mimo-course301011

Tel Aviv and Ben Gurion Universities

MIMO-OFDM LECTURE NOTES

Dr. Doron Ezri

iPreface

MIMO and OFDM technologies are gaining ever growing interest in modern com-

munications systems. These technologies provide a powerful tool for enhancing the

wireless link with an emphasis on increased spectral efficiency. MIMO and OFDM

have already been incorporated as major building blocks into existing standards and

are considered the bridge to fourth generation (4G) broadband wireless access systems

and technologies.

The course is made of two parts. In the first part the basic concepts of different

MIMO modes are presented together with an analysis of their performance. The

second part discusses OFDM and concludes with the fusion of OFDM and MIMO

in practical systems and standards (with an emphasis on LTE). This course targets

graduate students in Electrical Engineering and communications engineers who need

an efficient introduction to these all important subjects.

The course is devised to be self-contained. However, some of the material in

the MIMO part is covered in [13], while some of the OFDM part is covered in [20].

Additional references for further reading are listed in the Bibliography section.

Dr. Doron Ezri OFDM MIMO

ii

Contents

I Basic MIMO Concepts 1

1 The SISO Case 2

1.1 System Model and ML Receiver . . . . . . . . . . . . . . . . . . . . . 2

1.2 Evaluation of the Error Probability . . . . . . . . . . . . . . . . . . . 3

2 Receive Diversity - MRC 7



3 Transmit Diversity - STC 12



3.3 Transmit and Receive Diversity - STC+MRC . . . . . . . . . . . . . 16

3.3.1 System Model and ML Receiver . . . . . . . . . . . . . . . . . 16

3.3.2 Evaluation of the Error Probability . . . . . . . . . . . . . . . 18

4 Transmit Beamforming 20

4.1 System Model and Optimal Transmission . . . . . . . . . . . . . . . . 20


4.3 Maximal Ratio Transmission . . . . . . . . . . . . . . . . . . . . . . . 25


Contents iii

5 Spatial Multiplexing 26



5.3 The Sphere Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4 Linear MIMO Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.5 Successive Interference Cancelation . . . . . . . . . . . . . . . . . . . 33

5.6 The Diversity-Multiplexing Tradeoff . . . . . . . . . . . . . . . . . . . 35

6 Closed Loop MIMO 38

6.1 System Model and Optimal Transmission . . . . . . . . . . . . . . . . 38

6.2 Implications of Closed Loop MIMO . . . . . . . . . . . . . . . . . . . 40

7 Space Division Multiple Access 43

7.1 System Model and Basic Solution . . . . . . . . . . . . . . . . . . . . 43

7.2 More Advanced Solutions and Considerations . . . . . . . . . . . . . 45

II Practical OFDM-MIMO 48

8 The Wireless Channel 49

8.1 Propagation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.1.1 Path Loss and Shadowing . . . . . . . . . . . . . . . . . . . . 49

8.1.2 The Physics of Multipath . . . . . . . . . . . . . . . . . . . . 51

8.1.3 Delay Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8.1.4 Doppler Spread . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.2 Channel Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.2.1 Modeling Path Loss and Shadowing . . . . . . . . . . . . . . . 56

8.2.2 Modeling Mobile Channels . . . . . . . . . . . . . . . . . . . . 57


Contents iv

8.3 Simulating Mobile Channels . . . . . . . . . . . . . . . . . . . . . . . 59

8.4 Extension to the MIMO Case . . . . . . . . . . . . . . . . . . . . . . 60

8.4.1 The MIMO Channel . . . . . . . . . . . . . . . . . . . . . . . 60

8.4.2 Modeling MIMO Channels . . . . . . . . . . . . . . . . . . . . 62

8.4.3 Simulating MIMO Channels . . . . . . . . . . . . . . . . . . . 64

9 OFDM Basics 65

9.1 The Basic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

9.2 Pilots and Channel Estimation . . . . . . . . . . . . . . . . . . . . . . 70

9.3 Guards in Time and Frequency . . . . . . . . . . . . . . . . . . . . . 71

9.4 The Effects of Time and Frequency Offsets . . . . . . . . . . . . . . . 73

9.5 The OFDM Parameters Tradeoff . . . . . . . . . . . . . . . . . . . . 75

9.6 The PAPR Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

10 OFDMA and SC-FDMA 78

10.1 From OFDM to OFDMA . . . . . . . . . . . . . . . . . . . . . . . . . 78

10.2 SC-FDMA as a Variant of OFDMA . . . . . . . . . . . . . . . . . . . 80

11 Practical MIMO OFDM 83

11.1 The Fusion of OFDM and MIMO . . . . . . . . . . . . . . . . . . . . 83

11.2 Pilots Patterns in MIMO OFDM . . . . . . . . . . . . . . . . . . . . 84

11.3 Obtaining Channel Knowledge at the Transmitter . . . . . . . . . . . 86

11.3.1 Reciprocity Methods . . . . . . . . . . . . . . . . . . . . . . . 86

11.3.2 Feedback Methods . . . . . . . . . . . . . . . . . . . . . . . . 88

11.4 Future Directions in MIMO-OFDM . . . . . . . . . . . . . . . . . . . 90

Bibliography 90


Contents v

A Complex Normal Multivariate Distribution 95

B Log Likelihood Ratio 97

C Derivatives w.r.t a Vector and LS 99

D Some Results For Chapter 4 101

D.1 Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

D.2 Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

E DO and AG in 2 2 ZF 104

F The Baseband Channel 106

G The Impact of Correlation on MRC 108


1Part I

Basic MIMO Concepts


2Chapter 1

The SISO Case

1.1 System Model and ML Receiver

We begin with the simplest case of single input single output system (SISO) endowed

with single Tx and Rx antennas, depicted in Fig. 1.1. The received signal y satisfies

y = h s+ n, (1.1)

where the channel h is a zero mean complex Normal1 random variable (RV) with unit

variance (E{hh} = E{|h|2} = 1), s is a QPSK symbol bearing 2 bits (see Fig. 1.2), is the noise intensity, and n is a zero mean complex Normal RV with unit variance.

Thus, the signal to noise ratio (SNR) in this case is 1/2.

Figure 1.1: SISO communications system.

The aim of the receiver is to estimate the transmitted symbols (or bits) using the

1For the definition of circularly symmetric complex Normal distribution see AppendixA.


1. The SISO Case 3

measurement y. We also assume h is known at the receiver.

The Maximum a-Posteriori (MAP) receiver computes the the most probable sym-

bol s given the the measurement y. Assuming equally probable symbols we get the

Maximum Likelihood (ML) estimator

s = argmaxsQPSK

p(y|s). (1.2)

Using the conditional density of y2 the ML estimator takes the form

s = argmaxsQPSK

exp

(|y hs|

2

2

). (1.3)

Since the exponential is a monotone function, the ML estimator may be rewritten as

s = argminsQPSK

|y hs|2

= argminsQPSK

|s s|2, (1.4)

where s =y

h. The ML estimator (1.4) implies that we estimate each symbol according

to the constellation point which is nearest to s. We note the following.

The division by h plays the role of equalization (compensating for theeffect of the channel) in this simple example.

In coded systems the ML estimated (hard decision) symbol is not useful.Here the log likelihood ratio (LLR) which is a soft decision metric is

computed for each transmitted bit (see Appendix B).

1.2 Evaluation of the Error Probability

We turn now to the evaluation of the error probability. We note that

s = s+

hn, (1.5)

2Note that conditioned on h and s, y is complex Normal with mean hs and variance2. For the p.d.f. of a complex Normal distribution see Appendix A.


1. The SISO Case 4

Figure 1.2: QPSK, 16QAM and 64QAM modulations. Note that in a nor-malized QAM, with n bits/symbol, dmin =

12n16

.

so the probability of error given h is bounded by

Pr {error|h} Pr{

hn > dmin

2

h}

= Pr

{|n| > |h| dmin

2

h}

=

|h|dmin/2

2z exp(z2) dz,

(1.6)

where z = |n| is Rayleigh distributed3 with 2 = 12, and dmin =

2 in QPSK (see

Fig. 1.2). The integral in (1.6) may be evaluated analytically, so the error probability

given h reads

Pr {error|h} exp(|h|

2

22

)= exp

(SNR(h)

2

), (1.7)

3The absolute value z of a complex Normal RV x+jy where x and y are zero mean realvalued i.i.d. Gaussian RVs each with variance 2 is Rayleigh distributed with parameter and pdf p(z) = 12 z exp

( z222

)for z 0.


1. The SISO Case 5

where SNR(h) is the instantaneous SNR.

In order to obtain the unconditional error probability we average (1.7) w.r.t the

complex Normal distribution of h, which gives

Pr {error} =hC

Pr {error|h} p(h) dh

hC

exp

(|h|

2

22

)1

piexp

(|h|2) dh=

1

pi

hC

exp

([1 +

1

22

]|h|2

)dh.

(1.8)

Using the equality (A.7), the bound (1.8) simplifies to

Pr {error} 11 +

1

22

=1

1 +SNR

2

. (1.9)

The error probability (1.8) in the case of Rayleigh fading reveals how significantly

the Rayleigh channel affects the performance. For means of comparison, we note that

in the case of white channel, h = 1, the error probability may be evaluated through

(1.7) as

Pr {error} exp( 122

)= exp

(SNR

2

). (1.10)

The symbol error rate (SER) curves for SISO in AWGN and Rayleigh are given in

Fig. 1.3.


1. The SISO Case 6

Figure 1.3: SER curves of SISO with AWGN and Rayleigh channels.


7Chapter 2

Receive Diversity - MRC


We begin with the single input multiple output (SIMO) case, in which the receiver

is endowed with N receive antennas, as depicted in Fig. 2.1. In this case, the

mathematical model for the measurements vector y is

y = hs+ n, (2.1)

where the elements hi of the channel vector h, are independent complex Normal

RVs with unit variance, and the elements ni of the noise vector n, are independent

complex Normal RVs with unit variance.

Figure 2.1: MRC configuration.

We proceed as in the SISO case to obtain the ML estimator s, assuming the


2. Receive Diversity - MRC 8

receiver possesses h. The expression for s in this case becomes

s = argmaxsQPSK

exp

(y hs

2

2

)= argmin

sQPSKy hs2. (2.2)

We note that the functional y hs2 in (2.2) attains its global minimum at theleast squares (LS) solution (see Appendix C)

s = (hh)1hy =hyh2 , (2.3)

and that the quadratic cost y hs2 may be rewritten as

y hs2 = y hs+ hs hs2

= (y hs) + (hs hs)2

= y hs2 + h(s s)2 + 2


The LS solution in the case of Rx diversity is also known as maximal-ratio-

combining (MRC)[15, 16]. In the case of 2 Rx antennas, Eq. (2.3) reads

s =h0y0 + h

1y1

|h0|2 + |h1|2 , (2.8)

which means that the signal from each antenna is de-rotated according to the phase

of the corresponding channel. Then, the de-rotated signals are weighted according to

the strength of the channel (per antenna SNR) and summed.


Substituting (2.1) in the MRC expression (2.3) we get

s =h(hs+ n)

h2

= s+ hnh2 . (2.9)

Note that the variance of the noise term is2

h2 so the post-processing SNR ish22

. Thus, the probability of error given h, similarly to (1.6), is

Pr {error|h} hdmin/2

2z exp(z2) dz

= exp

(h

2

22

). (2.10)

The error probability is obtained by averaging (2.10) w.r.t. the complex Normal

distribution of h, which gives

Pr {error} =hCN

Pr {error|h} p(h) dh

hCN

exp

(h

2

22

)1

piNexp

(h2) dh=

1

piN

hCN

exp

(h

{[1 +

1

22

]I

}h

)dh.

(2.11)



Using the equality (A.7), the bound (2.11) simplifies to1

Pr {error} 1(1 +

1

22

)N = 1(1 +

SNR

2

)N . (2.12)It is evident that Rx diversity decreases the error probability considerably. Intuitively,

this conclusion may be derived from the expression for E|s s|2 = 2/h2 impliedfrom (2.9). The estimation error variance depends on the absolute value of all channels

and not just one as in the SISO case. Moreover, in the case of white channels, MRC

simply means averaging the signals from the antennas, reducing the estimation error

variance by a factor of N .

At this point we introduce two important concepts in MIMO. The first is the

diversity order (DO)

DO = limSNR

loge Pr {error}loge SNR

, (2.13)

which is the slope of the error probability curve at high SNR. The second is the array

gain (AG), defined as the average increase in the post processing SNR

AG =E{post processing SNR}

SNR. (2.14)

An alternative, more meaningful definition (in terms of performance) for the AG,

which may not coincide with the previous definition, is the shift of the error probability

curve w.r.t. the curve (for QPSK)

1(1 +

SNR

2N

)N . (2.15)Note that in the case of Rx diversity, the DO and AG (according to both definitions)

are equal to N . The SER curves of MRC with 2 Rx and 4 Rx antennas are given in

Fig. 2.2.

1We use here the identity det {A} = N detA, where A is an N N matrix.



Figure 2.2: SER curves of MRC 1 2 and 1 4.


12

Chapter 3

Transmit Diversity - STC


In many cases placing many Rx antennas at the receiver is impractical, so a natural

question is wether we can obtain the DO and AG of MRC using multiple Tx antennas

instead. We begin with a naive scheme where the same symbol s is transmitted from 2

Tx antennas (with the appropriate scaling 1/2 to ensure unit Tx power) as depicted

in Fig. 3.1. The model for the signal received at the single Rx antenna is

y = h012s+ h1

12s+ n

=12(h0 + h1)s+ n. (3.1)

Note that in independent Rayleigh, the RV

h =12(h0 + h1) (3.2)

is complex normal with zero mean and unit variance. This means we get a model

that is identical to SISO, so we gain nothing. Another approach has to be invoked.

One of the most prominent methods for Tx diversity is Alamoutis space time

coding (STC) [2], which is applicable for two Tx antennas. Besides the use of the

spatial domain (as done in MRC), STC makes further use of the time domain.


3. Transmit Diversity - STC 13

Figure 3.1: Naive Tx diversity scheme.

In Alamoutis scheme, the transmission is done from two Tx antennas and in pairs

of time slots, as depicted in Fig. 3.2. Beginning with the case of single Rx antenna,

the mathematical model corresponding to the ith time slot is

y(i) = [ h0 h1 ]

[x0(i)x1(i)

]+ n, (3.3)

where y(i) is the measurement at the Rx antenna at time i, x0(i) is the transmitted

signal from Tx antenna 0 at time i, and x1(i) is the transmitted signal from Tx

antenna 1 at time i. The transmission scheme is[x0(0) x0(1)x1(0) x1(1)

]=

12

[s0 s1s1 s

0

], (3.4)

which means that only one data stream is transmitted from the Tx antennas and

the transmission rate is identical to that in SISO. The factor 12, makes sure that

the total transmission power remains identical to the SISO case. Assuming that the

channel vector is identical at both time slots, the aggregated received signal is[y(0)y(1)

]

y

=12

[h0 h1h1 h0

]

H

[s0s1

] s

+n. (3.5)

We denote the linear transformation operating on s by H and not simply H , toemphasize that it is not the physical channel, but rather the effective channel created



by the STC transmission scheme. Note that the columns of H are orthogonal 1[h0h1

] [h1h0

]= [h0, h1]

[h1h0

]= 0. (3.6)

Figure 3.2: STC 2 1 configuration.

We proceed as before to obtain the ML estimator, assuming the receiver possesses

H. The expression for the ML estimator s of s in this case becomes

s = argminsQPSK2

y Hs2. (3.7)

We note that the term y Hs2 in (3.7) may be rewritten as

[s s](HH)[s s], (3.8)

where s is the least squares (LS) estimator of s given the measurements y, satisfying

s =H+y, (3.9)

where H+ is the pseudo-inverse of H defined as (HH)1H. We further note thatin Alamoutis STC, H is a scaled unitary matrix. That is

HH = |h0|2 + |h1|22

I. (3.10)

1This means in AWGN each of the symbols may be decoded individually (in MRCfashion) regardless to the other symbol, simlilarly I and Q components modulated withorthogonal Sine and Cosine.



This property of H is perhaps the most crucial part of the Alamouti scheme. Using(3.10), turns (3.8) to the form

|h0|2 + |h1|22

s s2, (3.11)

so the ML estimator becomes

s = argminsQPSK2

s s2

= argminsQPSK2

(|s0 s0|2 + |s1 s1|2) , (3.12)or simply

s0 = argmins0QPSK

|s0 s0|2

s1 = argmins1QPSK

|s1 s1|2, (3.13)

which means that in STC, the ML receiver sums up to LS estimation followed by

regular SISO processing for each of the symbols s0, s1 independently.


Proceeding as in the previous sections, we substitute (3.5) in the LS equation (3.9)

for s and obtain

s = (HH)1H [Hs+ n]

= s+2

|h0|2 + |h1|2Hn, (3.14)

which means that the covariance matrix of the noisy term is

22

|h0|2 + |h1|2I, (3.15)

which is identical to that in the MRC case ((2.9) and line under), except for the factor

2 which means a 3dB decrease in the AG. In STC the AG is 1.



Since the expression for the variance of the noisy part after the LS is identical to

that in MRC, up to the constant 2, the error probability takes the form

Pr {error} 1(1 +

1

42

)2 = 1(1 +

SNR

4

)2 , (3.16)which implies second order diversity, as in the case of MRC with 2 Rx antenna, but

AG of 1, which means no AG. The SER curve of STC 2 1 is given in Fig. 3.3. TheSER curve of MRC 1 2 is also given to show the 3dB difference between the curves,and the identical DO.

Figure 3.3: SER curves of STC 2 1 and MRC 1 2.

3.3 Transmit and Receive Diversity - STC+MRC

3.3.1 System Model and ML Receiver

The previous sections show that STC with 2 Tx antennas, provides DO of 2 and no

AG. They also show that MRC with N Rx antennas provides DO N and AG N . A

natural expansion of the ideas above would be the fusion of STC transmission and



Rx diversity. We consider here a MIMO array with 2 Tx antennas transmitting STC

and N Rx antennas. This MIMO system is given in Fig. 3.4.

Figure 3.4: STC with Rx diversity configuration.

The model for the received signal at the nth Rx antenna aggregated over twotime slots is identical to (3.5)[

yn(0)yn(1)

]

yn

=12

[hn,0 hn,1hn,1 hn,0

]

Hn

[s0s1

] s

+nn. (3.17)

Thus, the whole system model isy0y1...

yN1

y

=

H0H1...

HN1

H

[s0s1

] s

+n. (3.18)

Here y and n are vectors of length 2N . We note that here too the columns of H are

orthogonal

H H =[ H0 H1 HN1 ]

H0H1...

HN1

=

N1n=0

HnHn =1

2

N1n=0

(|hn,0|2 + |hn,1|2)I, (3.19)



so the ML receiver implies SISO processing on the output of the LS solution. We

further note that using the LS solutions sn at each of the antennas, the LS solution

takes the form

s = (H H )1H y

=1

1

2

N1n=0

(|hn,0|2 + |hn,1|2)

[ H0 H1 HN1 ]

y0y1...

yN1

=

N1n=0

(|hn,0|2 + |hn,1|2)snN1n=0

(|hn,0|2 + |hn,1|2), (3.20)

which means combining the LS solutions at the antennas in an MRC fashion. The

weight assigned to the LS solution of the n th antenna is |h0,n|2 + |h1,n|2, which isproportional to its post processing SNR.

3.3.2 Evaluation of the Error Probability

The LS solution s takes the form

s = (H H )1H [H s+ n]

= s+2

N1n=0

(|hn,0|2 + |hn,1|2)H n (3.21)

which means that the covariance matrix of the noisy term is

22

N1n=0

(|hn,0|2 + |hn,1|2)I, (3.22)



implying an AG of value N . The error probability is evaluated as in previous sections

and reads

Pr {error} 1(1 +

1

42

)2N = 1(1 +

SNR

4

)2N , (3.23)which means DO 2N . Intuitively, the STC provided DO 2 and the MRC provided

DO N , so the total DO is 2N . The SER curve of STC 2 2 is given in Fig. 3.5.Note the difference in DO compared with STC 2 1.

Figure 3.5: SER curves of STC 2 2 and STC 2 1.


20

Chapter 4

Transmit Beamforming

In the previous chapters, we assumed perfect channel knowledge at the receiver,

but no channel knowledge at the transmitter. In this chapter we concentrate on

methods for transmission and reception assuming perfect channel knowledge at both

sides. Practical methods for obtaining channel knowledge at the transmitter will be

discussed in Chapter 11.

4.1 System Model and Optimal Transmission

We consider a MIMO array endowed with M transmit and N receive antennas, de-

picted in Fig. 4.1. We assume that the transmitter possesses perfect knowledge of

the channels matrix. The question at hand is how to exploit the channel knowledge

at the transmitter to transmit the information symbol s using the M transmit anten-

nas, in a manner that optimizes the link performance. Hereafter we restrict ourselves

to linear precoding, where the transmitted signal x satisfies x = ws and w is the

precoding weight vector. This method of precoding in also known as beamforming,

and w is dubbed the beamformer. The mathematical model for the received signal

when beamforming is applied is

y =Hx+ n =Hws+ n. (4.1)


4. Transmit Beamforming 21

Note that when the beamformer w is applied, Hw may be viewed as the equivalent

channel. We further note that the post-processing SNR at the receiver (after applying

MRC with the N Rx antennas) is Hw2/2.

Figure 4.1: Tx beamforming configuration with M Tx and N Rx antennas.

When the optimization criterion is maximal SNR at the receiver, the optimal

beamforming problem may be written as

w = argmax:2=1

H2. (4.2)

The unity magnitude constraint = 1 is used to make sure that the transmissionpower remains equal to that in the SISO case. The optimization problem (4.2) may

be solved using Lagrange multipliers. We define the Lagrangian

L = H2 (2 1), (4.3)

differentiate w.r.t (assuming the real-valued case, see Appendix C), and equate to

zero

L

=

[HH ( 1)]

= 2HH 2 = 0. (4.4)

Applying conjugate transposition and returning to w leads to

(HH)w = w. (4.5)



This means thatw is an eigenvector of the rectangular matrixHH . Bearing in mind

that we are seeking the solution that maximizes Hw2, we reach the conclusion thatw is the eigenvector ofHH corresponding its the largest eigenvalue (the result holds

in the complex-valued case).

The eigenvectors of HH may also be found by the singular value decomposition

(SVD) of the (not necessarily rectangular) matrix H . The SVD decomposes H into

a product of three matrices

H = UDV , (4.6)

where U and V are unitary matrices satisfying U U = I, V V = I, and D is a

diagonal matrix with real positive singular values on its diagonal. The entries of D

are the square roots of the eigenvalues of HH or HH. The columns of V are the

singular vectors of H which are the eigenvectors of HH , and the columns of U are

the eigenvectors of HH.


As noted above, the receiver applies MRC to the signals at the N Rx antennas, so

the post processing SNR is

post processing SNR =Hw2

2=12

=d212, (4.7)

where 1 is the largest eigenvalue of HH and d1 is the largest singular value of H .

Using the fact that rotation matrices do not change the Frobenius norm1 of a matrix,

we get

H2F = UDV 2F = D2F =

d2i , (4.8)

1The Frobenius norm AF of an N M matrix A is defined as the square root ofthe sum of the absolute squares of its elements, AF =

n

m |An,m|2. Thus, A2F

measures the energy of the matrix.



where di is the ith singular value of H . Obviously, d21 is upper bounded by

d2i =

H2F . Moreover, since d1 is the largest singular value, d21 is lower bounded by

d2i

divided by the number of singular values. Thus, we have

H2Fmin{M,N} d

21 H2F . (4.9)

The post processing SNR =d212

is bounded by

H2Fmin{M,N}2 post processing SNR

H2F2

, (4.10)

which leads to bounds on the AG by applying an expectation and dividing by the

average SNR =1

2MN

min{M,N} AG MN. (4.11)

This means that in the case of 2 2, the AG is up to 3dB better than that inSTC+MRC.

An upper bound for the error probability is given by

Pr {error} 1(1 +

1

2min{M,N}2)MN = 1(

1 +SNR

2min{M,N})MN , (4.12)

which implies that in the case of 2 2, the DO is identical to that in STC+MRC.The SER curves of eigen beamforming 2 2 and 4 2 are given in Fig. 4.2. Notethat the DO in the 2 2 case is identical to STC 2 2 and the AG is better by lessthan 3dB.

In some cases the distribution of 1 is known explicitly, so the bounds may be

replaced with an explicit expression for the AG and error probability. For example,

in the 2 2 case, the distribution of 1 is [3, 17]

p(1) = exp(1)[21 21 + 2

] 2 exp (21) . (4.13)Dr. Doron Ezri OFDM MIMO


so the average error probability is bounded by

Pr {error} = 0

Pr {error|1} p(1) d1

0

exp

( 122

)p(1) d1

=

0

exp

( 122

){exp(1)

[21 21 + 2

] 2 exp (21)} d1=

32(2 +

1

2

)3(4 +

1

2

) 1(1 +

3.36 SNR2 4

)4 , (4.14)which implies AG of 3.36 (or 5.27dB) which is a 2.27dB advantage over the 2 2STC+MRC scheme (due to channel knowledge at the transmitter).

Figure 4.2: SER curves for eigen beamforming 2 2 and 4 2.

The DO and AG are asymptotic (at high SNR) metrics, so it may suffice to

approximate the error probability curve at high SNR, instead of seeking an accurate

expression at all SNR values. We note the average error probability bound in (4.14)

takes the form of the (one sided) Laplace transform of p (1)

Pr {error} 0

exp

( 122

)p(1) d1

=

0

exp(S1

)p(1) d1, S =

SNR

2, (4.15)



so the asymptotic error probability curve may be computed through the initial value

theorem. The asymptotic error curve then depends on the density of 1 near the

origin [21]. Specifically, p (1) is expanded about the origin as

p(1) = exp(1)[21 21 + 2

] 2 exp (21)=

1

331 + higher order terms, (4.16)

so the asymptotic expression of the error probability is 0

exp

( 122

)1

331 d1 =

32

SNR4, (4.17)

which gives the same solution (DO=4, AG=3.36).

4.3 Maximal Ratio Transmission

A special case of Tx beamforming is the case of a single Rx antenna, N = 1. This is a

common case since in many systems the receiver is to be low cost. Here the channels

matrix H reduces to a row vector h = [h0 . . . , hM1], and the matrix HH in (4.5)

turns to hh, which has rank 1.

In this simplified case the computation of the SVD reads

h = 1 h (h

h)

. (4.18)

Thus, the optimal beamformer in the case of single Rx antenna reads (remember

H = h)

w =h

h wi =hih . (4.19)

This method is known as maximal ratio transmission (MRT) [9]. Note that the SER

curve of MRT with M Tx antennas and one Rx antenna is identical to that in MRC

with one Tx antenna and M Rx antennas.


26

Chapter 5

Spatial Multiplexing

In the previous chapters, we exploited the MIMO configuration to enhance the link

properties, with the underlying assumption that a better link (higher SNR, less fad-

ing) means the ability to transmit more information by using less robust modulation

schemes (say switching from QPSK to 16QAM) conveying more information. In this

chapter we consider a different approach to exploit the MIMO configuration, in which

different information streams are transmitted from the Tx antennas. This approach

is known as spatial multiplexing.


In spatial multiplexing (SM), independent information streams are transmitted through

the Tx antennas. We consider a MIMO array with M transmit and N receive an-

tennas where N M , depicted in Fig. 5.1. The transmitted vector is 1Ms where

s = [s0, s1, . . . , sM1]T is a vector of M independent symbols. The factor 1M is in-

troduced in order to maintain unity transmission power. The mathematical model

for the received signal is

y =1MHPHY H

s+ n. (5.1)


5. Spatial Multiplexing 27

Thus, in SM we increase the throughput explicitly (assuming that the receiver is

capable of decoding the information).

Figure 5.1: Spatial multiplexing configuration with M Tx and N Rx anten-nas.

The ML estimator s of s reads

s = argminsQPSKM

y Hs2. (5.2)

Note that in SM, in contrast to all diversity schemes we discussed before, HH is not

a diagonal matrix. This means that the LS solution followed by SISO processing is no

longer optimal. Actually, no further simplification of the ML estimator (5.2) exists.

Moreover, Eq. (5.2) implies that optimal ML decoding in SM requires exhaustive

search in multiple dimensions. The problem becomes more severe when high modu-

lations (say 64QAM) or large number of Tx antennas are employed. For instance, in

M=4 and 64QAM, the computation of the ML estimator requires exhaustive search

over 644 16 106 options.The fact that ML reception in SM requires exhaustive search is troublesome,

so suboptimal schemes have to be devised when either the number of constellation

points or number of Tx antennas is high. In the sequel we consider some of the more

prominent suboptimal solutions.




In order to evaluate the error probability we assume that the symbol s was transmitted

and define the minimizing vector

s = argmin QAMM

J(), (5.3)

where J() = yH2. Focusing on an error event in s0, the error probability maybe interpreted as

Pr {error in s0} = Pr {s0 6= s0} . (5.4)

We begin by conditioning the error event on H and s, to get

Pr {error in s0|H , s} = Pr s:s0 6=s0

J(s) J(s)H , s

, (5.5)which may be bounded by the union bound

Pr {error in s0|H , s} s:s0 6=s0

Pr {J(s) J(s)|H , s} . (5.6)

The conditional probability on the r.h.s. of (5.6) is calculated in Section D.1 as

Pr {J(s) J(s)|H , s} = Q(H(s s)

2

), (5.7)

so Equation (5.6) turns to

Pr {error in s0|H , s} s:s0 6=s0

Q

(H(s s)2

), (5.8)

which may be further simplified using the bound Q(x) 12exp

(x

2

2

)to yield

Pr {error in s0|H , s} 12

s:s0 6=s0

exp

(H(s s)

2

42

). (5.9)



Averaging w.r.t s gives

Pr {error in s0|H} 12LM

sQAMM

eA0(s)

exp

(He

2

42

), (5.10)

where L is the number of points in the QAM constellation, and A0(s) is the set of

error vectors s s such that s0 6= s0.The unconditional error probability is obtained by averaging (5.10) w.r.t to the

Rayleigh distribution ofH . Specifically, the expectation of the term exp

(He

2

42

)w.r.t H is calculated in Section D.2 as

EH

{exp

(He

2

42

)}=

1[1 +

e24M2

]N , (5.11)which leads to the unconditional error probability

Pr {error in s0} 12LM

sQAMM

eA0(s)

1[1 +

e24M2

]N . (5.12)The expression (5.12) is dominated by the error vectors with minimal norm. Focusing

on QPSK, each symbol s has 2 dominant vectors e with e2 = 2, such that s0 6= s0.Thus, (5.12) may be approximated by

Pr {error in s0} 12 2 1[

1 +2

4M2

]N = 1[1 +

SNR

2M

]N , (5.13)which means DO N and AG N/M . We emphasize here that in SM the throughput

is M times that in SISO1.

5.3 The Sphere Decoder

The exponential complexity of the exhaustive search ML suggests that other methods

are ought to be sought. Sphere decoding [5] is an iterative method for the computation

1Note that nowhere in this derivation we assumed that N M . Actually the derivationhold even for N = 1 (but the performance is very poor).



of the ML solution. The method provides the optimal ML solution with significantly

smaller complexity.

Focusing on the case of 2 Tx and N 2 Rx antennas, the ML cost functionalmay be rewritten as

y Hs2 = C + (s s)HH(s s). (5.14)

Note that since HH is a positive definite symmetric matrix, it can be decomposed

into U U = HH where U is an upper triangular matrix with real diagonal (The

matrix U may be obtained through the QR decomposition of H). Thus, the cost

functional to be minimized turns to (we omitted the constant C)

(s s)U U (s s) = U s2, (5.15)

where s = s s. Using the special structure of U we write (5.15) explicitly as

|u11s1 + u12s2|2 + |u22s2|2

= u222 |s2 s2|2 + u211s1 s1 + u12u11 (s2 s2)

2 . (5.16)We begin with searching for points s for which the cost functional (5.16) is smaller

than an arbitrary r2. Taking only the first term in the sum (5.16) we obtain a

necessary (but not sufficient) condition for a point s to have a cost smaller than r2

as

u222 |s2 s2|2 < r2 |s2 s2|2 2, the SIC algorithm exhibits

somewhat superior performance [11].

5.6 The Diversity-Multiplexing Tradeoff

Equipped with the understanding of diversity and multiplexing, we reach the con-

clusion that in many cases there exists a tradeoff between the two. Considering a

physical configuration of 2 2 without channel knowledge at the transmitter, wehave 2 prominent approaches, STC and SM. In this case SM will deliver twice the

throughput with 2nd order diversity, while STC gives 4th order diversity.

A natural question that arises is which is better? In order to answer this question,

we need to compare equal throughput schemes (apples to apples), so we construct the

SER curve of STC 22 employing 16QAM modulation (this curve is identical to thatof QPSK, except an approx. 7dB shift in SNR due to the decrease in dmin). The SER

curves of STC 22 16QAM and SM 22 QPSK, featuring identical throughput, aregiven in Fig. 5.3. Note that the STC curve is superior to that of the SM (assuming

we are targeting SER of 104).

However, when comparing the SER curves of SM 2 2 16QAM with the equalthroughput competitor STC 2 2 256QAM as shown in Fig. 5.4, the situationreverses, and the SM is superior. This means that there is no simple answer to the

question above, and there is a strong dependency on channel condition. As a rule

of thumb, we say that diversity methods are superior in the regime of small SNR,



Figure 5.3: SER curves for SM 2 2 QPSK and STC 2 2 16QAM.

whereas multiplexing is superior in high SNR [27, 12]. Bearing in mind that as the

SNR increases more SNR is needed to double the rate (3dB to move from BPSK to

QPSK, but 7dB to move from QPSK to 16QAM etc.), it is clear that at large SNR

schemes that explicitly double the rate become more attractive.

Figure 5.4: SER curves for SM 2 2 16QAM and STC 2 2 256QAM.

Another important factor which we have not addressed yet is correlation between



the spatial channels. In order to demonstrate the effect of correlation we turn to

the limit of fully correlated channels, so all entries in the channels matrix H are

identical (but yet random). In this case the DO of all diversity schemes decreases to

1. However, in SM the schemes collapse and decoding is impossible (e.g. in the ZF

decoder, the matrix is not invertible). As a rule of thumb, we say that SM methods

are more sensitive to spatial correlation than diversity methods [4]. The impact of

spatial correlation on MRC is given in Appendix G.


38

Chapter 6

Closed Loop MIMO

In the previous chapter we considered SM and assumed no channel knowledge at the

transmitter. In this chapter we consider SM with perfect channel knowledge at the

transmitter, a technique that is known as closed loop (CL) MIMO. This technique may

be viewed as fusion between SM and beamforming. Practical methods for obtaining

channel knowledge at the transmitter will be discussed in Chapter 11.

6.1 System Model and Optimal Transmission

We consider here an array with M transmit and N receive antennas, and assume

perfect channel knowledge at the transmitter and receiver. We already saw that

using this system it is possible to transmit a single precoded stream and obtain MN -

th order diversity. The question at hand here, is how to exploit the channel knowledge

at the transmitter in order to concurrently transmit K min(M,N) streams.One solution to this problem is closely related to the concept of single stream

beamforming and SVD. We consider the following transmitted signal which may be

viewed as an extension of single stream beamforming

x =K1i=0

aivisi, (6.1)

where vi is the i-th singular vector ofH and ai is a positive power allocation coefficient


6. Closed Loop MIMO 39

with

a2i,i = 1 to maintain unity transmission power. This precoding scheme is

attractive since the singular vectors are orthogonal (in contrast to eigenvectors) and

remain orthogonal after multiplication with H

(Hvi)(Hvj) = vi (H

Hvj) = vi (vj) = 0 for i 6= j. (6.2)

which means orthogonal at the receiver.

In order to further investigate the scheme we rewrite the transmitted signal as

x = VAs, (6.3)

where V is the right hand unitary matrix in the SVD of channels matrix H and

AMK is a diagonal matrix with the entries ai on its diagonal. The resulting received

signal is

y =HVAs+ n. (6.4)

Note that now, the equivalent channel matrix HVA satisfies

(HVA)HVA = AV HHVA

= AV V DU H

UDV H

VA

= (DA)DA, (6.5)

where (DA)DA is a diagonal matrix. Thus, in this transmission mode ZF is optimal

(orthogonal transmission). An equivalent reception method decodes the received

signal with the matrix U so we arrive at the signal

z = U y = U HVAs+ U n. (6.6)

Using the expression the SVD and the fact that U is unitary, Eq. (6.6) turns to

z =DAs+ n, (6.7)

where DA is a diagonal matrix.



6.2 Implications of Closed Loop MIMO

Equation (6.7) for the decoded signal z leads to the following important implications.

The MIMO link is transformed into K parallel SISO links. This meansthat ZF decoding is optimal, and receiver complexity is significantly

reduced.

No noise amplification occurs since U is a unitary matrix that does notintroduce gain. This is in contrast to the noise amplification in ZF due

to the multiplication with the pseudo-inverse of H .

The singular values on the diagonal of D may significantly differ inmagnitude, so (unless compensated by A, and usually A does not

compensate for this effect, as discussed in the sequel) we have streams

with different SNR. The pdf of the singular values in an uncorrelated

4 4 Rayleigh channels matrix is given in Fig. 6.1.

Figure 6.1: The singular values pdf in an uncorrelated 44 Rayleigh channelsmatrix.



The matrix A determines the number K min(M,N) of spatialstreams transmitted according to the number of nonzero entries on

its diagonal. In particular, when A is set to

A =

1 0 . . . 00 0...

. . .

0 0

, (6.8)the transmission is reduced to single stream transmit beamforming dis-

cussed in Chapter 4.

The DO in the various streams is also different (as well as the AG). Infact recent results show that the DO of the kth stream (k = 0 . . . , K1) is (M k)(N k) [25]. This means that in the M M case, whilethe first stream enjoys DO of M2, the last experiences DO one (this

asymmetry is difficult to balance).

We note that we do not have a complete solution yet, since the matrix A has not

been determined. The intuition of some would lead them to the conclusion that A

should be proportional to D1, so that DA in (6.7) renders a scaled identity matrix

leading to parallel streams with equal SNR.

In order to investigate this issue an optimality criterion for the case of multiple

streams is required. Moreover, it is obvious that the maximal SNR criterion we used

this far does not suffice for this matter. For this case, the sum capacity criterion [13]

a = argmaxa2=1

log (1 + SNR of stream i)

= argmaxa2=1

log

(1 + a2i

d2i2

), (6.9)

is more suitable. In the case of two stream the optimization problem (6.9) has an

analytical solution and the optimal value of the power allocated to the first stream, a21,



is given in Fig. 6.2. Several SNRs of the two streams (corresponding tod2i2, i = 1, 2)

are considered.

Figure 6.2: The optimal power allocation in the sense of sum capacity.

The figure shows that the stream featuring higher SNR is allocated with more

(or all) power, and as the SNR increases the allocation is more equal. Thus, in the

sense of sum capacity the intuition pointing at allocating more power to the weaker

streams is misleading.


43

Chapter 7

Space Division Multiple Access

7.1 System Model and Basic Solution

Space division multiple access (SDMA) is a technique very similar to CL MIMO,

since in both techniques, multiple beamformed streams are transmitted concurrently,

and perfect channel knowledge at the transmitter is assumed. The difference between

the two techniques lies in the fact that in SDMA the Rx antennas belong to different

receivers/users. To simplify matters, we assume that every receiver is endowed with

a single Rx antenna. An SDMA configuration is given in Fig. 7.1.

The model for the received signals in an SDMA system withM transmit antennas

and N M receive antennas is y0...yN1

y

=HW

s0...sN1

s

+n, (7.1)

where yi is the signal received at the antenna of the ith user, and si is the informationsignal transmitted to the ith user. The main difference between CL MIMO andSDMA is now evident. In CL MIMO, the Rx antennas belong to a single receiver, and

it uses all of them to reconstruct the multiple information streams (the ith streamis decoded with uiy). In SDMA the situation is different and each receiver uses its


7. Space Division Multiple Access 44

Figure 7.1: SDMA Configuration in which each receiver is endows with asingle Rx antenna.

single antenna to reconstruct the single information stream addressing it. Thus, in

SDMA, the precoding matrix W has to be devised such that HW is diagonal or

nearly diagonal. Otherwise, multi-user interference (MUI) is introduced.

Assuming that the SNR is high and MUI is the main concern, the beamforming

matrix W should satisfy

HW = D, (7.2)

whereD is a diagonal matrix, and is a scaling factor. The precoding matrix should

also meet the unity power constraint

EWs2 = 1 W F = 1. (7.3)

Thus, a straight-forward solution meeting both requirements is the ZF beamformer

W =H+D

H+DF . (7.4)

The physical interpretation of SDMA is the following. For the ith receiver,SDMA uses wi to create a beam that amplifies si at the direction of that receiver,



and attenuates si at the directions of all other N 1 receivers (spatial nulls). Thisinterpretation is given in Fig. 7.2. The equation for the SDMA beamforming matrix

(7.4) also implies that an array of M Tx antennas can create up to M 1 nulls.

Figure 7.2: The beams shape in SDMA is such that the beam for the nthUT nulls out at the directions of all other users.

The matrix D which actually determines the power allocation to each stream

is determined according to the multi-user transmission strategy. If the strategy is

to maximize the sum capacity, the entries of D will be determined by a procedure

similar to (6.9). Note that the SNR of ith stream (when each stream is allocatedwith identical power) is proportional to

1

qi2where Q =H+.

7.2 More Advanced Solutions and Considerations

In the common case, the SDMA transmitter is a base station (BS). The BS endowed

with M antennas communicates with Nu user terminals (UTs). Usually, Nu >> M ,

so the BS cannot transmit simultaneously in an SDMA fashion to all UTs. Thus,

when SDMA transmission is employed, the BS needs to divide the UTs into sets of

up to N M users, to which it will transmit simultaneously.A natural algorithm to divide UTs into sets would be the grouping of UTs with

orthogonal channel vectors into the same set. In this case, MRT may be applied to



each UT independently, and no MUI is introduced. In this case, MRT would also be

the optimal solution. Obviously, in real scenarios perfect orthogonality is not to be

found, so sets with minimal correlation are desirable.

In the previous section, we assumed that the SNR is high and MUI is the main

concern. However, SDMA is also applied in medium and low SNR, so another criterion

such as maximal signal to noise and interference ratio (SINR) should replace the zero

MUI criterion we adopted [7, 18].

Another extension of the ideas demonstrated above is the application of SDMA

to the case where each receiver is endowed with multiple receive antennas [19, 24]. In

this case, the received signal takes the form y0...yN1

y

=HW

s0...sN1

s

+n, (7.5)

where yi = [yi,0, . . . , yi,Ni1]T is the signal vector arriving at the Ni antennas of the

i th UT, and si = [si,0, . . . , si,Ni1]T is the information vector addressing the i-th UT. Assuming that M iNi, it is possible to transmit up to Ni informationstreams to the ith UT in an SDMA fashion. This means up to iNi concurrentinformation stream to all UTs. In this case, the zero MUI approach requires that

HW is not diagonal, but block diagonal.

HW =

[B0] 0 . . . 00 [B1]...

. . .

0 [BNi1]

. (7.6)This approach leads to superior performance over the case of

iNi UTs with a single

Rx antenna.



The main problem with all of the approaches discussed above is the sensitivity of

the scheme to channel state information. In fact, when the channel state information

is not perfect (and this is always the case in practical systems), the near zero MUI

approach fails. The LTE and 802.16m standards are considering a different SDMA

concept, in which each UT is equipped withNe > 1 antennas, andK Ne streams aretransmitted concurrently (say one for each UT). This way each UT performs regular

SM processing and disregards the streams addressing other UTs. In this approach

precoding is optional, but obviously leads to superior performance.


48

Part II

Practical OFDM-MIMO


49

Chapter 8

The Wireless Channel

In previous chapters we assumed that each entry of the channels matrix H was a

single RV, independent of the frequency axis. This corresponds to the assumption

of flat fading. In this chapter we describe channel models that are more suitable for

wireless propagation.

8.1 Propagation Effects

The propagation effects are usually divided into three distinct types of models as

illustrated in Fig. 8.1 . These are mean path loss, slow variation about the mean due

to shadowing and scattering, and the rapid variation in the signal due to multipath

effects. The first two, which are also known as large scale fading, are usually consid-

ered frequency independent (about the carrier), while the last, known as small scale

fading, is frequency dependent.

8.1.1 Path Loss and Shadowing

The path loss L describes the mean attenuation in the radio channel primarily due to

physical separation between the transmitter and receiver. The free space path loss,


8. The Wireless Channel 50

Figure 8.1: Propagation effects.

which lower bounds any practical path loss, is given (in far field) by

FSPL(dB) = 20 log10(R) + 20 log10(f) + 32.5, (8.1)

where R is the transmitter-receiver separation (km) and f is the carrier frequency1

(MHz).

However, the free space path loss is not suitable for real life scenarios, so path

loss parameters are usually based on empirical evidence (measurement campaigns

in different physical scenarios). For instance the ITU-R [1] adopts the following

expression for the outdoor to indoor and the pedestrian environments

L(dB) = 40 log10R + 30 log10 f + 49, (8.2)

where f is the carrier frequency (MHz) in the vicinity of 2000MHz. Note that this

1Note that the path loss increases with the frequency. This is not due to the impact offree space, but rather due to the receive antenna aperture.



equation is usually not valid for other scenarios (e.g vehicular), or other frequency

bands (e.g. 5.8GHz).

Obstacles between the transmitter and receiver also attenuate the signal. The

overall phenomenon is known as shadowing. The effect of showing is usually slow

(seconds to minutes). The large scale fading components given above do not include

the important impact of multipath (small scale fading) which is considered in the

next sections.

8.1.2 The Physics of Multipath

In case of perfect line-of-sight (LOS) between the transmitter and receiver, the noise-

less version of the baseband signal, y(t), arriving at the receiver is simply a scaled

delayed version of the transmitted baseband signal s(t), reading

y(t) = a s(t ), (8.3)

where a is a complex valued factor2 and is the delay.

However, in multipath, the transmitted signal is reflected from numerous scatter-

ers creating multiple paths of propagation to the receiver (the multipath phenom-

enon). Each path results in a different attenuation and a different delay i (due to a

different path length). This scenario, known as non line-of-sight (NLOS) is depicted

in Fig. 8.2.

In multipath, the received signal is a superposition of the contributions of the K

different paths

y(t) =K1i=0

ais(t i), (8.4)

2For more information on baseband signals see Appendix F.



Figure 8.2: A multipath scenario.

so the corresponding baseband channel h() takes the form

h() =K1i=0

ai( i). (8.5)

When either the transmitter, receiver, or scatterers are in motion, the received

signals are subject to the Doppler effect. For example, in the case of LOS, motion in

absolute velocity v and relative velocity v cos() results in the frequency shift

fd = fm cos(), (8.6)

where fm is the maximal doppler shift

fm =v

=vfcC

, (8.7)

is the wavelength, and C is the speed of light.

Thus, when we have both multipath and Doppler, the received baseband signal is

[compare with (8.4)]

y(t) =K1i=0

ai exp [j2pifm cos(i) t] s(t i), (8.8)

where i is the angle corresponding to the ith path. We understand that eachpath may shift (spread) the original signal in both delay and frequency. The shifts



associated with each path may be different as each path has it own delay and own

Doppler, due to different angle. A measure for the spread in the delay is the delay

spread

= max i min i, (8.9)

and a measure for the spread in frequency is the Doppler spread

fd = max {fm cos(i)} min {fm cos(i)} . (8.10)

8.1.3 Delay Spread

Focusing on the case of zero Doppler (alternatively we can assume zero Doppler

spread), a channel with impulse response of the form (8.5), has a frequency response

H(f) =K1i=0

ai exp (j2pii f) (8.11)

that may significantly vary in frequency, so such channels are dubbed frequency se-

lective. An example for the frequency response of a frequency selective channel is

given in Fig. 8.3

In order to illustrate the way multipath impacts frequency selectivity we consider

the following simple channel with two paths

h() = a1() + a2( 2). (8.12)

The channel magnitude in the frequency domain is selective

|H(f)| = |a1 + a2 exp(j2pi2 f)|, (8.13)

as depicted in Fig. 8.4. Moreover, we understand that the rapidness of the channel

in frequency depends on 2 as depicted in Fig. 8.5, while similar gains imply deeper

fades as shown in Fig. 8.6.



Figure 8.3: An example for the frequency response of a selective fading chan-nel.

Figure 8.4: The magnitude of a simple two paths channel.

It is now clearer that the delay spread defined (8.9) may be misleading as it does

not account for power of each path. To solve this problem we define the root-mean-

square (RMS) delay spread which is the RMS value of the delay of reflections,

weighted proportionally to the energy in the reflected waves

= 2 2, (8.14)



Figure 8.5: The larger the delay spread, the more rapid the channel in fre-quency.

Figure 8.6: Similar paths gain lead to deeper fades.

where

=

K1i=0

|ai|2iK1i=0

|ai|2; 2 =

K1i=0

|ai|2 2iK1i=0

|ai|2. (8.15)

8.1.4 Doppler Spread

Focusing on the case of zero delay (alternatively we can assume zero delay spread)

the received signal (8.8) takes the form

y(t) = s(t)K1i=0

ai exp [j2pifm cos(i) t]

= s(t) g(t), (8.16)



where g(t) is the time varying process

g(t) =K1i=0

ai exp [j2pifm cos(i) t] . (8.17)

Note the complete duality between g(t) in (8.17) and H(f) in (8.11). In g(t), the

Doppler shifts fm cos(i) play the role of the delays i in H(f).

8.2 Channel Modeling

When designing a wireless communications systems we need benchmark channel mod-

els so we can design and test our algorithms and mechanisms. It is important that the

models capture the most important characteristics of the channel (e.g., delay spread,

Doppler spread and pdf of the fades). However, It is also important that these models

are simple to allow simple definition (in terms of number of parameters) and simple

simulation.

There are two major approaches to channel modeling. The first is the ray tracing

approach which draws the position and velocity of all entities (transmitter, receiver,

scatterers, etc.) and traces the propagation of the rays. The second approach is

stochastic, in which the channels are defined by their statistics (as random variables

and processes). We will concentrate in this section on the latter approach which is

usually simpler.

8.2.1 Modeling Path Loss and Shadowing

Modeling the path loss usually means adopting a pathloss expression similar to (8.1)

or (8.2). Considering the pathloss model alone, the channel from the transmitter to

the receiver depends only on the distance (for a fixed carrier frequency), and all points

on a circle centered at the transmitter will experience the same reception power.



Adding the shadowing effect makes things a bit more realistic and allows equally

distant receivers to experience different receive power. The randomness of the environ-

ment is captured by modeling the density of obstacles and their absorption behavior

as random numbers. For instance, The ITU-R augments the path loss model (8.2)

with a log-normal distributed RV for the shadowing. The standard deviation chosen

for the log-normal RV is 10dB.

8.2.2 Modeling Mobile Channels

In many scenarios, each of the K paths is composed of numerous subpaths reflected

from scatterers, so the central-limit theorem suggest the process g(t) in (8.17) may

be approximated by Gaussian process. Further assuming that the scatterers are

uniformly distributed leads to the classical U-shaped PSD given in Fig. 8.7. This

PSD, denoted Doppler PSD, is approximated by the Jakes model [8]3

S(f) =

1

pifm

11

(f

fm

)2 |f | < fm0 otherwise,

(8.18)

Following this approach, we may write the channel corresponding to (8.8) as

h( ; t) =K1i=0

ai(t)( i), (8.19)

and assume each of the processes ai(t) is a (usually independent) Gaussian random

process with a given PSD (up to scaling that corresponds to different per path power

2i ). This also means that at any time instance t, the channel coefficients are inde-

pendent Gaussian with variances 2i .

3Note that constant frequency shift in direction translates to h( ; t) =() exp(2pijfmt cos) so the autocorrelation function given is exp(2pijfmt cos).This way the unconditioned autocorrelation leads to a zero order Bessel function of theform

2pi0

exp(jx sin)d, with U shaped Fourier transform.



Equipped with the the Gaussian approximation we define thecoherence bandwidth

Bc which is a measure for the frequency selectivity of the channel. The coherence

bandwidth is defined as the frequency interval over which the channel magnitude is

highly correlated. A common value of the correlation is 0.5 which gives

Bc =1

2pi. (8.20)

The fact that Bc is inversely proportional to the RMS delay spread4 arises from

Fourier theory. Similarly, the coherence time, Tc, is defined as the time interval over

which the channel is highly correlated. Setting the correlation to 0.5, gives

Tc =9

16pifm 1

5fm. (8.21)

Figure 8.7: The classical Doppler PSD

This means that on top of the pathloss and shadowing models, a wireless mobile

channel may be defined with the number of paths K, the paths delays i, the paths

average power 2i , the paths PSD type (e.g., Jakes), and the maximal Doppler fre-

quency fm (implied from the carrier frequency fc and the velocity). An example for

4When dealing with Gaussian channels replace |ai|2 in (8.15) with 2i .



a table of parameters defining a benchmark mobile channel, taken from [1] is given

in Fig. 8.8

Figure 8.8: An example for mobile channel parameters [1].

8.3 Simulating Mobile Channels

The simulation of mobile channels consists of the generation of the independent sto-

chastic processes ai(t) for the different paths. Bearing in mind that each process

is Gaussian with known variance and normalized PSD, the process ai(t) may be

generated by passing a complex white Gaussian sequence wi(t) through the filter

G(f) =S(f). The output of the filter is a stochastic process with PSD equal

to |G(f)|2 = S(f). The output of the filter is then multiplied by i to adjust thevariance. The procedure of creating the stochastic processes ai(t) is depicted in Fig.

8.9.



Figure 8.9: A procedure for creating a dynamic channel.

8.4 Extension to the MIMO Case

8.4.1 The MIMO Channel

When we have multiple antennas, the channel respective to each antenna may differ.

To illustrate this we begin with static scenario with a single Tx antenna and N Rx

antennas. We assume the the Rx antennas form a linear array, which means they are

equally distant with separation d on a straight line. The linear array is shown in Fig.

8.10. In case the transmitter is far, it is safe to assume that the signal reaches all

Rx antennas with an identical angle of arrival , measured from the broadside of the

array. This way, the differential path length x between two consecutive antennas is

x = d sin , (8.22)

and the received signal at the kth antenna is

yk(t) = a1 exp

(j 2pi

kx

)s

(t 1 kx

C

). (8.23)



Figure 8.10: Wavefront from a far transmitter approaching a linear Rx array.

Assuming the signal bandwidth is much smaller than the coherence bandwidth in-

duced by the (maximal) differential delay between the antennas

BW


is then

yk(t) =i

ai exp (jpik sin i) s(t i), (8.28)

which corresponds to the frequency domain channel

Hk(f) =i

ai exp (jpik sin i) exp (j2pif i) . (8.29)

Note that here even if the paths differ only in angle (same gain and same delay)

Hk(f) = a1 exp (j2pif 1)i

exp (jpik sin i) , (8.30)

the channels magnitude may be totally different

|Hk(f)| = |a1|

i

exp (jpik sin i) . (8.31)

This implies that multipath creates spatial diversity. This gives rise to stochastic

modeling of the MIMO channel which includes spatial correlation that depends on

the geometry of the array and the multipath characteristics. Obviously, we can con-

sider the more involved case of MIMO channel with mobility (Doppler). In this case

each path is associated with a different frequency shift (which is common to all Rx

antennas).

8.4.2 Modeling MIMO Channels

In MIMO there are MN physical channels. If these channels are independent, then

each of them may be viewed as a SISO channel depicted in the previous section.

Moreover, it is usually assumed that the channels are identically distributed, so the

procedure in Fig. 8.9 should simply be performed for each channel independently.

However, in many cases, MIMO channels are assumed identically distributed but

correlated.



We begin by stating that the cross correlation between MN channels is defined

by an MN MN matrix R. This way it is possible to define different correlationvalues between different pairs of channels. Another common extension of this idea

is assigning a different correlation matrix to each path. The physical incentive mo-

tivating this approach is that usually the shorter paths, which are near LOS, are

more correlative than longer paths that are assumed to encounter multiple scatterers.

In this approach the MIMO channel is defined with R0, . . . ,RK1, where Ri is the

MN MN correlation matrix associated with the ith path.Assuming we have the per path correlation matrices, the question that needs to be

answered is how to create correlated stochastic processes that correspond to the paths

gains. We argue that correlated processes may be created by linear transformation

over independent processes. To illustrate this we assume x is a vector of i.i.d Gaussian

RVs, such that

E {xx} = I, (8.32)

Note that the covariance of the product Cx is

E {(Cx)(Cx)} = CC. (8.33)

This means that if we want to create a vector with covariance R we simply need to

find a matrix C such that CC = R. The matrix C may be easily found using the

SVD of R, which for symmetric matrices takes the form

R = V DV =(VD)(VD)

, (8.34)

whereD is the per element square root of D. Thus, it is readily seen that C =

VD.



8.4.3 Simulating MIMO Channels

The simulation of mobile MIMO channels may be achieved by the generation of MN

SISO channels and the introduction of correlation between their paths processes via

linear manipulations. To set ideas straight, we consider the 2 2 dynamic MIMOchannel

H( ; t) =

[h0(, t) h1(, t)h2(, t) h3(, t)

], (8.35)

where each entry is a SISO channel satisfying

hm(, t) =K1i=0

bm,i(t)( i), (8.36)

where bm,i(t) is the process of the ith path in the mth channel. We focus onthe 4 processes b0(t) = [b0,0(t), . . . , b3,0(t)]

T corresponding to the first path. These

processes satisfy

E {b0(t)b0(t)} = R0. (8.37)

Thus, the processes are generated by the productb0,0(t)b1,0(t)b2,0(t)b3,0(t)

= C0a0,0(t)a1,0(t)a2,0(t)a3,0(t)

(8.38)where C0 satisfies C0C

0 = R0, and a0,0(t), . . . , a3,0(t) are the gain processes of the

first path generated by the 4 independent SISO channel generators depicted in Fig.

8.9.


65

Chapter 9

OFDM Basics

9.1 The Basic Concept

In the previous chapter we discussed wireless channels that are selective both in

frequency and time. In this chapter we present orthogonal frequency division multi-

plexing (OFDM) which is a technology devised to mitigate such channels. OFDM is a

multi-carrier technique, in which a single high data rate stream is transmitted across

a large number of lower data rate subcarriers. One of the main reasons to use OFDM

is its ability to effectively deal with frequency selective channels or a narrow-band

interference.

Classical multi-carrier techniques divide the available bandwidth into a set of

non-overlapping, equally spaced subcarriers, onto which the modulated data is then

multiplexed. The spacing between subcarriers would be chosen so as to eliminate

the inter-channel interference; guard bands between subcarriers could be used as

an example. These techniques, however, do not use the available bandwidth very

efficiently. A more efficient technique would create an overlap between the used

subcarriers without increasing the inter-channel interference, which implies creating

orthogonality between the subcarriers.

Let us design such an orthogonal multi-carrier technique. We begin by choosing


9. OFDM Basics 66

Figure 9.1: The pulse and the frequency shifted pulse in the frequencydomain.

a rectangular pulse g(t) that is time-limited to the interval [0, T ]

g(t) =

{1 0 t T0 otherwise,

(9.1)

with frequency response G(f). Obviously G(f) will take the form of a sinc() functionwith the first null at 1/T as depicted in Fig. 9.1. We can devise a simple single carrier

transmission scheme using the base band signal

s(t) =m

amg(tmT ), (9.2)

where am is a series of information bearing QAM symbols. Thus, for T = 100s and

QPSK modulation, we transmit 2 10, 000 bits per second. We note that if we havea second carrier that has a frequency exactly fk = k/T higher than the first, and

modulate it with the same symbol rate, it turns out that both signals are orthogonal,

as depicted in Fig. 9.1. To illustrate this we shift the frequency response G(f) by

k/T

Gk(f) = G(f k/T ), (9.3)


9. OFDM Basics 67

which implies that in the time domain pulse gk(t) satisfies

gk(t) = exp

(j2pikt

T

)g(t). (9.4)

This means that gk(t) has the same support as g(t), and they are orthogonal as given

by T0

g(t)gk(t) dt = T0

g(t)g(t) exp(j 2pikt

T

)dt

=

T0

exp

(j 2pikt

T

)dt = 0. (9.5)

The orthogonality in the frequency domain is evident about the peaks of the sinc()functions. Using this approach we can simultaneously transmit over N carriers spaced

exactly 1/T away from each other and achieve very high spectral efficiency. The

problem with the simple-minded approach is that it takes lots of local oscillators,

each locked to the others, such that frequencies at exact multiples are attained. This

difficult and expensive scheme is given in Fig. 9.2.

Figure 9.2: Simple multicarrier scheme.

The above mentioned multicarrier approach may be realized efficiently by means

of digital signal processing (DSP). Concentrating on the first symbol in the interval


9. OFDM Basics 68

[0, T ], time domain multicarrier symbol takes the form

s(t) =N1k=0

akgk(t)

=N1k=0

ak exp

(j2pikt

T

)g(t)

=N1k=0

ak exp

(j2pikt

T

), (9.6)

where ak is the QAM symbol to be sent on the kth subcarrier. We further notethat sampling s(t) with period T/N leads to

sn = s

(nT

N

)=

N1k=0

ak exp

(j2pikn

N

), (9.7)

which is simply the inverse fast Fourier transform (IFFT) of the sequence ak, k =

0, . . . , N 1. OFDM takes exactly this approach. In OFDM, N QAM symbolsare transmitted simultaneously over N subcarriers in an orthogonal manner. The

transmitted OFDM symbol in the time domain is constructed by applying IFFT to

the sequence of QAM symbols followed by digital to analog converter (DAC). At the

receiver, the received signal in the time domain is sampled, and divided into blocks

of length N , such that each block corresponds to a single OFDM symbol. Then the

samples undergo an FFT operation. The structure of the transmitter and receiver is

given in Fig. 9.3. A two dimensional plot of the OFDM signal is given in Fig. 9.4. The

figure shows that all subcarriers are orthogonal. Although there is an overlap between

subcarriers, when the sampling point is chosen to be the peak of each subcarrier, then

all other subcarriers give no contribution at that point.

The special structure of the OFDM symbol in the frequency domain, transforms

the frequency selective channel into multiple flat fading channels. This is since in

principle, the transmitted signal S(f) in the frequency domain, is multiplied with the


9. OFDM Basics 69

Figure 9.3: Simplified OFDM transceiver structure.

channel H(f)1 so the received signal Y (f) satisfies

Y (f) = S(f)H(f). (9.8)

In particular, at the subcarrier frequencies f = k1

T= kfsc, the received signal

bk = Y (kfsc) is

bk = S(kfsc)H(kfsc)

= akH(kfsc). (9.9)

Thus, it is evident that every QAM symbol ak is multiplied with a complex valued

number H(kfsc), which is the frequency response of the channel at the frequency

of the subcarrier on which ak is modulated. This means that on a subcarrier level,

OFDM may be viewed as a flat fading system and the analysis of Chapter 1 is valid.

1We will address this issue in more detail in Section 9.3


9. OFDM Basics 70

Figure 9.4: Two dimensional plot of the OFDM signal.

9.2 Pilots and Channel Estimation

As a result of the last section, equalization in OFDM is rather easy. The equalizer

only has to compensate for a simple constant complex number. Some of the used

subcarriers could be used for transmitting known pilots, which will be used for chan-

nel estimation. The estimated channel, in turn, is used as the equalizer gain and

phase values. Fig. 9.5 shows an OFDM symbol in the frequency domain, which

has undergone frequency selective fading. It is easy to see that using only few pi-

lots, spread across the frequency axis, the entire frequency selective channel could be

approximated (using linear interpolation, for example).

The pilots are spread in a two-dimensional array, both along the frequency and

time axes. The density of pilots in each dimension should be set so as to be able

to track the channel changes. Specifically, when the delay spread of the channel

is large, implying small coherence bandwidth, the number of the pilots along the

frequency axis should be high enough to track the channel. In a similar way, when


9. OFDM Basics 71

Figure 9.5: An OFDM symbol in the frequency domain, affected by a selec-tive fading channel.

the coherence time is small, denser pilots in the time domain are needed. An example

of two dimensional pilots grid, with 3 subcarriers separation along the frequency axis

and 4 symbols separation along the time axis is given in Fig. 9.6.

9.3 Guards in Time and Frequency

In OFDM there are two guards - the guard band (GB) and the guard interval (GI).

The GBs are two frequency bands at the left most and right most parts of the occupied

bandwidth. In these bands the subcarriers are set to zero, in order to ensure that the

out-of-band emission is small enough to prevent interference to neighboring frequency

bands. The GBs force the sinc() functions to decay to such an extent that the out-of-band emission requirements (usually about -40dB) are met. The size of the bands

is usually 10% of the subcarriers on each side. An OFDM symbol endowed with GBs

is given in Fig. 9.7.


9. OFDM Basics 72

Figure 9.6: An example of a two dimensional pilots grid.

The GI is a temporal guard that is applied to eliminate inter symbol interference

(ISI). When an OFDM symbol with duration T propagates through a multipath

channel with maximal delay spread of max, the duration of the symbol at the output

of the channel is increased to T + max (a basic property of convolution). Thus, if this

effect is not accounted for, the symbols at the output of the channel overlap and ISI is

introduced. At first glance, it seems that a trivial solution to this problem would be

the insertion of a quiet GI (in which no transmission is made) between consecutive

symbols. As long as the length of the GI is larger than the maximal delay spread, no

ISI is introduced.

In OFDM, instead of inserting a quiet GI, the GI contains a cyclic extension

of the OFDM symbol, also known as cyclic prefix (CP). This means that if the GIs

duration is 1/8 of the symbol duration, then the last 1/8 portion of the symbol is

copied into the GI. At the receiver, the GI is removed prior to the FFT operation.

The process of GI insertion and removal is depicted in Fig. 9.8, and the correspond-

ing transceiver structure is given in 9.9. The CP ensures that the OFDM symbol is


9. OFDM Basics 73

Figure 9.7: An OFDM symbol with guard bands.

cyclicly convolved with the impulse response of the channel, maintaining orthogonal-

ity between subcarriers and making sure that for sufficiently long CP (longer than

the delay spread of the channel) Eq. (9.9) holds.

9.4 The Effects of Time and Frequency Offsets

The effect of uncompensated frequency offset in an OFDM receiver is simple and

dramatic. As noted above, the orthogonality of the OFDM subcarriers relies on ac-

curately sampling the frequency domain exactly at the subcarriers frequencies. Thus,

since frequency offset f actually means that the reference points in the frequency

domain are shifted, the receiver samples the frequency domain away from the optimal

sampling points. The shifted sampling points are kfsc+f , so the orthogonality of

the subcarriers is compromised. Usually, the receiver would tolerate frequency offset

of no more than a few percents of the subcarrier spacing fsc.

The CP converts timing offsets, due to synchronization errors, to cyclic shifts of

the symbol in the time domain (actually, this is true only in one direction). Thus,


9. OFDM Basics 74

Figure 9.8: The GI insertion and removal process.

after the FFT operation at the receiver, this shift is transformed into the introduction

of linear phase in the frequency domain. Specifically, an offset of n samples in the

time domain, turns the original symbol bk in the frequency domain into

bk = bk exp(j 2pinkN

). (9.10)

Note that using the expression for bk (9.9), Eq. (9.10) may be rewritten as

bk = akH(kfsc) exp(j 2pinkN

) Heq(kfsc)

, (9.11)

where Heq(kfsc) is the equivalent channel including the effect of the temporal offset.

Thus, for small enough values of n, temporal shift is transparent to the receiver,

the equivalent channel is estimated at the receiver and compensated for.


9. OFDM Basics 75

Figure 9.9: Transceiver structure with CP insertion and removal.

9.5 The OFDM Parameters Tradeoff

We review here some of the primitive OFDM parameters and discuss the related

tradeoffs. We begin with noting that the sampling frequency fs in OFDM is N/T =

Nfsc which is approximately the occupied bandwidth (including GBs). Assuming

we set the GI duration fixed in terms of the OFDM symbol duration (say 10%), the

question that remains to be answered concerns the number of subcarriers to be used

(the length of the FFT).

The answer to this question reveals the following interesting tradeoff. The larger

the value of N (more subcarriers, smaller subcarrier spacing), the larger the duration

of the symbol T = N/fs. This means that the GI is longer and the transmission can

mitigate larger delay spreads without ISI. However, this also means that the trans-

mission is more sensitive to frequency shifts, since the subcarrier spacing is smaller.


9. OFDM Basics 76

We further note that the throughput is independent of N . Thus, in actual systems,

the number of subcarriers is determined according to the deployment scenario, taking

both the expected delay spread and frequency shift into account.

9.6 The PAPR Problem

Being a multi-carrier technique, OFDM suffers from high peak to average power ratio

(PAPR). The use of a large number of subcarriers creates a highly varying envelope,

and high temporal peaks, due to occasional constructive combining of subcarriers.

Thus, OFDM imposes some difficult requirements on the front end power amplifier

(PA), in terms of linearity over a large range. An example of a time domain OFDM

symbol is given in Fig. 9.10. Note the large temporal peak about the sample number

250.

Figure 9.10: An example of a time domain OFDM symbol.


9. OFDM Basics 77

When the temporal peaks exceed the linearity range of the PA, loss of orthogonal-

ity of the subcarriers is introduced, leading to the degradation of bit error rate (BER).

Another effect of the nonlinear amplifier is the spectral spreading and out-of-band

interference, affecting adjacent frequency bands. High power efficiency is of utmost

importance in mobile radios; however, operating near the saturation point of the PA

will result in the unwanted nonlinear interference and may outweigh the advantages

of the OFDM system. Thus, PAPR reduction is an important issue in OFDM system,

especially on the UT side, where the PA is to remain low in cost. A good survey of

prominent PAPR reduction techniques is given in [20].


78

Chapter 10

OFDMA and SC-FDMA

In the previous chapter we discussed OFDM technology. In this chapter we con-

sider a generalization of OFDM known as orthogonal frequency division multiple

access (OFDMA). We then discuss single carrier frequency division multiplexing (SC-

FDMA) which is a variant of OFDMA adopted for the LTE uplink.

10.1 From OFDM to OFDMA

In most modern communications systems, the link between the BS and the UTs

is bidirectional. The BS transmission towards the UTs is dubbed downlink (DL),

and the UT transmission towards the BS is dubbed uplink (UL). The DL and UL

transmissions must be separated in some domain. In time division duplex (TDD)

the DL and UL are transmitted in the same frequency band, but at different time

instances. In frequency division duplex (FDD) the DL and UL are transmitted in

disjoint frequency band.

In TDD OFDM, transmission is often done in sets of symbols known as frames.

The frame is composed of DL symbols (DL subframe) followed by UL symbols (UL

subframe). Each DL symbol may address multiple UTs (in a broadcast fashion), and

each UL symbol is transmitted by a single UT. An example for a TDD OFDM frame


10. OFDMA and SC-FDMA 79

structure is given in Fig. 10.1.

Figure 10.1: TDD OFDM frame structure. In the DL each colored columnrepresents an OFDM symbol. In the UL each colored column represents anOFDM symbol transmitted by a different UT.

OFDMA may be viewed as a generalization of OFDM, in the sense that in the DL

and UL, allocations are not made of entire symbols. In OFDMA, the allocations are

rectangular in time and frequency. On top of the fact that this approach allows more

flexibility in the process of allocating resources to different UTs (known as schedul-

ing), OFDMA holds another prominent advantage for distant UTs. In OFDMA,

distant UTs that need more power to arrive at the BS with sufficient signal to noise

ratio (SNR), may transmit via allocations that use a small number of subcarriers

spread over a large number of symbols. This way the distant user concentrates all its

energy on a small frequency band, and the BS receives the users transmission with

significantly enhanced SNR. We emphasize the significance of this result by the fol-

lowing example. Considering an OFDMA system with FFT size of 1024, and minimal

allocation size of 4 subcarriers (in the frequency domain), the SNR gain following the

OFDMA approach is up to 10 log10(1024/4) 24dB! An example for an OFDMA


10. OFDMA and SC-FDMA 80

frame structure with an allocation of a distant UT is given in Fig. 10.2.1

Figure 10.2: TDD OFDMA frame structure.

10.2 SC-FDMA as a Variant of OFDMA

While the 802.16e adopted OFDMA as its transmission methods for both DL and UL,

the LTE chose SC-FDMA, a variant of OFDMA for the UL. In SC-FDMA, at each

OFDM symbol the N active subcarriers (containing N QAM constellation points)

undergo an N < M point DFT operation prior to the regular M point IFFT of

OFDM. The DFT operation may be viewed as precoding. The transceiver structure

in SC-FDMA is given in Fig. 10.3. Note that besides the short DFT operation at

the transmitter and the dual short IDFT at the receiver, the transceiver remains the

same (IFFT, CP, FFT etc).

In case the N active subcarriers are consecutive at some frequency band (localized

mode), the concatenation of the shorter DFT and IFFT results in interpolation, so

1Note that in some of the IEEE802.16

mimo-course301011

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