Orthogonal Frequency-Division Multiplexing and Multiple Access
4.2 Fundamental Principles of OFDM Signaling
OFDM is a spectrally efficient and a robust transmission technique for broadband digital communication systems that is particularly effective
against many channel adversities, including frequency selective fading and impulse noise. Perhaps, the key motivation behind the widespread adoption of OFMD signaling in mobile communication is its ability to convert a wideband frequency selective fading channel into a series of approximatelyflat fading narrowband channels. In other words, OFDM is effective in removing the frequency-dependent aspect of fading without requiring sophisticated equalization procedures; however, it should be noted that OFDM does not eliminate the fading effect itself.
A major advantage of OFDM over conventional FDM is its spectral efficiency. Figure 4.1 shows the division of spectrum of conventional FDM into the secondary channels, and the same for the OFDM scheme. The figure clearly demonstrates the spectral efficiency advantages offered by
Figure 4.1 Spectrum of FDM (a) and OFDM (b). The comparison of the secondary channel spacing demonstrates the spectral efficiency advantage of OFDM.
OFDM. First, as Figure 4.1 illustrates, the OFDM secondary channels’
spectral spacing is much narrower than that of conventional FDM;
enabling, for a given bandwidth allocation, the creation of many more subcarriers and secondary channels for OFDM. Second, the OFDM secondary channels not only spectrally overlap, but are also orthogonal.
4.2.1 Parallel Transmission, Orthogonal Multiplexing, Guard Time, and Cyclic Extension
In conventional serial transmission systems, the data bits (or symbols) modulate a single carrier waveform, where the modulated signal occupies the entire channel bandwidth. In contrast to this, parallel signaling partitions the traffic data into N symbol (or bit) streams each with a symbol duration of Ts. Each symbol stream modulates a subcarrier; the sum of these waveforms constitutes a multicarrier (MCM) signal. Clearly, each of the subsignals occupy a portion of the available bandwidth. When the frequencies of neighboring subcarriers are spaced exactly by recipro-cal of the symbol rate, that is, 1=TsHz, it can be readily shown that they exhibit orthogonality over the symbol interval. In other words, tone frequencies are of the form k=Ts where k is an integer. These tones are orthogonal simply because of the following computation.
1 Ts∫
Ts
0
cos 2 πk1t=Tscos 2πk 2t=Tsdt 1 if k1 k2
0 if k1 ≠ k2
(4.1)
Under this orthogonality circumstance, the MCM signal constitutes an OFDM waveform.
More often than not, OFDM is accompanied by channel coding. The resulting signaling technique is referred to as Coded-OFDM; COFDM.
Assume that the input bit stream to the OFDM modulator is given by sequence f g. This sequence is converted into N parallel symbolak
streams, which is represented by Sf g. Each of the streams modulate ak
subcarrier of the OFDM system. Note that unless the subcarrier modu-lations are binary, Sf g is a complex sequence. As Equation 4.1 indicates,k
by selecting the frequencies of subcarriers as multiple integer of the inverse of the actual symbol duration, orthogonality of secondary chan-nels is guaranteed. Subcarrier waveforms orthogonality in OFDM signal-ing can be interpreted from two different perspectives:
1) In the time domain, orthogonality implies that individual signals may be separated through correlation processes without imposing inter-symbol interference on each other. This is an important feature of OFDM signaling.
2) From the frequency domain point of view, orthogonality of OFDM subsignals implies that the signals satisfy thefirst Nyquist criterion in the Fourier domain. In other words, when the spectrum of one subsignal is at its maximum value the spectra of all other subsignals cross the frequency axis. A direct consequence of this property is that the spectrum of OFDM signal becomes almost a flat one. This is shown in Figure 4.1.
4.2.1.1 Cyclic Prefix and Guard Time
Since OFDM, through parallel transmission, spreads out the effect of frequency selective fading over a number of secondary channels, it can essentially convert the channel into a series of approximatelyflat fading channels, and thus mitigating the effect of ISI to a great extent. To eliminate ISI in the channel almost completely, a guard time is added to each OFDM symbol. In other words, the guard time is inserted to eliminate the possible residual ISI caused by frequency selective fading channel, most of which has already been removed by channel split process that is inherent in OFDM signaling. Guard time is normally selected to be larger than the channel delay spread (denoted byστas per Chapter 2), or larger than maximum channel delay spread (σmax), which can also be considered as the time duration of the multipath channel impulse response. Should the guard interval be composed of merely baseline, the subcarrier signals would not remain orthogonal, which would give rise to intercarrier (actually intersubcarrier) interference (ICI)2problem. To circumvent this problem, the OFDM signal is cyclically extended within the guard interval, this is referred to as cyclic prefix (CP), since the guard band is normally added to the beginning of the OFDM symbol. The time duration of the CP or guard band is denoted by Tcp. Note that the guard time is not a part of useful signal duration and will be discarded after the detection process. The relationship between useful symbol period, guard interval, and transmission time duration T is given by T Ts Tcp. The ratio of the guard interval Tcp to Ts (useful symbol duration), Tcp=Ts, which reflects the bandwidth overhead required for the guard time is, by and large, application dependent. For IEEE 802.16e Std.-based networks (WiMAX and AeroMACS), Tcp=Tsmay be selected from values 1=2, 1=4, 1=8, and 1=16. In classical OFDM applications, however, since the insertion of guard interval reduces the data throughput, Tcp=Tsis usually selected to be less than 1=4, which includes the ratio of 1=32 as well [8].
2 In the interest of preventing ambiguity, the following two terms are defined. ISI refers to intersymbol interference between various OFDM signals. The term ICI (intercarrier interference) is used to denote cross talk or interference between subcarrier signals within a single OFDM waveform.
The proper selection of CP time duration maintains subcarrier orthogo-nality, ensures that the delayed echoes of the OFDM signal have integer number of cycles within the transmission time duration (essentially con-verts linear convolution to cyclic convolution), therefore, multipath signals with delays less than guard time do not cause intersymbol interference, ISI.
Thus, the CP allows the complete reception of one OFDM symbol before the next symbol is received. The insertion of CP, also, increases the receiver tolerance to symbol time synchronization error [4].
4.2.2 Fourier Transform-Based OFDM Signal
For radio communications OFDM creates a doubly modulated signal. First, the parallel streams of data modulate the subcarriers whose frequencies are determined by k fs, where fs 1=Ts. The subcarrier signals are then added and multiplexed to the RF signal of carrier frequency fc. Thus, the OFDM symbol, in the mth transmission interval, may be expressed by
sm t XN 1
k0
Re S m 1Nkexp j2π kf s fc
t
m 1
T t mT
0 otherwise
(4.2) Hence, the OFDM general signal that extends from 1 to 1 can be written as follows.
sOFDM t X1
m 1
XN 1
k0
Re S m 1NkΠ t 2m 1T=2 T
exp j2π kf s fc
t
(4.3) HereΠ t t0
T
is a rectangular function that is equal to 1 over t0 T=2 t t0 T=2 and it is equal to zero outside of this interval. The equivalent complex baseband signal over the fundamental OFDM signal interval (m= 1) derived from Equation 4.2 is given in Equation 4.4.
s t XN 1
k0
Skexp j2πkf st
f or 0 t T (4.4)
Clearly, s t is the inverse Fourier transform of N complex symbols Sf g.k
Taking samples of this signal that are apart by Ts=N provides the result given in Equation 4.5.
s nTs XN 1
k0
Skexp j2 πkfsnTs=N
(4.5)
A comparison is now made between this latter equation and the standard inverse Fourier transform equation for N samples of a general signal x(t), as given in Equation 4.6.
x nTs 1 N
X
N 1
k0
X k
N Ts
exp j2 πnk=N (4.6)
Ignoring the constant factor of 1=N in front of Equation 4.6, since Sks are actually samples of Fourier transform of s t , the two equations are identical under the following condition:
fsTs N 1
N ! fs 1 Ts
(4.7) This is in fact the same condition that is required for orthogonality.
Therefore, one consequence of maintaining orthogonality is that the OFDM signal can be defined by Fourier transform [9]. In summary, the complex symbols Sf g are Fourier transform of the modulated signal,k which is intuitively obvious since these symbols will be multiplied by their corresponding complex exponential functions to realize the modulation process. To put Equation 4.5 into standard inverse discrete Fourier transform form, by making use of Equation 4.7, and as it is customary, dropping Tsfrom the equation, that is, letting s nT s sn, we conclude
snXN 1
k0
Skexp j2 πkn=N (4.8)
Equation 4.8 implies that the discrete version of the OFDM signal may be generated by computing the IDFT3of the complex input symbols at the transmitter, and conversely the complex symbols are obtained (or esti-mated) through DFT calculation of the down-converted signal at the receiver. In practice, for sufficiently large value of N, these computations are implemented using fast Fourier transform (FFT) algorithms. If N is not a power of 2, the symbols are padded with zeros such that total number of symbols is a power of 2. It is also noted that the number of subcarriers is equal to the size of FFT. That is to say, for instance, a FFT size of 512 corresponds to 512 available subcarriers.
4.2.3 Windowing, Filtering, and Formation of OFDM Signal
The OFDM signal formed by performing IDFT and adding guard interval is equivalent to a number of unfiltered modulated (BPSK, QPSK, QAM,
3 As it was mentioned earlier, the standard equation for IDFT contains a factor of 1/N in front of the summation sign. The effect of this factor can be reflected in the computation of DFT at the receiver side.
etc.) signals whose out-of-band spectrum falls off rather slowly. To force the spectrum to decay more rapidly around the boundary and outside of the allocated band, in order to comply with regulatory spectrum masking, two techniques of windowing and filtering are available. These two techniques may use the same mathematical equations, for example, raised cosine, root raised cosine, and so on. However, windowing refers to multiplication in the time domain, or equivalently convolution in the frequency domain; whereas filtering entails convolution in the time domain and multiplication in the frequency domain. As such computa-tional complexity of their implementation is different. In general, the implementation of a window, in the context of DSP realization, is less complex than that of afilter with the same characteristics.
In practice, the OFDM signal is generated according to the following algorithmic order.
1) The complex symbols are padded with sufficient zeros to get proper number of samples for IFFT computation.
2) The computation of IFFT sample values is carried out.
3) The samples of IFFT are inserted at the start of the OFDM symbol, and thefirst samples of IFFT are appended to the end of the OFDM symbol (CP).
4) The symbol is then multiplied by a raised cosine window, orfiltered by a raised cosinefilter.
The OFDM signal so generated will be added to the previous OFDM signal with a delay of T Ts Tcp[4].
Time–Frequency Resource in OFDM Signaling
It is insightful to view OFDM bandwidth resource from the perspective of time–frequency framework, as shown in Figure 4.2.
Figure 4.2 OFDM symbols illustrated in frequency–time grid.
Figure 4.2 illustrates a frequency–time resource grid for eight OFDM symbols. Each small square represents the basic resource unit, which is a subcarrier (tone) signal made available at certain point in time. The available bandwidth, in this example, is divided into eight secondary channels. The frequency spacing separating adjacent subcarriers is equal to the inverse of OFDM symbol time duration Ts, which guarantees subcarrier orthogonality.
4.2.4 OFDM System Implementation
In practice, the OFDM transmitter and receiver are implemented as a digital system using FFT algorithm, however, in the interest of gaining more insight, we examine the analog implementation of the OFDM transmitter circuit. Figure 4.3 illustrates a block diagram for an OFDM transmitter system in the analog domain. The serial stream of traffic data symbols, which might originate from a single source or a multiple of sources, are primarily converted into N parallel data streams, each of which modulates a sinusoidal subcarrier signal. The subcarrier signals, as shown in Figure 4.3, are generated by a bank of oscillator circuits. The subcarrier signals (secondary signals) are added next. Guard time, through cyclic extension, is arranged for the resulting signal. The signal is then multiplied by a, perhaps, raised cosine windowing function. The last phase is the RF modulation of the OFDM signal, which is the process of up-converting the signal to any required radio frequency band.
The oscillation frequencies of the bank of oscillators shown in the circuit diagram of Figure 4.3 are determined based on a total available OFDM channel bandwidth BT, the starting frequency of the band f0, and the number of required subcarriers N, as shown in Figure 4.4.
The available bandwidth is divided into N bands to create N secondary OFDM channels, therefore, BT=N 1=Ts. The frequency of tones (sub-carriers) are, hence, given by Equation 4.9.
f1 f0BT
2N f0 1
2Ts
f2 f03BT
2N f0 3
2TS
... ... ... ... ...
fN f0 2N 1BT
2N f0 2N 1 2Ts
8>
>>
>>
>>
>>
>>
><
>>
>>
>>
>>
>>
>>
:
(4.9)
At the OFDM receiver side, following signal detection, what essentially takes place is that all signal processing operations performed at the transmitter are reversed.
4.2.5 Choice of Modulation Schemes for OFDM
Modulation schemes for OFDM signals are normally selected from the families of MPSK and QAM digital modulation techniques. In general, a modulation scheme is selected for a given application based on evaluation of various performance measures of the technique, and how closely it matches the characteristics of the transmission channel, and the require-ments set forth by the user [10]. Important criteria for this selection are bandwidth efficiency and spectral behavior, power/error performance (BER), noise and interference immunity, and hardware/software com-plexity of the modulator and detector schemes. For OFDM, the choice of modulation schemes for various subchannels is dictated by a compromise between data rate requirements and transmission robustness. An inter-esting feature of OFDM is that different modulation techniques may be exploited for different subcarriers. This would facilitate layered services in a wireless network such as WiMAX and AeroMACS [11].
Figure 4.4 Dividing the OFDM bandwidth into N secondary channels (subcarriers).
Figure 4.3 An analog implementation for OFDM transmitter system.
4.2.6 OFDM Systems Design: How the Key Parameters are Selected The selection of OFDM parameters involves a process of compromise between available resources, user’s requirements, and channel char-acteristics. The primary resource available to the system designer is the available channel bandwidth. The user requirements are normally expressed in terms of data rate and data reliability, that is, bit error rate (BER). Channel statistics such as noise PSD for AWGN (additive white Gaussian noise) channels, or fading characteristics as well as delay spread and coherence bandwidth for frequency selective fading channels are needed as a starter for the design of OFDM system. Normally, available bandwidth, data throughput, and tolerable delay spread are given. As a rule of thumb, the guard interval should be selected two to four times longer than delay spread, where the exact value depends on modulation and coding schemes applied in the design [4]. The choice of useful signal duration is a critical one. To minimize SNR loss caused by the guard time, it is desirable to have the useful signal duration much longer than the guard interval. However, longer useful symbol duration results in an increase in the number of subcarriers and smaller carrier spacing, which implies more sensitivity to phase noise and carrier frequency error, as well as larger size FFT algorithm [12]. In practice, how closely two subcarriers can be placed depends on tolerable phase noise and carrier offset. For mobile reception of OFDM signal, the subcarrier spacing must be large enough to make the Doppler shift negligible. The useful symbol duration should be selected such that channel impulse response does not change during symbol time period [13].