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Orthogonal Frequency Division Multiplexing
PhD First Progress Seminar Report
Submitted in the partial fulfillment of the requirements
for the degree of
Doctor of philosophy
by
Santosh V Jadhav
Roll No. 02407802
under the guidance of
Prof. Vikram M Gadre
Department of Electrical Engineering
Indian Institute of Technology, Bombay
August 2003
1
Acknowledgement I would like to express my sincere gratitude to my guide, Prof. Dr. Vikram M
Gadre for his help and guidance during the course of my first progress seminar.
Thanks to Dr K L Asanare, Director, FAMT for his continuous encouragement
and support. Also thanks to mathematics professor Dr Ch V Ramanamurthy for
helping me during this seminar preparation.
Santosh V Jadhav
August 2003
2
Contents Abstract
1. Single carrier to multi-carrier and multi-carrier to OFDM
1.1 Single Carrier modulation
1.2 Multicarrier approach
1.3 Orthogonal Frequency Division Multiplexing
2. Band-Limited Orthogonal Signals for Multichannel Data Transmission 2.1 Need of orthogonality in time-functions
2.2 Synthesis of OFDM signals for multichannel data transmission
2.2.1 Orthogonal multiplexing using band-limited signals
2.2.2 Transmission filter characteristics
2.3. Receiver structure for multichannel orthogonal signal
3. OFDM: FFT Based Approach
3.1 The Discrete-Time Model
3.2 Modulation of OFDM data using IDFT
3.3 Orthogonality
3.4 The cyclic prefix
3.5 Demodulation using the DFT
4. The Downsides of OFDM
5. Future Directions
References
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Abstract Orthogonal frequency division multiplexing (OFDM) is used for transmitting a
number of data messages simultaneously through a linear band-limited transmission
medium at a maximum data rate without interchannel and intersymbol interferences.
Chapter-1 gives an idea about how OFDM is evolved from single carrier and
multicarrier modulation schemes. Chapter-2 discusses a general method for
synthesizing an infinite number of classes of band-limited orthogonal time functions
in a limited frequency band. Stated in practical terms, the method permits the
synthesis of a large class of practical transmitting filter characteristics for an
arbitrarily given amplitude characteristics of the transmission medium. Chapter-3
explains how Fourier transform data communication system acts as a realization of
OFDM in which discrete Fourier transform (DFT) and inverse DFT are computed as
part of the modulation and demodulation processes respectively. Chapter-4 and 5
gives drawbacks of OFDM and how endless appetite to achieve capacity, maximum
bit rate and minimum bit error rate (BER) gives rise to the future research scopes in
this field.
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Chapter 1.
Single carrier to multi-carrier and multi-carrier to OFDM
Nonideal linear filter channels introduce intersymbol interference (ISI), which
degrades performance compared with the ideal channel. The degree of performance
degradation depends on the frequency-response characteristics. Furthermore, the
complexity of the receiver increases as the span of the ISI increases. Given a
particular channel characteristic, the communication system designer must decide
how to efficiently utilize the available channel bandwidth in order to transmit the
information reliably within the transmitter power constraint and receiver complexity
constraints.
1.1 Single Carrier modulation : For a nonideal linear filter channel, one option is to employ a single carrier (SC)
modulation system in which the information sequence is transmitted serially at some
specified rate R symbols/s. In such a channel, the time dispersion is generally much
greater than the symbol rate, and, hence, ISI results from the non-ideal frequency-
response characteristics of the channel. If the level of interference is sufficient to
noticeably deteriorate the radio link, then an equalizer is used at the receiver to
remove the effects of ISI.
1.2 Multicarrier approach : An alternative approach to the design of a bandwidth-efficient communication
system in the presence of channel distortion is to subdivide the available channel
bandwidth into a number of subchannels, such that each subchannel is nearly ideal.
To elaborate, suppose that H(f) is the frequency response of a non-ideal, band-limited
channel with a bandwidth B, and that the power spectral density of the additive
Gaussian noise is φnn(f). Then, divide the bandwidth B into N=B/∆f subbands of width
∆f, where ∆f is chosen sufficiently small that |H(f)|2/φnn(f) is approximately a constant
within each subband. Furthermore, we select the transmitted signal power to be
distributed in frequency as P(f), subject to the constraint that
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∫B P(f) df ≤ Pav (1.1)
where Pav is the available average power of the transmitter. Then, we transmit the data
on these N subchannels.
Recall that the capacity of an ideal, band-limited, AWGN channel is
Pav
C = B log2[1 + ] (1.2) BN0
where C is the capacity in bits/s, B is the channel bandwidth, and Pav is the average
transmitted power. In a multicarrier system, with ∆f sufficiently small, the subchannel
has capacity
∆f P(fi) |H(fi)|2
Ci = ∆f log2[1+ ] (1.3) ∆f φnn(fi)
Hence, the total capacity of the channel is
N N P(fi) |H(fi)|2
C =∑ Ci = ∆f ∑ log2[1+ ] (1.4) i=1 i=1 φnn(fi)
This suggests that multicarrier modulation (MCM) that divides the available
channel bandwidth into subbands of relatively narrow width ∆f=B/N provides a
solution that could yield transmission rates close to capacity. The signal in each
subband may be independently coded and modulated at a synchronous symbol rate of
1/∆f, with the optimum power allocation P(f). If ∆f is small enough, then H(f) is
essentially constant across each subband, so that no equalization is necessary because
the ISI is negligible due to effectively reduced symbol rate in each subband. The
dominant noise in transmission over subscriber lines is crosstalk interference from
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signals carried on other telephone lines located in the same cable. The power
distribution of this type of noise is also frequency-dependent, which can be taken into
consideration in the allocation of the available transmitted power.
1.3 Orthogonal Frequency Division Multiplexing : Orthogonal Frequency Division Multiplexing (OFDM), a special form of MCM
with densely spaced subcarriers and overlapping spectra was patented in the U.S. in
1970 [1]. OFDM abandoned the use of steep bandpass filters that completely
separated the spectrum of individual subcarriers, as it was common practice in older
Frequency Division Multiplex (FDMA) systems (e.g. in analogue SSB telephone
trunks), in Multi-Tone telephone modems and still occurs in Frequency Division
Multiple Access radio. In stead, OFDM time-domain waveforms are chosen such that
mutual orthogonality is ensured even though subcarrier spectra may overlap. This
feature of OFDM is explained in detail in next chapter. It appeared that such
waveforms can be generated using a Fast Fourier Transform at the transmitter and
receiver [2, 3]. For a relatively long time, the practicality of the concept appeared
limited. Implementation aspects such as the complexity of a real-time Fourier
Transform appeared prohibitive, not to speak about the stability of oscillators in
transmitter and receiver, the linearity required in RF power amplifiers and the power
back-off associated with this. After many years of further intensive research in the
1980's, e.g. [4, 5, 6], today we appear to be on the verge of a breakthrough of MCM
techniques. Many of the implementational problems appear solvable [7] and MCM
has become part of several standards.
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Chapter 2.
Band-Limited Orthogonal Signals for Multichannel Data
Transmission
2.1 Need of orthogonality in time-functions :
Consider a set of orthogonal functions φ1(θ), φ2(θ), … , φi(θ) in the region
–½ ≤ θ ≤ +½. From the definition of orthogonal functions[8] follows-
+½
∫ φl(θ) φm(θ) dθ = K for l = m (2.1) –½ = 0 for l ≠ m
K is assumed to normalized to 1.
If the functions φ(θ) are multiplied by 1 or 0 and added, one obtains a function f(θ).
For example, if φ1(θ), φ2(θ), φ3(θ) are multiplied by 1 and all other φ(θ) by 0, the
result is the function-
f(θ) = φ1(θ) + φ2(θ) + φ3(θ) (2.2)
Because of the orthogonality of the functions φ(θ) one may readily decompose f(θ)
to determine which of the functions φ1(θ) … φi(θ) have been multiplied by 1.
Application of (2.1) yields-
+½
∫ f(θ) φl(θ) dθ = 1 for l =1,2,3 (2.3) –½ = 0 for l =4,5,…
Figure 2.1 shows a diagram of a communication system which uses the normalized
orthogonal system of sine and cosine functions 1, √2Sin 2kπθ, √2Cos 2kπθ; θ = t/τ,
k=1,2,… The first five of these functions are shown in figure 2.2. Assume that the
five digits character 11001 is to be transmitted. The switches (multipliers) in the
transmitter let through 1, √2Sin 2πθ and √2Cos 4πθ, but not √2Cos 2πθ and
√2Sin 4πθ. The adder adds the voltages and yield –
f(θ) = 1 + √2Sin 2πθ + √2Cos 4πθ (2.4)
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Figure 2.1: Communication system using set of orthogonal functions
Figure 2.2: Set of orthogonal trigonometric functions
The function f(θ) varies continuously with time and is similar to a voice signal. It
may be transmitted by amplitude or frequency modulation, or any other type of
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modulation suitable for the transmission of continuously varying time functions. In
the receiver f(θ) is multiplied by the same five functions which were used in the
transmitted, the integral over the period τ is taken and the character 11001 multiplied
by τ is recovered
+½
τ ∫ f(θ)(1) dθ = τ –½
+½
τ ∫ f(θ) √2Sin 2πθ dθ = τ –½
+½
τ ∫ f(θ) √2Cos 2πθ dθ = 0 –½
+½
τ ∫ f(θ) √2Sin 4πθ dθ = 0 –½
+½
τ ∫ f(θ) √2Cos 4πθ dθ = τ –½
Here only trigonometric functions are considered but there is no difficulty other than
computational in applying other sets of orthogonal functions.
Bandwidth required for the communication system may be derived as follows:
f(θ) is the sum of several sine and cosine terms; hence its frequency spectrum is the
sum of the frequency spectra of the sine and cosine terms. The normalized frequency
spectra for 1, √2Sin 2kπθ and √2Cos 2kπθ, (-½ ≤ θ ≤ +½), may be readily computed
+½
Ø(ν 1) = ∫ (1) e-J2πνθ dθ = Sin πν / πν (2.5) –½ 1 Sin π(ν-k) Sin π(ν+k)
Ø(ν √2Sin 2kπθ) = ( - ) (2.6) √2 π(ν-k) π(ν+k)
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1 Sin π(ν-k) Sin π(ν+k)
Ø(ν √2Cos 2kπθ) = ( + ) (2.7) √2 π(ν-k) π(ν+k) [* J denotes the imaginary number √-1, while j is used as an index] Figure 2.3 shows the frequency spectra of the first five orthogonal trigonometric
functions. It can be seen that original signals are recovered even though the individual
spectrums of orthogonal time signals overlaps. Also most of the energy is
concentrated at low frequencies. For the comparison with standard PCM systems the
dot frequency fd is introduced
fd = 1 / 2T = n / 2τ (2.8)
T is the average time available for the transmission of one digit, τ the average time
available for the transmission of one character, and n the number of digits in one
character. For five digit characters follows fdτ = νd = 2.5, it can be seen from figure
2.3 that a low pass filter with a cut-off frequency ν=3, which is only 20% larger than
the dot frequency, will let through almost all energy.
Figure 2.3: Frequency spectra of the functions of fig. 2.2
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2.2 Synthesis of OFDM signals for multichannel data transmission : In this section it is shown that by using a new class of band-limited orthogonal
signals, the channels can transmit through a linear band-limited transmission medium
at a maximum possible data rate without interchannel and intersymbol interferences.
A general method is given for synthesizing an infinite number of classes of band-
limited orthogonal time-functions in a limited frequency band. This method permits
one to synthesize a large class of transmitting filter characteristics for arbitrarily given
amplitude and phase characteristics of the transmission medium. The synthesis
procedure is convenient. Furthermore, the amplitude and the phase characteristics of
the transmitting filters can be synthesized independently, i.e., the amplitude
characteristics need not be altered when the phase characteristics are changed, and
vice versa. The system can be used to transmit not only binary digits or m-ary digits,
but also real numbers, such as time samples of analog information sources. As will be
shown, the system satisfies the following requirements.
(i) The transmitting filters have gradual cutoff amplitude characteristics.
Perpendicular cutoffs and linear phases are not required.
(ii) The data rate per channel is 2fs bauds, where fs is the center frequency difference
between two adjacent channels. Overall data rate of the system is [N/(N+1)] Rmax,
where N is the total number of channels and Rmax, which equals two times the overall
baseband bandwidth, is the Nyquist rate for which unrealizable rectangular filters with
perpendicular cutoffs and linear phases are required. Thus, as N increases, the overall
data rate of the system approaches the theoretical maximum rate Rmax, yet rectangular
filtering is not required.
(iii) When transmitting filters are designed for an arbitrary given amplitude
characteristic of the transmission medium, the received signals remain orthogonal for
all phase characteristics of the transmission medium. Thus, the system (orthogonal
transmission plus adaptive correlation reception) eliminates interchannel and
intersymbol interferences for all phase characteristics of the transmission medium.
(iv) The distance in signal space between any two sets of received signals is the same
as if the signals of each channel were transmitted through an independent medium and
intersymbol interference in each channel were eliminated by reducing data rate. The
same distance protection is therefore provided against channel noise (impulse and
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Gaussian noise). For instance, for band-limited white Gaussian noise, the receiver
receives each of the overlapping signals with the same probability of error as if only
that signal were transmitted. The distances in the signals space are also independent of
the phase characteristics of the transmitting filters and the transmission medium.
(v) When signaling intervals of different channels are not synchronized, at least half
of the channels can transmit simultaneously without interchannel and intersymbol
interference.
2.2.1 Orthogonal multiplexing using band-limited signals :
Consider N AM data channels sharing a single linear transmission medium which
has an impulse response h(t) and a transfer function H(f) eJη(f) as shown in figure 2.4.
H(f) and η(f) will be referred to, respectively, as the amplitude and the phase
characteristic of the transmission medium. Since this analysis treats only transmission
media having linear properties, the question of performance on real channels subject
to such impairments as nonlinear distortion and carrier frequency offset is not
considered here.
Figure 2.4: N data channels transmitting over one transmission medium
Consider a single channel first (say the ith channel). Let b0, b1, b2, …, be a
sequence of m-ary (m ≥ 2) signals digits or a sequence of real numbers to be
transmitted over the ith channel. As is well known, b0, b1, b2, … can be assumed to be
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represented by impulses with proportional heights. These impulses are applied to the
ith transmitting filter at the rate of one impulse every T seconds (data rate per channel
equals 1/T bauds). Let ai(t) be the impulse response of the ith transmitting filter, then
the ith transmitting filter transmits a sequence of signals as
b0 ai(t) + b1 ai(t-T) + b2 ai(t-2T) + …, (2.9)
The received signals at the output of the transmission medium are
b0 ui(t) + b1 ui(t-T) + b2 ui(t-2T) + …, (2.10)
where
+∞
ui(t) = ∫ h(t-τ) ai(τ) dτ (2.11) –∞
These received signals overlap in time, but they are orthogonal if
+∞
∫ ui(t) ui(t-kT) dt = 0, k = ±1, ±2, … (2.12) -∞
As is well known, orthogonal signals can be separated at the receiver by correlation
techniques; hence, intersymbol interference in the ith channel can be eliminated if
(2.12) is satisfied.
Next consider interchannel interference. Let c0, c1, c2, … be the m-ary signal digits
or real numbers transmitted over the jth channel which has impulse response aj(t). It
has been assumed that the channels transmit at the same data rate and that the
signaling intervals of different channels are synchronized, hence the jth transmitting
filter transmits a sequence of signals,
c0 aj(t) + c1 aj(t-T) + c2 aj(t-2T) + …, (2.13)
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The received signals at the output of the transmission medium are
c0 uj(t) + c1 uj(t-T) + c2 uj(t-2T) + …, (2.14)
These received signals overlap with the received signals of the ith channel, but they
are mutually orthogonal (no interchannel interference) if
+∞
∫ ui(t) uj(t-kT) dt = 0, k = 0, ±1, ±2, … (2.15) -∞
Thus, intersymbol and interchannel interferences can be simultaneously eliminated if
the transmitting filters can be designed (i.e., if the transmitted signals can be
designed) such that (2.12) is satisfied for all i and (2.15) is satisfied for all i and j
(i ≠ j).
Denote Ui(f) eJµi(f) as the Fourier transform of ui(t). One can rewrite (2.12) as
+∞
∫ U i 2(f) e-J2πfkT df = 0, k = ±1, ±2, … i=1,2, … ,N, (2.16)
-∞
and rewrite (2.15) as
+∞
∫Ui(f) eJµi(f) Uj(f) e-Jµj(f) e-J2πfkT df = 0, -∞ k = 0, ±1, ±2, … i,j =1,2, … ,N, i ≠ j (2.17)
Let Ai(f) eJαi(f) be the Fourier transform of ai(t). The transfer function of the
transmission medium is H(f) eJη(f). Equation (2.16) becomes
+∞
∫ Ai2(f)H2(f) e-J2πfkT df = 0, k = ±1, ±2,… i=1,2, … ,N, (2.18)
-∞
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alternatively
+∞
∫ Ai2(f)H2(f) [Cos 2πfkT – J Sin 2πfkT ] df = 0,
-∞ k = ±1, ±2,… i=1,2, … ,N,
+∞
∫ [Ai2(f)H2(f) Cos 2πfkT df– J Ai
2(f)H2(f) Sin 2πfkT ] df = 0, -∞ k = ±1, ±2,… i=1,2, … ,N,
+∞ +∞
∫ Ai2(f)H2(f) Cos 2πfkT df– J ∫ Ai
2(f)H2(f) Sin 2πfkT df = 0, -∞ -∞ k = ±1, ±2,… i=1,2, … ,N,
Here imaginary part of LHS of above equation is ZERO, as it is the integration of odd
symmetric function (Ai2(f)H2(f) Sin 2πfkT) from -∞ to +∞.
+∞
∴ ∫ Ai2(f)H2(f) Cos 2πfkT df = 0, k = ±1, ±2,… i=1,2, … ,N,
-∞ As function under above integration (Ai
2(f)H2(f) Cos 2πfkT) is even, this equation can
be written as –
∞
∫ Ai2(f)H2(f) Cos 2πfkT df = 0, k = 1, 2,… i=1,2, … ,N, (2.19)
0
and equation (2.17) becomes
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+∞
∫Ai(f)Aj(f) H2(f) eJ[αi(f) - αj(f) - 2πfkT] df = 0, -∞ k = 0, ±1, ±2, … i,j =1,2,… ,N, i ≠ j (2.20)
This equation (2.17) can be written as
+∞
∫Ai(f)Aj (f) H2(f) Cos[αi(f) - αj(f) - 2πfkT] df + -∞ +∞
J ∫Ai(f)Aj(f) H2(f) Sin[αi(f) - αj(f) - 2πfkT] df = 0 -∞ k = 0, ±1, ±2, … i,j =1,2, … ,N, i ≠ j
Here imaginary part of LHS of above equation is ZERO, as it is the integration of odd
symmetric function (Ai(f) Aj(f) H2(f) Sin[αi(f)-αj(f)-2πfkT]) from -∞ to +∞. Applying
symmetry arithmetic and trigonometric identity to real part we have,
∞
∫Ai(f)Aj (f) H2(f) Cos[αi(f) - αj(f)].Cos 2πfkT df 0 ∞
+ ∫Ai(f)Aj (f) H2(f) Sin[αi(f) - αj(f)].Sin 2πfkT df = 0 0 k = 0, ±1, ±2, … i,j =1,2, … ,N, i ≠ j
So as to obtain a well plausible strong solution, it is essential that individually both
terms of LHS of above equation to be zero, then
∞
∫Ai(f)Aj (f) H2(f) Cos[αi(f) - αj(f)].Cos 2πfkT df = 0 (2.21) 0 and ∞
∫Ai(f)Aj (f) H2(f) Sin[αi(f) - αj(f)].Sin 2πfkT df = 0 (2.22) 0 k = 0, 1, 2, … i,j =1,2, … ,N, i ≠ j
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It will be recalled that the transmitting filters and the data processors operate in
baseband. Let fi, i=1,2,…,N denote the equally spaced baseband center frequencies of
the N independent channels. One can choose
f1 = (h + ½)fs (2.23)
where h is any positive integer (including zero), and fs is the difference between
center frequencies of two adjacent channels. Thus,
fi = f1 + (i-1)fs = (h + i - ½)fs (2.24)
Carrier modulation will translate the baseband signals to a given frequency band for
transmission.
Each AM data channel transmits at the data rate 2fs bauds. Hence,
T = 1 / 2fs seconds (2.25)
For a given amplitude characteristic H(f) of the transmission medium, band-limited
transmitting filters can be designed (i.e band-limited transmitted signals can be
designed) such that (2.19), (2.21), (2.22) and (2.25) are simultaneously satisfied (no
intersymbol and interchannel interference for a data rate of 2fs bauds per channel). In
addition, the five requirements in Section 2.2 are also satisfied. A general method of
designing these transmitting filters is given in the following theorem.
For a given H(f), in order satisfy equation (2.19), let Ai(f), i = 1,2,…,N, be shaped
such that
Ai2(f)H2(f) = Ci + Qi(f) > 0, fi - fs < f < fi + fs
= 0 f < fi-fs, f > fi + fs (2.26)
where Ci is an arbitrary constant and Qi(f) is a shaping function having odd
symmetries about fi + (fs/2) and fi - (fs/2), i.e.,
Qi[(fi+fs/2) + f′] = -Qi[(fi+fs/2) - f′], 0 < f′ < fs/2 (2.27)
Qi[(fi-fs/2) + f′] = -Qi[(fi-fs/2) - f′], 0 < f′ < fs/2 (2.28)
Furthermore, the function [Ci + Qi(f)]⋅[Ci+1 + Qi+1(f)] is an even function about
fi + (fs/2), i.e.,
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[Ci + Qi(fi + fs/2+f′)]⋅[Ci+1 + Qi+1(fi + fs/2+f′)]
= [Ci + Qi(fi + fs/2-f′)]⋅[Ci+1 + Qi+1(fi + fs/2-f′)]
0 < f′ < fs/2 i = 1,2, … .N-1 (2.29)
By considering even symmetry of Cos 2πfkT and odd symmetry of Sin 2πfkT about
fi + (fs/2) for all k, from equation (2.21) and (2.22) let the phase characteristic
αi(f), i = 1,2, … N, be shaped such that
αi(f) - αi+1(f) = ± π/2 + γi(f), fi < f < fi + fs
i = 1,2, …, N-1, (2.30)
where γi(f) is an arbitrary phase function with odd symmetry about fi + (fs/2).
If Ai(f) and αi(f) are shaped as in (2.26) through (2.30) and f1 is set according to
(2.23), then (2.19), (2.21), (2.22) and (2.25) are simultaneously satisfied (no
intersymbol or interchannel interference for a data rate of 2fs bauds per channel).
Furthermore, the five requirements in section 2.2 are also satisfied.
2.2.2 Transmission filter characteristics:
Consider first the shaping of the amplitude characteristics Ai(f) of the transmitting
filters. Equations (2.26), (2.27) and (2.28) can be easily satisfied. Equation (2.29) can
be satisfied in many ways. For instance, a simple, practical way to satisfy (2.29) is
given below.
Under the simplifying condition that
(i) Ci should be the same for all i
(ii) Qi(f), i = 1,2, …, N, should be identically shaped, i.e.,
Qi+1(f) = Qi(f-fs), i = 1, 2, …, N-1, (2.31)
Equation (2.29) holds when Qi(f) is an even function about fi, i.e.,
Qi(fi + f′) = Qi(fi - f′), 0 < f′< fs. (2.32)
Two examples are given for illustration purpose. The first example is illustrated in
figure 2.5 where Qi(f) is chosen to be
Qi(f) = ½·Cos(π.(f - fi)/fs), fi – fs < f < fi + fs, i=1,2, …,N,
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Figure 2.5: Shaping the amplitude characteristics Ai(f) of the transmitting filters
This simple choice satisfies (2.27), (2.28), (2.31) and (2.32). Let Ci be ½ for all i, then
(2.27), (2.28) and (2.29) are all satisfied. From (2.26)
Ai 2(f)H2(f) = Ci + Qi(f)
= ½ + ½⋅ Cos((π.(f - fi)/fs)
and
Ai(f)H(f) = Cos((π.(f - fi)/(2fs)), fi – fs < f < fi + fs, i=1,2, …,N.
This Ai(f)H(f) is similar to the amplitude characteristic of a standard duobinary
filter (except shift in center frequency). The second example is illustrated in fig. 2.6
where Qi(f) is chosen such that Ai(f)H(f) has a shape similar to that of a multiple
tuned circuit.
Figure 2.6: Another example of shaping the amplitude characteristic Ai(f) of the transmitting filter.
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Example of amplitude characterization (Ai(f)H(f)) of first five channels with Ci=½ is shown in figure 2.7
Figure 2.7: Amplitude characterization of first five channels It can be seen from these two examples that there is a great deal of freedom in
choosing the shaping function Qi(f). Consequently, Ai(f)H(f) can be easily shaped into
various standard forms. Ai(f) would have the same shape as Ai(f)H(f), if H(f) is flat in
the frequency band fi-fs to fi+fs of the ith channel. If H(f) is not flat in this individual
band, Ai(f) can be obtained from Ai(f)H(f)/H(f), provided that H(f) ≠ 0 for any f in the
band.
It is also noted from the preceding that if Ci is chosen to be the same for all i and if
Qi(f), i = 1, 2, … ,N, are chosen to be identically shaped (i.e., identical in shape
except shifts in center frequencies), the Ai(f)H(f), i=1,2, … ,N, will also be identically
shaped. Consequently, Ai(f), i=1,2, … ,N, will be identically shaped if H(f) is flat or is
made flat. An advantage of having identically shaped filter characteristics is that each
filter can be realized by using an identical shaping filter plus frequency translation.
H(f) can be made flat by using a single compensating network which compensates
the variation of H(f) over the entire band. As an alternative note that Ai(f) exists only
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from fi-fs to fi+fs. Hence, for the ith receiver, the integration limits of (2.19), (2.21) and
(2.22) can be changed to fi-fs and fi+fs. Therefore, the signal at the ith receiver only has
to satisfy the theorem in the limited frequency band fi-fs to fi+fs. This permits one to
design the transmitting filters for flat H(f) and then compensate the variation of H(f)
individually at the receivers, i.e., use an individual network at the ith receiver to
compensate only for the variation of H(f) in the limited frequency band fi-fs to fi+fs.
Finally, note that if the channels are narrow, each channel will usually be
approximately flat. In these cases, one may design the transmitting filters for flat H(f)
without using compensating networks. This design should lead to only small
distortion.
Consider next the shaping of the phase characteristics αi(f) of the transmitting
filters. It is only required that (2.30) be satisfied. However, if it is desired to have
identically shaped transmitting filter characteristics, one may consider a simple
method such as follows –
Under the simplifying condition that αi(f), i = 1,2, … ,N, be identically shaped, i.e.,
αi+1(f) = αi(f-fs) i = 1,2, … ,N-1 (2.33)
(2.30) holds when
f - fi f – fi f - fi αi(f) = hπ + ϕ0 + ∑ ϕm Cos mπ + ∑ ψn Sin nπ 2fs m fs n fs
m=1,2,3,4,5, … n=2,4,6, … fi – fs < f < fi+fs (2.34)
where h is an arbitrary odd integer and the other coefficients (ϕ0, ϕm, m=1,2,3,4….; ψn,
n=2,4,6, … ) can all be chosen arbitrarily.
Note that if the index n in equation (2.34) were not required to be even, αi(f) would
be completely arbitrary ( aFourier series with arbitrary coefficients). This shows that
there is a great deal of freedom in shaping αi(f) even if the additional constraint of
ideal shaping is introduced. The linear term hπ(f-fi)/2fs is introduced not only to give
the term ±π/2 in (2.30), but also because a linear component is usually present in filter
phase characteristics. A simple example is given in figure 2.8 to illustrate (2.34). For
22
clarity, the arbitrary Fourier coefficients are all set to zero, except ψ2 = 0.3 and h is set
to -1.
Figure 2.8: Example of h = -1, ψ2 ≠ 0, all other coefficient zero.
As can be seen in the theorem, the requirement on αi(f) is independent of the
requirements on Ai(f). Hence, the amplitude and the phase characteristics of the
transmitting filters can be synthesized independently. This gives even more freedom
in designing the transmitting filters. A simple set of Ai(f) and αi(f) is sketched in
figure 2.9 for four adjacent channels. This illustrates that the frequency spectrum of
each channel is limited and overlaps only with that of the adjacent channel. H(f) is
assumed flat and transmitting filters are identically shaped. As mentioned previously,
these filters can be realized either by different networks or simply by using identical
shaping filters plus frequency translations.
23
Figure 2.9: Transmitting filter characteristics for orthogonal multiplexing
data transmission
Now consider the five requirements in Section 2.2. The first requirement is
satisfied since the transmitting filters designed are of standard forms. Perpendicular
cutoffs and linear phase characteristics are not required.
As for the second requirement, it can be seen from figure 2.9 that the overall
baseband bandwidth of N channels is (N+1)fs. Since data rate per channel is 2fs bauds,
the overall data rate of N channels is 2fsN bauds. Hence,
Overall data rate = [2N/(N+1)] . Overall baseband bandwidth
= [N/(N+1)] . Rmax ,
Where Rmax, which equals two times overall baseband bandwidth, is the Nyquist rate
for which unrealizable filters with perpendicular cutoffs and linear phases are
required. Thus, for moderate values of N, the overall data rate of the orthogonal
multiplexing data transmission system is close to the Nyquist rate, yet rectangular
filtering is not required. This satisfies second requirement.
24
Now consider the third requirement. As has been shown, the received pulses are
orthogonal if (2.19), (2.21) and (2.22) are simultaneously satisfied. Note that the
phase characteristic η(f) of the transmission medium does not enter into these
equations. Hence, the received signals will remain orthogonal for all η(f), and
adaptive correlation reception can be used no matter what the phase distortion is in the
transmission medium. Also note that so far as each receiver is concerned, the phase
characteristics of the networks in each receiver (including the bandpass filter at the
input of each receiver) can be considered as part of η(f), and hence has no effect on
the orthogonality of the received signals.
In the case of the fourth requirement, let
bki, k = 0, 1, 2, …; i = 1, 2, … ,N,
and
cki, k = 0, 1, 2, …; i = 1, 2, … ,N,
be two arbitrary distinct sets of m-ary signal digits or real numbers to be transmitted
by the N channels. The distance in signal space between the two sets of received
signals
∑ ∑ bki ui(t - kT)
i k and
∑ ∑ cki ui(t - kT)
i k is
∞
d = [ ∫ [∑ ∑ bki ui(t - kT) - ∑ ∑ ck
i ui(t - kT) ]2 dt ]½
-∞ i k i k
In an ideal case where transmitting the signals of each channel through an
independent medium eliminates interchannel and intersymbol interferences and
slowing down data rate such that the received signals in each channel do not overlap,
25
the distance d can be written as signals in each channel do not overlap, the distance d
can be written as
∞
dideal = [∑ ∑ ∫ (bki - ck
i)2 ui2(t - kT) dt ]½
i k -∞
In this study, the N channels transmit over the same transmission medium at the
maximum data rate (T= ½fs). If the transmitting filters were not properly designed, the
distance d could be much less than dideal and the system would be much more
vulnerable to channel noises (impulse and Gaussian noises). However, if the
transmitting filters are designed in accordance with the results in Section 2.2.1, the
received signals will be orthogonal and d = dideal. Thus, the distance between any two
sets of received signals is preserved and the same distance protection is provided
against channel noises. For instance, since d = dideal, it follows from maximum
likelihood detection principle that for band limited white Gaussian noise and m-ary
transmission the receiver will receive each of the overlapping signals with the same
probability of error as if only that signal is transmitted.
Note further that dideal can be written as
∞
dideal = [∑ ∑ (bki - ck
i)2 ∫ Ai2(f) H2(f) df]½
i k -∞
Thus dideal is independent of the phase characteristics αi(f) of the transmitting filters
and the phase characteristic η(f) of the transmission medium. Since d = dideal, it
follows that d is also independent of αi(f) and η(f) and the same distance protection is
provided against channel noises for all αi(f) and η(f).
Finally, consider the fifth requirement. It is assumed that signaling intervals of
different channels are synchronized. However, it is interesting to point out that the
frequency spectra of alternate channels (for instance, i = 1,3,5, …) do not overlap (see
figure 2.7). Hence, if one uses only odd or even-numbered channels, one can transmit
26
without interchannel and intersymbol interferences and without synchronization
among signaling intervals of different channels. The overall data rate becomes Rmax/2
for all N. A very attractive feature is obtained in that the transmitting filters may now
have arbitrary phase characteristics αi(f). [This is because αi(f) is not involved in
(2.19) and intersymbol interference is eliminated for all αi(f).] Thus, only the
amplitude characteristics of the transmitting filters need to be designed as in the
theorem and the transmitting filters can be implemented very easily.
Another case of interest is where part of the channels are synchronized. As a simple
example, assume that there are five channels and that channel 1 is synchronized with
channel 2; channel 4 is synchronized with channel 5; while channel 3 cannot be
synchronized with other channels. If the amplitude characteristics of the five channels
plus the phase characteristics of channels 1,2,4 and 5 are designed as in the theorem,
one can transmit simultaneously through channels, 1,2,4 and 5 or simultaneously
through channels, 1,2,4 and 5 or simultaneously through channels 1,3, and 5 without
interchannel or intersymbol interferences. The overall data rate is then between Rmax/2
and (N / N+1)Rmax.
2.3. Receiver structure for multichannel orthogonal signal: The receiver of a single channel (say, the fifth channel) is shown in figure
2.10(a). When viewed at point B toward the transmitter, the channels have amplitude
characteristics as shown in figure 2.10(b). The bandpass filter at the input of the fifth
receiver has a passband from f5 - fs to f5 + fs [figure 2.10(c). This filter serves the
important purpose of rejecting noises and signals outside the band of interest. Sharp
impulse noises with broad frequency spectra are greatly attenuated by this filter.
Signals in other channels are rejected to prevent overloading and cross modulation.
The product device translates the frequency spectra further toward the origin so
that the signal can be represented by minimum number of accurate time samples and
the adaptive correlator can operate in digital fashion. The transmitter can transmit a
reference frequency fs or a known multiple of fs to the receivers for deriving the
signals Cos[2π(i-1)fst + θi] for the product devices. It is important to note that the
transmitter can lock this frequency fs to the data rate 2fs so that the arbitrary phase
angle θi is time invariant and can be taken into account by adaptive correlation.
27
Furthermore, the receiver can also derive the sampling rate 2fs from this reference
frequency.
Figure 2.10: Reception of the signals in channel 5
28
When observed at point D, the channels have amplitude characteristics as shown
in figure 2.10(d). Note that the fifth channel now has a center frequency at 1.5fs
[satisfying (2.24)] and an undistorted amplitude characteristic; hence, the signals in
channel 5 remain orthogonal. The overlapping frequency spectra between channel 5
and channels 4 and 6 remain undistorted, and the phase differences α4(f) - α5(f) and
α5(f) - α6(f) are unchanged; therefore, the signals in channels 4 and 6 remain
orthogonal to those in channel 5. Other channels produce no interference since their
spectra do not overlap with that of channel 5.
Let b0u(t), b1u(t-T), b2u(t-2T), … be the signals in channel 5 at point D, where
b0, b1, b2, … are the information digits. These signals can be represented by vectors of
time samples as
b0u0, b1u1, b2u2, …
Since u(t), u(t – T), … differ only in time origin, it is only necessary to learn u0 for
correlation purposes. The received signal at point D can be written as
∑ bnun + v n where v represents the sum of the signals in other channels. Here due to orthogonality
uk′uj = λ k = j
= 0 k ≠ j
uk′v = 0.
Thus, the adaptive correlator can learn the vector u0 prior to data transmission and
then correlate the received signal with uk, k = 0,1,2, … to obtain the information digits
bk, k = 0,1,2, ….
29
Chapter 3.
OFDM: FFT Based Approach
For a large number of channels, the arrays of sinusoidal generators and coherent
demodulators required in parallel systems become unreasonably expensive and
complex [9]. However, it can be shown [10] that a multitone data signal is effectively
the Fourier transform of the original serial data train, and that the bank of coherent
demodulators is effectively an inverse Fourier transform generator. This point of view
suggests a completely digital modem built around a special-purpose computer
performing the fast Fourier transform (FFT). Fourier transform techniques, although
not necessarily the signal format described here. This innovation had a tradeoff in that
their system could not guarantee orthogonality between subcarriers over a dispersive
channel. In 1980, A Peled and A Ruiz solved the orthogonality problem with their
introduction of the cyclic prefix (CP) [11]. The cyclic prefix is a copy of the last part
of the OFDM symbol attached to the front of the transmitted symbol. When the CP
length is longer than the channel impulse response, the received signal is a cyclic
convolution of the transmitted signal and the channel impulse response. The cyclic
convolution results in the orthogonality of the subcarriers being maintained over a
dispersive channel.
3.1 The Discrete-Time Model : Figure 3.1 illustrates a block diagram of DFT based OFDM system. A serial-to-
parallel buffer segments the information sequence into frames of Nf bits. The Nf bits
in each frame are parsed into N′ groups, where the ith group is assigned ni′ bits, and
N′
∑ ni′ = Nf (3.1)
i=1
30
Figure 3.1: OFDM Digital System Model
Each group may be encoded separately, so that the number of output bits from the
encoder for the ith group is ni ≥ ni′.
It is convenient to view this multicarrier modulation as consisting of N′
independent QAM channels, each operating at the same symbol rate 1/T, but each
channel having a distinct QAM constellations, i.e. ith channel will employ Mi = 2ni
signal points. We denote the complex-valued signal points corresponding to the
information symbols on the subchannels by Xk = ak + bk. For example, if 2 bits were
used for each component of the complex number, then each component could take on
the value of (±1, ±3) and the result would be the 16-point QAM constellation shown
below in figure 3.2
31
Figure 3.2: 16-symbol QAM constellation
Each symbol transmitted by the OFDM system is composed of a set of N′ complex
numbers (N will be a power of 2, to allow for the use of the IFFT in modulation),
shown in figure 3.1 as X0 through XN-1. These complex numbers exist in the
frequency domain, however, and must be converted into the time domain. This
transformation is achieved by using an inverse Fast Fourier Transform (IFFT), which
is just a fast implementation of an Inverse Discrete Fourier Transform (IDFT). The
IDFT also serves to modulate the complex numbers, as will be shown in the next
section.
3.2 Modulation of OFDM data using IDFT :
However, if we compute the N′-point IDFT of {Xk}, we shall obtain a complex-
valued time series, which is not equivalent to N′ QAM-modulated subcarriers. Instead,
we create N = 2N′ information symbols by defining
XN-k = Xk*, k=1, ……. ,N′-1 (3.2)
32
And X0 = Re{X0}, and XN′ = Im(X0). Thus, the symbol X0 is split into two parts, both
real. Then the N-point IDFT yields the real-valued sequence
1 N-1
xn = ∑ Xk eJ2πnk / N , n = 0,1,2,…N-1 (3.3) √N k=0
where 1/√N is simply a scale factor.
The sequence {xn, 0 ≤ n ≤ N-1} corresponds to the samples of the sum x(t) of N′
subcarrier signals, which is expressed as
1 N-1
x(t) = ∑ Xk eJ2πkt / T , 0 ≤ t ≤ T (3.4) √N k=0
where T is the symbol duration. We observe that the subcarrier frequencies are
fk = k/T, k=0,1,2….N′. Furthermore, the discrete-time sequence {xn} in equation (3.3)
represents the samples of x(t) taken at times t=nT/N where n = 0,1,2, …, N-1.
The resulting baseband signal is then converted back into serial data and undergoes
the addition of the cyclic prefix (which will be explained in the next section). In
practice, the signal samples {xn} are passed through a digital-to-analog (D/A)
converter at time intervals T/N. Next, the signal is passed through a low-pass filter to
remove any unwanted high-frequency noise, whose output, ideally, would be the
signal waveform x(t). The resulting signal closely approximates the frequency
division multiplexed signal.
The last step in the transmission process is to convert the signals from baseband to
bandpass by stepping up each subcarrier frequency in the up converter.
Because of the rectangular windowing of the above function (0 ≤ t ≤ T), the spectrum
of each subcarrier (at carrier frequency fk) is a Sinc function as shown in figure 3.3.
33
Figure 3.3: Spectra of OFDM subchannels
3.3 Orthogonality : From equation (3.3) it can be shown that in resultant equation of xn, each Xk is
multiplied by
1 φk(n) = eJ2πnk / N n = 0,1,2,…N-1, ∀ k ∈ [0, N-1] √N It can be very easily shown that the inner product –
<φm(n), φn(n)> = 1 m = n
= 0 m ≠ n (3.5)
where m,n are any integers ranging from 0 to N-1. Thus orthogonality is maintained
in the exponentials associated with every set of different subchannels. This definition
can also be loosely applied to the frequency domain as well: Orthogonality of OFDM
34
subcarriers is maintained so long as all subcarriers are sampled at their peaks, where a
null exists for the spectra of all other subcarriers. That is, at any subcarrier peak the
product of that particular subcarrier’s spectrum and any other subcarrier’s spectrum
will be equal to zero, causing inner product to yield zero and satisfy the criteria for
orthogonality.
3.4 The cyclic prefix : Orthogonality of OFDM subcarriers is critical since it prevents interchannel
interference. As such, OFDM is highly sensitive to frequency dispersion caused by
Doppler shifts [12]. If an OFDM receiver is mobile and moving towards the
transmitter, the Doppler shift can cause a corresponding shift in the OFDM spectrum.
This frequency shift causes a subcarrier to be sampled at a frequency other than the
one corresponding to its peak. As a result, orthogonality is lost and there is a
reduction in the signal amplitude as well as intercarrier interference. The solution for
this is the cyclic prefix.
The cyclic prefix is a copy of the last part of the OFDM symbol which is
prepended to the beginning of the symbol (hence the term prefix in the name) as
shown in following figure 3.4
Figure 3.4: The Cyclic Prefix
The output of the channel is the waveform
r(t) = x(t) ∗ h(t) + n(t) (3.6)
where h(t) is the impulse response of the channel and ∗ denotes convolution. By
selecting the bandwidth ∆f of each subchannel to be very small, the symbol duration
T=1 / ∆f is large compared with the channel time dispersion. To be specific, let us
assume that the channel dispersion spans v+1 signal samples where v << N. One way
35
to avoid the effect of ISI is to insert a time guard band of duration vT/N between
transmission of successive blocks.
An alternative method that avoids ISI is to append a cyclic prefix to each block of
N signal samples {x0, x1, …, xN-1}. These new samples are appended to the beginning
of each block. Note that the addition of the cyclic prefix to the block of data increases
the length of the block to N+v samples, which may be indexed from n=-v, …, N-1,
where the first v samples constitute the prefix. Then, if {hn, 0 ≤ n ≤ v} denotes the
sampled channel impulse response, its convolution with {xn, -v ≤ n ≤ N-1} produces
{rn}, the received sequence. We are interested in the samples of {rn} for 0 ≤ n ≤ N-1,
from which we recover the transmitted sequence by using the N-point DFT for
demodulation. Thus the first v samples of {rn} are discarded.
Thus cyclic prefix serves two very important purposes. First, it acts as a guard space
between transmitted symbols. The length of the cyclic prefix is usually less than ¼ the
length of the symbol period, but is assumed to be longer than the RMS delay spread
of the channel. So when the cyclic prefix is removed in the receiver, the delay spread
contained within it is also removed. Thus, time dispersion and ISI are avoided.
Second, since the cyclic prefix is assumed longer than the channel impulse response,
r(t) appears to the channel as a periodic signal and the channel impulse response can
be cyclically convolved with the signal. Because of this cyclic convolution, the
orthogonality of the subcarriers is maintained [11] and ICI is avoided.
3.5 Demodulation using the DFT : From a frequency-domain viewpoint, when the channel impulse response is
{hn, 0 ≤ n ≤ v}, its frequency response at the subcarrier frequencies fk = k/N is
v Hk ≡ H(2πk / N) = ∑ hn e-J2πk / N (3.7)
n=0
Because of the cyclic prefix, successive blocks (frames) of the transmitted
information sequence do not interfere and, hence, the demodulated sequence may be
expressed as
Xk′ = HkXk + ηk, k = 0,1, …, N-1 (3.8)
36
Where {Xk′} is the output of the N-point DFT demodulator, and ηk is the additive
noise corrupting the signal. We note that by selecting N >> v, the rate loss due to the
cyclic prefix can be rendered negligible.
As shown in figure 3.1, the information is demodulated by computing the DFT of
the received signal after it has been passed through an analog-to-digital (A/D)
converter. As in the case of the modulator, the DFT computation at the demodulator is
performed efficiently by use of the FFT algorithm. It is simple matter to estimate and
compensate for the channel factors {Hk} prior to passing the data to the detector and
decoder. A training signal consisting of either a known modulated sequence on each
of the subcarriers or unmodulated subcarriers may be used to measure the {Hk} at the
receiver. If the channel parameters vary slowly with time, it is also possible to track
the time variations by using the decisions at the output of the detector or the decoder,
in a decision directed fashion. Thus, this kind of multicarrier system can be rendered
adaptive.
Other types of implementation besides the DFT are possible. For example, a
digital filter bank that basically performs the DFT may be substituted for the FFT-
based implementation when the number of subcarriers is small, e.g., N ≤ 32. For a
large number of subcarriers, e.g. N > 32, the FFT-based systems are computationally
more efficient.
This OFDM multicarrier QAM of the type described above has been
implemented for a variety of applications, including high-speed transmission over
telephone lines, such as digital subcarrier lines.
37
Chapter 4.
The Downsides of OFDM
OFDM systems have a number of drawbacks [13]
1. Cyclic Prefix Overhead-
A cyclic prefix at least as long as the channel response must be attached to each
OFDM symbol to prevent interference between symbols. The cyclic prefix represents
a significant overhead and the only way to improve efficiency (reduce the relative
overhead) is to increase the number of subcarriers.
2. Frequency Control-
OFDM relies on orthogonality between the overlapped sub-carriers achieved by
frequency control to within one percent of the subcarrier spacing. Frequency offset
errors mean that the subcarriers are no longer orthogonal, resulting in intercarrier
interference and severe degradation in performance.
For example, a total 3.5 MHz bandwidth shared between 512 sub-channels gives a
frequency spacing of 6.8 KHz between subcarriers, requiring frequency accuracy
better than 68 Hz. For this reason, OFDM systems are very sensitive to both
frequency offsets and phase noise, and require costly, high specification radio
components. The larger the number of subcarriers, the closer the frequency spacing,
and so frequency accuracy becomes more and more critical. In Broadband Wireless
Access (BWA) systems, vehicles moving through the beam induce sporadic Doppler
shifts, resulting in additional frequency offset.
3. Requirement for coded or adaptive OFDM -
A null in the frequency response of the channel results in one or more subcarriers
having a very low SNR, and these subcarriers will dominate the overall error rate. For
this reason, OFDM without either power/rate adaptation, or coding, gives worse
performance than single carrier (SC) modulation. The full potential of OFDM can
only be achieved if power and bit rate are optimal for each individual subcarrier for
each individual receiver. This is not possible for broadcast transmission on the
downlink.
38
Another way to protect individual sub-carriers from frequency nulls, is to use
error-control coding. Coded OFDM gives similar performance to equalized SC
transmission [14], at the expense of very low coding rates, typically in the range 0.5 to
0.75. As well as reducing system throughput, the error-control code also increases
receiver complexity, particularly for convolutional codes.
4. Latency and block based processing -
Fast Fourier Transform (FFT) processing for OFDM is performed on blocks. If
the number of of subcarriers is increased, the FFT block size is also increased. Block-
based processing imposes a minimum block size for each transmitted packet, resulting
in delays and loss of efficiency for very small data bursts. The latency of OFDM is
high because the smallest burst that can be sent is a full OFDM block. Contrary SC
transmission is very efficient, because the block length can be reduced to fit the data
payload.
5. Synchronization-
Synchronization for OFDM transmission is more difficult, and typically requires
reception of several OFDM symbols, each having a number of subcarriers dedicated
for pilot tones. This is feasible for broadcast OFDM (such as HDTV), where the
receiver has plenty of time to acquire synchronization. However, for short burst,
point-to-multipoint transmission (especially on the uplink), there is little time to
acquire synchronization, and so complicated synchronization schemes are
necessary[15].
6. Peak-to-average power ratio (PAR) -
A major problem with OFDM is the relatively high PAR that is inherent in the
transmitted signal. In general, large signal peaks occur in the transmitted signal when
the signals in many of the various subchannels add constructively in phase. Such large
signal peaks may result in clipping of the signal voltage in a D/A converter when
multicarrier signal is synthesized digitally, and/or it may saturate the power amplifier
and, thus, cause intermodulation distortion in the transmitted signal.
39
Various methods have been devised to reduce the PAR in multicarrier systems.
One of the simplest methods is to insert different phase shifts in each of the
subcarriers. These phase shifts can be selected pseudorandomly, or by means of some
algorithm, to reduce the PAR. For example, we may have a small set of N stored
pseudorandomly selected phase shifts which can be used when the PAR in the
modulated subcarriers is large. The information on which set of pseudorandom phase
shifts is used in any signal interval can be transmitted to the receiver on one of the N
subcarriers. Alternatively, a single set of pseudorandom phase shifts may be
employed, where this set is found via computer simulation to reduce the PAR to an
acceptable level over the ensemble of possible transmitted data symbols on the N
subcarriers.
It is possible to achieve a small reduction in PAR using coding to avoid data
sequences with high peaks, although this reduces the useful data rate.
40
Chapter 5.
Future Directions
Unending appetite to achieve capacity, high data rate, minimum bit error rate
(BER), spectral efficiency and minimum power requirements are the main driving
forces behind future research efforts in OFDM. Various approaches to cope up with
the drawbacks given in previous chapter produces infinite possibilities for future
research. Few of them are given here-
• Channel estimation with diversity techniques and/or coding to compensate the
effect(s) of channel with delay/Doppler shifts and try to achieve optimum
values of capacity, BER etc.
• Appropriate combination of waveform shaping and frame overlap in time
limited OFDM without destroying orthogonality.
• Dynamic selection of phase shifts, data sequences and signal clipping or either
of the combination of those to reduce PAR
• Design of channel estimator with time/frequency correlations with adaptation.
• Adaptive subcarrier, bit, and power allocation to each subband to achieve
optimum values of capacity, BER, minimum power requirement etc.
• Any other combined approaches like OFDM-CDMA, OFDM-SDMA.
• Blind estimation techniques for symbol timing and carrier frequency offset.
• OFDM application specific DSP architectures.
And last but not least, efforts to make it user-centered technology for improvement of
quality of life of the individual.
41
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