Rohling H. (Ed.) OFDM: Concepts for Future Communication Systems

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6 1 Introduction

multi-path delay τ

max

in order to avoid ISI completely which could occur in multi-

path channels in the transition interval between two adjacent OFDM symbols [17].

(t)=

⎧

⎨

⎩

j2πkΔft

∀t ∈ [−T

]

0 ∀t ∈ [−T

]

(1.5)

The guard interval is directly removed in the receiver after the time synchro-

nization procedure. From this point of view, the guard interval is a pure system

overhead. The total OFDM symbol duration is therefore T = T

+ T

.Itisan

important advantage of the OFDM transmission technique that ISI can be avoided

completely or can be reduced at least considerably by a proper choice of OFDM

system parameters.

The orthogonality of all subcarrier signals is completely preserved in the receiver

even in frequency-selective radio channels, which is an important advantage of

OFDM. The radio channel behaves linearly and in a short time interval of a few

OFDM symbols even time-invariant. Hence, the radio channel behavior can be

described completely by a Linear and Time Invariant (LTI) system model, charac-

terized by the impulse response h(t). The LTI system theory gives the reason for

this important system behavior that all subcarrier signals are orthogonal in the re-

ceiver even when transmitting the signal in frequency-selective radio channels. All

complex-valued exponential signals (e.g., all subcarrier signals) are eigenfunctions of

each LTI system and therefore eigenfunctions of the considered radio channel, which

means that only the signal amplitude and phase will be changed if a subcarrier signal

is transmitted over the linear and time-invariant radio channel.

The subcarrier frequency is not aﬀected at all by the radio channel transmission,

which means that all subcarrier signals are even orthogonal in the receiver and at

the output of a frequency-selective radio channel. The radio channel disturbs only

amplitudes and phases individually, but not the subcarrier frequency of all received

sub-channel signals. Therefore all subcarrier signals are still mutually orthogonal in

the receiver. Due to this important property, the received signal which is superim-

posed by all subcarrier signals can be split directly into the diﬀerent sub-channel

components by a Fourier transform and each subcarrier signal can be restored by a

single-tap equalizer and demodulated individually in the receiver.

At the transmitter side, each subcarrier signal is modulated independently and

individually by the complex valued modulation symbol S

n,k

, where the subscript

n refers to the time interval and k to the subcarrier signal number in the consid-

ered OFDM symbol. Thus, within the symbol duration time interval T the time-

continuous signal of the n-th OFDM symbol is formed by a superposition of all N

simultaneously modulated subcarrier signals.

(t)=

N−1



k=0

n,k

(t − nT ) (1.6)

The total time-continuous transmit signal consisting of all OFDM symbols se-

quentially transmitted on the time axis is described by Eq. (1.7):

1.2 Basics of the OFDM Transmission Technique 7

s(t)=

∞



n=0

N−1



k=0

n,k

j2πkΔf(n−nT )

rect



2(t − nT )+T

− T



(1.7)

The analytical signal description shows that a rectangular pulse shaping is applied

for each subcarrier signal and each OFDM symbol. Due to the rectangular pulse

shaping, the spectra of all the considered subcarrier signals are sinc-functions which

are equidistantly located on the frequency axis, e.g., for the k-th subcarrier signal,

the spectrum is described in Eq. (1.8):

(f)=T · sinc [πT(f −kΔf)] (1.8)

The typical OFDM spectrum shown in Fig. 1.6 consists of N adjacent sinc-

functions, which are shifted by Δf in the frequency direction.

()Sf

Figure 1.6: OFDM spectrum which consists of N equidistant sinc functions

The spectra of the considered subcarrier signals overlap on the frequency axis, but

the subcarrier signals are still mutually orthogonal, which means the transmitted

modulation symbols S

n,k

can be recovered by a simple correlation technique in each

receiver if the radio channel is assumed to be ideal in a ﬁrst analytical step:



(t)g

∗

(t)dt =

⎧

⎨

⎩

1 k = l

0 k = l

(1.9)

n,k



(t)g

∗

(t)dt =



(t)e

−j2πkΔft

dt (1.10)

where g

∗

(t) is the conjugate complex version of the subcarrier signal g

(t). Eq. (1.11)

shows the correlation process in detail:

Corr =



(t)g

∗

(t)dt =



N−1



m=0

n,m

(t)g

∗

(t)dt

N−1



m=0

n,m



(t)g

∗

(t)dt =

N−1



m=0

n,m

m,k

= S

n,k

(1.11)

In practical applications, the OFDM transmit signal s

(t) is generated as a time-

8 1 Introduction

discrete signal in the digital baseband. Using the sampling theorem while considering

the OFDM transmit signal inside the bandwidth B = NΔf, the transmit signal must

be sampled with the sampling interval Δt =1/B =1/N Δf . The individual samples

of the transmit signal are denoted by s

n,i

,i=0, 1,...,N − 1 and can be calculated

as follows (see Eq. (1.7)):

s(t)=

N−1



k=0

n,k

j2πkΔft

s(iΔt)=

N−1



k=0

n,k

j2πkΔfiΔt

n,i

N−1



k=0

n,k

j2πik/N

(1.12)

Equation (1.12) above exactly describes the inverse discrete Fourier transform

(IDFT) applied to the complex valued modulation symbols S

n,k

of all subcarrier

signals inside a single OFDM symbol.

Figure 1.7: OFDM system structure in a block diagram

The individually modulated and superimposed subcarrier signals are transmitted

in a parallel way over many narrowband sub-channels. Thus, in each sub-channel,

the symbol duration is large and can be chosen much longer than the maximum

multi-path delay of the radio channel. In this case, each sub-channel has the property

to be frequency non-selective.

Figure 1.7 shows the general OFDM system structure in a block diagram. The

basic principles of the OFDM transmission technique have already been described

in several publications, e.g., [19], [27]. In the very early and classical multi-carrier

system considerations like [20], [26], narrowband signals have been generated in-

dependently, assigned to various frequency bands, transmitted, and separated by

1.2 Basics of the OFDM Transmission Technique 9

analog ﬁlters at the receiver. The new and modern aspect of the OFDM trans-

mission technique is that the various subcarrier signals are generated digitally and

jointly by an IFFT in the transmitter and that their spectra strongly overlap on

the frequency axis. As a result, generating the transmit signal is simpliﬁed and the

bandwidth eﬃciency of the system is signiﬁcantly improved.

The received signal is represented by the convolution of the transmitted time

signal with the channel impulse response h(t) and an additive white Gaussian noise

term:

(t)=s

(t) ∗h

(t)+n

(t) (1.13)

Due to the assumption that the coherence time T

will be much larger than

the symbol duration T

, the received time-continuous signal r

(t) can be separated

into the orthogonal subcarrier signal components even in frequency-selective fading

situations by applying the correlation technique mentioned in Eq. (1.10):

n,k



(t)e

−j2πkΔft

dt (1.14)

Equivalently, the correlation process at the receiver side can be applied to the time-

discrete receive signal at the output of an A/D converter and can be implemented

as a DFT, which leads to Eq. (1.15):

n,k

N−1



i=0

n,i

−j2πik/N

(1.15)

In this case r

n,i

= r

(iΔt) describes the i-th sample of the received time-continuous

base-band signal r

(t)andR

n,k

is the received complex-valued symbol at the DFT

output of the k-th subcarrier.

If the OFDM symbol duration T is chosen much smaller than the coherence time

of the radio channel, the time-variant transfer function of the radio channel

H(f,t) can be considered to be constant within the time duration T of each mod-

ulation symbol S

n,k

for all subcarrier signals. In this case, the eﬀect of the radio

channel in multi-path propagation situations can be described analytically by only a

single multiplication of each subcarrier signal g

(t) with the complex transfer factor

n,k

= H(kΔf, nT). As a result, the received complex valued symbol R

,k at the

DFT output can be described analytically as follows:

(t)=s

(t) ∗h

(t)+n

(t)

n,i

= s

n,i

∗ h

n,i

+ n

n,i

n,k

= S

n,k

· H

n,k

+ N

n,k

(1.16)

where N

n,k

describes an additive noise component for each speciﬁc subcarrier gen-

erated in the radio channel. Eq. (1.16) shows the most important advantage of

applying the OFDM transmission technique in practical applications. It encom-

passes the complete signal transfer situation of the OFDM block diagram including

IDFT, guard interval, D/A conversion, up- and down-conversion in the RF part,

frequency-selective radio channel, A/D conversion and DFT process in the receiver,

10 1 Introduction

neglecting non-ideal behavior of any system components.

The transmitted Symbol S

n,k

can be recovered, calculating the quotient of the

received complex valued symbol and the estimated channel transfer factor

n,k

− N

n,k

= R

n,k

(1.17)

It is obvious that this one tap equalization step of the received signal is much easier

compared to a single carrier system for high data rate applications. The necessary

IDFT and DFT calculations can be implemented very eﬃciently using the Fast-

Fourier-Transform (FFT) Algorithms such as Radix 2, which reduces the system

and computation complexity even more. It should be pointed out that especially the

frequency synchronization at the receiver must be very precise in order to avoid any

Inter Carrier Interferences (ICI). Algorithms for time and frequency synchronization

in OFDM based systems will be described in a later chapter of this book.

Besides the complexity aspects, another advantage of the OFDM technique lies in

its high degree of ﬂexibility and adaptivity. Division of the available bandwidth into

many frequency-nonselective subbands gives additional advantages for the OFDM

transmission technique. It allows a subcarrier-speciﬁc adaptation of transmit pa-

rameters, such as modulation scheme (PHY mode) and transmit power (e.g., Water

Filling) in accordance to the observed and measured radio channel status. In a multi-

user environment the OFDM structure oﬀers additionally an increased ﬂexibility for

resource allocation procedures as compared to SC systems [21]. The important

system behavior that all subcarrier signals are mutually orthogonal in the receiver

makes the signal processing and the equalization process realized by a single tap

procedure very simple and leads to a low computation complexity.

1.3 OFDM Combined with Multiple Access

Schemes

A very high degree of ﬂexibility and adaptivity is required for new mobile commu-

nication systems and for the 4G air interface. The combination between multiple

access schemes and OFDM transmission technique is an important factor in this

respect. In principle, multiple access schemes for the OFDM transmission technique

can be categorized according to OFDM-FDMA, OFDM-TDMA, OFDM-CDMA and

OFDM-SDMA [1] [22]. Clearly, hybrid schemes can be applied which are based on

a combination of the above techniques. The principles of these basic multiple access

schemes are summarized in Fig. 1.8, where the time-frequency plane is depicted and

the user speciﬁc resource allocation is distinguished by diﬀerent colors.

These access schemes provide a great variety of possibilities for a ﬂexible user spe-

ciﬁc resource allocation. In the following, one example for OFDM-FDMA is brieﬂy

sketched (cf. [7]). In the case that the magnitude of the channel transfer function

is known for each user the subcarrier selection for an OFDM-FDMA scheme can be

processed in the BS for each user individually which leads to a multi user diversity

(MUD) eﬀect. By allocating a subset of all subcarriers with the highest SNR to each

user the system performance can be improved. This allocation technique based on

1.3 OFDM Combined with Multiple Access Schemes 11

Time

Subcarriers

Code, Space

FDMA

TDMA

CDMA, SDMA

Time

Subcarriers

Code, Space

FDMA

TDMA

CDMA, SDMA

Figure 1.8: OFDM transmission technique and some multiple access schemes

the knowledge of the channel transfer function shows a large performance advan-

tage and a gain in Quality of Service (QoS). Nearly the same ﬂexibility in resource

allocation is possible in OFDM-CDMA systems. But in this case the code orthog-

onality is destroyed by the frequency-selective radio channel resulting in multiple

access interferences (MAI), which reduces the system performance.

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12 1 Introduction

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H. Rohling (ed.), OFDM: Concepts for Future Communication Systems, 15

Signals and Communication Technology, DOI: 10.1007/978-3-642-17496-4_2,

2 Channel Modeling

M. Narandžić, A. Hong, W. Kotterman, R. S. Thomä, Ilmenau Univer-

sity of Technology, Germany

L. Reichardt, T. Fügen, T. Zwick, Karlsruhe Institute of Technology

(KIT), Germany

In the OFDM signalling concept, the wide-band radio-communication channel is

eﬀectively utilized as a collection of narrow-band channels. Basic system parameters

like the number of subcarriers N and symbol duration T are selected to mitigate

the key channel impairments: Inter-Symbol-Interference (ISI) induced by frequency

selectivity and loss of subcarrier orthogonality due to time selectivity [51]. There-

fore, a proper channel model is required both for system design and for performance

evaluation. Additionally, when Channel-State-Information (CSI) is available during

system operation, transmission characteristics, such as the signal constellation or the

allocated power, could be adaptively adjusted at transmitter per subcarrier in or-

der to maximize total throughput. Further improvements of the spectral eﬃciency

could be obtained by simultaneous transmission and/or reception from/by multi-

ple antenna elements. Additionally to time and frequency, this concept known as

MIMO (Multiple-Input-Multiple-Output) exploits the spatial propagation dimension

or, more speciﬁc, multiplicity of energy propagation paths. Since the reachable spec-

tral eﬃciency is tightly related to the signal correlation across the antenna array [15],

the proper representation of correlation levels becomes essential for the analysis of

MIMO systems. In order to obtain an antenna-independent representation of the

channel that implicitly comprises correlation properties, geometry-based models are

generally used.

In this chapter, the necessary concepts for representation of the multidimensional

radio-channel are summarized. Data collected during multidimensional channel

sounding and post-processed by high-resolution parameter estimation algorithms of-

fer the most detailed insight into radio-propagation mechanisms. In that way, joint

space-time-frequency representations being consistent with measurements can be ob-

tained (Section 2.1). On the other hand, when an appropriate description of the EM

environment is available (in the form of databases deﬁning geometry and material

properties), the EM ﬁeld could be predicted by use of the Geometrical or Uniform

Theory of Diﬀraction (GTD/UTD), as explained in Section 2.2. Note that for tuning

and veriﬁcation of ray tracing/launching procedures, sounding experiments are still

required. For system design and performance evaluation, site-independent modeling

with lower complexity is preferred. For this purpose, stochastic characterization of

diﬀerent radio-environment classes could be combined with geometry based propa-

gation aspects. This results in the class of Geometry-based Stochastic (GbS) channel

models that are described in Section 2.3. In order to properly reproduce space-time

channel evolution, this class of empirical models uses stochastic characterization of