In order to perform the mathematical analysis of OFDM signals, the most convenient strategy is to use the signal theory related to the orthogonal expansion of signals [Pro95]. In this theory, the signal is contained in a subspace and is represented by a linear combination of the base signals Yn(t) of that subspace. The signals conforming the
base have the property of being orthonormal, i.e.
Y (t Yn
and any
signal s(t) may be represented in the following
lineal form
N
st) kA Yk t ) (2.2 )
k
where the coefficients Ak are complex. As we already mentioned in 2.3.1, OFDM is a special case of MCM
(see Figure
2.8),
but with the
special
feature
that the
different sub carriers are allowed to overlap their
spectra. In the case of OFDM, the system is sketched in Figure 2.9. From Figure 2.9 it is straightforward to
identify the
kind of base
signals being used, i.e.
where w(t) is a window of length T.
The easiest choice for w(t) is the normalized rectangular window defined as
ejfDnm(T/ 2) e jfDnm(T/ 2)
j2
fD
(nm)T
)
For the case m = n,
the expression (2.23) directly
yields
e= 1. The orthogonality is achieved by setting
Df T, thus making e 0 for any m n. The window w(t) in (2.22) was selected to be the rectangular pulse only for simplicity. Considering this window, the Fourier transformation of the signal s(t) in (2.20) will be made up by a combination of N sincfunctions, each one separated by 1 T (Ak are considered deterministic constants),
i.e.
S( f )
sinc(x) sin(pxpx
Figure 2.9. General
scheme of an OFDM transmitter.
The total bandwidth occupied by the OFDM
signal may
be obtained
from (2.24). This bandwidth directly depends on the
selected window w(t). In this
particular case, the sinc function decreases as 1/f, thus
generating big secondary lobes that
extend
the bandwidth.
Hence, the total bandwidth may be reduced by considering other windows with smaller secondary lobes.
or equivalently,
The functions w(t) which accomplish the condition in (2.27) are those having a vestigial symmetry around t=T 2. The definition of vestigial symmetry is given graphically in Figure 2.10, where the integral in (2.27) has been represented for
the particular case of k=1. Nevertheless, due to
the even
symmetry of the cosine function around t = T 2 in (2.27), the integral will be zero for
any integer
value of k (k . There is an infinite number of functions satisfying the previous condition. Besides the rectangular window in (2.22), we will mention the linear pulse (2.2 ) and the raised cosine pulse (2.29). The two later windows are controlled
by a parameter ar called roll off
factor, 0 ar 1, which determines
the bandwidth of the window,
Figure 2.10. Graphical interpretation of the vestigial symmetry.
Note that the previous expressions represent the square value of the respective window. For the special case
of ar = 0, both windows (2.28) and (2.29) result in the rectangular one. The representation of these windows
is sketched in Figure 2.11, for the
particular case
of ar
Figure 2.11. Representation of normalized windows: rectangular,
raised cosine and linear (ar
The analysis of the spectrum of these windows in Figure 2.12 shows that all of them have zero crossings at integer values of 1 T, thus achieving orthogonality. In addition, the linear and raised cosine pulses achieve a better spectral efficiency because of their lower secondary lobes.
Figure 2.12. Comparison of the spectra for different normalized windows
(ar
During transmission of the OFDM
signal, the
symbols
will be multiplied by the
window
w(t) and not
by
w(t). The spectra of the pulses w(t) have no zero crossings at integer values of 1 T as it happens with w(t)
it only happens for the particular case of the rectangular window, because in this case w(t) = w(t). The orthogonality is achieved at the receiver side by multiplying the incoming signal by w (t). The information sequence Ak is obtained after multiplication by the corresponding complex conjugated phasor and integrating
over an interval T(1+ar), as shown in Figure
2.13. In Figure 2.14, the spectrum of an OFDM signal with Ak
k and N=32 is represented. There it can be better appreciated the dependency of the total bandwidth of the OFDM signal with respect to the selected window. Clearly, the root raised cosine (RRC) window results in
the better bandwidth efficiency among the proposed windows.
