.00136i 4 -0.00934- 0.00726i -0.03289 - 0.02495i -0.02330- 0.009009i 0.01166 - 0.02058i 5 0.00879 + 0.00057i -0.00280 - 0.00547i -0.00072 + 0.00785i -0.00044 + 0.05364i 1 -0.00254 + 0.01459i -0.00247 - 0.02514i 0.00051 - 0.01239i 0.034216 - 0.06769i 2 0.01244 + 0.00347i -0.01007 - 0.02681i -0.01971 - 0.08106i -0.003504 + 0.01975i 3 0.00665 + 0.00853i 0.00021 - 0.01303i 0.01479 + 0.00724i -0.08061 - 0.00688i 4 0.02305 - 0.009834i -0.00868 - 0.03118i 0.03563 + 0.03017i 0.019476 + 0.01758i 5 -0.00351- 0.00070i 0.00546 - 0.00642i -0.08181 + 0.04070i 0.00876 + 0.00964i 6 0.01351 + 0.00103i 0.07769 + 0.04163i 0.02180 - 0.00622i -0.00251 + 0.00760i 7 -0.06896 - 0.03703i 0.00003 + 0.00010i 0.01173 + 0.00265i 0.01042 + 0.01587i 8 0.01214 + 0.01653i -0.05697 + 0.06416i 0.00512 + 0.00583i 0.00654 + 0.00204i

Table 3-4 codebook for v = 8 km/hr

index codewords

Table 3-5 codebook for v = 10 km/hr

index codewords 1 0.02692- 0.02778i 0.05072 - 0.11008i 0.03964+ 0.09216i 0.07518- 0.10204i

2 -0.20369- 0.08143i -0.00424- 0.00608i 0.00546+ 0.01879i -0.01307+0.00941i 3 0.00951+ 0.00001i -0.01817- 0.02675i -0.20294- 0.03991i -0.023612- 0.07295i 4 0.01735- 0.023523i 0.01399- 0.01443i 0.10748- 0.18637i 0.04081- 0.01840i 5 0.03052- 0.02662i 0.14073+ 0.08023i 0.03114+ 0.01050i -0.10101+ 0.00621i 6 0.02303+ 0.21302i 0.00811+ 0.02800i 0.01418+ 0.00739i 0.01157- 0.01202i 7 0.01571- 0.01906i 0.00910+ 0.01425i -0.02651+ 0.02517i 0.11569+ 0.18245i 8 0.05089- 0.02973i -0.16741+ 0.02444i 0.01857+ 0.05147i -0.05736+ 0.01484i

Table 3-6 codebook for v = 15 km/hr

index codewords 1 -0.03156+ 0.01836i 0.23321- 0.27631i 0.00668+ 0.07667i 0.06117+ 0.07148i

2 -0.19797+ 0.29578i -0.09071+ 0.04809i -0.05678+ 0.07583i 0.06777- 0.02833i 3 0.05109+ 0.02838i -0.06168- 0.00099i 0.00341+ 0.03938i -0.37495+ 0.10158i 4 0.05547 - 0.01414i -0.10691- 0.03118i -0.21091- 0.28494i 0.08396+ 0.05085i 5 -0.18560- 0.32995i -0.08078 - 0.03426i -0.01932+ 0.04005i -0.00943- 0.1251i 6 0.27523- 0.02563i -0.04658+ 0.01468i -0.04051+ 0.18578i 0.08386- 0.11775i 7 -0.01383- 0.03967i 0.23025+ 0.34098i -0.03386- 0.05087i 0.04909+ 0.01794i 8 0.02344+ 0.03002i -0.04260+ 0.004195i 0.36572- 0.11770i 0.03537+ 0.02099i

From the above tables, we can observe that the magnitude of the codewords is larger in the higher speed codebook since for high speed users, the magnitude difference between two consecutive CSI is larger.

Following are three simulation results. The first simulation result in

Fig. 3-12 shows the proposed differential VQ successfully exploits the temporal

correlation of channel vectors and can achieve higher SINR compared to full VQ under 5 km/hr environment. If we extend the time of full CSI feedback, average number of bits decrease but SINR degrades. Fig. 3-14 shows the SINR for different velocities. Because higher velocity results in smaller temporal correlation, smaller correlation of CSI degrades the performance of differential quantization. Unlike

differential VQ which suffers from error accumulation, full VQ quantizes channel vector at every time instant independently, so the performance of full VQ is unvarying under different velocities.

2 3 4

Fig. 3-12 Differential vector quantization vs. Full vector quantization

2 3 4

Different Length (N) of Full CSI Feedback

Diff. VQ: N=100, R

_avg

=4.06 bits Diff. VQ: N=200, R

_avg

=3.53 bits Diff. VQ: N=300, R

_avg

=3.35 bits Full VQ: R=4 bits

Fig. 3-13 Differential vector quantization under different N T ( )

_f

2 3 4 -2

0 2 4 6 8 10 12 14 16 18

Number of Users

SI N R

Different Velocities Diff.VQ: v = 3 km/hr Diff.VQ: v =5 km/hr Diff.VQ: v=8 km/hr Diff.VQ: v=10 km/hr Diff.VQ: v=15 km/hr Full VQ: v=3 km/hr Full VQ: v=5 km/hr Full VQ: v=8 km/hr Full VQ: v=10 km/hr Full VQ: v=15 km/hr

Fig. 3-14 Differential vector quantization under different velocities

3.3.2 Extension to MIMO OFDM system

In LTE frame structure, one radio frame is 10 ms long and consists of 20 slots of length 5 ms. A subframe is defined as two consecutive slots, that is one radio frame is composed of 10 subframes. A physical resource block is defined as

N

_symb

( 7) =

consecutive OFDM/SC-FDMA symbols in the time domain and

N

_sc^RB ( 12)= consecutive subcarriers in the frequency domain. The smallest resource unit is denoted a resource element. Since there are many subcarriers (may up to 1024) in a time instant, it is impossible to feedback all the subcarriers information to the transmitter. Usually, k consecutive resource blocks share one codeword and k depends on different modes in LTE [25]. So only the subcarrier in the mid of k consecutive resource blocks is fed back.

Fig. 3-15 LTE resource grid

Subcarrier (frequency)

Table 3-7 Simulation Parameters

Parameter Value

Channel

Rayleigh fading channel

Number of transmit antennas (M)

4

Number of users

2-4

Codebooks (Full VQ)

LTE Release 8 (Table 2-1)

Codebooks (Differential VQ)

Self-Trained Codebooks are listed in

3 km/hr: Table 3-8 5 km/hr: Table 3-9 8 km/hr: Table 3-10 10 km/hr: Table 3-11 15 km/hr: Table 3-12

Table 3-8 codebook for v = 3 km/hr

index codewords 1 0.01850+ 0.01663i 0.15751+ 0.10612i -0.075845+ 0.09921i 0.12522- 0.18442i

2 0.40755 - 0.26827i 0.18229+ 0.02534i 0.07001+ 0.21347i 0.08667+ 0.07725i 3 0.10392+ 0.24114i 0.09846+ 0.10685i -0.11746+ 0.06245i -0.10948+ 0.12052i 4 0.03421+ 0.0204i -0.15147+ 0.14220i -0.06990- 0.09554i 0.15874+ 0.22629i 5 -2.00152- 1.08612i 2.39253+ 0.61204i -1.92461- 1.81288i 0.58650- 0.87042i 6 0.00092- 0.11588i 0.03269 - 0.18963i 0.15093+ 0.01060i -0.07942- 0.11052i 7 -2.81571- 0.071022i 0.79602- 0.96609i 1.46382- 0.36982i 0.92291+ 1.14374i 8 -0.19272+ 0.13480i -0.04540+ 0.07487i -0.08165- 0.09375i -0.11690- 0.13523i

Table 3-9 codebook for v = 5 km/hr

index codewords 1 0.30445+ 0.22635i 0.15751+ 0.10612i -0.07584+ 0.09921i 0.12522 - 0.18442i

2 0.40755- 0.26827i 0.18229+ 0.02534i 0.07001+ 0.21347i 0.08667 + 0.07725i 3 0.10392+ 0.24114i 0.09846+ 0.10685i -0.11746+ 0.06245i -0.10948+ 0.12052i 4 0.03421+ 0.02045i -0.15147+ 0.14220i -0.06990- 0.09554i 0.15874+ 0.22629i 5 -2.00152 - 1.08612i 2.39253+ 0.61204i -1.92416 - 1.81288i 0.58650- 0.87042i 6 0.00092- 0.11588i 0.03269- 0.18963i 0.15093 + 0.01060i -0.07942 - 0.110529i 7 -2.81571 - 0.07102i 0.79602 - 0.96609i 1.46382 - 0.36988i 0.92291+ 1.14374i 8 -0.19272+ 0.13480i -0.04540+ 0.07487i -0.08165- 0.09375i -0.11690- 0.13523i

Table 3-10 codebook for v = 8 km/hr

index codewords 1 -0.30445+ 0.22635i -0.08192 + 0.30640i -0.13432+ 0.199751i 0.41735 - 0.14419i

2 -0.15031+ 0.01269i 0.35441+ 0.11945i 0.04724+ 0.20804i 0.08446+ 0.14228i

3 -0.28148- 0.06025i -0.17323 - 0.00949i 0.17891- 0.19109i -0.08100- 0.23096i

4 0.10639+ 0.01218i 0.04763+ 0.06343i -0.0409 - 0.43678i 0.24865+ 0.29275i

5 0.017640- 0.07233i -0.22372+ 0.28215i 0.063256+ 0.00006i -0.22293+ 0.24969i

6 0.04519+ 0.19604i -0.00388- 0.10754i -0.06925+ 0.27971i -0.35732- 0.13852i

7 0.29571- 0.15511i -0.26035- 0.35558i 0.06717- 0.01712i -0.17588+ 0.02205i

8 0.03268+ 0.06397i 0.31217- 0.31186i -0.23381- 0.12263i 0.17470- 0.45801i

Table 3-11 codebook for v = 10 km/hr

index codewords 1 0.38819- 0.05640i 0.279081- 0.084417i -0.085771- 0.228474i 0.294287+ 0.50050i 2 -0.26705+ 0.09830i -0.10912+ 0.27842i -0.18124- 0.39648i -0.15851 + 0.34360i 3 0.27077+ 0.36627i 0.00919+ 0.09340i 0.27123+ 0.19372i -0.37589+ 0.29274i 4 0.033498 - 0.28856i -0.23501- 0.04900i 0.56626- 0.32576i -0.06516- 0.04615i 5 0.12185+ 0.30981i 0.02615+ 0.04254i -0.02207+ 0.25378i 0.43702- 0.31196i 6 -0.35221+ 0.03415i 0.22407- 0.03079i -0.00789+ 0.168390i -0.35287- 0.31445i 7 -0.06664 - 0.39172i -0.21855- 0.15347i -0.02613+ 0.39965i 0.00177- 0.04736i 8 -0.18204 - 0.47617i 0.13907- 0.05957i -0.29671- 0.264254i 0.34720- 0.24472i

Table 3-12 codebook for v = 15 km/hr

index codewords 1 0.31149- 0.51042i -0.84816- 0.35939i 0.00875- 0.03616i -0.55162- 0.09785i

2 0.14076+ 0.32218i -0.01956+ 0.94223i 0.29629+ 0.24979i 0.116174- 0.74335i 3 -0.11393+ 0.72280i -0.11386- 0.20279i -0.41209+ 0.70975i -0.06094+ 0.49142i 4 -0.62541- 0.14539i 0.50913- 0.50412i -0.37464 - 0.15517i -0.03885- 1.05382i 5 0.81650+ 0.38587i 0.38412+ 0.43180i -0.25944 - 0.55869i -0.15011+ 0.17496i 6 0.15015- 0.75979i 0.83052+ 0.15415i 0.12825+ 0.12149i 0.43432+ 0.31504i 7 -0.27572+ 0.48021i 0.31749- 0.67221i 0.68262 - 0.49347i 0.36509+ 0.07361i 8 -1.07468- 0.17525i -0.35833+ 0.58647i 0.20414- 0.10759i -0.38812+ 0.22483i

Similarly, the codebook for higher velocity users has the larger magnitude of codeword since the temporal correlation of channel is weaker. In MIMO OFDM systems, the temporal correlation is not obvious so the performance tends to degrade compare to that in MIMO systems. But our proposed differential quantization scheme can still attain higher SINR under highly correlated channel environments as shown in

Fig. 3-16, Fig. 3-17 and Fig. 3-18. Intuitively, the reason for differential VQ is

superior to full VQ is that the magnitude of codewords in differential VQ is smaller than those in full VQ. The same size of codebook can represent the differential CSI more precisely compared to full CSI. Therefore, the quantization error will be smaller in our proposed differential VQ scheme.

2 3 4

Fig. 3-16 Differential vector quantization vs. Full vector quantization

2 2.5 3 3.5 4

Different Length(N) of Full Quantization Feedback

Diff. VQ: N=50, R

_avg

=5.12 bits Diff. VQ: N=100, R

_avg

=4.06 bits Diff.VQ: N=150, R

avg

=3.7 bits Diff. VQ: N=200, R

_avg

=3.53 bits Full VQ: R=4 bits

Fig. 3-17 Differential vector quantization under different N T ( )

_f

2 3 4 -4

-2 0 2 4 6 8 10 12 14

Number of Users

SI N R

Different Velocities

Diff.VQ: v=3 km/hr Diff.VQ: v=5km/hr Diff.VQ: v=8 km/hr Diff.VQ: v=10 km/hr Diff.VQ: v=15 km/hr Full VQ: v=3 km/hr Full VQ: v=5 km/hr Full VQ: v=8 km/hr Full VQ: v=10 km/hr Full VQ: v=15 km/hr

Fig. 3-18 Differential vector quantization under different velocities

3.4 Summary

In this chapter, we apply the concept of differential quantization in data compression to limited feedback systems. The proposed differential vector quantization scheme is presented. We first incorporate the model of PVQ in the limited feedback system. Also, LMMSE predictor is used in our PVQ model. The codebook is trained by GLA with some modifications. The full CSI are periodically fed back in between differential CSI to correct the accumulated error. For full CSI, a large number of quantization bits are used while fewer quantization bits are used for differential CSI. The average feedback overhead is roughly the same as in full VQ. To sum up, the proposed differential vector quantization scheme can attain higher SINR at the expense of a slight increase in feedback overhead compared to full VQ.

Chapter 4

Discussion of Different Feedback Overheads and Spatial Correlation Matrix Feedback

In this chapter, two main topics are addressed; one is discussion of users’

performances under different feedback overheads and the other is feedback reduction for spatial correlation matrix via vector quantization. In the previous chapter, all users are permitted to feedback CSI under the same overhead but in the practical communication system, some users might be allowed to have higher feedback overhead. Therefore, the first part of this chapter is mainly to discuss the system performance in different feedback overhead scenarios.

Also, in Chapter 3, the proposed differential vector quantization has been shown to be an effective way to quantize CSI. So, the second part of this chapter aims to further find an effective method to quantize the spatial correlation matrix to reduce the number of quantization bits. Here, we propose diagonal-wise full vector quantization by exploiting the channel spatial correlation. The proposed scheme can achieve a smaller MSE compared to existing methods such as entry-wise differential scalar quantization.

4.1 Different Feedback Overheads in Multi-user Systems

In LTE systems, there are different reporting modes for CSI feedback. Some reporting modes are triggered by scheduling grants and allow mobile users to feedback CSI with higher overhead,. In this section, we are going to discuss about the influence on all mobile users under the scenario in which only some specific users are permitted to have higher feedback overhead. First, a simulation result demonstrates the performance for users with different feedback overheads. Then, discussion of the result will be given under a geometrical point of view.

4.1.1 Simulation Demonstration

In the simulation, we have two simple cases as shown in Table 4-1. Only the first user in Case 1 has higher feedback overhead. Other users in Case 1 have the same feedback overhead as the users in Case 2. The simulation parameters are defined in

Table 4-2. As we can see from Fig. 4-1, only first user in Case 1 benefits from the

higher cost of feedback overhead. Other users in Case 1 have the same SINR as those users in Case 2. This implies that user performance does not change if some other users in the system are allowed to feedback CSI with higher overheads.

Table 4-1 Two cases in simulation

Case 1 First user: 4 bits/ 12 subcarriers

Other users: 4 bits/ 48 subcarriers Case 2 All the users: 4 bits/ 48 subcarriers

Table 4-2 Simulation Parameters

Parameters Value

Quantization Scheme

Full vector quantization

Channel

Multipath Rayleigh fading channel

Tap

1 2 3 4 5 6

Relative delays (ms)

0 10 20 30 40 50

Number of transmit antennas

4

Carrier frequency

2.5G Hz

Velocity

5 km/hr

Fig. 4-1 Simulation demonstration of different feedback overheads for different users

4.1.2 Geometric Interpretations

In the following sections, we will analyze the simulation result in Fig. 4-1 under a geometrical point of view. The geometric representation for a two-user case is shown in Fig. 4-2, and the corresponding definitions of parameters are listed in

Table 4-3.

Fig. 4-2 Geometric representation for two users case

Table 4-3 Definitions of parameters

Parameters Definitions

Since the transmitter performs zero-forcing precoding, the beamforming vector

v is orthogonal to the quantized channel vector

j

h for

_i

i

≠

j (i.e.,

1. ⁱ 2 would happen when the system performs pseudo-inverse in zero-forcing precoding. If we exclude the singular case and

h and

_i

h

_j (i ≠ ) are nearly

j

orthogonal, we can reasonably assume that β_i is small.

In the following sections, the interferences

h v

^H_{i j} ² and precoding gain

h v

^H_{i i} ² in

4.1.3 Discussions of Interferences and Precoding Gain

a. Interferences

Fig. 4-3 shows the geometric representation of interferences where Ω is the

angle between

v and

_j

h . The interferences for user i result from

_i

h v

_{i j} ² for

Fig. 4-3 Geometric representation of interference

From the assumption that

From (4.3), we can conclude that the interferences for user i is only bounded by the sine of its own quantization angular error α_i and independent of other users’

quantization errors.

b. Precoding Gain

As for the precoding gain, its geometrical representation is illustrated in Fig. 4-4.

Now Ω is the angle between

v and

_i

h . The precoding gain for user i is

_i

2 cos2

H

i i = Ω

h v

 .

Fig. 4-4 Geometric representation of precoding gain

From the assumption that β_i is small and

α

+

β

<

π

i i 2 , we can infer that

α β

Ω≤ + .

_{i i} (4.4)

So

cos

²

Ω ≥ cos

²

( α

_i

+ β

_i

) ≈ cos

²

α

_i, and the precoding gain of user i has a lower bound as

2 cos2 .

H

i i ≥

α

i

h v

 (4.5)

Similar to the upper bound of interferences, we can conclude that the precoding gain of user i is only bounded by the cosine of its own quantization angular error α_i and independent of other users’ quantization errors.

α _i

v i

h i

h i

Ω

β _i

4.1.4 Conclusions

From above analysis of interferences and precoding gain, we can derive the lower bound of SINR as angular error and independent of other users’. Our analysis of SINR is consistent with the simulation results in Fig. 4-1. Only the mobile user with higher feedback overheads benefits, while other users’ performances remain the same.

4.2 Proposed Diagonal-wise Vector Quantization for Spatial Correlation Matrix Feedback

Generally, the codebook design is mainly based on typical communication environments. However, it is impossible to consider all cases. So, channel spatial correlation at the transmitter end can provide the current statistic information of the channel and therefore can be used to refine the codewords of the codebook, such that the refined codewords are more suitable for the current communication scenario.

Methods of refining the codebook by using the channel spatial correlation can be found in [28]. In [28], the average spatial correlation matrix as defined in (4.7) is fed back to the transmitter. The average spatial correlation matrix is

, ,

n th time slot. Since the spatial correlation matrix R is a Hermitian matrix, we need

only quantizing the upper triangular part of R .

In the following sections, the spatially and temporally correlated channel model is given first, followed by the introduction of one of the existing methods for spatial correlation matrix feedback: entry-wise differential scalar quantization [29]. Finally, the proposed a diagonal-wise vector quantization scheme which feeds back spatial correlation matrix by exploiting the spatial correlation of channels is presented.

4.2.1 Correlated Channel Model [30]

The temporally correlated channel model used in Chapter 3 is modified to a spatially and temporally correlated channel model for the following sections. The new correlated channel model is

{

^{1/ 2}

( ) }

, ,

s t =

unvec

s

vec

t

H R H

(4.8)

where the parameters are defined in Table 4-4. Generally, the temporal correlated channel is multiplied by the channel spatial correlation matrix

R and become a

_s spatially and temporally correlated channel.

Table 4-4 Parameter definitions

Parameters/functions Definitions (.)

vec

Colum-wise stacking

(.)

unvec

Reverse operation of

vec

(.)

H

t Temporally correlated channel defined in 2.3

,

H

s t Spatially and temporally correlated channel

R

s

R

_s =

R

_BS ⊗

R ( ⊗ is Kronecker product)

_MS

R and

BS

R in Table 4-4 are the spatial correlation seen from BS and MS.

_MS Their corresponding ( , )

p q entries, { R

BS

}

pq and

{ R

MS

}

pq , represent the antenna spatial correlation between the p th and q th antennas at the BS and MS respectively and are approximated using 20 subpaths as [30]

( )

{

( )

( ) }

and

θ respectively. The angular offsets of k th subpath are determined by

_MS

Fig. 4-5 Angle parameters in MIMO channel model [30]

Table 4-5 Values of

Δ [30] _k known, the spatially and temporally correlated channel can be determined by multiplying the original temporally correlated with spatial correlation

R .

_s

4.2.2 Existing Solution: Entry-wise Differential Scalar Quantization [29]

The basic idea of entry-wise differential scalar quantization is to quantize differential spatial correlation matrix

( R

n −

R

n−1

)

entry-wise. The mathematical representation to update the quantized spatial correlation matrix at the transmitter end is

1 ,

n =

γ

n₋ + n

R R C (4.11)

where

R is the quantized matrix at time slot n ,

_n

C is the differential updating

_n matrix, and γ is forgetting factor. The elements of

C are determined by the

_n difference of current matrix

R and previous quantized matrix

_n

R

_n−1 (i.e.,

1

n − n−

R R

). The real and imaginary parts of each entry in

C are individually

_n quantized. Since the entries in the main diagonal of

R are all real, the number of

_n quantization bits is half for the main diagonal entries compared to those off-diagonal entries. The main diagonal entries are quantized as follows

{ } { }

Entry-wise differential SQ uses 4 bits per real/imaginary component for the initial full quantization while 1 bits per real/imaginary component for differential quantization. The differential quantization scheme is mainly to exploit temporal correlation of the channel. In the next section, we propose another method to quantize the spatial correlation matrix

R more effectively.

_n

4.2.3 Proposed Diagonal-wise Full Vector Quantization

The quantization scheme proposed in 4.2.2 requires a large number of bits since it is an entry-wise quantization method. Our first attempt to quantize the spatial correlation matrix is column-wisely vector quantization as shown in Fig. 4-6, but the result is not very effective.

Fig. 4-6 column wise vector quantization

We further investigate the structure of the spatial correlation matrix,

1, ', 2, ', , ', 1, ', 2, ', , ', are located far away, the spatial correlation is smaller. So as one moves from the main diagonal to the top off-diagonal, the magnitudes of the entries will decrease. Putting those entries which have similar magnitude together as a vector, we propose diagonal-wise full vector quantization as shown in Fig. 4-7.

Fig. 4-7 Diagonal-wise full vector quantization

The proposed scheme can successfully exploit the property of spatial correlation matrix that the diagonal entries have similar magnitudes and hence it can quantize the matrix effectively with a lower cost of feedback overhead. Take a 3× spatial 3 correlation matrix as example, the system requires three sets of codebooks which are trained by GLA separately. The bit allocation for each codebook can be further optimized, but in our simulation, we simply allocate more bits for the vector which has more real variables.

Also, the number of quantization bits of the proposed method can be further reduced by decreasing the feedback frequency as shown in Fig. 4-8. Only the first matrix in the N matrices sequence is quantized and fed back, and the remaining

N − time slots use the first quantized matrix at the transmitter end.

1

Fig. 4-8 Reducing the feedback frequency

N

Quantized matrix

R

₁

…

…

2 =...= _N = 1

R R R

4.2.4 Comparisons Between Entry-wise Differential SQ and Proposed Diagonal-wise Full VQ

Before the demonstration of computer simulations, we compare the number of quantization bits between entry-wise differential SQ and the proposed diagonal-wise VQ. In our simulation environment, the system is 3× MIMO. We allocate 5, 6, and 3 3 bits for the three codebooks respectively to make the largest average quantization bit (when

N = ) for a matrix to be 14 bits which is still smaller than the average

1 quantization bits (14.4 bits) for entry-wise differential SQ.

Table 4-6 Comparisons of diagonal-wise full VQ and enty-wise differential SQ

Diagonal-wise Full VQ Entry-wise Differential SQ

Number of

4.2.5 Computer simulations

In this section, the computer simulations are based on two different environments:

time-invariant channel and time-varying channel. For time-varying channel, the angle of departure and arrival ( AOD and AOA ) related to the spatial correlation of channel shown in (4.9) is updated every 0.1 second according to a linear equation

1

The following simulations consider 3× MIMO OFDM systems. The 3 simulation parameters are listed in Table 4-7. At the receiver end, the CSI is estimated every 1 ms. The fed back average spatial correlation is

40 48

which takes average over 40 ms and 48 subcarriers. Our performance metric is the normalized MSE as follows

Table 4-7 Simulation parameters

Parameters value

Channel

Multipath Rayleigh fading channel

Tap

1 2 3 4 5 6

Relative delays (ms)

0 10 20 30 40 50

Spatial correlation parameters _AS

_BS

_{= 10 , ° AS}

_MS

_{= 22 °}

,

θ

_BS

= ° 0 , θ

_MS

= ° 0 Antenna spacing λ

/2=

f

_c/(2× ,

c

)

c = ×

3 10⁸

Number of transmit antennas

3

Number of receive antennas

3

N

1-3

Sampling time

1 ms

Number of subcarriers

512

Carrier frequency f

_c 2.5 G Hz

Codebooks for diagonal-wise full VQ

For time-invarying channel environment:

Table 4-8, Table 4-9, Table 4-10

For time-varying channel environment:

Table 4-8, Table 4-9, Table 4-10

For time-varying channel environment:

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