Chapter 1

The principles of OFDM have been developed about 25 years ago (see [21]), 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 [1], [2], [9] and [16]. In its most general form, the lowpass equivalent OFDM signal can be written as a set of modulated carriers transmitted in parallel, as follows:

(1.1) | ||||

with | (1.2) | |||

and | (1.3) |

where

- C
_{n,k}is the symbol transmitted on the k^{th}subcarrier in the n^{th}signaling interval, each of duration T_{s} - N is the number of OFDM subcarriers
- f
_{k}is the k^{th}subcarrier frequency, with f_{0}being the lowest frequency to be used.

We define the n^{th} OFDM frame as the transmitted
signal for the n^{th} signaling interval of duration equal to one
symbol period T_{s}, and denote it by F_{n}(t). By substituting
F_{n}(t) in equation (1.1) instead of the term in parenthesis which
corresponds to the n^{th} OFDM frame , the relation can be rewritten
as

(1.4) |

and thus, F_{n}(t) corresponds to the set of symbols
C_{n,k}, k=0...N-1, each transmitted on the corresponding subcarriers
f_{k}.

Demodulation is based on the orthogonality of the carriers
g_{k}(t), namely:

(1.5) |

and therefore the demodulator will implement the relation:

(1.6) |

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/T_{s}, and assuming f_{0}=0 (that is the carrier
frequency is equal to the lowest subcarrier frequency), the OFDM frame
can be expressed as:

(1.7) | ||||

which yields | (1.8) |

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:

(1.9) |

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/T_{s}.
Then, the sampling rate of N/T_{s} 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/T_{s}.

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:

(1.10) |

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 h_{i},
i=0...L, with L being the delay spread of the channel, relation (1.10)
becomes

(1.11) |

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

(1.12) | ||||

where | (1.13) |

After discarding the prefix, the received frame becomes

(1.14) |

where (m-i)_{N} represents the modulo N subtraction. After DFT
demodulation we get

(1.15) |

where H_{k} is the channel's transfer function at the subcarrier
frequency f_{k} 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 H_{k}, 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:

- Makes efficient use of the spectrum by allowing overlap.
- By dividing the channel into narrowband flat fading subchannels, OFDM is more resistant to frequency selective fading than single carrier systems are.
- Eliminates ISI and IFI through use of a cyclic prefix.
- Using adequate channel coding and interleaving one can recover symbols lost due to the frequency selectivity of the channel.
- Channel equalization becomes simpler than by using adaptive equalization techniques with single carrier systems.
- It is possible to use maximum likelihood decoding with reasonable complexity, as discussed in [2].
- OFDM is computationally efficient by using FFT techniques to implement the modulation and demodulation functions. Also, for multiple communication channels, as is the case in digital audio broadcasting (DAB) systems, partial FFT algorithms can be used in order to implement program selection and decimation.
- In conjunction with differential modulation there is no need to implement a channel estimator.

- Is less sensitive to sample timing offsets than single carrier systems are.
- Provides good protection against cochannel interference and impulsive parasitic noise (see [7]).

In terms of drawbacks we mention the following:

- The OFDM signal has a noise like amplitude with a very large dynamic range, therefore it requires RF power amplifiers with a high peak to average power ratio.
- It is more sensitive to carrier frequency offset and drift than single carrier systems are due to leakage of the DFT.

copyright

© 1997, László Házy