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ISI Analysis in Network MIMO OFDM Systems

with Insufficient Cyclic Prefix Length

Vincent Kotzsch, Wolfgang Rave and Gerhard Fettweis

Vodafone Chair Mobile Communications Systems, TU-Dresden, Germany

Email: {vincent.kotzsch, rave, fettweis}@ifn.et.tu-dresden.de

Abstract—We consider a cellular network MIMO OFDM sys-

tem where cooperating base stations apply joint signal processing

to the receive signals of several users. In this situation, due

to geometry, the problem of unavoidable differences in time of

arrival between the users’ signals occurs. As a result, symbol

timing offsets can be larger than the cyclic prefix, which leads

to OFDM inter-symbol interference. In addition to the multi-

user interference, a coupling between adjacent subcarriers and

consecutive OFDM symbols is induced in such scenarios. In this

paper, we analyze the multi-user joint detection performance in

systems with distance dependent asynchronous interference and

pathloss environments for different channel delays as well as

fixed cyclic prefix length. To this end we use an exact expression

for the post equalization SINR that is used for the evaluation in

numerical simulations. The system level cell setup considers two

user positioning models based on a hexagonal cell geometry with

varying cell radii.

I. INTRODUCTION

In cooperating space division multiple access (SDMA)

systems which are often referred to as network multiple-input

multiple-output (MIMO) systems, groups of user terminals

(UT) transmit their data on the same time and frequency

resources to geographically separated base stations (BS) which

are connected to each other in order to perform joint signal

processing (e.g. [1], [2]). As we can see in the left-hand side

of Fig. 1 we have different distances d between each UT and

BS that cause different signal path delays τd:

τd=

d

clight

(1)

as well as pathloss ψd:

ψd= Ψ

?d

d0

?−η

(2)

on each wireless communication link where clightis the speed

of light, η is used as pathloss exponent, d0 as the reference

distance and Ψ as an environment propagation factor here.

It has been shown that the orthogonal frequency division

multiplex (OFDM) modulation scheme in SDMA systems is

a promising candidate to fulfill requirements for achieving

high spectral efficiency at adequate computational effort (e.g.

[3]). One advantage of OFDM is the frequency domain data

transmission on orthogonal subcarriers on which flat channels

can be assumed such that common MIMO signal processing

algorithms can be utilized. Since orthogonality is destroyed if

inter-symbol interference (ISI) due to multi-path channels with

impulse response of length τchfrom previous OFDM symbols

occurs, a cyclic prefix (CP) of duration TCP is used (e.g.

[4]). The aforementioned path delays lead to a misplacement

of the timing point for the receiver window of the discrete

Fourier transform (DFT) that is used for the time frequency

transformation in OFDM. To overcome such misalignments

synchronization procedures are used to compensate those path

delays (e.g. [5] [6]). These techniques are well established for

single user transmission. In network MIMO systems it is only

possible to be synchronized to a single transmitter such that

we have always time differences of arrivals (TDOA) between

the synchronized user and the others.

A possible timing scenario is depicted in the right-hand side

of Fig. 1 where the signals of three users are received by three

base stations taking the links to base station one (BS#1) as an

example. The desired user (UT#1) is synchronized to the base

τd2 , ψd2

t=0

UT #1:

(desired)

UT #2:

UT #3:

Rx DFT Window

Symbol i

Symbol (i-1)

t

CP

CP

Inter-symbol interference (ISI) from previous symbol

UT#3

BS#1

τd3, ψd3

τd1, ψd1

UT#2

UT#1

BS#2

BS#3

Fig. 1: Timing Scenario in Network MIMO OFDM Systems

station while the others are delayed in such a way that the

TDOA of user two lies within the CP but the ISI already leaks

into the DFT window. The third user even violates the CP limit

such that a portion of the previous OFDM symbol lacks into

the DFT span. In the existing literature that kind of interference

is often referred to as asynchronous interference (e.g. [7]). The

upper plot of Fig. 2 depicts the joint distribution of occurring

timing delays after synchronization for different cell radii R,

for the case that three users are uniformly distributed within

a hexagonal cell area that is jointly served by three base

stations. As a reference the duration of the 3GPP/LTE short

CP (TCP = 4.7µs) as well as the long CP (TCP = 16.7µs)

is indicated within the figure by vertical lines. The multi-user

detection (MUD) after asynchronous signal transmission is e.g.

treated in [8], [9] and [10] but without considering cellular

environments where also the distance dependent pathloss (see

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0

0.2

0.4

0.6

0.8

1.0

0246810 12141618 20

τd [µs]

cdf( τd )

R=1000m

TCP (LTE Short/Long)

R=2000m

R=3000m

R=4000m

0

0.2

0.4

0.6

0.8

1.0

1015 20 25 3035 404550 5560

SNR [dB]

cdf( SNR )

R=4000m

R=3000m

R=2000m

R=1000m

Fig. 2: Distance dependent TDOA and SNR distributions in

hexagonal cells for different cell radii and triplets of uniformly

distributed users

Eq. (2)) plays a significant role. In the bottom plot of Fig. 2 we

depict the distribution of occurring signal to noise ratios (SNR)

that only depends on large scale fading based on the pathloss

parameters from Tab. I (fC=0.8GHz) and can be calculated

by:

SNR =

pk,maxΨUT,AGΨBS,AG

VψdΨBS,NF ΨFFM ΨIIM

σ2

(3)

The contribution of this paper is the analysis of post

equalization signal to interference plus noise ratios (SINR) in

cooperating base station systems under ISI, inducing the effect

of pathloss dependent differences of receive power levels. In

this context we investigate the following channel effects on

the MUD performance:

• Impact of the channel length L, the cyclic prefix length

NCP, the pathloss exponent η and the carrier frequency

fCon the MUD performance for different cell radii R.

• Comparison of single and multi user detection perfor-

mance.

The paper is organized as follows: In section II we derive

our network MIMO OFDM system model for asynchronous

interference channels. A multi-user detector as well as its

SINR expression is derived in section III that is used in section

IV for providing numerical simulation results for two user

positioning senarios. Section V summarizes the main results.

Notation: Boldface letters denote matrices and underlined

letters vectors respectively. We use [.] for indexing an element

of a vector or matrix. Lowercase letters describe variables in

the time domain and uppercase letters variables in frequency

domain respectively. (.)His used as conjugate transpose

operator. The operator E(.) expresses the expectation value.

II. SYSTEM MODEL

We assume an uplink system model with K active users

each with one transmit antenna and M active base stations

each with one receive antenna. We use OFDMA modulation

and transmit complex symbols X ∈ CN×1only one a set of

subcarriers D where X[l] ?= 0 ∀ l ∈ D and X[l] = 0 ∀ l / ∈ D.

By using a cyclic prefix extension the NB= N+NCPsamples

of OFDM symbol i in time domain xk

by a DFT of size N for user k .

?

l∈D

The number of samples used for the CP w.r.t the symbol period

TS is given as NCP = TCP/TS. After transmission over a

channel specified by the vector hm,kfor the link between

mobile k and base station m, the signal at the m-th receiver

is obtained as:

i∈ CNB×1are generated

xk

i[n] =

1

√N

Xk

i[l] e

j2πln

N

, −NCP≤ n ≤ N −1 (4)

ym

i[n] =

K

?

k=1

L

?

λ=1

?hm,k[λ]xk

v) incorporates spatially and temporally

uncorrelated complex white Gaussian noise of variance σ2

The parameter µm,kexpresses an integral (in terms of the

sampling interval) timing offset on the link. The elements λ =

1...L of the link specific channel impulse responses (CIR) h

are modeled as h[λ] ∼ NC(0,σ2

represents the discrete channel length. The received signal in

frequency domain is obtained by the DFT operation applied

to the received samples ym

i

i[n − λ − µm,k]?+ vm

i[n]. (5)

Here vm

i[n] ∼ NC(0,σ2

v.

h[λ]ψ) where L = ⌊τch/TS⌋

at the m-th base station:

Ym

i[l] =

1

√N

N−1

?

n=0

ym

i[n] e

−j2πln

N

(6)

In [11] we derived a closed form expression for the frequency

domain transmission on one subcarrier in fully asynchronous

OFDM systems that particularly includes the effects of ar-

bitrary symbol timings. If we define Zi,q ∈ CM×Kas

the representation of the interference channel on the q-th

subcarrier in frequency domain with the elements Zi,q[m,k] =

f(hm,k,τm,k

d

,ψm,k,N,NCP), we can summarize the trans-

mission of the user symbols Xi,q∈ CK×1to the receivers

Yi,l∈ CM×1as follows:

Yi,l= Zi,l

?

+

Zi,q

+

√Pi,lXi,l

???

MUI

+Vi,l

?

∀q∈D\l

?

√Pi,qXi,q

???

ICI

?

∀q∈D

?

Ui,l

Zi−1,q

√Pi−1,qXi−1,q

???

ISI

?

???

(7)

where we also introduced a transmit power scaling through

the diagonal matrices P ∈ CK×K. In that case we assume

E?XXH?

symbols i. As we can observe in this expression, we now

= ΦXX = IK for all subcarriers q and OFDM

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have a coupling to adjacent subcarriers (ICI) and consecutive

OFDM symbols (ISI) in addition to the coupling between users

(MUI).

III. INTERFERENCE AWARE MULTI-USER EQUALIZATION

The goal of the multi-user equalization is the reconstruction

of the desired user signals Xi,lin Eq. (7) by using the observed

receive vector Yi,l. Therefore we have to employ a receive

filter matrix G ∈ CK×Mthat can be used for the user

equalization on the l-th subcarrier:

?Xi,l=Gi,lYi,l

(8)

Unless the subcarrier and OFDM symbol indices are explicitly

given, we assume the l-th subcarrier and the i-th OFDM sym-

bol for our equations in the following. The error covariance

matrix of the mean squared error (MSE) between the received

and transmitted symbols Φee= E

Φee= (GZ − I)P?ZHGH− I?+ G(ΦUU+ ΦV V)GH

with ΦV V = E?V VH?= σ2

ΦUU=

Zi,qPi,qZH

i,q+

????X −?X

???

2?

is given by:

(9)

vIM and:

?

∀q∈D\l

?

∀q∈D

Zi−1,qPi−1,qZH

i−1,q

(10)

We can derive an expression for the post equalization SINR of

the k-th user after the joint BS signal processing as the ratio

of the desired signal power and the portion of MUI, ICI and

ISI plus noise as:

SINRk=

pk?G

H

kZkZH

k?Gk

?G

H

k

??K

r=1,r?=kprZrZH

r+ ΦUU+ ΦV V

?

?Gk

(11)

where?Gkdenotes the k-th column of GH. Note that Zk

mentioning that the used filter matrix only aims at canceling

the multi-user interference for the desired subcarrier with

the knowledge of the colored noise U (see Eq. (7)). The

interference aware MUD receive filter can then be obtained

by minimizing the sum mean squared error (SMSE):

represents the k-th column of Zi,l in Eq. (7). It is worth

G = argmin

G

{tr{Φee}} = argmin

G

?

E

????X −?X

???

2

2

??

(12)

The minimum argument is obtained by setting the derivative

of the SMSE to zero (e.g. [12]). In this way we obtain:

G = PZHΦ−1

Y Y

(13)

with ΦY Y = E?Y YH?and Y = Yi,las denoted in Eq. (7).

If we use the filter defined in Eq. (13) the post equalization

SINR for the k-th user yields:

SINRk=

ZH

kΦ−1

Y YZk

kΦ−1

1/pk− ZH

Y YZk

(14)

TABLE I: Simulation Parameters

ParameterValue

FFT Size N / Used Subcarrier NSC

CP Length NCP= TCP/TS

System Bandwidth BS= 1/TS

Subcarrier Bandwidth BSC

Carrier Frequency fC

Target SNR γ

UT Maximum Transmit Power pk,max

UT antenna gain ΨUT,AG

BS Noise Figure ΨBS,NF

BS antenna gain ΨBS,AG

Noise Power per Subcarrier σ2

Fast Fading Margin ΨFFM

Inter-cell Interference Margin ΨIIM

256 / 256

18

3.84 MHz

15 kHz

0.8 GHz , 2.68 GHz

20 dB

23 dBm

-1 dBi

4 dB

15 dBi

-132 dBm

2 dB

3 dB

1.69 dB @0.8 GHz

11.93 dB @2.68 GHz

3.86

V

Propagation Env. Factor 10lg(Ψ)

Pathloss Coefficient η

In the SUD case each base station wants to detect the users

in its own serving cell without knowledge of the others. In

this case the serving base station is defined to be the one with

the strongest link between the k-th mobile and all M involved

base stations. The equalizing vector Gkis then obtained by:

IV. NUMERICAL SINR ANALYSIS

In our simulations we evaluate the SINR expression of Eq.

(14) for different channel conditions. Unless it is otherwise

stated the parameter set of Tab. I is used for all simulations,

which is based on a 3GPP/LTE setup and is mainly derived

from [13]. We used the non line of sight models for the

pathloss calculations. Furthermore we assumed an average

building height of 5m and a BS antenna height of 35m for

the rural macro model. The height for a mobile terminal is

assumed to be 1.5m and the street width is chosen as 20m. In

order to vary the channel length L between the simulations we

use a parametrizable exponential channel power delay profile

with the following tap variances:

1

σλ

with?L

the first one by setting the RMS delay spread to σλ= −

Note that we assume a timing synchronization places the

receiver DFT window in such a way that the timing delay of

the shortest link is compensated. Furthermore, power control

is applied such that the strongest link is controlled to achieve

the target SNR γk. The transmit power values in that case can

be obtained by:

γkσ2

?

Gk[m] =

Zk[m]−1

argmax

m

otherwise

???Zm,k??2?

0

(15)

σ2

h[λ]=

e

−(λ−1)

σλ

, λ = 1...L .

(16)

λ=1σ2

h[λ]= 1. We define the last tap to be 10dB below

L−1

ln(0.1).

pk=

v

max

m

|Zm,k|2?

(17)

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0

2

4

6

8

10

12

14

00.20.40.60.81.0

rC / R

Average SINR Loss [dB]

SUD

0/18/10

0/18/1

3.86/77/1

3.86/18/10

3.86/18/1

Fig. 3: SINR loss in a symmetric user configuration scenario

for R=3000m, fC=0.8GHz, γk=20dB and different values for

η, NCP and L (Notation: η / NCP / L)

A. Symmetric User Configurations

In a first simple scenario we let the users move from the cell

center to their serving base stations in a so called symmetric

user configuration setup. We observe the average SINR in the

case that the user circle radius rCis increased. In such scenario

one only needs to distinguish between the diagonal and off-

diagonal elements of the channel matrix which simplifies the

analysis.

The first experiment is intended to get an idea about

the impact of the ISI. Therefore as reference we assume a

system with a CP length that covers the whole cell area

(NCP=

pathloss environment. We chose a cell radius of R=3000m and

a carrier frequency of fC=800MHz as well as an unlimited

transmit power as fixed parameters for this simulation. The

results in terms of the SINR loss w.r.t the target SNR are

depicted in Fig. 3. As it can be observed the signal attenuations

due to the pathloss lead to a decoupling of the channel matrix

between the diagonal an off-diagonal entries. In the cell center

the single user power control scheme is not able to control the

transmit powers to achieve the target SNR since the multi-

user interference is not considered. In a next scenario we

limit the CP as denoted in Tab. I. Now we can see a SINR

degradation at the point where the CP limit is exceeded but

as we can also observe the ISI power is attenuated due to

pathloss. In a more realistic case we assume a channel length

of L = 10 in order to investigate the impact of the ISI

due to the channel decay from the previous OFDM symbol.

Now we notice that the SINR degradation starts at the half

of the CP limit since we have chosen a channel length of

L ≈ NCP/2. For comparison we plotted the case without

pathloss where we see that the SINR degradations due to

ISI power increases monotonically. The maximum cooperation

2R

clightTS= 77) and a flat channel (L = 1) in a

0

2

4

6

8

10

00.20.40.60.81.0

rC / R

Average SINR Loss [dB]

SUD

R=4000m

R=3000m

R=2000m

R=1000m

Fig. 4: SINR loss in a symmetric user configuration scenario

for η=3.86, fC=0.8GHz, NCP=18, γk=20dB for different R

and L

radius rCfor both cases can also be analytically obtained by

solving the requirement that the delay differences must not

exceed the CP limit

w.r.t rC. In that case we get

rC,max

R

≈0.14 for L=10.

It can be observed that the single user detection (SUD)

performance is equal to the MUD performance if the users

are close to their serving base stations where they assumed to

be decoupled.

In a second experiment we show the SINR loss for different

cell radii R and limited transmit power for a flat as well as

for a non flat channel (see Fig. 4). At this we set the cyclic

prefix and pathloss exponent to fixed values as specified in

Tab. I. As we can observe for a cell radius of R=1000m

no ISI occurs since all TDOAs are covererd by the CP. In

the case that we use cell radii larger than R ≥ 2000m the

impact of the ISI increases too till the pathloss lead to the

user decoupling whereas also the ISI is attenuated. It should

be noticed that on the left-hand side the initial SNR loss at

the cell center increases at larger cell radii due to the transmit

power limitation.

?R2+ rCR + r2

C−R+rC≤ clightTCP

rC,max

R

≈ 0.29 for L=1 and

B. Arbitrary User Configurations

In a more interesting scenario occurs we investigate setups

with arbitrary user configurations in which the users are

equally distributed within the entire cooperating cell. The goal

of these simulations is to get a more general performance

prediction and an idea about the SINR behaviour under several

channel conditions. While for the symmetric user configu-

rations the channel matrix is always diagonally dominated

we predominantly expect row dominated channel matrices for

arbitrary user positioning. In the first experiment we analyze

the impact of the CP length. Therefore we used a fixed channel

length of L = 10 as well as a limited transmit power and

pathloss with η=3.86. In Fig. 5 we plotted the cumulative

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0246810 12 1416 1820

SINR [dB]

cdf( SINR )

SUD

Short CP

Long CP

R=1000m

R=3000m

Fig. 5: SINR distribution in a arbitrary user configuration

scenario for η=3.86, L=10, fC=0.8GHz, γk=20dB for different

R and NCP

distribution function of the occurring post equalization SINRs

for different cell radii R, once for a CP length of NCP= 18

and once for NCP=

covered by the CP. As we can observe from the SUD curve,

only 5% of the users have an SINR loss lower than 3dB. In the

case of MUD 30% of the users have a 3dB SINR loss if we

use the long CP. By using the short CP we get a degradation

that is caused by additional ISI.

In a second experiment we used basically the same param-

eter set as in the former setup but with a fixed CP length

of NCP=18. In Fig. 6 we compare the SINR results for

different carrier frequencies fCand different cell radii R. As

expected we get better SINR results in the case of lower carrier

frequency since we have less pathloss attenuations. As we can

observe for R=1000m, in the MUD case 35% of the users have

only a 3dB loss for fC=2.68GHz and 40% for fC=0.8GHz

respectively. For R=3000m these values change to 10% for

fC=2.68GHz and 20% for fC=0.8GHz.

2R

clightTS, where the entire cell area is

V. CONCLUSION

In this paper we presented an analysis of the impact of inter-

symbol interference in CP limited network MIMO OFDM

systems with large cell radii and under pathloss conditions.

Based on a derived expression for the post equalization SINR

for transmissions with arbitrary symbol timings we investi-

gated the combined effect of ISI and pathloss in a simple

symmetric user configuration scenario as well as in a scenario

with arbitrary user positioning. We have shown that due to

the pathloss the ISI is attenuated such that it has only an

impact in certain regions where users are still coupled via

the channel matrix. Therefore we analyzed channel effects by

using system parameters which are similar to state-of-the-art-

wireless communication systems.

0

0.2

0.4

0.6

0.8

1.0

02468 10121416 1820

SINR [dB]

cdf( SINR )

SUD

R=3000m

R=1000m

fC=0.8 GHz

fC=2.68 GHz

Fig. 6: SINR distribution in a arbitrary user configuration

scenario for η=3.86, L=10, NCP=18, γk=20dB for different

R and fC

VI. ACKNOWLEDGEMENT

The authors acknowledge the excellent cooperation with all

project partners within the EASY-C project and the support

by the German Federal Ministry of Education and Research

(BMBF).

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