ECE 5664 Project: Orthogonal Frequency Division Multiplexing OFDM!: "utorial an# $nalysis
Dece%&er ''( )**'
Erich Cos&y
[email protected] Virginia Tech. Northern Virginia Center
1
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OFDM OFDM Overvie1// Overvie1////// ///////// ///////// ///////// ////////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ////////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// /////////// ////////////// /////////// /// 0
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Preli%in Preli%inary ary Concepts/// Concepts/////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ////////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ////////// ///////// ///////// ///////// ///////// /////////// //////// 4
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De2initi De2inition on o2 Carriers/ Carriers///// //////// //////// ///////// ///////// ///////// ///////// ///////// ////////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ////////// ///////// ///////// ///////// ///////// ///////////// /////////// /// 5
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-eceptio -eception n an# an# De%o#ul De%o#ulatio ation/// n/////// //////// //////// ///////// ///////// ///////// ////////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ////////// ///////// ///////// ///////////// ///////////// ///// 3
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Multipa Multipath th Charac Characteri teristics stics///// ////////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ////////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ////////// ////////// ////////// ///// '0
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Physical Physical +%ple%en +%ple%entati tation// on////// //////// //////// ///////// ///////// ///////// ///////// ///////// ////////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ////////// ///////// ///////// ///////// ///////// /////// // '4
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$pplica $pplication tions//// s//////// ///////// ////////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ////////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ////////// ///////// ///////// //////////// ////////////// /////// '4
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1. 1
+,"-OD.C"+O,
Pur pose
Efficient use of radio spectrum includes placing modulated carriers as close as possible without causing nter!Carrier nterference "C#. $ptimally% the bandwidth of each carrier would be ad&acent to its neighbors% so there would be no wasted spectrum. n practice% a guard band must be placed between each carrier bandwidth to provide a space where a filter can attenuate an ad&acent carrier's signal. These guard bands are wasted bandwidth. n order to transmit high data rates% short symbol periods must be used. The symbol period is the inverse of the baseband data rate "T ( 1)*#% so as * increases% T must decrease. n a multi!path environment% a shorter symbol period leads to a greater chance for nter!+ymbol nterference "+#. This occurs when a delayed version of symbol ,n' arrives during the processing p eriod of symbol ,n-1'. $rthogonal re/uency 0ivision ultipleing "$0# addresses both of these problems. $0 provides a techni/ue allowing the bandwidths of modulated carriers to overlap without interference "no C#. t also provides a high date rate with a long symbol duration% thus helping to eliminate +. $0 may therefore be considered as a candidate modulation techni/ue in a broadband% multi!path environment. The purpose of this report is to provide the following information concerning $03 theory of operation analysis of important characteristics implementation eample "matlab#
• • •
1. 2
OFDM Ove r vi ew
$0 is a modulation techni/ue where multiple low data rate ca rriers are combined by a transmitter to form a composite high data rate transmission. 0igital signal processing ma4es $0 possible. To implement the multiple carrier scheme using a ban4 of parallel modulators would not be very efficient in analog hardware. 5owever% in the digital domain% multi!carrier modulation can be done efficiently with currently available 0+6 hardware and software. Not only can it be done% but it can also be made very fleible and programmable. This allows $0 to ma4e maimum use of available bandwidth and to be able to adapt to changing system re/uirements. Each carrier in an $0 system is a sinusoid with a fre/uency that is an integer multiple of a base or fundamental sinusoid fre/uency. Therefore% each carrier is li4e a ourier series component of the composite signal. n fact% it will be shown later that an $0 signal is created in the fre/uency domain% and then transformed into the time domain via the 0iscrete ourier Transform "0T#. Two periodic signals are orthogonal when the integral of their product% over one period% is e/ual to 7ero. This is true of certain sinusoids as illustrated in E/uation 1. Equation ' : De2inition o2 Orthogonal
Continuous Time 3 T
∫ cos"2
8 t #
π nf
8
× cos" 2π mf 8 t #dt =
8
"n
≠ m#
0iscrete Time 3 N !1
∑ cos 2 N n × cos 2 Nm = 8
4
=
π k
π k
"n
≠ m#
8
The carriers of an $0 system are sinusoids that meet this re/uirement because each one is a multiple of a fundamental fre/uency. Each one has an integer number of cycles in the fundamental period.
9
)
OFDM OPE-$"+O,
2. 1
Pr el i mi nar yConcept s
:hen the 0T "0iscrete ourier Transform# of a time signal is ta4en% the fre/uency domain results are a function of the time sampling period and the number of samples as shown in igure 1. The fundamental fre/uency of the 0T is e/ual to 1)NT "1)total sample time#. Each fre/uency represented in the 0T is an integer multiple of the fundamental fre/uency. The maimum fre/uency that can be represented by a time signal sampled at rate 1)T is f ma ( 1)2T as given by the Ny/uist sampling theorem. This fre/uency is located in the center of the 0T points. ;ll fre/uencies beyond that point are images of the representative fre/uencies. The maimum fre/uency bin of the 0T is e/ual to the sampling fre/uency "1)T# minus one fundamental "1)NT#. The 0T "nverse 0iscrete ourier Transform# performs the opposite operation to the 0T. t ta4es a signal defined by fre/uency components and converts them to a time signal. The parameter mapping is the same as for the 0T. The time duration of the 0T time signal is e/ual to the number of 0T bins "N# times the sampling period "T#. t is perfectly valid to generate a signal in the fre/uency domain% and convert it to a time domain e/uivalent for practical use<. This is how modulation is applied in $0. < The fre/uency domain is a mathematical tool used for analysis. ;nything usable by the real world must be converted into a real% time domain signal. s"t#
1
2
9
/ / / / / / / /
N "number of samples#
T "sample period#
t
NT "total time used for the 0T is the product of the sample period times the number of samples#
0T
0T
> +"f# >
/ / / / / / / /
8 1)NT 2)NT 9)NT ????
////////
1)2T "Ny/uist bin#
???.. "N!1#)NT "N)NT ( 1)T ( sampling fre/uency#
0T bins representing discrete fre/uency components of f"t#.
=
f
Figure ' : Para%eter Mapping 2ro% "i%e to Frequenc y 2or the DF"
n practice the ast ourier Transform "T# and T are used in place of the 0T and 0T% so all further references will be to T and T.
2 . 2
De fini t i onofCa r r i e r s
The maimum number of carriers used by $0 is limited by the si7e of the T. This is determined as follows in E/uation 23 Equation ) : OFDM Carrier Count
N carriers
≤
N carriers
≤
IFFTsize 2 IFFTsize
−2
"real ! valued time signal#
−1
"comple ! valued time signal#
n order to generate a real!valued time signal% $0 "fre/uency# carriers must be defined in comple con&ugate pairs% which are symmetric about the Ny/uist fre/uency "f ma#. This puts the number of potential carriers e/ual to the T si7e)2. The Ny/uist fre/uency is the symmetry point% so it cannot be part of a comple con&ugate pair. The 0C component also has no comple con&ugate. These two points cannot be used as carriers so they are subtracted from the total available. f the carriers are not defined in con&ugate pairs% then the T will result in a time domain signal that has imaginary components. This must be a viable option as there are $0 systems defined with carrier counts that eceed the limit for real!valued time signals given in E/uation 2. *eference A1B describes a system with T si7e 2 and carrier count 21. This design must result in a comple time waveform. urther processing would re/uire some sort of /uadrature techni/ue "use of parallel sine and cosine processing paths#. n this report% only real!value time signals will be treated% but in order to obtain maimum bandwidth efficiency from $0% the comple time signal may be preferred "possibly an analagous situation to D6+ vs. F6+#. E/uation 2% for the comple time waveform% has all T bins available as carriers ecept the 0C bin. Foth T si7e and assignment "selection# of carriers can be dynamic. The transmitter and receiver &ust have to use the same parameters. This is one of the advantages of $0. ts bandwidth usage "and bit rate# can be varied according to varying user re/uirements. ; simple control message from a base station can change a mobile unit's T si7e and carrier selection.
2. 3
Modul at i on
Finary data from a memory device o r from a digital processing stream is used as the modulating "baseband# signal. The following steps may be carried out in order to apply modulation to the carriers in $03 • •
• • • •
combine the binary data into symbols according to the number of bits)symbol selected convert the serial symbol stream into parallel segments according to the number of carriers% and form carrier symbol se/uences apply differential coding to each carrier symbol se/uence convert each symbol into a comple phase representation assign each carrier se/uence to the appropriate T bin% including the comple con&ugates ta4e the T of the result
This is the same modulation techni/ue described in *eference A9B. The *eference A2B matlab program carries out these steps and provides detailed commentary and eamples for each one. $0 modulation is applied in the fre/uency domain. igure 2 and igure 9 give an eample of modulated $0 carriers for one symbol period% prior to T. or this eample% there are = carriers% the T bin si7e is =% and there is only 1 bit per symbol. The magnitude of each carrier is 1% but it could be scaled to any value. The phase for each carrier is either 8 or 1G8 degrees% according to the symbol being sent. The phase determines the value of the symbol "binary in this case% either a 1 or a 8#. n the eample% the first 9 bits "the first 9 carriers# are 8% and the = th bit "= th carrier# is a 1.
OFDM Ca r r i e rFr e q u en c yMa g n i t u d e 1. 5
1
e d u t i n0 . 5 g a M
0
0 . 5 0
10
20
30 I FFTBi n
40
50
60
50
60
Figure ): OFDM Carrier Magnitu#e prior to +FF"
OFDM Car r i erPhase 200 150 100 ) 0 s 5 e e r g e d ( 0 e s a h P50
1 0 0 1 5 0 2 0 0 0
10
20
30 I FFTBi n
40
Figure 0 : OFDM Carrier Phase prior to +FF"
H
Note that the modulated $0 signal is nothing more than a group of delta "impulse# functions% each with a phase determined by the modulating symbol. n addition% note that the fre/uency separation between each delta is proportional to 1)N where N is the number of T bins. The fre/uency domain representation of the $0 is described in E/uation 9. Equation 0 : OFDM Frequency Do%ain -epresentation one sy%&ol perio#!
S "k #
N N = e j θ m δ k − m − + e − j θ m δ k + m − 2 2
single "real# $.0 modulated carrier
= fre/uency "8 to N ! 1# m = $.0 carrier fre/uency N = ..T bin si7e 4
=
S "k # ofdm
c last m
j θ m N N − j θ e δ k − m − 2 + e m δ k + m − 2 c first
∑ =
composite "real# $.0 modulated carriers
c = $.0 carrier "first through last#
;fter the modulation is applied% an T is performed to generate one symbol period in the time domain. The T result is shown in igure =. t is clear that the $0 signal has a varying amplitude. t is very important that the amplitude variations be 4ept intact as they define the content of the signal. f the amplitude is clipped or modified% then an T of the signal would no longer result in the original fre/uency characteristics% and the modulation may be lost. This is one of the drawbac4s of $0% the fact that it re/uires linear amplification. n addition% very large amplitude pea4s may occur depending on how the sinusoids line up% so the pea4!to!average power ratio is high. This means that the linear amplifier has to have a large dynamic range to avoid distorting the pea4s. The result is a linear amplifier with a constant% high bias current resulting in very poor power efficiency. or a detailed treatment of the pea4!to!average power ratio problem in $0% see *eference A=B. igure is provided to illustrate the time components of the $0 signal. The T transforms each comple con&ugate pair of delta functions "each carrier# into a real!valued% pure sinusoid. igure shows the separate sinusoids that ma4e up the composite $0 waveform given in igure =. The one sinusoid with 1G8 phase shift is clearly visible as is the fre/uency difference between each of the = sinusoids. Note that this figure is ,7oomed' i.e. all = point of the T are not shown. n addition% note that the waveform plots OF DM Ti methere Si gnal ,On eSy mbolP er i od per cycle for any of the sinusoids. are not very smooth. This is because are not many samples 0 . 0 8
The time domain representation of the $0 signal is given in E/uation =. Equation0 40 :6 OFDM "i%e Do%ain -epresentat ion one sy%&ol perio#! .
0 . 0 4 s" n #
c last
= m
N
−1
2 cos ∑ ∑ = = c first n
8
π mn
N
+ θ m
0 . 0 2 e d 0 u t i l p m 0 . 0 2 A
0 . 0 4
n = time sample m = $.0 carrier N = ..T bin si7e
= phase modulation for $.0 carrier "m# c first % c last = $.0 carriers "first and last#
θ m
0 . 0 6 0 . 0 8
G 0 . 1 0
10
20
30
40 Ti me
50
60
70
Figure 4 : OFDM ignal( ' y%&ol Perio#
Se pa r a t e dTi meWa v e f o r msCa r r i e r s 0 . 0 3
0 . 0 2
0 . 0 1 e d u t i l p m A
0
0 . 0 1
0 . 0 2
0 . 0 3 0
2
4
6
8 10 Ti me
12
Figure 5 : eparate# Co%ponents o2 the OFDM "i%e 8ave2or%
I
14
16
18
2. 4
Tr ansmi ssi on
The 4ey to the uni/ueness and desirability of $0 is the relationship between the carrier fre/uencies and the symbol rate. Each carrier fre/uency is separated by a multiple of 1)NT "57#. The symbol rate "*# for each carrier is 1)NT "symbols)sec#. The effect of the symbol rate on each $0 carrier is to add a sin"#) shape to each carrier's spectrum. The nulls of the sin"#) "for each carrier# are at integer multiples of 1)NT. The pea4 "for each carrier# is at the carrier fre/uency 4)NT. Therefore% each carrier fre/uency is located at the nulls for all the other carriers. This means that none of the carriers will interfere with each other during transmission% although their spectrums overlap. The ability to space carriers so closely together is very bandwidth efficient. igure H shows the spectrum for of an $0 signal with the following characteristics3 1 bit ) symbol • 188 symbols ) carrier "i.e. a se/uence of 188 symbol periods# • • = carriers = T bins • spectrum averaged for every 28 symbols "188)28 ( averages# • *ed diamonds mar4 all of the available carrier fre/uencies. Note that the nulls of the spectrums line up with the unused fre/uencies. The four active carriers each have pea4s at carrier fre/uencies. t is clear that the active carriers have nulls in their spectrums at each of the unused fre/uencies "otherwise% the nulls would not eist#. ;lthough it cannot be seen in the figure% the active fre/uencies also have spectral nulls at the ad&acent active fre/uencies. igure shows the $0 time waveform for the same signal. There are 188 symbol periods in the signal. Each symbol period is = samples long "188 = ( =88 total samples#. Each symbol period contains = carriers each of which carries 1 symbol. Each symbol carries 1 bit. Note that igure again illustrates the large dynamic range of the $0 waveform envelope. t is not currently practical to generate the $0 signal directly at * rates% so it must be upconverted for transmission. To remain in the discrete domain% the $0 could be upsampled and added to a discrete carrier fre/uency. This carrier could be an intermediate fre/uency whose sample rate is handled by current technology. t could then be converted to analog and increased to the final transmit fre/uency using analog fre/uency conversion methods. ;lternatively% the $0 modulation could be immediately converted to analog and directly increased to the desired * transmit fre/uency. Either way% the selected techni/ue would have to involve some form of linear ; "possibly implemented with a mier#.
2. 5
Recept i onandDemodul at i on
The received $0 signal is downconverted "in fre/uency# and ta4en from analog to digital. 0emodulation is done in the fre/uency domain "&ust as modulation was#. The following steps may be ta4en to demodulate the $03 • • • • • •
partition the input stream into vectors representing each symbol period ta4e the T of each symbol period vector etract the carrier T bins and calculate the phase of each calculate the phase difference% from one symbol period to the net% for each carrier decode each phase into binary data sort the data into the appropriate order
The *eference A2B matlab program carries out these steps and provides detailed commentary and eamples for each one. igure G and igure I show the magnitude and spectrum of the T for one received $0 symbol period. or this eample% there are = carriers% the T bin si7e is =% there is 1 bit per symbol% and the signal was sent through a channel with ;:JN having an +N* of G dF. The figures show that% under these conditions% the modulated symbols are very easy to recover. Note in igure I that the unused fre/uency bins contain widely varying phase values. These bins are not decoded% so it does not matter% but the result is of interest. Even if the noise is removed from the channel% these phase variations still occur. t must be a result of the T)T operations generating very small comple values "very close to 8# for the unused carriers. The phases are a result of these values. 18
OFDM Ti meSi gna l 0 . 1 5
0. 1
0 . 0 5 ) s t l o v (
0
e d u t i l p 0 . 0 5 m A
0 . 1
0 . 1 5
0 . 2 0
1000
2000
3000 4000 T i me( s a mp l e s )
5000
6000
7000
Figure 6 : OFDM "i%e 8ave2or%
OFDM Si gnalSpect r um
10
0 ) B d ( e 10 d u t i n g a M
20
30
40 0
0. 05
0. 1
0. 15 0. 2 0. 25 0. 3 0. 35 No r ma l i z edFr e qu en cy( 0 . 5=f s / 2 )
Figure : OFDM pectru%
11
0. 4
0. 45
0. 5
OFDM Recei v eSpect r um,Magni t ude 1. 5
1
e d u t i n0 . 5 g a M
0
0 . 5 0
10
20
30 FFTBi n
40
50
60
Figure ;: OFDM Carrier Magnitu#e 2ollo1ing FF"
OFDM Recei v eSpect r um,Phase 200 150 100 ) 0 s 5 e e r g e d ( 0 e s a h P50
1 0 0 1 5 0 2 0 0 0
10
20
30 FFTBi n
40
Figure 3: OFDM Carrier Phase 2ollo1in g FF"
12
50
60
0
3. 1
$,$+
Guar dPer i od
$0 demodulation must be synchroni7ed with the start and end of the transmitted symbol period. f it is not% then + will occur "since information will be decoded and combined for 2 ad&acent symbol periods#. C will also occur because orthogonality will be lost "integrals of the carrier products will no longer be 7ero over the integration period#% *eference AB. To help solve this problem% a guard interval is added to each $0 symbol period. The first thought of how to do this might be to simply ma4e the symbol period longer% so that the demodulator does not have to be so precise in pic4ing the period beginning and end% and decoding is always done inside a single period. This would fi the + problem% but not the C problem. f a complete period is not integrated "via T#% orthogonality will be lost. n order to avoid + and C% the guard period must be formed by a cyclic etension of the symbol period. This is done by ta4ing symbol period samples from the end of the period and appending them to the front of the period. The concept of being able to do this% and what it means% comes from the nature of the T)T process. :hen the T is ta4en for a symbol period "during $0 modulation#% the resulting time sample se/uence is technically periodic. This is because the T)T is an etension of the ourier Transform which is an etension of the ourier +eries for periodic waveforms. ;ll of these transforms operate on signals with either real or manufactured periodicity. or the T)T% the period is the number of samples used. *eference AB provides an ecellent eplanation of the ourier +eries and its etensions. :ith the cyclic etension% the symbol period is longer% but it represents the eact same fre/uency spectrum. ;s long as the correct number of samples are ta4en for the decode% they may be ta4en anywhere within the etended symbol. +ince a complete period is integrated% orthogonality is maintained. Therefore% both + and C are eliminated. Note that some bandwidth efficiency is lost with the addition of the guard period "symbol period is increased and symbol rate is decreased#.
Fase symbol period% e.g = samples
Etension% e.g 18 samples
Fase symbol period% e.g = samples
ntegrate "T# any = samples within the etended period
Etended symbol period% e.g =-18(H= samples
Figure '* : 7uar# Perio# via Cyclic Extensio n
19
3. 2
Wi ndowi ng
The $0 signal is made up of a series of Ts that are concatenated to each other. ;t each symbol period boundary% there is a signal discontinuity due to the differences between the end of one period and the start of the net. These discontinuities can cause high fre/uency spectral noise to be generated "because they loo4 li4e very fast transitions of the time waveform#. To avoid this% a window function "5amming% 5anning% Flac4man% ?# may be applied to each symbol period. The window function would attenuate the time waveform at the start and the end of each period% so that the discontinuities are smaller% and the high fre/uency noise is reduced. 5owever% this attenuation distorts the signal and some of the desired fre/uency content is lost. *eference A9B recommends using a window function% and one is included in the *eference A2B matlab program% but it is commented out. t is not clear whether the window function helps more than it hurts. f it does not add sufficient benefit% it should not be included as it is an etra processing step that would increase the load on the $0 generation process "an etra multiply operation on each symbol period vector#.
3 . 3
Mul t i pa t hCha r a c t e r i s t i c s
$0 avoids fre/uency selective fading and + by providing relatively long symbol periods for a given data rate. This is illustrated in igure 11. or a given transmission channel and a given source data rate% $0 can provide better multipath characteristics than a single carrier. ingle Carrier bps data rate "*# T* ( 1)* ( 1H.G usec -
Out#oor "rans%ission Channel σ ( 18.8 usec "*+ delay spread# Fc ( 1)σ ( 28 57 "Coherence Fandwidth#
Multiple Carrier OFDM! bps data rate "*# 112 carriers * C;**E* ( 88 bps T C;**E* ( 2888 usec -
Out#oor "rans%ission Channel σ ( 18.8 usec "*+ delay spread# Fc ( 1) σ ( 28 57 "Coherence Fandwidth#
-
-
K T* )18
σ
K T* )18 18.8 K 2888)18 18.8 K 28 "no# Flat Fa#ing reduced + a dd guard pe riod and completely eliminate +
σ
note3 rule of thumb% if σ K T* )18% then channel has fre/uency selective fading Figure '': OFDM vs/ ingle Carrier( Multipath Characte ristic Co%parison
5owever% since the $0 carriers are spread over a fre/uency range% there still may be some fre/uency selective attenuation on a time!varying basis. ; deep fade on a particular fre/uency may cause the loss of data on that fre/uency for a given time% but the use of orward Error Coding can fi it. f a single carrier eperienced a deep fade% too many consecutive symbols may be lost and correction coding may be ineffective. or more detail on $0 error coding techni/ues% see *eference AB.
3. 4
Bandwi dt h
; comparison of * transmit bandwidth between $0 and a single carrier is shown in igure 12 "using the same eample parameters as in igure 11#. The calculations show that $0 is more bandwidth efficient than a single carrier. Note that another efficient aspect of $0 is that a single transmitter's bandwidth can be increased incrementally by addition of more ad&acent carriers. n addition% no bandwidth buffers are needed between transmit bandwidths of separate transmitters as long as orthogonality can be maintained between all the carriers.
1=
ingle Carrier bps data rate "*# T* ( 1)* ( 1H.G usec *aised Cosine ilter pulse shaping% F:* ( *"1- α# ( bps "1-8.9# F:* ( H2.G 57 -
Multiple Carrier OFDM! bps data rate "*# 112 carriers * C;**E* ( 88 bps T C;**E* ( 2888 usec N ( T bin count T ( base period NT ( symbol period ( T C;**E* ( 2888 usec n ( number of carriers ( 112 F:* ( "n-1#)NT F:* ( "112-1#)2888 usec F:* ( . 57 -
( 8.9
α
$0 is 2I more bandwidth efficient for this eample "H2.G). ( 1.2GG#
Figure '): OFDM 9an#1i#th E22icienc y
3 . 5
Ph ys i c a lI mpl e me nt a t i on
+ince $0 is carried out in the digital domain% there are many ways it can be implemented. +ome options are provided in the following list. Each of these options should be viable given current technology3 1.
;+C ";pplication +pecific ntegrated Circuit# ;+Cs are the fastest% smallest% and lowest power way to implement $0 • Cannot change the ;+C after it is built without designing a new chip •
2.
Jeneral!purpose icroprocessor or icroController 6ower6C H=88 or other processor capable of fast vector operations • 5ighly programmable • • Needs memory and other peripheral chips • Lses the most power and space% and would be the slowest
9. ield!6rogrammable Jate ;rray "6J;# • ;n 6J; combines the speed% power% and density attributes of an ;+C with the programmability of a general purpose processor. ;n 6J; could be reprogrammed for new functions by a base station to meet future • "currently un4nown re/uirements#. • This should be the best choice
3 . 6
Appl i c at i ons
$0 applications include the f ollowing AH%GB3 1.
0igital ;udio Froadcasting "0;F#% wireless C0!/uality sound transmission
2.
0igital Video Froadcasting "0VF#% specifically% 0igital Terrestrial Television Froadcasting "0TTF#
9. :ireless M;N "EEE G82.11a# =.
;0+M ";symmetric 0igital +ubscriber Mine#% also called 0T "0igital ulti!Tone#
1
4
-EFE-E,CE
A1B :ide!band $rthogonal re/uency ultipleing ":!$0#% :hite 6aper by :ireless 0ata Communications nc.% www.wi!lan.com. A2B $0 +imulation "matlab#% Erich Cosby% 2881 "based on *eference A9B#% this matlab file is included as +ection of this report. A9B The suitability of $0 as a modulation techni/ue for wireless telecommunications% with a C0; comparison% Eric Mawrey% 1IIH% www.eng.&cu.edu.au)eric)thesis)Thesis.htm. A=B $rthogonal re/uency 0ivision ultipleing "$0# O ;pplications for :ireless Communications with Coding% un4nown author% www.comlab.hut.fi)opetus)911)ofdmPmod.pdf . AB The how and why of C$0% Q. 5. +tott% 1IIG% www.bbc.co.u4)rd)pubs)papers)pdffiles)ptrevP2HG! stott.pdf . AB :ho s ourierR% The Transnational College of Me% 1II% Manguage *esearch oundation "available at ama7on.com#. AHB $rthogonal re/uency 0ivision ultipleing "$0# Eplained% agis Networ4s% nc.% ebruary G% 2881% www.magisnetwor4s.com . AGB $0 Tutorial% :ave *eport% www.wave!report.com)tutorials)$0.htm .
1
5
M$"$9
The matlab file allows the user to specify $0 parameters and analy7e the resulting waveforms. There are embedded plot se/uences that can be ,uncommented' to analy7e the $0 processing path at different points in the se/uence.
!!!!!!!!1!!!!!!!!!2!!!!!!!!!9!!!!!!!!!=!!!!!!!!!!!!!!!!!!!!!!!!!!!H!!!!!!!!!G $0 +imulation ! ECE = 6ro&ect ! Erich Cosby ! 0ecember =% 2881 ! Virginia Tech% Nothern Va. Center Fased on the matlab wor4 of Eric Mawrey in his 1IIH F+EE thesis% SThe suitability of $0 as a modulation techni/ue for wireless telecommunications% with a C0; comparisonS% refer to www.eng.&cu.edu.au)eric)thesis)Thesis.htm. clear all close all Fasic $0 system parameters ! choice of defaults or user selections fprintf "U$0 ;nalysis 6rogramnnU# defaults ( input"UTo use default parameters% input S1S% otherwise input S8S3 U# if defaults (( 1 TPbinPlength ( 182= T bin count for T% T bin count for * carrierPcount ( 288 number of carriers bitsPperPsymbol ( 2 bits per symbol symbolsPperPcarrier ( 8 symbols per carrier +N* ( 18 channel signal to noise ratio "dF# else TPbinPlength ( input"UT bin length ( U# carrierPcount ( input"Ucarrier count ( U# bitsPperPsymbol ( input"Ubits per symbol ( U# symbolsPperPcarrier ( input"Usymbols per carrier (U# +N* ( input"U+N* ( U# end 0erived parameters basebandPoutPlength ( carrierPcount < symbolsPperPcarrier < bitsPperPsymbol carriers ( "13carrierPcount# - "floor"TPbinPlength)=# ! floor"carrierPcount)2## con&ugatePcarriers ( TPbinPlength ! carriers - 2 !!!!!!!!1!!!!!!!!!2!!!!!!!!!9!!!!!!!!!=!!!!!!!!!!!!!!!!!!!!!!!!!!!H!!!!!!!!!G T*;N+T KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK Jenerate a random binary output signal3 ! a row of uniform random numbers "between 8 and 1#% rounded to 8 or 1 ! this will be the baseband signal which is to be transmitted. basebandPout ( round"rand"1%basebandPoutPlength##
1H
Convert to Umodulo NU integers where N ( 2WbitsPperPsymbol ! this defines how many states each symbol can represent ! first% ma4e a matri with each column representing consecutive bits from the input stream and the number of bits in a column e/ual to the number of bits per symbol ! then% for each column% multiply each row value by the power of 2 that it represents and add all the rows ! for eample3 input 8 1 1 8 8 8 1 1 1 8 bitsPperPsymbol ( 2 convertPmatri ( 8 1 8 1 1 18818 moduloPbaseband ( 1 2 8 9 2 convertPmatri ( reshape"basebandPout% bitsPperPsymbol% length"basebandPout#)bitsPperPsymbol# for 4 ( 13"length"basebandPout#)bitsPperPsymbol# moduloPbaseband"4# ( 8 for i ( 13bitsPperPsymbol moduloPbaseband"4# ( moduloPbaseband"4# - convertPmatri"i%4#<2W"bitsPperPsymbol!i# end end !!!!!!!!1!!!!!!!!!2!!!!!!!!!9!!!!!!!!!=!!!!!!!!!!!!!!!!!!!!!!!!!!!H!!!!!!!!!G +erial to 6arallel Conversion ! convert the serial modulo N stream into a matri where each column represents a carrier and each row represents a symbol ! for eample3 serial input stream ( a b c d e f g h i & 4 l m n o p parallel carrier distribution ( C1)s1(a C2)s1(b C9)s1(c C=)s1(d C1)s2(e C2)s2(f C9)s2(g C=)s2(h C1)s9(i C2)s9(& C9)s9(4 C=)s9(l . . . . . . . . carrierPmatri ( reshape"moduloPbaseband% carrierPcount% symbolsPperPcarrier#U ;pply differential coding to each carrier string ! append an arbitrary start symbol "let it be 8% that wor4s for all values of bitsPperPsymbol# "note that this is done using a vertical concatenation AyB of a row of 7eros with the carrier matri% sweetX# ! perform modulo N addition between symbol"n# and symbol"n!1# to get the coded value of symbol"n# ! for eample3 bitsPperPsymbol ( 2 "modulo =# symbol stream ( 9 2 1 8 2 9 start symbol ( 8 coded symbols ( 8 - 9 ( 9 9 - 2 ( 11 ( 1 1-1(2 2-8(2 2 - 2 ( 18 ( 8 8-9(9 coded stream ( 8 9 1 2 2 8 9 !!!!!!!!1!!!!!!!!!2!!!!!!!!!9!!!!!!!!!=!!!!!!!!!!!!!!!!!!!!!!!!!!!H!!!!!!!!!G
1G
carrierPmatri ( A7eros"1%carrierPcount#carrierPmatriB for i ( 23"symbolsPperPcarrier - 1# carrierPmatri"i%3# ( rem"carrierPmatri"i%3#-carrierPmatri"i!1%3#%2WbitsPperPsymbol# end Convert the differential coding into a phase ! each phase represents a different state of the symbol ! for eample3 bitsPperPsymbol ( 2 "modulo =# symbols ( 8 9 2 1 phases ( 8 < 2pi)= ( 8 "8 degrees# 9 < 2pi)= ( 9pi)2 "2H8 degrees# 2 < 2pi)= ( pi "1G8 degrees# 1 < 2pi)= ( pi)2 "I8 degrees# carrierPmatri ( carrierPmatri < ""2
1I
6M$T F;+C *EDLENCZ 0$;N *E6*E+ENT;T$N !!!!!!!!1!!!!!!!!!2!!!!!!!!!9!!!!!!!!!=!!!!!!!!!!!!!!!!!!!!!!!!!!!H!!!!!!!!!G figure "1# stem"83TPbinPlength!1% abs"TPmodulation"2%13TPbinPlength##%Ub
28
;pply a :indow unction to each time waveform ! N$TE T5;T :N0$:NJ + CL**ENTMZ C$ENTE0 $LT% i.e. N$ :N0$:NJ ! each time waveform "row of timePwavePmatri# represents one symbol period for all carriers ! the T result has discontinuities at each end ! when the time waveforms are seriali7ed "concatenated#% the discontinuites will introduce unwanted fre/uency components ! the window function deemphasi7es the signal at the end points "at the discontinuites# ! this reduces the effects of the discontinuities ! it also distorts the desired fre/uency response "undesired side effect# ! between Flac4man% 5anning% and 5amming3 5amming introduces less distortion ! note that the transpose of the 5amming function is used "because a row vector is needed# +ince all imaginary values of timePwavePmatri are practically e/ual to 7ero% only the real part is retained for windowing. for i ( 13symbolsPperPcarrier - 1 windowedPtimePwavePmatri"i%3# ( real"timePwavePmatri"i%3## .< hamming"TPbinPlength#U windowedPtimePwavePmatri"i%3# ( real"timePwavePmatri"i%3## end +eriali7e the modulating waveform ! se/uentially ta4e each row of windowedPtimePwavePmatri and construct a row vector ! the row vector will be the modulating signal ! note that windowedPtimePwavePmatri is transposed% this is to account for the way the atlab UreshapeU function wor4s "reshape ta4es the columns of the target matri and appends them se/uentially# ofdmPmodulation ( reshape"windowedPtimePwavePmatriU% 1% TPbinPlength<"symbolsPperPcarrier-1## 6M$T $0 +JN;M "time# tempPtime ( TPbinPlength<"symbolsPperPcarrier-1# figure "# plot"83tempPtime!1%ofdmPmodulation# grid on ylabel"U;mplitude "volts#U# label"UTime "samples#U# title"U$0 Time +ignalU# 6M$T $0 +JN;M "spectrum# symbolsPperPaverage ( ceil"symbolsPperPcarrier)# avgPtempPtime ( TPbinPlength
21
EN06M$T !!!!!!!!1!!!!!!!!!2!!!!!!!!!9!!!!!!!!!=!!!!!!!!!!!!!!!!!!!!!!!!!!!H!!!!!!!!!G Lpconversion to * or this model% the baseband will be inserted directly into the channel without conversion to * fre/uencies. TPdata ( ofdmPmodulation !!!!!!!!1!!!!!!!!!2!!!!!!!!!9!!!!!!!!!=!!!!!!!!!!!!!!!!!!!!!!!!!!!H!!!!!!!!!G C5;NNEM (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( The channel model is Jaussian ";:JN# only ! *ayleigh fading would be a useful addition TPsignalPpower ( var"TPdata# linearP+N* ( 18W"+N*)18# noisePsigma ( TPsignalPpower)linearP+N* noisePscalePfactor ( s/rt"noisePsigma# noise ( randn"1% length"TPdata##
22
title"U$0 *eceive +pectrum% 6haseU# EN0 $ 6M$TTNJ
!!!!!!!!1!!!!!!!!!2!!!!!!!!!9!!!!!!!!!=!!!!!!!!!!!!!!!!!!!!!!!!!!!H!!!!!!!!!G Etract the carrier T bins ! only 4eep the fft bins that are used as carriers ! ta4e the transpose of the result so that each column will represent a carrier ! this is in preparation for using the diff" # function later to decode differential encoding ! format following this operation is3 C1!s1 C2!s1 C9!s1 ... C1!s2 C2!s2 C9!s2 ... C1!s9 C2!s9 C9!s9 ... . . . . . . ! 6$*T;NT ;TM;F N$TE C$NCE*NNJ T*;N+6$+NJ ;N0 C$NQLJ;T$N ! it appears that each time a matri is transposed% the con&ugate of each value is ta4en ! if an even number of transposes are done% then it is transparent ! obviously% this does not affect real numbers *Pcarriers ( *Pspectrum"carriers%3#U !!!!!!!!1!!!!!!!!!2!!!!!!!!!9!!!!!!!!!=!!!!!!!!!!!!!!!!!!!!!!!!!!!H!!!!!!!!!G 6M$T E;C5 *ECEVE0 +ZF$M figure "I# *PphaseP6 ( angle"*Pcarriers# *PmagP6 ( abs"*Pcarriers# polar"*PphaseP6% *PmagP6%UbdU# EN0 6M$T ind the phase "angle# of each T bin "each carrier# ! convert from radians to degrees ! normali7e phase to be between 8 and 9I degrees *Pphase ( angle"*Pcarriers#<"1G8)pi# phasePnegative ( find"*Pphase [ 8# *Pphase"phasePnegative# ( rem"*Pphase"phasePnegative#-98%98# Etract phase differences "from the differential encoding# ! the matlab diff" # function is perfect for this operation ! again% normali7e the result to be between 8 and 9I degrees *PdecodedPphase ( diff"*Pphase# phasePnegative ( find"*PdecodedPphase [ 8# *PdecodedPphase"phasePnegative# ( rem"*PdecodedPphase"phasePnegative#-98%98# !!!!!!!!1!!!!!!!!!2!!!!!!!!!9!!!!!!!!!=!!!!!!!!!!!!!!!!!!!!!!!!!!!H!!!!!!!!!G Convert phase to symbol ! calculate the base phase which is the phase difference between each consecutive symbol ! for eample% if there are 2 bits per symbol% base phase is I8 and the symbols are represented by 8% I8% 1G8% and 2H8 degrees
29
! calculate the maimum deviation from the base phase that will still be decoded as base phase ! for eample% if base phase is I8% then delta phase is =% and anything within = degrees of base phase is accepted as base phase ! continuing the above eample% a symbol represented by 1G8 will be decoded as 1G8 as long as it is within the range 1G8-= and 1G8!= ! generate a symbol matri where the results of the phase decode will be placed ! note that since the matri is created as a 7ero matri% then the 7ero values do not have to be decoded ! 7ero is therefore the default value after decoding% if a value is not decoded as anything else% then it is 7ero ! this actually save alot of trouble since the 7ero phase spans the lowest and highest phase values and therefore re/uires special processing ! it is also efficient in that it eliminates a pass through the loop basePphase ( 98)2WbitsPperPsymbol deltaPphase ( basePphase)2 *PdecodedPsymbols ( 7eros"si7e"*PdecodedPphase%1#%si7e"*PdecodedPphase%2## for i ( 13"2WbitsPperPsymbol ! 1# centerPphase ( basePphase
2=
