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Orthogonal Frequency Division Multiplexing ∗
Rethnakaran.P and Herbert Dawid
1 November 2003
Abstract
Orthogonal Frequency Division Multiplexing (OFDM) is
1 Multi Carrier Modulation
Multi carrier modulation as the name suggests is a modulation technique in which multiple numberof carriers are used for modulating the information signals (data bits), as against a single carrierused in the most common single carrier modulation schemes such as BPSK and QPSK etc. Thusmulti carrier modulation can be defined as the principle of transmitting data by dividing the streaminto several parallel bit streams, each of which has a much lower bit rate, and by using these substreams to modulate several carriers. The first systems using MCM were military HF radio links inthe late 1950s and early 1960s. The most popular of the MCM scheme is the well known OrthogonalFrequency Division Modulation also known as OFDM in brief.
Serialtoparallel
modulator
modulator
modulator
xn
xn,1
xn,2
xn,N
φ1(t)
φN(t)
φ2(t)
ψn,1(t)
ψn,2(t)
ψn,N(t)
s(t)∑
Figure 1: Multi carrier modulation scheme
A functional block diagram of MCM scheme is shown in Fig.(1). The serial data bits carryinginformation are first converted to parallel bit streams. So every block of N data bits entering theMCM will be multiplexed onto N channels where each of these bits are modulated by a differentcarrier signal φ. As illustrated N carriers φ1, φ2, . . . , φN are used. The carrier signals are carefullychosen subject to various conditions and they differ from one scheme to another. We will herefocus on OFDM which chooses these carrier signals to be orthogonal in time. We will restrict ourinvestigation of MCM to OFDM only.∗Digital Communication Solutions, Synopsys Inc
1
2 Orthogonal Frequency Division Multiplexing (OFDM)
Orthogonal Frequency Division Multiplexing (OFDM), a special form of MCM with densely spacedsub carriers and overlapping spectra was patented in the U.S. in 1970 [1]. By definition, OFDMexpects the sub carriers to be orthogonal. In olden days the orthogonality was assumed in frequency.For that reason typically the OFDM systems of that era used steep bandpass filters that completelyseparate the spectrum of individual sub carriers. In that context this sound very much akin toFrequency Division Multiplex (FDMA) systems. These systems evolved and the orthogonality itselfredefined in the form of time orthogonality. Thus OFDM time-domain waveforms are chosen suchthat mutual orthogonality is ensured even though sub carrier spectra may overlap. It appeared thatsuch waveforms can be generated using a Fast Fourier Transform at the transmitter and receiver.For a relatively long time, the practicality of the concept appeared limited. Implementation aspectssuch as the complexity of a real-time Fourier Transform appeared prohibitive, not to speak aboutthe stability of oscillators in transmitter and receiver, the linearity required in RF power amplifiersand the power back-off associated with this. With the invention (re-invention!) of Fast FourierTransform algorithm to compute DFT, a new direction on the practicality of an efficient OFDMsystem evolved. Thus the FFT based OFDM was born. Clearly the FFT based OFDM came withan easier implementation point of view. But first let us look at the mathematics behind OFDMitself. We will then discuss how DFT (and FFT) can own a place in meeting the requirement of anOFDM system of equations.
We have seen that OFDM is a block modulation scheme where data symbols are transmitted inparallel by employing a large number of orthogonal sub-carriers. A block of N serial data symbols,each of duration Ts, is converted into a block of N parallel data symbols, each of duration T = NTs.The N parallel data symbols modulate N (different of-course) orthogonal sub carriers. How do wechose these orthogonal carriers? As mentioned earlier these are signals orthogonal in time. Whatdoes time orthogonality mean in the first place? As a matter of fact, we can chose any set of Nsignals which are time orthogonal. Gram Schmidt will tell us how to pick to the orthogonal signalsets[ratna:give the ref] and we won’t go into that. But we will look into the orthogonality conditionsof a set of functions (signals) which are elegant and useful in simplifying the overall structure of theOFDM system. We will consider a s set of functions φk(t) = ej2πkt/T . where 0 ≤ k ≤ N . We willcheck the time orthogonality of this set through the following stages. We have,
< sk(t), sl(t) > =∫ T
0
sk(t)s∗l (t)dt
=∫ T
0
ej2πktT e
−j2πltT dt
=∫ T
0
ej2π(k−l)t
T dt
=
∣∣∣∣∣ej2π(k−l)t
T
j2π(k−l)T
∣∣∣∣∣T
0
= Tδk,l (1)
From Eq.(1) we see that the sub carriers are orthogonal in the symbol interval [0, T ]. It mustbe remembered however that the support of the sub carrier functions are not restricted to [0, T ].Hence the orthogonality conditions are not met outside the interval t ∈ [0, T ]. To maintain theorthogonality everywhere (t ∈ (−∞,∞)), the sub carrier functions are required to be windowedby the rectangular pulse function uT . This can be effectively realized by using a rectangular pulseshaping function in the transmitter (ha(t) = uT (t)). The sub carriers are separated in frequency by1/T , which respond for their orthogonality.
2
Going backward, we can arrive at the minimum separation of the sub carrier frequencies. Letone of the sub carrier signal be φk(t) = exp(j2πfkt) with the sub carrier frequency fk. Let anothersub carrier signal be φ1(t) = exp(j2πfk + ∆ft) where the corresponding sub carrier frequency fl is∆f away from fk. Orthogonality in [0, T ] is achieved if < φk(t), φl(t) >= 0
< φk(t), φl(t) > =∫ T
0
φk(t)φ∗l (t)dt
=∫ T
0
ej2πfkte−j2πfltdt
=ej2π(fk−fl)T − 1j2π(fk − fl)
← k 6= l
= ejπ(fk−fl)T[
sinπ(fk − fl)Tπ(fk − fl)
]← k 6= l (2)
The condition for right hand side of Eq.(2) to be of zero value is 2π(fk − fl) = mπ where m isany integer value. Thus, for two sub carrier signals to be orthogonal, their frequencies should beseparated by integer multiples of 1/2T . We can see that this is only half of the frequency separationused in OFDM (see Eq.(1)). This is to maintain orthogonality of the sub carriers even when thereexists some random phase in the sub carrier modulated signals. This will become clear through thefollowing steps.
Let the two sub carriers φk and φl considered for investigation has random phases θk and θl. Wewill derive the condition for orthogonality in such a scenario.
< φk(t), φl(t) > =∫ T
0
φk(t)φ∗l (t)dt
=∫ T
0
ej2πfkt+jθke−j2πflt−jθldt
=ej(2π(fk−fl)T+[θk−θl]) − e[θk−θl]
j2π(fk − fl)← k 6= l (3)
When 2π(fk − fl)T is a multiple of 2π, the right hand side of Eq.(3) will be zero for any valueof θk − θl. Thus, when random phase offset is present, the required condition for orthogonality is tohave the sub carrier frequencies separated by 1/T in frequency. This is the reason for choosing subcarrier spacing equal to 1/T .
Fig.(5) shows the generation of OFDM modulated waveform. In practice, a fixed frequency offsetis added so that the bandpass signal is centered about the center frequency. This frequency offsetterm addition is not shown in Fig.(5). Now we are ready to give the mathematical representationof an OFDM modulated signal.
The complex envelope of an OFDM signal is given by
s(t) = A∑n
b(t− nT,xn) (4)
where
b(t,xn) = ha(t)N−1∑k=0
xn,kej2π(k−N−1
2 )tT (5)
The data block xn where n is the block index is the data symbol block at epoch n.xn consists ofn data symbols. We can thus write
xn = xn0, xn1, . . . , xnN−1 (6)
3
e0
...
...
xn
xn,0
xn,1
xn,N−1
ha(t)
N−1∑k=0
ej2πt/T
ej2π(N−1)t/T
b(t,xn)
Figure 2: OFDM Modulation
The data symbols xnk are complex symbols, often chosen from QAM or PSK constellation. Ingeneral any 2D signal constellation can be used. The frequency offset
exp(−jπ(N − 1)t
T
)shifts the band pass signal to the center frequency as discussed earlier.
2.1 Comment on the pulse shaping function ha(t)
While discussing the orthogonality, we have seen that, when a rectangular shaping pulse ha(t) = AuTis chosen, then 1/THz frequency separation of the sub carriers ensures that they are orthogonalregardless of the random phases imparted due to data modulation. But the rectangular pulse doesnot really suit the power density spectrum desired. To minimize the effect of adjacent channelinterference, the power radiated into the adjacent band (sub band) should be far below (typically 60to 80 dB) that in the desired band. Hence it is desired to have modulation techniques with narrowmain lobe and fast roll off of side lobes. The rectangular pulse thus does not help in realizing acompact power spectral density spectrum.
On the other hand, choosing a good ha(t) that meet the power spectral density spectrum require-ment violates the sub carrier orthogonality principle and thereby degrade the error rate performance.We will discuss the power spectrum of the OFDM system in a separate section. See section [ratna:givelink to the section].
3 FFT based OFDM
One of the key bullet in the OFDM propaganda is the existence of an efficient and simple implemen-tation of transmitter and receiver. This structure is realized through the Discrete Fourier Transform(DFT) and its inverse (IDFT). The computation of DFT and IDFT are themselves are performedby fast Fourier transform (FFT) techniques. In this section we will see how the OFDM system isrealized through DFT and IDFT. We will start with the OFDM signal structure once again.
4
3.1 OFDM Signal Description
The OFDM complex envelope s(t) (Eq.(4)) and bandpass signal s(t) are
s(t) = A
∞∑n=−∞
N−1∑k=0
xn,kej2πk−N−1
2 t
NTs uT (t) (7)
s(t) = <e[s(t)ej2πfct
](8)
The complex baseband equivalent signal in the symbol interval [0, T ) is given1 by
sb(t) = AN−1∑k=0
xkej2π(k−N−1
2 )tNTs uT (t) (9)
where
n ← block indexN ← block size, typically a power of 2
uT (t) = u(t)− u(t− T ) ← unit amplitude lebgth-T rectangular pulseT = NTs
3.2 OFDM through DTFT
The base band equivalent representation of OFDM signal given in Eq.(9) has a familiar form. Re-moving the common term (the fixed delay of (N − 1)/2 in the argument), we can write 2.
sb(t) = AN−1∑k=0
xkej2πktNTs uT (t)
= AuT (t)× IDT FT x ← x , [x0, x1, . . . , xN−1] (10)
where IDT FT is the Inverse Discrete Time Fourier Transform of theN sample set x , [x0, x1, . . . , xN−1].It must be remembered that the DT FT & IDT FT will always result in continuous and periodicsignals. Multiplication by uT (t) will window the signal to one single period T . From this neatinterpretation, the OFDM transmitter job is just to do an IDT FT !. Fig3 will illustrate this.
3.3 OFDM through DFT (FFT algorithms)
Computing IDT FT and DT FT are conceptually simple and straightforward, but they cannot berealized in its entirety in a digital system. We require the discrete clone of DT FT which is knownas the DFT . Computing DFT is easy and implementable solution exist in digital form. More overthe fast Fourier transform aid in doing DFT faster. But there is penalty in realizing DTFT throughDFT . We will see that shortly.
Loosely speaking, DFT consists of discrete samples of the continuous and yet periodic DFTsignal. From the ideal Nyquist sampled sets we can theoretically construct the DTFT signal. Nyquistsampling of sb(t) involves Nyquist sampling of two signals. Namely
N−1∑k=0
xkej2πktNTs & uT (t)
1the block index n is dropped from the expressions from here on, unless explicitly specified. Thus xn,k = xk.2Without loss of generality the common term exp(−jπ(N − 1)) is omitted for all analytical treatment from here
on
5
xkx0 x1 xN−2 xN−1. . .
x
IDTFT
N−1∑k=0
xkej2πkt/N
sb(t)
AuT (t)
Figure 3: OFDM Transmitter using IDT FT
Sampling the first waveform at 1/Ts ensure Nyquist sampling and that will result in the samples,
Xm =N−1∑k=0
xkej2πkmN ← m = 0, 1, . . . , N − 1 (11)
which is nothing but the Discrete Fourier Transform of the the data symbols x , [x0, x1, . . . , xN−1]According to Shannon’s sampling theorem, infinite number of samples are required to exactly rep-resent uT (t), which means, that by representing uT (t) by finite number of samples the rectangularnature is distorted.
uT (t) =∞∑
k=−∞
sinc(t
Ts− k)← Ideal Sampling, infinite sample sum
=N−1∑k=0
sinc(t
Ts− k)
+ ε(t)← ε(t) , error (12)
= ha(t) + ε(t)← ha(t) ,N−1∑k=0
sinc(t
Ts− k)
(13)
where
ha(t) ,N−1∑k=0
sinc(t
Ts− k)
(14)
is the distorted rectangular pulse function. The distorted pulse waveform is shown in Fig.(4).From Eq.(11) and Eq.(12) we see that, with finite number (N) of samples, the signal sb(t) can only
be approximated but not exactly represented. The rectangular pulse AuT (t) distort its rectangularshape to become ha as in Eq.(14), due to which the sub carrier orthogonality gets distorted. But,if we can tolerate the moderate amount of sub carrier orthogonality, the OFDM signal sb(t) can be
6
−100 −80 −60 −40 −20 0 20 40 60 80 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
t/Ts
h a(t)
Distorted Pulse
ha(t)
Figure 4: Distorted rectangular pulse
represented by the samples Xm given by
Xm =N−1∑k=0
xkej2πkmN ← m = 0, 1, 2, . . . , N − 1 (15)
Eq.(15) is nothing but the Inverse Discrete Fourier Transform (IDFT). (I)DFT can be computed ef-ficiently by the popular (I) FFT algorithms. The OFDM transmitter structure using FFT algorithmis shown in Fig(5).
4 Combating ISI with OFDM
One of the key advantage cited for OFDM is its ability to mitigate ISI. We will see how this isachieved in OFDM system. To prevent ISI between OFDM blocks a guard interval is insertedbetween the blocks.
We will consider a linear discrete time channel 3. The channel input-output relationship can thusbe written as
Rn =∞∑−∞
gmXn−m , gn ∗Xn (16)
for n = i, i+ 1, . . . , i+ L− 1 where i is the arbitrary first index of a window of width L.Assume that we know that the impulse response gm is zero except for indices in the interval
0, 1, . . . , L− 1. In general, we don’t know the exact value of L but we can easily4 come up with an3The combination of D/A converter, waveform channel g(t), anti aliasing filter and the A/D converter yields a
equivalent discrete time channel with sampled impulse response gmLm=0.4This can be done since we know the speed of light and from the surrounding topology we can upper bound the
longest path along which the signal may travel between the transmit and receive antennas.
7
xkx0 x1 xN−2 xN−1. . .
x
. . .X0 X1 XN−2 XN−1
Xm =N−1∑k=0
xkej2πkm/N
DAC
shb (t) = ha(t)sb(t)
ha(t) =N−1∑k=0
sinc(t/Ts − k)IFFT
Figure 5: FFT based OFDM
L so that gn = 0 ∀ n ≥ L. Hence we may write Eq.(17) as
Rn =L−1∑
0
gmXn−m , gn ∗Xn (17)
From Eq.(17) we see that the value of Rn depends on the L non zero values of the impulseresponse gm and on Xn−L+1, Xn−L+2, . . . , Xn−1, Xn. Eq.(17) tell us that the channel output isnothing but linear convolution of the channel impulse response g , [g0, g1, . . . , gL−1] with inputsample vector X , [X0, X1, . . . , XN−1].
Xk Rkg
Rn = gn ∗Xn =L−1∑m=0
gmXn−m
Figure 6: Linear Discrete Time Channel
Let us write down the expression for the N channel output symbols for the block k. The symbol
8
Rki denotes the ith symbol of the block k.
Rk0 =L−1∑m=0
gmXk−m
= g0Xk0 +
L−1∑1
gmXk+1L−m︸ ︷︷ ︸
ISI
Rk1 =L−1∑m=0
gmXk1−m
=1∑
m=0
gmXk1−m +
L−1∑m=2
gmXk+1L+1−m︸ ︷︷ ︸
ISI
Rk2 =L−1∑m=0
gmXk2−m
=2∑
m=0
gmXk2−m +
L−1∑m=3
gmXk+1L+2−m︸ ︷︷ ︸
ISI
......
...
RkL−2 =L−1∑m=0
gmXkL−2−m
=L−2∑m=0
gmXkL−2−m + gL−1X
k+11︸ ︷︷ ︸
ISI
RkL−1 =L−1∑m=0
gmXkL−1−m ← No ISI (18)
Fig.(7) shows the effect of ISI on the first L− 1 symbols of a block. On the kth block, the firstL− 1 channel outputs are ISI prone.
To avoid the ISI in the first L − 2 symbols of every block of channel output, L Guard symbols(0’s) may be inserted. This effectively removes the ISI terms in Eq.(18).
In FFT based OFDM, at the receiver the first task (after ADC) is to do an FFT to get thetransmitted symbol Xn. Of-course the noise will have its role to play here, due to which the receivercan only make an estimate of the transmitted symbol Xn. But adopting IFFT & FFT (IDFT &DFT) in our OFDM has a problem though. This is partly because of the channel not being strictlydigital. Let us write the equations
To remove ISI we decided to add 0 guard symbols. The channel output after removing the Guardsymbols become
Rg=0n = Xn ∗ gn n = 0, 1, . . . , N − 1 (19)
At the receiver performing FFT (demodulation) will result in the symbol estimate Xn
Xn = DFT (Xn ∗ gn) (20)
Performing DFT on a digitized linear convoluted signals does not offer much in terms of simplicity.Instead, if it were a cyclic convoluted set, on DFT they become just product of individual DFTs.
9
g g g
Rk0 Rk
1RkL−2
L
Xk+10 , Xk+1
1 , . . . , Xk+1N−1 Xk
0 , , Xk1 , . . . , X
kN−1
Figure 7: Inter symbol interference
That is
DFT (a~ b) = DFT (a)×DFT (b) (21)DFT (a ∗ b) 6= DFT (a)×DFT (b) (22)
Can we some how make the linear convoluted channel some how cyclic convoluted one? Theanswer is yes and the trick used is called cyclic prefix (or suffix).
4.1 Cyclic Prefix (or suffix)
Instead of adding 0s as guard symbols, if we use cyclically extended version of the symbols fromthe same block, not only the ISI can be eliminated, but also, the channel output (after removingthe added symbols) can be made look like cyclic convolution of channel coefficients gn with inputsymbols Xn.
4.1.1 Cyclic Suffix
The cyclic suffix guard symbols (G of them) are added onto the channel input as
Xgn = Xmod(n,N) n = 0, 1, 2, . . . , N +G− 1 (23)
It must be noted that adding guard symbols (cyclic prefix or suffix etc) will reduce the transmis-sion rate because
(N +G)T gs = NTs
10
Suppose that the sequence Xgn is transmitted over the DLTI channel with impulse response
g = gnL−10 , the output become
Rgn =L−1∑m=0
gmXgn−m (24)
The receiver first (after ADC) removes the guard symbols according to the formula
Rn = RgG+mod(n−G,N) (25)
If we choose G > L− 1, then the received samples after the removal of Guard interval become
Rn =L−1∑m=0
gmXmod(n−m,N)
= gn ~Xn ← ~ = cyclic convolution (26)
XnN−1n=0
Xgn Rg
n
Rn
Rn
Xgn = X mod (n,N) Rg
n = g ∗Xgn
Guard ChannelremoveGuard
Rn = gn ~Xn
Rn = RgG+ mod (n−G,N)
Xn
Figure 8: Cyclic suffix realizes the circular convolution
Now when the OFDM demodulator perform the DFT (FFT) on the vector R = RnN−1n=0 , the
demodulated sequence become,
Zi = FFT (Rn)= FFT (g ~X)= FFT (g)FFT (X)= FFT (g)FFTIFFT (x)
= xiFFT (g)← FFT (g) =L−1∑m=0
gme−j2πmi/N
= ηxi ← η =L−1∑m=0
gme−j2πmi/N (27)
Note: In the above expressions the pulse amplitude and the IFFT scaling factor (1/N) to be in-corporated. Also the noise in the channel is not yet considered. In the presence of noise a suitableestimator will follow the FFT in the receiver.
11
4.1.2 Cyclic Prefix
The cyclic prefix guard symbols (G of them) are realized by adding the guard symbols as follows.
Xgn = Xmod(n+N−G,N) n = 0, 1, 2, . . . , N +G− 1 (28)
This symbol Xgn (of length (N + G) will go into the FIR channel and output Rgn the linear
convolution with the channel impulse response gmL−10 . This process is shown in Eq.(29).
Rgn =L−1∑m=0
gmXgn−m (29)
Guard symbols are removed from the symbol Rgn based on the mapping Eq(30) to produce Rn.
Rn = RgG+n n = 0, 1, 2, . . . , N − 1 (30)
From Eq.(29) and Eq.(30) we can write
Rn = RgG+n n = 0, 1, 2, . . . , N − 1
=L−1∑m=0
gmXgn+G−m
=L−1∑m=0
gmXmod(n+G−m+N−G,N)
=L−1∑m=0
gmXmod(n−m+N,N)
=L−1∑m=0
gmXmod(n−m,N)
= gn ~Xn (31)
Thus cyclic prefix also helps to realize the circular convolution trick, as does the cyclic suffixdiscussed earlier.
4.2 OFDM Transmitter
A block diagram of Fast Fourier Transform Technique (FFT) based OFDM Transmitter is shown inFig.(9).
4.3 OFDM Receiver
A block diagram of Fast Fourier Transform Technique (FFT) based OFDM receiver is shown inFig.(10).
5 OFDM Impairments
5.1 Channel variations due to Doppler
Channel variations within a block causes loss of orthogonality between sub channels (ICI). Assumethe channel has impulse response hm,k at position m and time instant k (for LTI hm,k = hmindependent of k) where m = 0, 1, 2, . . . , L− 1. The receiver removes the guard symbol according to
Rk = RgG+mod(k−G,N (32)
12
xkx0 x1 xN−2 xN−1
IFFT
D/A
D/A
XN−2 XN−1X0 X1 . . .
. . .
Gaurd
xn
Xn
s(I)(t)
s(Q)(t)
X(Q),gn
X(I),gn
Figure 9: FFT based OFDM transmitter
r(I)(t)
r(Q)(t)
ADC
ADC
removeguard
EstimatexZ
R0 R1 . . . RN−1
Z0 Z1 ZN−1. . .
FFT
Figure 10: FFT based OFDM receiver
13
and performs an FFT on the resulting sequence to get Zl
Zl =1N
N−1∑n=0
Rne−j2πlnN
=1N
N−1∑n=0
RgG+mod(n−G,N)e−j2πlnN
=1N
N−1∑n=0
e−j2πlnN
L−1∑m=0
hm,G+mod(n−G,N)XgG−m+mod(n−G,N)
=1N
N−1∑n=0
e−j2πlnN
L−1∑m=0
hm,G+mod(n−G,N)
[N−1∑i=0
xi exp(j2πi(G−m+ mod (n−G,N))
N
)]
=1N
N−1∑n=0
L−1∑m=0
N−1∑i=0
xie−j2πlnN hm,G+mod(n−G,N) exp
(j2πi(G−m+ mod (n−G,N))
N
)
=1N
N−1∑i=0
L−1∑m=0
N−1∑n=0
xie−j2πlnN hm,G+mod(n−G,N) exp
(j2πi(G−m+ mod (n−G,N))
N
)
=N−1∑i=0
L−1∑m=0
xiej2πi(G−m)
N1N
N−1∑n=0
e−j2πlnN hm,G+mod(n−G,N) exp
(j2πi( mod (n−G,N))
N
)
=N−1∑i=0
L−1∑m=0
xie−j2πim
N1N
N−1∑n=0
e−j2πlnN hm,G+mod(n−G,N)e
j2πn(i−l)N︸ ︷︷ ︸
,Hm(i−l)
=N−1∑i=0
L−1∑m=0
xie−j2πim
N Hm(i− l)
=L−1∑m=0
xle−j2πim
N Hm(0) +N−1∑i 6=l
L−1∑m=0
xie−j2πim
N Hm(i− l)
= xl
L−1∑m=0
e−j2πim
N Hm(0)︸ ︷︷ ︸η
+N−1∑i 6=l
L−1∑m=0
xie−j2πim
N Hm(i− l)︸ ︷︷ ︸cl
= ηxl + cl (33)
From Eq.(33) we see that channel variation causes a random ICI term cl. When no time variationsduring block period (hm,k = hm), we can see that Hm(i− l) = hmδil as shown below.
Hm(i− l) ,1N
N−1∑n=0
e−j2πlnN hm,G+mod(n−G,N)e
j2πn(i−l)N
=1N
N−1∑n=0
hme−j2πn(i−l)
N ← hm,k = hm
= hm1N
N−1∑n=0
e−j2πn(i−l)
N
= hmδil (34)
14
5.2 Residual Carrier Frequency Offset
Residual carrier frequency offset also causes Inter channel Interference (ICI). We will see how thisis .
Let ∆f be the carrier frequency offset. The channel input will be
Xk =N−1∑n=0
xnej2π(nkN +k∆fTs) k = 0, 1, . . . , N +G− 1 (35)
The demodulated sequence (in the absence of noise) can be written as
Zl = FFTXn
= xlsin[π(NTs∆f)π(NTs∆f)
ejπNTs∆f︸ ︷︷ ︸,η
+∑n 6=l
sin[π(n− l +NTs∆f)]n− l +NTs∆f
ejπ(n−l+NTs∆f)
︸ ︷︷ ︸,H(n,l)
= ηxl + cl (36)
We can see that carrier frequency offset has two effects
1. It reduces the useful energy by the factor(sin[π(NTs∆f)π(NTs∆f)
)2
2. Introduces an additional noise term cl
6 Why OFDM? Clues from Information Theory
The OFDM system typically operates over non-ideal channel with transfer function T (f), such thatthe amplitude response |T (f)| is not constant across the channel bandwidth. According to Shannon,the capacity for a non AWGN channel is achieved by water pouring such a way that
Ωt(f) =
K − Snn(f)
|T (f)|2 f ∈ W
b otherwise(37)
where K is a constant chosen to satisfy the power constraint∫W
Ωt(f)df ≤ Ωav
One method to achieve capacity is to slice the spectrum to make it some thing like parallelGaussian channels.
7 PowerSpectral Density
S(f) = A2Tfσ2x
N−1∑k=0
|Ha
(f − 1
Tk − N − 1
2
)|2 (38)
To be discussed later
15
8 Cyclic Prefix OFDM: A system analysis
In this section we will look into the cyclic prefix based OFDM system. Fig.(11) shows the conceptualdiscrete time representation of a CP-OFDM transmission. The DAC, Power amplifier and otheranalog partitions not considered. We will follow a linear algebraic formulation of the system.
xN(m)FHN
XgN(m) L−1∑
l=0
hlXgn−l
WGN
RgN(m)
XN(m)
XCP (m)
P/S
Figure 11: CP-OFDM
The input data stream xN (m) for the mth block entering the IFFT block is an N element vector5
which can be written as,
xN (m) =
x0(m)x1(m)x2(m)x3(m)
...xN−2(m)xN−1(m)
(39)
The IFFT output vector XN (m) can also be written in similar form as
XN (m) =
X0(m)X1(m)X2(m)X3(m)
...XN−2(m)XN−1(m)
(40)
It is well known and obvious that IFFT matrix is the hermitian of the Fourier matrix. Wewill denote the IFFT matrix thus as FHN , where H denotes the Hermitian operation and FN is theFourier matrix of order N ×N . Fourier matrix can be written explicitly in matrix form as,
FN =
1 1 1 . . . . . . 1 11 W W 2 . . . . . . WN−2 WN−1
1 W 2 W 4 . . . . . . W 2(N−2) W 2(N−1)
1 W 3 W 6 . . . . . . W 3(N−2) W 3(N−1)
......
......
......
......
. . . . . . . . . . . . . . ....
......
......
......
...1 WN−2 W 2(N−2) . . . . . . W (N−2)(N−2) W (N−2)(N−1)
1 WN−1 W 2(N−1) . . . . . . W (N−2)(N−1) W (N−1)(N−1)
(41)
5We will follow our favourite and perhaps more elegant column space approach
16
The matrix FHN will now assume the form,
FHN =
1 1 1 . . . . . . 1 11 W−1 W−2 . . . . . . W−(N−2) W−(N−1)
1 W−2 W−4 . . . . . . W−2(N−2) W−2(N−1)
1 W−3 W−6 . . . . . . W−3(N−2) W−3(N−1)
......
......
......
......
. . . . . . . . . . . . . . ....
......
......
......
...1 W−(N−2) W−2(N−2) . . . . . . W−(N−2)(N−2) W−(N−2)(N−1)
1 W−(N−1) W−2(N−1) . . . . . . W−(N−2)(N−1) W−(N−1)(N−1)
(42)
The two vectors XN (m) and xN (m) are related by the linear matrix relationship,
XN (m) = FHNxN (m) (43)
That is,
X0(m)X1(m)X2(m)X3(m)
...XN−2(m)XN−1(m)
=
1 1 1 . . . . . . 11 W−1 W−2 . . . . . . W−(N−1)
1 W−2 W−4 . . . . . . W−2(N−1)
1 W−3 W−6 . . . . . . W−3(N−1)
.... . . . . . . . . . . .
...1 W−(N−2) W−2(N−2) . . . . . . W−(N−2)(N−1)
1 W−(N−1) W−2(N−1) . . . . . . W−(N−1)(N−1)
x0(m)x1(m)x2(m)x3(m)
...xN−2(m)xN−1(m)
(44)
Adding cyclic prefix is essentially duplicating the lower G rows of XN (m) and appending it ontop as G rows. We can therefore write the following matrix equations.
XgN (m) =
Xg0 (m)
Xg1 (m)
Xg2 (m)...
XgN−1(m)XgN (m)
XgN+1(m)
...XgN+G−2(m)
XgN+G−1(m)
=
XN−G+1(m)XN−G+2(m)XN−G+3(m)
...XN−1(m)X0(m)X1(m)
...XN−2(m)XN−1(m)
(45)
We can write the relationship as,
17
XN−G+1(m)XN−G+2(m)XN−G+3(m)
...XN−1(m)X0(m)X1(m)X2(m)X3(m)
...XN−2(m)XN−1(m)
=
1 W−(N−G+1) W−2(N−G+1) . . . . . . W−(N−G+1)(N−1)
1 W−(N−G+2) W−2(N−G+2) . . . . . . W−(N−G+2)(N−1)
1 W−(N−G+3) W−2(N−G+3) . . . . . . W−(N−G+3)(N−1)
......
......
......
1 W−(N−1) W−2(N−1) . . . . . . W−(N−1)(N−1)
1 1 1 . . . . . . 11 W−1 W−2 . . . . . . W−(N−1)
1 W−2 W−4 . . . . . . W−2(N−1)
1 W−3 W−6 . . . . . . W−3(N−1)
.... . . . . . . . . . . .
...1 W−(N−2) W−2(N−2) . . . . . . W−(N−2)(N−1)
1 W−(N−1) W−2(N−1) . . . . . . W−(N−1)(N−1)
x0(m)x1(m)x2(m)x3(m)
...xN−2(m)xN−1(m)
(46)Or simply written as,
XgN (m) = FH,gN xN (m) (47)
where FH,gN is the (N + G) ×N matrix obtained by cyclic prefixing the matrix FHN with the lowerG rows of itself, as illustrated below.
FH,gN ,
1 W−(N−G+1) W−2(N−G+1) . . . . . . W−(N−G+1)(N−1)
1 W−(N−G+2) W−2(N−G+2) . . . . . . W−(N−G+2)(N−1)
1 W−(N−G+3) W−2(N−G+3) . . . . . . W−(N−G+3)(N−1)
......
......
......
1 W−(N−1) W−2(N−1) . . . . . . W−(N−1)(N−1)
1 1 1 . . . . . . 11 W−1 W−2 . . . . . . W−(N−1)
1 W−2 W−4 . . . . . . W−2(N−1)
1 W−3 W−6 . . . . . . W−3(N−1)
.... . . . . . . . . . . .
...1 W−(N−2) W−2(N−2) . . . . . . W−(N−2)(N−1)
1 W−(N−1) W−2(N−1) . . . . . . W−(N−1)(N−1)
=[FgCP(m)
FHN
](48)
Where FgCP(m) is a G×N matrix formed by the lower G rows of the inverse fourier matrix FHN .The N +G element6 channel output vector Rg
N (m) (after the inclusion of AWGN noise) can bewritten as,
RgN (m) = HXg
N (m) + HIBIXgN (m− 1)︸ ︷︷ ︸IBI
+ nN+G(m)︸ ︷︷ ︸Noise
(49)
6Note that the Guard symbols added in the form of cyclic prefix is not yet removed. After removing the guardsymbols the vector will assume the nominal N element vector
18
The channel matrices H and HIBI are given by,
H ,
h0 0 0 0 . . . 0 0 . . . 0 0h1 h0 0 0 . . . 0 0 . . . 0 0h2 h1 h0 0 . . . 0 0 . . . 0 0...
......
......
... . . . 0 0 0hL−2 hL−3 . . . h1 h0 0 0 . . . 0 0hL−1 hL−2 . . . h2 h1 h0 0 . . . 0 0
0 hL−1 hL−2 . . . h2 h1 h0 . . . 0 0...
......
......
......
......
......
......
......
......
......
...0 . . . 0 . . . hL−1 hL−2 . . . h1 h0 00 . . . 0 . . . 0 hL−1 hL−2 . . . h1 h0
(50)
HIBI ,
0 . . . 0 hL−1 hL−2 . . . h2 h1
0 . . . . . . 0 hL−1 . . . h3 h2
0 . . ....
......
......
...0 . . . . . . 0 0 . . . hL−1 hL−2
0 . . . . . . 0 0 . . . 0 hL−1
0 . . . . . . 0 0 . . . 0 0
0 . . ....
......
......
...0 . . . . . . 0 0 . . . 0 0
(51)
The matrices H and HIBI are (N+G)×(N+G). Removing the guard bits is essentially choppingthe top G rows of the two matrices. Thus we can express the channel output (noise omitted herefor clarity) after the guard symbol removal through the following equation. For L = G,
RN (m) =
hL−1 hL−2 . . . . . . h1 h0 0 . . . 0 00 hL−1 . . . . . . h2 h1 h0 . . . 0 0...
......
......
......
......
...0 . . . 0 . . . 0 hL−1 hL−2 . . . h1 h0
0 0 0 0 . . . 0 0 . . . 0 00 0 0 0 . . . 0 0 . . . 0 00 0 0 0 . . . 0 0 . . . 0 0
XN−G+1(m)...
XN−2(m)XN−1(m)X0(m)X1(m)X2(m)X3(m)
...XN−G+1(m)
...XN−2(m)XN−1(m)
(52)
19
For the general case with L > G the above equation becomes a special case as,
RN (m) =
0 . . . hL−1 . . . h1 h0 0 . . . 0 00 . . . 0 . . . h2 h1 h0 . . . 0 0...
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...0 . . . 0 . . . hL−1 hL−2 . . . h1 h0 00 . . . 0 . . . 0 hL−1 hL−2 . . . h1 h0
XN−G+1(m)...
XN−2(m)XN−1(m)X0(m)X1(m)X2(m)X3(m)
...XN−G+1(m)
...XN−2(m)XN−1(m)
(53)
We can see that the above expression has a circulant matrix property using which we can rewritethe expression as follows
RN (m) =
h0 . . . 0 . . . 0 hL−1 hL−2 . . . h2 h1
h1 h0 0 . . . 0 0 hL−1 . . . h3 h2
h2 h1 h0 . . . 0 0 0 . . . h4 h3
h3 h2 h1 . . . 0 0 0 . . . h5 h4
......
......
......
......
......
hL−2 hL−3 hL−4 . . . h0 0 0 . . . 0 hL−1
hL−1 hL−2 hL−3 . . . h1 h0 0 . . . 0 0...
......
......
......
......
...0 . . . . . . hL−1 hL−2 hL−3 hL−4 . . . h0 00 . . . 0 0 hL−1 hL−2 hL−3 . . . h1 h0
X0(m)X1(m)X2(m)X3(m)
...XN−G+1(m)
...XN−2(m)XN−1(m)
(54)
That is,
RN (m) = CN (h)XN (m) + nN (m)= CN (h)FHNxN (m) + nN (m)︸ ︷︷ ︸
Noise
(55)
8.1 Equalization of CP-OFDM
Equalization of CP-OFDM is quite simple, thanks to the property of the circulant matrix CN (h).It is well known that every circulant matrix can be diagonalized by pre (post) multiplication by (I)FFT matrices. (Fourier meets Shannon here!!). Let us first write down the math here.
xN = FNCN (h)FHNxN (m) + FNnN (m)= diag(H0,H1, . . . ,HN−1)xN (m) + FNnN (m)
9 Zero Padded OFDM (ZP-OFDM)
The ZP-OFDM system is conceptually shown in Fig.(12). The only difference with CP-OFDM isthat the CP is replaced by G trailing zeroes that are padded at each precoded block XN (m). Wecan thus write the input-output relationship as follows.
20
XgN(m) ∑
WGN
RgN(m)
P/S
xN(m) FHN
0
XN(m)
Figure 12: ZP-OFDM
The input data stream xN (m) for the mth block entering the IFFT block is an N element vector7
which can be written as,
xN (m) =
x0(m)x1(m)x2(m)x3(m)
...xN−2(m)xN−1(m)
(56)
The IFFT output vector XN (m) can also be written in similar form as
XN (m) =
X0(m)X1(m)X2(m)X3(m)
...XN−2(m)XN−1(m)
(57)
It is well known and obvious that IFFT matrix is the hermitian of the Fourier matrix. Wewill denote the IFFT matrix thus as FHN , where H denotes the Hermitian operation and FN is theFourier matrix of order N ×N . Fourier matrix can be written explicitly in matrix form as,
FN =
1 1 1 . . . . . . 1 11 W W 2 . . . . . . WN−2 WN−1
1 W 2 W 4 . . . . . . W 2(N−2) W 2(N−1)
1 W 3 W 6 . . . . . . W 3(N−2) W 3(N−1)
......
......
......
......
. . . . . . . . . . . . . . ....
......
......
......
...1 WN−2 W 2(N−2) . . . . . . W (N−2)(N−2) W (N−2)(N−1)
1 WN−1 W 2(N−1) . . . . . . W (N−2)(N−1) W (N−1)(N−1)
(58)
7We will follow our favourite and perhaps more elegant column space approach
21
The matrix FHN will now assume the form,
FHN =
1 1 1 . . . . . . 1 11 W−1 W−2 . . . . . . W−(N−2) W−(N−1)
1 W−2 W−4 . . . . . . W−2(N−2) W−2(N−1)
1 W−3 W−6 . . . . . . W−3(N−2) W−3(N−1)
......
......
......
......
. . . . . . . . . . . . . . ....
......
......
......
...1 W−(N−2) W−2(N−2) . . . . . . W−(N−2)(N−2) W−(N−2)(N−1)
1 W−(N−1) W−2(N−1) . . . . . . W−(N−2)(N−1) W−(N−1)(N−1)
(59)
The two vectors XN (m) and xN (m) are related by the linear matrix relationship,
XN (m) = FHNxN (m) (60)
That is,
X0(m)X1(m)X2(m)X3(m)
...XN−2(m)XN−1(m)
=
1 1 1 . . . . . . 11 W−1 W−2 . . . . . . W−(N−1)
1 W−2 W−4 . . . . . . W−2(N−1)
1 W−3 W−6 . . . . . . W−3(N−1)
.... . . . . . . . . . . .
...1 W−(N−2) W−2(N−2) . . . . . . W−(N−2)(N−1)
1 W−(N−1) W−2(N−1) . . . . . . W−(N−1)(N−1)
x0(m)x1(m)x2(m)x3(m)
...xN−2(m)xN−1(m)
(61)
In ZP-OFDM, the zero padding is nothing but adding G rows of XN (m) by zeroes.
XgN (m) =
Xg0 (m)
Xg1 (m)
Xg2 (m)...
XgN−1(m)XgN (m)
XgN+1(m)
...XgN+G−2(m)
XgN+G−1(m)
=
X0(m)X1(m)
...XN−2(m)XN−1(m)
000...0
(62)
We can write the relationship as,
22
X0(m)X1(m)X3(m)X4(m)
...XN−2(m)XN−1(m)
000...0
=
1 1 1 . . . . . . 11 W−1 W−2 . . . . . . W−(N−1)
1 W−2 W−4 . . . . . . W−2(N−1)
1 W−3 W−6 . . . . . . W−3(N−1)
.... . . . . . . . . . . .
...1 W−(N−2) W−2(N−2) . . . . . . W−(N−2)(N−1)
1 W−(N−1) W−2(N−1) . . . . . . W−(N−1)(N−1)
0 0 0 . . . . . . 00 0 0 . . . . . . 00 0 0 . . . . . . 0...
......
......
...0 0 0 . . . . . . 0
x0(m)x1(m)x2(m)x3(m)
...xN−2(m)xN−1(m)
(63)
Or simply written as,XgN (m) = FH,gN xN (m) (64)
where FH,gN is the (N +G)×N matrix obtained by adding G lower zero rows to the matrix FHN asillustrated below.
FH,gN ,
1 1 1 . . . . . . 11 W−1 W−2 . . . . . . W−(N−1)
1 W−2 W−4 . . . . . . W−2(N−1)
1 W−3 W−6 . . . . . . W−3(N−1)
.... . . . . . . . . . . .
...1 W−(N−2) W−2(N−2) . . . . . . W−(N−2)(N−1)
1 W−(N−1) W−2(N−1) . . . . . . W−(N−1)(N−1)
0 0 0 . . . . . . 00 0 0 . . . . . . 00 0 0 . . . . . . 0...
......
......
...0 0 0 . . . . . . 0
=[
FHNFgZP(m)
]=[FHN0
](65)
Where FgZP(m) is a G×N 0 matrix and FHN is the inverse fourier matrix.The N +G element8 channel output vector Rg
N (m) (after the inclusion of AWGN noise) can bewritten as,
RgN (m) = HXg
N (m) + HIBIXgN (m− 1)︸ ︷︷ ︸IBI
+ nN+G(m)︸ ︷︷ ︸Noise
(66)
8Note that the Guard symbols added in the form of 0’s are not yet removed. After removing the guard symbolsthe vector will assume the nominal N element vector
23
The channel matrices H and HIBI are given by,
H ,
h0 0 0 0 . . . 0 0 . . . 0 0h1 h0 0 0 . . . 0 0 . . . 0 0h2 h1 h0 0 . . . 0 0 . . . 0 0...
......
......
... . . . 0 0 0hL−2 hL−3 . . . h1 h0 0 0 . . . 0 0hL−1 hL−2 . . . h2 h1 h0 0 . . . 0 0
0 hL−1 hL−2 . . . h2 h1 h0 . . . 0 0...
......
......
......
......
......
......
......
......
......
...0 . . . 0 . . . hL−1 hL−2 . . . h1 h0 00 . . . 0 . . . 0 hL−1 hL−2 . . . h1 h0
(67)
HIBI ,
0 . . . 0 hL−1 hL−2 . . . h2 h1
0 . . . . . . 0 hL−1 . . . h3 h2
0 . . ....
......
......
...0 . . . . . . 0 0 . . . hL−1 hL−2
0 . . . . . . 0 0 . . . 0 hL−1
0 . . . . . . 0 0 . . . 0 0
0 . . ....
......
......
...0 . . . . . . 0 0 . . . 0 0
(68)
The matrices H and HIBI are (N +G)× (N +G).Since G ≥ L, we can see that the IBI term will vanish. We will show it through the neat matrix
formulation.
HIBIXgN (m− 1) ,
0 . . . 0 hL−1 hL−2 . . . h2 h1
0 . . . . . . 0 hL−1 . . . h3 h2
0 . . ....
......
......
...0 . . . . . . 0 0 . . . hL−1 hL−2
0 . . . . . . 0 0 . . . 0 hL−1
0 . . . . . . 0 0 . . . 0 0
0 . . ....
......
......
...0 . . . . . . 0 0 . . . 0 0
X0(m− 1)X1(m− 1)X3(m− 1)X4(m− 1)
...XN−2(m− 1)XN−1(m− 1)
000...0
= 0 (69)
Hence the channel output can be written as,
RgN (m) = HXg
N (m) + nN+G(m)︸ ︷︷ ︸Noise
(70)
24
where
HXgN (m) ,
h0 0 0 0 . . . 0 0 . . . 0 0h1 h0 0 0 . . . 0 0 . . . 0 0h2 h1 h0 0 . . . 0 0 . . . 0 0...
......
......
... . . . 0 0 0hL−2 hL−3 . . . h1 h0 0 0 . . . 0 0hL−1 hL−2 . . . h2 h1 h0 0 . . . 0 0
0 hL−1 hL−2 . . . h2 h1 h0 . . . 0 0...
......
......
......
......
......
......
......
......
......
...0 . . . 0 . . . 0 hL−1 hL−2 . . . h1 h0
0 0 0 0 . . . 0 0 . . . 0 00 0 0 0 . . . 0 0 . . . 0 00 0 0 0 . . . 0 0 . . . 0 0
X0(m)X1(m)X3(m)X4(m)
...XN−3(m)XN−2(m)XN−1(m)
000...0
(71)
Removing the guard bits in ZP-OFDM is essentially chopping the bottom G ≥ L rows of thematrix HXg
N (m) to get the (N +G)×N matrix H0. Thus we can express the channel output (noiseomitted here for clarity) after the guard symbol removal through the following equation.
RN (m) , H0 =
h0 0 0 0 . . . 0 0 . . . 0 0h1 h0 0 0 . . . 0 0 . . . 0 0h2 h1 h0 0 . . . 0 0 . . . 0 0...
......
......
... . . . 0 0 0hL−2 hL−3 . . . h1 h0 0 0 . . . 0 0hL−1 hL−2 . . . h2 h1 h0 0 . . . 0 0
0 hL−1 hL−2 . . . h2 h1 h0 . . . 0 0...
......
......
......
......
......
......
......
......
......
...0 . . . 0 . . . 0 hL−1 hL−2 . . . h1 h0
0 0 0 0 . . . 0 0 . . . 0 00 0 0 0 . . . 0 0 . . . 0 00 0 0 0 . . . 0 0 . . . 0 0
X0(m)X1(m)X3(m)X4(m)
...XN−3(m)XN−2(m)XN−1(m)
000...0
(72)
For L = G,
RN (m) =
hL−1 hL−2 . . . . . . h1 h0 0 . . . 0 00 hL−1 . . . . . . h2 h1 h0 . . . 0 0...
......
......
......
......
...0 . . . 0 . . . 0 hL−1 hL−2 . . . h1 h0
0 0 0 0 . . . 0 0 . . . 0 00 0 0 0 . . . 0 0 . . . 0 00 0 0 0 . . . 0 0 . . . 0 0
X0(m− 1)X1(m− 1)X3(m− 1)X4(m− 1)
...XN−2(m− 1)XN−1(m− 1)
000...0
(73)
25
For the general case with L > G the above equation becomes a special case as,
RN (m) =
0 . . . hL−1 . . . h1 h0 0 . . . 0 00 . . . 0 . . . h2 h1 h0 . . . 0 0...
......
......
......
......
......
......
......
......
......
...0 . . . 0 . . . 0 hL−1 hL−2 . . . h1 h0
0 0 0 0 . . . 0 0 . . . 0 00 0 0 0 . . . 0 0 . . . 0 00 0 0 0 . . . 0 0 . . . 0 0
X0(m− 1)X1(m− 1)X3(m− 1)X4(m− 1)
...XN−2(m− 1)XN−1(m− 1)
000...0
(74)
which can be written in a simplified version (thanks to the many 0’s present in the column vector)as,
We can see that the above expression has a circulant matrix property using which we can rewritethe expression as follows
RN (m) =
h0 . . . 0 . . . 0 hL−1 hL−2 . . . h2 h1
h1 h0 0 . . . 0 0 hL−1 . . . h3 h2
h2 h1 h0 . . . 0 0 0 . . . h4 h3
h3 h2 h1 . . . 0 0 0 . . . h5 h4
......
......
......
......
......
hL−2 hL−3 hL−4 . . . h0 0 0 . . . 0 hL−1
hL−1 hL−2 hL−3 . . . h1 h0 0 . . . 0 0...
......
......
......
......
...0 . . . . . . hL−1 hL−2 hL−3 hL−4 . . . h0 00 . . . 0 0 hL−1 hL−2 hL−3 . . . h1 h0
X0(m− 1)X1(m− 1)X3(m− 1)X4(m− 1)
...XN−2(m− 1)XN−1(m− 1)
000...0
(75)
That is,
RN (m) = CN (h)XN (m) + nN (m)= CN (h)FHNxN (m) + nN (m)︸ ︷︷ ︸
Noise
(76)
10 Acknowledgement
I would like to thank Dr. Herbert Dawid for the useful discussions on clarifying my doubts.
References
[1] R.W. Chang, Orthogonal Frequency Division Multiplexing, U.S. Patent 3,488,445, filed 1966,issued Jan. 6, 1970
[2] L. Hanzo, W. Webb and T. Keller, Single and Mult-carrier Quadrature Amplitude Modulation,Wiley International, 2000
26
[3] G. Stuber,Principles of Mobile Communication 2nd Edition
[4] J. A. C. Bingham, ”Multicarrier Modulation for Data Transmission: An Idea Whose Time HasCome,” IEEE Communications Magazine, pp. 5-14, May 1990.
[5] S. B. Weinstein and P. M. Ebert, ”Data Transmission by Frequency Division Multiplexing Usingthe Discrete Fourier Transform,” IEEE Trans. Commun.,Vol. 19 No. 5, pp. 628-633, October 1971
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