The principles of OFDM have been developed about 25 years ago (see ), however, practical interest has only increased recently, due in part to advances in signal processing and microelectronics. In the past, as well as in the present, this same modulation scheme is referred to as multitone, multicarrier, Fourier transform, and orthogonal frequency division multiplex communication. Throughout this document we shall use the latter name and abbreviate it as OFDM.
The main idea behind OFDM is to split the data stream to be transmitted into N parallel streams of reduced data rate and to transmit each of them on a separate subcarrier. These carriers are made orthogonal by appropriately choosing the frequency spacing between them. Therefore, spectral overlapping among subcarriers is allowed, since the orthogonality will ensure that the receiver can separate the OFDM subcarriers, and a better spectral efficiency can be achieved than by using simple frequency division multiplex. Next, we give a mathematical description of the OFDM signal and we present a typical OFDM system.
The following description of the OFDM signal and the OFDM communication system is mainly based on references , ,  and . In its most general form, the lowpass equivalent OFDM signal can be written as a set of modulated carriers transmitted in parallel, as follows:
We define the nth OFDM frame as the transmitted signal for the nth signaling interval of duration equal to one symbol period Ts, and denote it by Fn(t). By substituting Fn(t) in equation (1.1) instead of the term in parenthesis which corresponds to the nth OFDM frame , the relation can be rewritten as
and thus, Fn(t) corresponds to the set of symbols Cn,k, k=0...N-1, each transmitted on the corresponding subcarriers fk.
Demodulation is based on the orthogonality of the carriers gk(t), namely:
and therefore the demodulator will implement the relation:
The block diagram of an OFDM modulator is given in Figure 1.1, while the demodulator is shown in Figure 1.2, where, for simplicity, we have ignored the filters inherent in all communication systems.
In order to make an OFDM system practical, a more economical implementation of the modulator and demodulator is required, since according to Figure 1.1 and Figure 1.2 a large number of identical modulator/demodulator blocks would be needed. This can be accomplished through discrete time signal processing and by making use of the filtering properties of the discrete Fourier transform (DFT). By sampling the low pass equivalent signal of (1.1) and (1.4) at a rate N times higher than the subcarrier symbol rate 1/Ts, and assuming f0=0 (that is the carrier frequency is equal to the lowest subcarrier frequency), the OFDM frame can be expressed as:
Next, we point out the difference between OFDM and FDM (frequency division multiplex). Let us consider the power spectrum density for the two systems with binary phase shift keying (BPSK) data on all carriers. Further, let the data streams originate from one, rate R, BPSK stream through an appropriate serial to parallel (S/P) conversion. Figure 1.3 illustrates the two spectra indicating the occupied bandwidth W as function of the number of carriers N.
From this figure one can see that the OFDM signal requires less bandwidth as the number of carriers is increased, and in the limit we have:
This is possible since there is spectral overlapping, which is then resolved making use of the orthogonality of the subcarriers, as stated in equations (1.5) and (1.6).
By performing the sampling as indicated, the OFDM signal is subject to no loss. This is so, since, in view of relation (1.9), the two sided bandwidth of the lowpass equivalent OFDM signal (neglecting sidelobes due to the outer subcarriers) is W=N/Ts. Then, the sampling rate of N/Ts is exactly the corresponding Nyquist rate, and hence there will be no frequency domain aliasing. For illustrative purposes, Figure 1.4 shows the typical power spectrum of an OFDM signal, where the frequency axis is normalized to the inter-carrier spacing 1/Ts.
In conclusion, up to a constant multiplying factor of N, the sampled OFDM frame can be generated using an inverse DFT (modulation function), and hence the transmitted data can be recovered from the OFDM frame through DFT (demodulation function). A block diagram of the digital OFDM system employing DFT is given at the end of the next section (Figure 1.6), after discussing the need for a cyclic prefix.
When a signal s(t) which is passed through a channel with impulse response h(t), the received signal is given by the convolution:
and if the channel is not ideal, there will be inter symbol interference (ISI). It is convenient to view the OFDM signal in terms of data frames, so we can appreciate that the channel will produce ISI within the frame, and will also produce inter frame interference (IFI) among adjacent frames. Considering the discrete time equivalent signal and the channel hi, i=0...L, with L being the delay spread of the channel, relation (1.10) becomes
Figure 1.5 shows this convolution sum for the particular case of L=2. From this graphical representation it can be seen that the introduction of a guard interval of length equal to the delay spread L of the channel between two adjacent frames will "absorb" the channel delay and hence remove IFI.
This may be accomplished by inserting L leading zeros in each frame at the transmitter and removing them at the receiver. However, in order to also eliminate ISI from within the frame, it is better to use a cyclic prefix instead of an all zero guard interval. In this case, after dumping the prefix at the receiver, one would get the periodic (cyclic) convolution of the transmitted data frame and the channel. The cyclically extended frame can then be written as
After discarding the prefix, the received frame becomes
where (m-i)N represents the modulo N subtraction. After DFT demodulation we get
where Hk is the channel's transfer function at the subcarrier frequency fk from relation (1.3). Therefore, by using a cyclic prefix the effect of the channel is transformed into a complex multiplication of the data symbols with the channel coefficients Hk, and all ISI and IFI is removed. In view of these, the block diagram of a basic OFDM system is as shown in Figure 1.6.
After having introduced the OFDM signaling scheme in the previous section, we give here its major advantages and disadvantages.
Advantages of OFDM signaling:
In terms of drawbacks we mention the following: