192 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 3, FEBRUARY 1, 2009
Fig. 4. Time domain sequence of OFDM symbols showing the cyclic prefix.
the same average power. This simplifies the analysis of many
OFDM functions. The insets in Fig. 3 show an example of the
signals at the input and the output of the IFFT for 4 QAM mod-
ulation and
. The input to the IFFT is a vector of
random values from the 4 QAM constellation
. The output is the corresponding time domain vector
. While the components of take only a few discrete values,
the probability distribution of
is not obvious from the dia-
gram. In fact for
the real and imaginary components of
an OFDM time domain signal are approximately Gaussian. For
wireless OFDM systems which have already been standardized,
values of
ranging from 64 in wireless LAN systems to 8096
in digital television systems have been used. The terminology
throughout the OFDM literature is not consistent. In this paper
the term ‘symbol’ is used to describe the time domain or fre-
quency domain sequence associated with one IFFT operation.
(In some papers this is described as block or frame.)
At the receiver the FFT performs a forward transform on the
received sampled data for each symbol
for
(3)
where
is the vector representing the
sampled time domain signal at the input to the receiver FFT and
is the discrete frequency domain
vector at the FFT output. Note that only
samples are required
per OFDM symbol (excluding CP). To understand the function
of the IFFT, first consider what would happen if there were no
noise or distortion in the channel or the transmitter and receiver
front ends, then because the FFT and IFFT are transform pairs,
.
If additive white Gaussian noise (AWGN) is added to the
signal, but the signal is not distorted then
(4)
where
is a sample of white Gaussian noise, substituting (4)
in (3) and rearranging gives
(5)
where
for
(6)
is the noise component of the th output of the receiver
FFT. Because each value of
is the summation of in-
dependent white Gaussian noise samples,
, it too is an in-
dependent white Gaussian noise process. Even if the time do-
main noise,
, does not have a Gaussian distribution, in most
cases, because of the central limit theorem, the frequency do-
main noise
will be Gaussian. This, combined with the use
of FEC, means that usually the performance of OFDM systems
depend on the average noise power, unlike conventional serial
optical systems where it is the peak values of the noise which
often limit performance.
B. Sequences of Symbols and the Cyclic Prefix
The description above showed how the IFFT generates each
OFDM symbol. The transmitted signal consists of a sequence
of these OFDM symbols. To denote different OFDM symbols
when a sequence of symbols rather than a single symbol is
being considered we need to extend the notation to include
a time index. Let
be the output of the IFFT in the th symbol period. In most
OFDM systems, a CP is added to the start of each time do-
main OFDM symbol before transmission. In other words
a number of samples from the end of the symbol is ap-
pended to the start of the symbol. So instead of transmitting
the sequence
(7)
is transmitted; where
is the length of the cyclic prefix. Al-
though the CP introduces some redundancy, and reduces the
overall data rate, we will show that the use of the CP elimi-
nates both ISI and intercarrier interference (ICI) from the re-
ceived signal and is the key to simple equalization in OFDM.
Fig. 4. shows the time domain sequence of OFDM symbols.
C. Individual OFDM Subcarriers
Considerable insight into the operation of an OFDM system
can be obtained by considering what happens to individual sub-
carriers as they pass through the system. However, it is also
important to note that in an OFDM system because the IFFT