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Mathematical Model of the OFDM Signal

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Mathematical Model of the OFDM Signal



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 Y)n

-

 

and any signal s(t) may be represented in the following lineal form

 

N -1

s(t) = kA Yk (t ) , (2.20)

k =0

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

 

=

 

e j 2 fD (n-m )(T / 2) - e - j 2 fD (n-m )(T / 2)

 

=

 

j2

 

fD

 

(n - m)T

 

(2.23)

 

For the case m = n, the expression (2.23) directly yields e^ = 1. The orthogonality is achieved by setting

Df = 1/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 sinc1 functions, each one separated by 1/T (Ak are considered deterministic constants), i.e.

 

S ( f ) =

 

1 sinc(x) sin(px)/px

 

19

 


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)|2 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 0). There is an infinite number of functions satisfying the previous condition. Besides the rectangular window in (2.22), we will mention the linear pulse (2.28) and the raised cosine pulse (2.29). The two later windows are controlled by a parameter ar called roll-off factor, 0 a r 1, which determines the bandwidth of the window,




 

20

 


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 =0.4.

 

Figure 2.11. Representation of normalized windows: rectangular, raised cosine and linear (ar =0.4).

 

21

 


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 = 0.4).

 

During transmission of the OFDM signal, the symbols will be multiplied by the window w(t) and not by

w2(t). The spectra of the pulses w(t) have no zero-crossings at integer values of 1/T as it happens with w2(t)

-it only happens for the particular case of the rectangular window, because in this case w(t) = w2(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=1

'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.

 

Figure 2.13. General scheme of an OFDM receiver.

 

22

 








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