E&CE 411, Spring 2005, Handout 3, G. Gong
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Multicarrier Modulation and OFDM
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Description of Multicarrier Modulation and OFDM
In Chater 4 (Chapter 6 in the text), we considered digital transmission through bandlimited
channels and observed that such channels cause intersymbol interference. In such a case, a
channel equalizer is employed at the receiver to compensate for the channel distortion. If the
channel is a bandpass channel with a specified bandwidth, the information-bearing signal may
be generated at the baseband and then translated in frequency to the passband of the channel.
Thus, the information-bearing signal is transmitted on a signal carrier.
An alternative approach to the design of a bandwidth-efficient communication system in
the presence of channel distortion or equivalently ISI, is to divide the available channel bandwidth into a number of equal-bandwidth subchannels, where the bandwidth of each channel is
sufficiently narrow so that the frequency response characteristics of the subchannels are nearly
equal. Such a division of the overall bandwidth into smaller subchannels is illustrated in Figure
1.
|C(f )|
0
W
∆f
Figure 1: Subdivision of the channel bandwidth W into narrow band subchannels of equal
width ∆f
Thus, we create N = W/∆f subchannels, where different information symbols can be
transmitted simultaneously into N subchannels. Consequently, the data symbol is transmitted
by frequency-division multiplexing (FDM). This is known as a multicarrier modulation system.
That’s, each subchannel is associated with a carrier fi given by
fi = f0 + (i − 1)∆f, i = 1, · · · , N
(1)
which is the mid-frequency in the ith subchannel.
If the subcarriers are orthogonal over the symbol duration Ts , then it is referred to as
orthogonal frequency-division multiplexing (OFDM). A general OFDM modulator is shown in
Figure 2.
E&CE 411, Spring 2005, Handout 3, G. Gong
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m1 bits
f1
m2 bits
f2
Serial to
OFDM signal out
data
parallel
converter
mk bits
fk
Figure 2: A General Model of an OFDM System
The OFDM concept utilizes parsing of the input data bit stream (which may be due to
a single source or several sources multiplexed) into N symbol streams, each with symbol rate
1/Ts , and each of which in turn is used to modulate parallel, synchronous subcarriers. With an
OFDM system having N subchannels, the symbol rate on each subcarrier is reduced by a factor
of N relative to the symbol rate on a single carrier system that employs the entire bandwidth
W and transmits data at the same rate as OFDM.
If the modulation scheme used on the ith subcarrier can accommodate ki bits per symbol,
and the subcarriers are spaced by T1s Hz, where Ts is the symbol duration of the subcarriers,
i.e, ∆f = T1s (recall this condition makes these subcarriers orthogonal), then Tof dm , the symbol
duration of the OFDM system and Ts , the symbol duration of the subcarriers are related by
Ts = N Tof dm .
(2)
By selecting N to be sufficiently large, the symbol interval Ts of the subcarriers can be made
significantly larger than the time duration of the channel-time dispersion. Hence, ISI can be
made arbitrarily small by selection of N . In other words, each subchannel appears to have a
fixed frequency response C(fk ), k = 0, 1, · · · , N − 1. A typical OFDM spectrum is illustrated in
Figure 3 for rectangular pulse shaping and separation of subcarriers by T1s .
As long as we maintain time synchronization among the subcarriers, OFDM allows us to
transmit a different number of bits/symbol on each carrier. Therefore, subcarriers that yield
a higher SNR due to a lower attenunation can be modulated to carry more bits/symbol than
subchannels that yield a lower SNR (high attenuation).
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DFT and IDFT Implementated OFDM
Let {Xk } be a symbol sequence which is used to modulate N simultaneously transmitted
subcarriers. From (1), we have
fk = f0 +
k−1
Hz,
Ts
k = 1, · · · , N − 1.
Let
T 0 = Ts + δ
E&CE 411, Spring 2005, Handout 3, G. Gong
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20
10
Power/frequency (dB/Hz)
0
−10
−20
−30
−40
−50
20
25
30
Frequency (Hz)
35
40
Figure 3: Spectrum of an OFDM signal
where Ts is the symbol duration of the subcarriers and δ is a guard time interval, assumed to
occur at the start of a new signaling period, which is used to minimize ISI effects. Then, the
OFDM signal in the complex baseband signalling can be written as
N −1
1 X
Xi exp(j2πit/Ts ),
x(t) = √
N i=0
0 ≤ t < Ts + δ.
The time variable sampled at t = kTof dm yields
N −1
1 X
Xi exp(j2πikTof dm /Ts )
x(kTof dm ) = √
N i=0
Since Tof dm /Ts = 1/N from (2), let xk = x(kTof dm ), we have
N −1
1 X
xk = √
Xi exp(j2πik/N ), k = 0, 1, · · · , N − 1
N i=0
(3)
which is inverse discrete Fourier transform (IDFT). Thus, we can generate x(t) by the IDFT
in (3). From (3), we know that the sequence corresponds to samples of the multicarrier OFDM
signal x(t), consisting of N carriers. The signal samples {xk } generated by computing IDFT
(i.e., (3)) are passed through a digital-to-analog (D/A) converter, where output, ideally, is the
OFDM signal waveform x(t). This is illustrated in Figure 4.
E&CE 411, Spring 2005, Handout 3, G. Gong
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Serial to
data
parallel
IDFT
D/A
converter
Channel
A/D
converter
DFT
converter
Parallel
to
serial
Estimated data
converter
Figure 4: OFDM implementation by IDFT and DFT
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Some Discussions
A major problem with the multicarrier modulation in general and OFDM system in particular is the high peak-to-average power ratio (PAR) that is inherent in the transmitted signal.
Large signal peaks occur in the transmitted signal when the signals in the N subchannels
add constructingly in phase. Such large signal peaks may saturate the power amplifier at the
transmitter and thus, cause intermodulation distortion in the transmitted signal.
Multicarrier OFDM using QAM modulation on each of the subcarrier has been implemented
for a variety of applications, including high-speed transmission over telephone lines, such as
digital subcarrier lines. This type of OFDM modulator has also been called discrete multitone
(DMT) modulator. OFDM is also used in digital audio broadcasting in Europe and other parts
of the world and in digital cellular communication systems.