This document provides an overview of OFDM and the downlink physical layer design in LTE. It discusses why OFDM is necessary for high data rates in LTE, describing how OFDM avoids intersymbol interference through the use of multiple orthogonal subcarriers. It then covers OFDM signal structure and modulation, including the transmitter and receiver designs based on the inverse discrete Fourier transform and discrete Fourier transform. The document also introduces the concept of a guard interval to eliminate intersymbol interference and provides a matrix representation of multicarrier systems using cyclic prefix and DFT/IDFT.

1. Seminar
Ausgewählte Kapitel der Nachrichtentechnik, WS 2009/2010
LTE:
Der Mobilfunk der Zukunft
OFDM and Downlink Physical
Layer Design
Shahram Zarei
11. November 2009
Abstract In the last years the tendency to have higher data rates in cellular mo-
bile phone networks, has been growing very rapidly. LTE (Long Term Evolution)
is the current standard, which provides very high data rates having Orthogonal
Frequency Multiplexing (OFDM) as a key feature. In this work rst is the ques-
tion why OFDM is neccessary for LTE downlink is answered and then topics like
OFDM receiver and transmitter structures and OFDM parameter dimensioning
are introduced. In the second part the physical layer in the downlink is analyzed.
Signal structure in the time domain, resource management, signal generation chain
and Multiple-Input Multiple-Output (MIMO) technique are topics of the second
part.
1 Introduction
Digital cellular communications beginning with GSM called as 2nd generation were serving
only speech communication at the very early versions. Adding GPRS and EDGE as data
packet services gaining higher spectral eciency were the rst steps to make cellular networks
capableoftransportingdatapackets. Ifwelookatthedevelopmentofthelatergenerationslike
UMTS (3G) or extensions like EGDE, EGPRS, EGPRS2 (extension of EDGE with 16QAM
instead of 8PSK and Turbo encoder), HSDPA and HSUPA, we will observe that the main
trendistoachievehigherdatarates. LTE(LongTermEvolution)isthecurrentelementinthe
development chain of digital cellular communication systems with following key design targets:
• Signicantly higher peak data rates than older standards (e.g. 100 Mbps in 20 MHz
bandwidth Downlink and 50 Mbps in 20 MHz bandwidth Uplink)

2. 2 Shahram Zarei
• Scalable bandwidth for more spectrum exibility: 1.25, 1.6, 2.5, 5, 10, 15, 20 MHz
• MIMO (Multiple-Input Multiple-Output)
• Low latency (round trip delay 10 ms)
• Packet oriented data transmission (all IP network)
• High UE (User Equipment) mobility conditions: Up to 350 or even 500 km/h
2 OFDM
2.1 Why OFDM?
One of the central goals of LTE is the signicantly higher peak data rate compared to the
older standards like GSM/EDGE or UMTS. For example in the downlink, employing MIMO
(Multiple-Input Multiple-Output), data rates of up to 100 Mbps are reachable.
Inasinglecarriersystem,wheretheinformationismodulatedonasinglecarrier,thedatasym-
bolswillhaveveryshortdurationcomparedtothedelaycausedbythemultipathpropagation,
which causes ISI (intersymbol interference).
This eect can be seen in Fig. 1.
Figure 1: Single carrier transmission with short symbols (according to [3])
AveryintuitivesolutiontoavoidISIwouldbeusinglongersymbols. Butifwewanttotransfer
the same data rate, we have to use another resource dimension, for example frequency. This
leads to the concept of multicarrier communication, which can be seen in Fig. 2.
The concept of multicarrier communication is based on the fact, that data transmission occurs
in two resource dimensions: time and frequency. Data is transported in form of symbols in
the time domain and each symbol in the time domain consists of several subcarriers in the
frequency domain, each carrying part of the information to be transported.

3. OFDM and Downlink Physical Layer Design 3
Figure 2: Multicarrier transmission with long symbols (according to [3])
AswecanseeintheFig.2,usinglongersymbolsISIcanbereducedenormously. Inmulticarrier
systemsthemaindatastreamissplittedintodatastreamswithlowerbit-ratesandthuslonger
symbols. Eachsub-datastreamismodulatedonasubcarrier. Duetothefactthatdatasymbols
are longer, ISI would be much smaller. One very interesting eect in multicarrier systems is
thatthewholefrequencybandisdividedintomuchsmallersubbandsandthereforethechannel
in each subband can be considered as a at fading channel because of a very small bandwidth
which means the channel behaves as a constant complex factor. In this case the channel can
be equalized in the receiver by simply multiplying the samples by the inverse of the DFT
coecients of the channel impulse response corresponding to each subband. This eect can be
seen in Fig. 3.
2.2 OFDM signal structure
OFDM stands for Orthogonal Frequency Division Multiplexing and is a special form of multi-
carrier modulation, where the subcarriers are orthogonal to each other.
The OFDM signal in the time domain has the form:
SOFDM(t) =
1
√
T
N2
n=0
ak[n]ej2πnt
+
1
√
T
N−1
n=N−N1
ak[n]ej2πnt
, in the kth interval : kT t (k+1)T
(1)
Note, that T is the duration of the modulation symbol and inverse of subcarrier spacing.
We can formulate orthogonality as following:
(k+1)T
kT
e−j2π(fn−fν )t
dt = 0, for n = ν (2)

4. 4 Shahram Zarei
Figure 3: Equalization in multicarrier systems (according to [3])
AninterestingpointconcerningmodulationinOFDMsystemsis,thateachsubcarriercanhave
an individual modulation scheme and an individual data rate, depending on the signal quality
(e.g. signal-to-noise ratio (SNR) ) in the considered sub-band. That means there are some
algorithms, which can dedicate part of the whole data rate on each carrier, depending on its
SNR: The bigger the SNR, the bigger the data rate dedicated.
Due to the orthogonality of the subcarriers to each other, there isn't any problem regarding
theexistingoverlappingpartbetweenthesubbandsandthereforethereisn'tanyneedforsome
guard-bandsbetweensubcarriers. ThismakestheOFDMconceptmoreecientconcerningthe
bandwidth eciency.
2.3 OFDM signal in the frequency domain
Inthefollowingdiscussionitisassumed,thatthesymbolsarei.i.d. (independentandidentically
distributed).
We assume, that the modulation pulse is a rectangular pulse, then we have:
grect(t) =



1/T, for 0 ≤ t ≤ T
0, otherwise
(3)
The power spectral density of a rectangular pulse is:
|Grect(f)|2
=
T sin2
(πfT)
(πfT)2
(4)
The rectangular pulse in the time domain and its power spectral density can be seen in Fig. 4
and Fig. 5.

5. OFDM and Downlink Physical Layer Design 5
Figure 4: Rectangular pulse in the time domain (according to [3])
Figure 5: Power spectral density of the rectangular pulse (according to [3])
Due to the fact that symbols are i.i.d., the power spectral density of the OFDM signal will be
the sum of shifted versions of the power spectral density of the rectangular pulse:
GOFDM(f) =
N2
n=0
|Grect(f −
n
T
)|2
+
N−1
n=N−N1
|Grect(f −
n
T
)|2
(5)
In Fig. 6 the PSD (power spectral density) of an OFDM signal can be seen.
Figure 6: Power spectral density of an OFDM signal (according to [3])
2.4 OFDM transmitter based on IDFT
The early versions of multicarrier modulators in the 60ies were using oscillator banks. Inverse
discreteFouriertransform(IDFT)istodaythestandardmethodforgeneratingOFDMsignals.
If the number of subcarriers is a power of two, then a fast Fourier transform (FFT) can be
used, which is a computationally ecient (fast) variant of the DFT.

6. 6 Shahram Zarei
In the Fig. 7 the general structure of an OFDM transmitter based on the IDFT can be seen.
At the very rst step data symbols coming from earlier modules, e.g. channel encoder, are
parallelized using a seria-to-parallel converter. This is done because of the fact that IDFT
works blockwise. After serial-to-parallel converter, comes the mapping module, which maps
the incoming symbols on complex-valued samples. The mapping schemes in LTE downlink
are QPSK, 16QAM and 64QAM. Then the samples are fed into the IDFT module to get the
time domain samples. An interesting point here is, that usually some carriers are left unused,
which is done by simply putting zeros into the IDFT at the corresponding subcarrier position.
Using less carriers has the advantage that the ltering of the OFDM shoulders is much easier.
After the IDFT there is a parallel-to-serial converter followed by the DAC (Digital-to-Analog
Converter), which converts the signal from digital to analog form. In order to lter out the
shouldersoftheOFDMsignalwhichproduceout-of-bandemissions,thereisaspectralshaping
lter after the DAC. Finally an upconverter module consisting of a mixer and a local oscillator
mixes the generated signal from baseband into the RF domain.
Figure 7: OFDM transmitter based on IDFT (according to [3])
2.5 OFDM receiver based on DFT
In the OFDM receiver, after mixing down in the baseband or low IF, the analog signal is
rst matched ltered and then converted to digital samples with an ADC (Analog-to-Digital
Converter). A very important module here is the synchronization block, which synchronizes
the local oscillator frequency and clock frequency of the ADC using information gained from
the received samples. Channel estimation and frame synchronization are further tasks of this
module. Asweseelater,frequencysynchronizationisamustinOFDMsystems,becauseoftheir
very high sensitivity against frequency shifts. The samples are then parallelized and processed
blockwise by the DFT module. Assuming a frequency-selective channel the equalization in the
receiverissimplydonebymultiplyingthesampleswiththeinverseofthecorrespondingchannel
coecients. This is one of the most important attractions of OFDM. After equalization there

7. OFDM and Downlink Physical Layer Design 7
is a decision module, which is a hard desicion slicer here. In scenarios with channel coding,
a soft decision is often used, which works with LLRs (Log Likelihood Ratios). A demapping
module recovers the bits from symbol delivered by the decision module. The bits are nally
converted from parallel to serial form. Fig. 8 shows the structure of an OFDM receiver based
on the DFT.
Figure 8: OFDM receiver based on DFT (according to [3])
2.6 Concept of Guard Interval
As we already know, if the data symbols are long enough compared to the longest delay in the
channel, then the ISI would be small but it has not completely vanished. In order to eliminate
the ISI properly, one can use some kind of guard space between adjacant symbols. There are
several methods to do so, for example putting zeros in the guard space. However the most
popularwayistouseaCyclicPrex(CP),whichmeansthelastsamplesoftheOFDMsymbol
are copied and pasted into the front of it. The number of copied samples or the length of the
Guard Interval are important design parameters, which depend on the channel characteristics
and the largest delay existing in the channel. Fig. 9 shows the functionality of the Guard
Interval.
The Cyclic Prex has following properties:
• Converts linear convolution to circular convolution
• Orthogonality is maintained

8. 8 Shahram Zarei
Figure 9: Guard interval in multicarrier systems (according to [3])
2.7 Matrix representation of multicarrier systems based on DFT and
CP
The model discussed here consists of IDFT in the transmitter, the channel which is described
by a channel coecient matrix, additive noise, which for simplicity is not considered here, and
a DFT block at the receiver.
a [n] are the complex-valued samples from the mapper, which are processed blockwise by the
IDFT module. The mathematical description of the whole process is as follows:
The vector with data symbols is given by:
a = [a[1], ..., a[N]]T
. (6)
After the IDFT we get the samples in the time domain:
x = [x[1], ..., x[N]]T
=
1
√
N
WH
a. (7)
The DFT matrix is dened as:
W =







1 1 · · · 1
1 ω · · · ωN−1
... ... ... ...
1 ωN−1
· · · ω(N−1)(N−1)







, (8)
with ω = e−j 2π
N . (9)

9. OFDM and Downlink Physical Layer Design 9
Appending the Cyclic Prex with the length of N0 to the samples results in:
xcp = [x[N − N0 + 1], ..., x[N], x[1], ..., x[N]]T
, (10)
and the relation between the signal at the input and the output of the channel is:
ycp = Hxcp, (11)
where H is the channel matrix and qh the order of the channel impulse response:
H =












h[0] 0 0 · · · 0 0
h[1] h[0] 0 · · · 0 0
h[2] h[1] h[0] · · · 0 0
... ... ... · · ·
... ...
h[qh] h[qh − 1] h[qh − 2] · · · h[0] 0
0 h[qh] h[qh − 1] · · · h[1] h[0]












. (12)
After the removal of the Cyclic Prex we get:
y = Hcx, (13)
with Hc as the cyclic channel matrix:
Hc =












h[0] 0 h[qh] · · · h[2] h[1]
h[1] h[0] 0 · · · h[3] h[2]
h[2] h[1] h[0] · · · h[4] h[3]
... ... ... · · ·
... ...
h[qh] h[qh − 1] h[qh − 2] · · · h[0] 0
0 h[qh] h[qh − 1] · · · h[1] h[0]












. (14)
After the DFT we get:
a =
1
√
N
Wy =
1
√
N
WHcx =
1
√
N
WHc
1
√
N
WH
a =
1
N
WHcWH
a. (15)
A very interesting property of the cyclic channel matrix is:
WHcWH
= WWH
Λ, (16)
which leads to:
a =
1
N
WWH
Λa =
1
N
NIΛa = Λa, (17)
or simply:

10. 10 Shahram Zarei
a = Λ a, (18)
Λ = diag(λ1, λ2, . . . , λN ), (19)
λi =
qh
µ=0
h[µ] e−j
2πµ(i−1)
N , with i ∈ {1, 2, ..., N}. (20)
The λi are the DFT coecients of the channel or the eigenvalues of the channel coecient
matrix.
At this point we have derived a very important feature of ODFM with Cyclic Prex, namely
that the communication chain reduces to a diagonal matrix with the DFT of the channel
coecients as diagonal elements, which means, if we want to equalize the channel, we have to
multiply by the inverse of this diagonal matrix, which is also a diagonal matrix with inverted
diagonal elements. In other words, we can simply multiply each output of the DFT module by
its inverted channel coecient.
2.8 OFDM system parameter dimensioning
AnOFDMsystemhassomekeyparameters, whichhavetobedesignedcorrectlydependingon
the considering scenario. The design parameters of OFDM are:
• ∆f: Subcarrier spacing
⇒ Demand in order to keep Doppler caused ICI low: ∆f fdmax , fdmax : Max. Doppler
shift
• TCP : Length of the Cyclic Prex
⇒ To prevent ISI: TCP ≥ Td, Td: Length of the channel impulse response
⇒ Demand for high spectral eciency: TCP T, T: OFDM symbol duration
• N: Number of subcarriers
⇒ N B∆f, B: OFDM signal bandwidth
The physical layer parameters of the LTE downlink are summarized in Fig. 10.
The technology used in LTE downlink is OFDMA, which stands for Orthogonal Frequency
Division Multiple Access. In OFDMA, in contrast to OFDM, dierent subcarriers can be
assigned to dierent users.

11. OFDM and Downlink Physical Layer Design 11
Figure 10: LTE downlink physical layer parameters
2.9 OFDM drawbacks
OFDM has two main disadvantages:
• PAPR (or crest factor): Stands for Peak-to-Average Power Ratio and can be formulated
as follows:
PAPR =
max{|x[n]|2
}
E{|x[n]|2}
(21)
Power ampliers for signals with high PAPR should be highly linear over a broad range.
This makes the transmitters expensive. There are several solutions to overcome this
problem. The rst and the simplest one is to use power ampliers with large back o
e.g. using a 1 kW amplier for 100 W output power. This, however, is quite inecient.
The second solution is to use algorithms, which reduce the PAPR without disturbing the
main information content of the signal. Of course there is a certain signal processing
complexity with PAPR reduction algorithms.
• Sensitivity to frequency osets: The second disadvantage of OFDM systems is their high
sensitivitytofrequencyosets. ThiseectcanbeseeninFig.11and12. Aswecanseein
Fig.12,ifthereisafrequencyosetbetweentransmitterandreceiver,thesubcarriersare
notorthogonaltoeachotheranymoreandthiscausesICI(InterCarrierInterference). To
avoidthisthelocaloscillatorinthereceivershouldbewellsynchronizedtothetransmitter
frequency.

12. 12 Shahram Zarei
Figure 11: Zero ICI
Figure 12: Nonzero ICI if some frequency oset is present

13. OFDM and Downlink Physical Layer Design 13
3 Physical layer in downlink
3.1 LTE signal in time domain: Generic frame structure
The LTE signal in time domain is based on frames, which are 10 ms long and consist of 10
subframes each of 1 ms duration. The subframes are divided further into two slots each 0.5 ms
long. In each slot 7 or 6 OFDM symbols are contained depending on whether normal or short
Cyclic Prex is used, cf. [1]. The time domain frame structure of the LTE downlink can be
seen in the Fig. 13.
Figure 13: LTE downlink signal structure in time domain
3.2 Resource management in LTE downlink physical layer
LTE uses a three dimensional space to manage the resources time, frequency and space (an-
tennas). In Fig. 14 only time and frequency dimensions are shown. The smallest unit is the
so-called Resource Element (RE), which consists of a time interval of duration of one OFDM
symbol and one subcarrier. Seven OFDM symbols (in case of normal CP length) or 6 symbols
(in case of long CP length) build a time slot. The area consisting of 12 subcarriers (180 kHz)
and one time slot is called Resource Block and contains 12x7=84 Resource Elements in case of
normal Cyclic Prex.
Figure 14: Resource management in LTE downlink

14. 14 Shahram Zarei
3.3 Downlink reference symbols
In each Resource Block four so-called reference symbols are transmitted. The position of the
reference symbols can be seen in the Fig. 15, cf. [7]. The main task of the reference symbols
can be summarized as given below:
• Cell search and initial aquisition
• Channel estimation
• Coherent detection
• Channel quality estimation
Figure 15: Reference symbols in LTE downlink
3.4 LTE downlink signal generation chain
In Fig. 16 the components of the downlink physical layer signal generation chain can be seen.
First a 24 bit CRC (Cyclic Redundancy Check) eld is introduced to detect errors in the
receiver. After the CRC module comes a turbo encoder as forward error correction (FEC)
channel coder. The LTE downlink turbo encoder has R = 1/3 as basic code rate and is (can
be) with puncturing. There is an HARQ module following the turbo encoder. HARQ stands
for Hybrid Automatic Repeat Request and is a mechanism based on stop and wait ARQ which
transmits the packets again in case of errors detected by the CRC. In order to achieve more
coding gain, a scrambling module based on a 31 bit Gold sequence is used after the HARQ
module. The nal module is the modulator, which can use QPSK, 16QAM or 64QAM in LTE
downlink, cf. [1].

15. OFDM and Downlink Physical Layer Design 15
In LTE there is the possibility to use MIMO. In this case an antenna mapping module decides
on which antenna the packets are to be sent. In LTE downlink up to 4 antennas can be used.
Finally the resource management module selects the appropriate resources, which are in LTE
time slots and subcarriers to transmit the packet.
Figure 16: Signal generation chain in LTE downlink physical layer
3.5 MIMO in LTE
OneoftheimportanttechnologiesusedinLTEdownlinkisMIMO.MIMOstandsforMultiple-
Input Multiple-Output. In MIMO systems, independent parts of a data block are sent over
uncorrelated antennas (with a minimum distance greater than e.g. 10λ to ensure uncorrelated-
ness). The number of transmit and receive antennas can vary from 1 to 4. For example a 4x4
system MIMO has four TX and four RX antennas. MIMO exploits the independency of the
scattered signal components in a radio channel und makes some data rate gain possible. The
greater the independencies between dierent paths in a channel with scattering, the higher the
achieved data rate gains.
Example: Downlink peak data rates (64QAM):
• MIMO (2x2): 172.8 Mbps
• MIMO (4x4): 326.4 Mbps

16. 16 Shahram Zarei
Figure 17: MIMO in LTE
3.6 Summary
• LTE is a high performance technology for mobile broadband services.
• OFDMAisthekeycoreoftheLTEdownlinkphysicallayerandmakessignicantlyhigher
data rates possible.
• With higher order modulation schemes (up to 64QAM), greater spectrum eciency is
achievable .
• With the MIMO feature, the scattering behavior of the channel is used to increase data
rate even more.
References
[1] E. Dahlman, S. Parkvall, J. Sköld, P. Beming: 3G Evolution: HSDPA und LTE for
mobile broadband, 2007, Academic Press.
[2] S. Sesia, I. Touk, M. Baker: LTE - The UMTS Long Term Evolution: From Theory
to Practice, 2009, John Wiley Sons.
[3] W.Koch: LecturescriptFundamentalsofMobileCommunications,2008,University
of Erlangen-Nuremberg.
[4] R. Fischer, J. Huber: Skriptum zur Vorlesung Digitale Übertragung, 2009, Univer-
sity of Erlangen-Nuremberg.
[5] U. Barth: 3GPP Long Term Evolution / System architecture evolution overview,
2006, Alcatel.
[6] J. Zyren, W. McCoy: Overview of the 3GPP Long Term Evolution physical layer
[7] E. Seidel, V. Pauli: Nomor 3GPP Newsletter, Overview LTE PHY, Nomor Research