In practice the Fast Fourier Transform (FFT) and IFFT are used in place of the DFT and IDFT, so all further
references will be to FFT and IFFT.
2.2 Definition of Carriers
The maximum number of carriers used by OFDM is limited by the size of the IFFT. This is determined as
follows in Equation 2:
Equation 2 : OFDM Carrier Count
In order to generate a real-valued time signal, OFDM (frequency) carriers must be defined in complex
conjugate pairs, which are symmetric about the Nyquist frequency (f
max
). This puts the number of potential
carriers equal to the IFFT size/2. The Nyquist frequency is the symmetry point, so it cannot be part of a
complex conjugate pair. The DC component also has no complex conjugate. These two points cannot be
used as carriers so they are subtracted from the total available. （奈奎斯特频点和直流点被减去）
If the carriers are not defined in conjugate pairs, then the IFFT will result in a time domain signal that has
imaginary components. This must be a viable option as there are OFDM systems defined with carrier counts
that exceed the limit for real-valued time signals given in Equation 2. Reference [1] describes a system with
IFFT size 256 and carrier count 216. This design must result in a complex time waveform. Further
processing would require some sort of quadrature technique (use of parallel sine and cosine processing
paths). In this report, only real-value time signals will be treated, but in order to obtain maximum bandwidth
efficiency from OFDM, the complex time signal may be preferred (possibly an analagous situation to QPSK
vs. BPSK). Equation 2, for the complex time waveform, has all IFFT bins available as carriers except the DC
bin.
Both IFFT size and assignment (selection) of carriers can be dynamic. The transmitter and receiver just
have to use the same parameters. This is one of the advantages of OFDM. Its bandwidth usage (and bit
rate) can be varied according to varying user requirements. A simple control message from a base station
can change a mobile unit’s IFFT size and carrier selection.
2.3 Modulation
Binary data from a memory device or from a digital processing stream is used as the modulating (baseband)
signal. The following steps may be carried out in order to apply modulation to the carriers in OFDM:
combine the binary data into symbols according to the number of bits/symbol selected
convert the serial symbol stream into parallel segments according to the number of carriers, and
form carrier symbol sequences
apply differential coding to each carrier symbol sequence
convert each symbol into a complex phase representation
assign each carrier sequence to the appropriate IFFT bin, including the complex conjugates
take the IFFT of the result
This is the same modulation technique described in Reference [3]. The Reference [2] matlab program
carries out these steps and provides detailed commentary and examples for each one.
OFDM modulation is applied in the frequency domain. Figure 2 and Figure 3 give an example of modulated
OFDM carriers for one symbol period, prior to IFFT. For this example, there are 4 carriers, the IFFT bin size
is 64, and there is only 1 bit per symbol. The magnitude of each carrier is 1, but it could be scaled to any
value. The phase for each carrier is either 0 or 180 degrees, according to the symbol being sent. The phase
determines the value of the symbol (binary in this case, either a 1 or a 0). In the example, the first 3 bits (the
first 3 carriers) are 0, and the 4
th
bit (4
th
carrier) is a 1.
5
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