We’ve seen important and popular standards that use direct spreading sequences on user data, thus implementing a CDMA scheme. We now review another technique called Orthogonal Frequency Division Multiplex (OFDM), which is increasingly popular and adopted by standards like WiFi, WiMAX, and LTE.
OFDM techniques consist in splitting a user data stream into several substreams, which are sent in parallel on several subcarriers. These substreams and subcarriers benefit from a number of properties that we now review in details.
Recall the classic example of continuous wave to encode information: the carrier frequency in itself in not capable of encoding information. The quantity of information s(t) is encoded by changes or modulation of the wave, and affects the amount of spectrum required Δf_{c} as shown on figure 9.1.
One can of course use several carriers f_{i},i {1,2,…,N_{c}}, and filter them separately. That is a common approach and is used extensively in FDMA systems: in particular multiple network operators who own licenses over a same area must take care not to exceed allowed levels of adjacent channel interference into oneanother’s bands.
OFDM improves on the idea by using orthogonal properties of functions to increase spectral efficiency by choosing a specific interval Δf = f_{i+1} f_{i} between subcarriers. Multiple parallel signal streams are used: s_{i}(t) = exp(jω_{i}t) (where ω_{i} = 2πf_{i}), and in frequency domain: S_{i}(f) = δ(f  f_{i}).
In fact time signals are limited in a time window, and a user information symbol has a time interval for transmission: [0, T_{s}], so
 (9.1) 
(where u_{i} is a user information symbol) and the frequency domain representation of the signal is modified from a perfect Dirac function δ(f  f_{i}) to a sinc function^{1}:
 (9.2) 
This last expression is derived from Fourier transform, using definitions from the next section.
Now recall the duality between time domain and frequency domain, with Fourier transform (and inverse Fourier transform) to switch from one domain to the other. Several definitions exist; let us the following definition of Fourier transform
 (9.3) 
and inverse Fourier transform
 (9.4) 
(Other definitions exist, with different signs under the exponent and different 2π factors, so it is important to always specify what definitions are used.) With this definition, the reader can readily derive formula (9.2):
 (9.5) 
OFDM is a multicarrier modulation in which a user bit stream (of rate R_{u}) is transmitted over N_{c} subcarriers, each having a symbol rate R_{s} = , or a symbol duration T_{s} = = . The advantage of that parallel transmission is that the symbol time may be increased, which mitigates intersymbol interference.
Each symbol stream is multiplied by a function φ_{k} from a family of orthogonal functions {φ_{k}},k {0,…,N_{c}  1}. In CDMA, these functions were Walsh codes, in OFDM, they are windowed complex exponentials (or possibly cosine functions):
 (9.6) 
So in a similar manner to the CDMA forward link presented in section 8.1, multiple channels are multiplexed and combined, using exponential functions instead of Walsh code sequences:
 (9.7) 
(where u_{i} = s_{i}g_{i}; s_{i} represents the information symbol (+1 or 1), g_{i} is the individual channel gain) or for many successive bits m = 0,1,2,..., etc.
 (9.8) 
That sequence is manipulated further and sent over the air; on the receiver side, that sequence may decoded by using orthogonal properties of {φ_{k}}.
 (9.9) 
where δ_{kl} is the Kronecker symbol (i.e. 0 if k = l, 1 otherwise), and the asterisk (*) denotes the complex conjugate of the expression. Equations (9.6) and (9.9) gives us the orthogonal condition for subcarriers’ spacing: the right hand side equals zero if and only if (ω_{k} ω_{l})T_{s} = 2πn, for n nonzero integer. This leads to the condition Δf = n∕T_{s}, and 1∕T_{s} is the smallest separation between two subcarriers.
On the receiving side, the sequence may be decoded by simply integrating for each channel; for channel k, the information bit is retrieved from the sign of the integral:
 (9.10) 
Although in this case an additional trick is used, and direct and inverse Fourier transforms are used for decoding.
If we then examine the Fourier transform of our functions given in equation (9.6), we obtain a sinc function of pseudoperiod T_{s}, which means that in the frequency domain subcarriers are spaced exactly such that the peak of the next one corresponds to the previous one’s first zero – see figure 9.3.
The overall envelope looks a bit like a spread spectrum signal, and may be tapered further to reduce out of band spectral power density.
The above choice of orthogonal basis functions has another useful property, relating to Fourier transform. Indeed sampling S_{tot} turns the above expressions into the usual discrete Fourier Transform (DFT), and therefore, instead of multiplying, summing, and then integrating for decoding, OFDM allows to simply carry out a DFT and its inverse (IFT), which are very efficient operations.
Looking back at the Fourier transform (9.3), and sampling the time function and its Fourier transform (with N samples), one may define the following notations: u_{k} = u(t_{k}),t_{k} = kτ,k {0,1,…,N  1}, and U_{n} = U(f_{n}),f_{n} = ,n {N∕2,…,N∕2}. And one obtains the discrete Fourier transform
 (9.11) 
and the inverse discrete Fourier transform
 (9.12) 
Now comparing these discrete transforms to above S′_{tot} with the particular OFDM orthogonal functions, one sees that the s_{k,m} ⋅g_{k,m} coefficients are Fourier transforms ^{2} of the complex amplitude of the subcarriers.
Consequently, encodingdecoding of an OFDM signal is practically not done with integration like (9.10), but by simple FFT. The transmitter builds: {S(ω_{n})} = DFT{s_{k,m} ⋅ g_{k,m}}. And the receiver decodes the received spectral signal: S_{tot} = IDFT{S(ω_{n})}.
If sampling is made as a power of 2 (N = 2^{p}), DFT (and IFT) algorithms are in O(Nlog_{2}N), referred to as fast Fourier transform (FFT) and very efficient to implement. OFDM schemes are therefore based on the nearest power of two, and when fewer subcarriers are users, the same order N = 2^{p} is used for the FFT algorithms, but with zero entries for a subset N_{z} = N  N_{c}.
Note that increasing the number of subcarriers in a given band of spectrum does not increase capacity but provides a useful parameter to optimize: there is an interesting tradeoff between number of subcarriers N_{c} and subcarriers symbol time T_{c}. The more subcarriers are used, the longer their symbol rate is, which means that the overall rate of information remains the same, but a longer symbol rate is useful for multipath mitigation (recall conditions when equalizers are required). Consequently subcarrier spacing is a fundamental parameter to chose for an OFDM standard like WiFi or WiMAX.^{3}
A few more standard techniques are used in combination with the above OFDM definition in practical radio systems. [104]
OFDM systems are therefore well suited to resolve rich multipath situations and slow time varying channels, which explains their popularity for standards like WiFi. They are however not ideal for Doppler shift and phase noise.
WiFi is a standard for interoperable equipment, certified by the WiFi alliance, and based on various iterations of IEEE 802.11, which uses OFDM for its highest throughput profiles. WiFi has been the most successful local area network standard, and it is worth spending some time examining some of its OFDM parameters.
Details of the 802.11 air interface can be found in a number of references, and recent books have good overview of the latest efforts [105], including good overview of 802.11n [106]. We only examine here some aspects of 802.11 as they relate to OFDM in order to provide some insight on performance goals and limitations.
General Parameters: 802.11a/g uses an N = 64 point FFT in a 20MHz channel, δ_{f} = 1∕T_{s} = 312.5kHz, T_{s} = 3.2μs, 4 μs time block is used, with cyclic, 52 of the 64 subcarriers are populated, 4 are pilots (for phase and frequency training and tracking), 48 actually carry data.
The 802.11g packet structure includes the following:
802.11n is a high throughput amendment to 802.11 containing improvements over 802.11a/g. So what exactly does improve in this high throughput amendment? We review its major improvements in the physical and MAC layers.
A number of improvements in the physical (PHY) layer were designed to increase throughput in some situations, although these improvements may come at a cost, which will be pointed out.

The media access control (MAC) layer deals with multiple element addressing, channel access prioritization, and control. It transmits among other things beacons with regulatory and management information (such as country code, allowed channels, max power), and scans channels for beacons. Scanning is usually done passively but when regulations allow it, active probe requests can be sent for specific SSIDs or BSSIDs. MAC improvements for 802.11n include:
A number of further development are in the work for 802.11. They produce new amendments to the specification with the following goals:
In addition, a similar standard was recently produced to deal with longer links in TV white space. 802.22 addresses Wireless Regional Area Networks (WRAN), PHY MAC, policies and procedures for operations in TV white spaces (TVWS); the standard was published 7/2011, and was widely reported on in the press, nicknamed super WiFi. Given the TVWS spectrum landscape, 802.22 defines 6, 7, or8 MHz channels, it uses 2048FFT, up to 64QAM, 200300μs symbol time, which adapts well to wider area delay spreads.
IEEE 802.16 is a standard for wide area wireless networks [107]. The group focuses from the beginning on important service providers’ requirements for service reliability. Consequently 802.16 standardizes important features such as quality of service (QoS), security, flexible and scalable operations in many RF bands. WiMAX goes one step further and narrows down some implementation choices of 802.16 in order to achieve interoperation between equipment manufacturers. WiMAX still standardizes several air interfaces and several profiles in different frequency bands. Of course, performance varies with frequency, channel bandwidth, and other profile characteristics; and conformance between products and suppliers exist only in a given profile. [108]
Two very different families of WiMAX systems exist: fixed and mobile WiMAX. In addition, a regional initiative, WiBro, which resembles mobile WiMAX, has been standardized in Korea.
Although the standard community is focusing on mobile WiMAX, fixed WiMAX applications still have a small role to play, especially in less dense areas. Small and large service providers worldwide have conducted over 200 fixed WiMAX trials, and analysts once estimated some growth potential for fixed wireless market. ^{4} All in all, fixed wireless access remains usually a fairly small scale offering, led by small carriers, and do not achieve the order of magnitude of mobile wireless carriers. (Recall figure 1.4 from chapter 1.)
Fixed WiMAX is based on the 802.16d standard and has the following properties:
OFDM is primarily used for fixed access. For mobility WiMAX uses a method for providing multiple user access in different simultaneous OFDM subchannels. This Orthogonal Frequency Division Multiple Access (OFDMA) is the true focus of 802.16 and WiMAX standards. Figure 9.4 shows how groups of subcarriers form subchannels, which are allocated to different users (as well as pilot and control channels). [113] [114]
Mobile WiMAX is based on the 802.16e standard and has the following properties:


WiMAX frame structures are flexible in terms of use of subcarriers, which can be allocated to different subscriber units according to their needs (figure 9.5). The number of subcarriers is used as a mean to establish frequency reuse schemes. Recall from §2.1.1 that the reuse factor has a strong impact on spectrum efficiency, and that one of the strength of CDMA is to allow a reuse factor of one whereas TDMA schemes needed higher reuse factors. Mobile WiMAX and OFDMA use fractional reuse to optimize spectrum in different areas: the concept is simple, use all subcarriers near the center of the cell (full use of subcarriers, or FUSC), but only make partial use of subcarriers (PUSC) in areas where they would interfere.
Further work in 802.16m will provide the 4G evolution in a backward compatible way (including MIMO and OFDMA); 4G improvements are inserted in reserved fields that can be ignored from legacy 802.16e gear, but utilized by future 802.16m equipment.
The goal of LTE is to provide 3GPP with further evolutions, improving its architecture, throughput, and spectrum efficiency. LTE can:
LTE’s air interface, like other 4G standards, revolves around OFDMA. MIMO is used to either enhance data rates or increase data integrity (diversity and MRC). And the other usual tools are used as well: convolutional and turbo codes, and adaptive modulation (QPSK, 16QAM, 64QAM). LTE offers a flexible range of channel bandwidth (1.4, 3, 5, 10, or 20 MHz), which is well adapted to the current cellular and PCS bands.

Subsequent releases of 3GPP LTE have been published:
LTE uses OFDMA for the downlink, with a fairly simple frame structure, and SCFDMA for the uplink.
LTE FDD uses 10ms frames, divided into 20 subframes or slots (of 0.5ms each). Each subframe uses 7 OFDM symbols, each with a cyclic prefix. Subchannels separation is Δf = 1∕T =15kHz, where T is the OFDM symbol period. (For multimedia broadcast multicast service MBMS dedicated cell, reduced carrier spacing can be used in the downlink Δf=7.5kHz). A cyclic prefix (CP) is used to duplicate part of the symbol: total symbol duration T_{s} = T_{u} + T_{cp}. For normal 15kHz subcarrier spacing, the normal CP is 7 OFDM symbols per slot, which works well in typical urban multipath (T_{u} = 66.7μs, and T_{cp} = 5.21μs for first symbol, 4.7μs for the following symbols). An extended CP for larger cells or heavy multipath is available: T_{cp} = 16.67μs.
This splits radio resources into time and frequency elements, called resource blocks. On the frequency scale a resource block is 12 subcarriers wide (180kHz), on the time scale it is one slot (0.5ms).

There are three downlink channels in the physical layer, shared, control, and common control. And there are two uplink channels, the shared and the control channel. Modulation techniques used for uplink and downlink are QPSK, 16 QAM, 64 QAM while the broadcast channel uses only QPSK.

The uplink standard is departing from the usual OFDMA approach: it uses single carrier FDMA (SCFDMA). SCFDMA is a type of frequency domain equalization (FDE). In SCFDMA, a bit stream is converted into single carrier symbols, then a Discrete Fourier transform (DFT) is applied to it, subcarriers are mapped to these DFT tones and an inverse DFT (IDFT) is performed to convert back for transmission. Much like in OFDMA, the signal has a cyclic prefix to limit ICI, and pulse shaping is used to limit ISI.
Similar parameters are used as for downlink: subcarrier spacing 15kHz, CP normal or extended (Note that CP is the same for all UE in cell, and the same as downlink). The uplink uses the same symbol period and resource elements as in the downlink. Resource blocks are defined in the same manner, with N_{SC}^{RB} = 12 subcarriers and N_{RB} depends on bandwidth: 6, 15, 25, 50, 75, or 100.
The reasons for preferring SCFDMA over OFDMA are mainly that transmitting mobile units have strict limitations on the transmit power, and that peaktoaverage power ratios (PAPR) are high for OFDMA. [118]

LTE physical layer throughput calculations are easily derived from the 3GPP specifications: 1 Radio Frame has 10 subframes, each subframe has 2 timeslots, each timeslot is 0.5 ms long, 1 timeslot has 7 modulation symbols or OFDMA symbols (when normal CP length is used). Each modulation symbol = 6 bits at 64 QAM (note that these are physical layer bits, not actual user information).
A Resource Block (RB) uses 12 Subcarriers. Assume 20 MHz channel bandwidth (100 RBs), normal CP. The number of bits in a 1ms subframe is 100RBs x 12 subcarriers x 2 slots x 7 modulation symbols x 6bits=100800 bits. So the data rate is 100.8 Mbps. For 4x4 MIMO the peak data rate is simply four time that, or 403Mbps. (Of course, a more robust FEC coding, lowers the bitrate to 336Mbps at 64QAM 5/6, or 302Mbps at 64QAM 3/4.)
Note that the above accounts for every resource block, which has to carry overhead signaling, reference signals, etc. Practically, looking at resource elements in a resource block for one (1ms) subframe, some resource elements are reserved (for instance with control frame indicator CFI=2).

Out of the 12x14 RE, 36 are used for control (PDCCH) and reference signals, so only 132 can carry data. (See figure 9.12). So 20% of the physical layer data rate is reserved. So the maximum physical layer data rate is 80.64Mbps (or 322.56Mbps in 4x4 MIMO).
Commonly cited numbers are 75Mbps uplink, and 300Mbps downlink for LTE, this because layer 2 has additional transport block size (TBS) restrictions and frame overhead – typically around 910%, leading to 75Mbps and 300Mbps rates (for 4x4 MIMO in 20MHz).

Many other wireless standards exist and are continually in development for a wide range of applications. Figure 9.14 shows a summary of the most popular ones with their typical throughput, range, and domain of applications.
Such table is difficult to keep uptodate as standards work focuses on new needs and new opportunities; and incorporate the latest technology innovations into them as needed. Some of them become extremely successful and nearly omnipresent, while others miss their window of opportunity and nearly die on the vine. They range over a wide industry interest from wireless cellular and LAN’s to smart grids, RFID tagging, entertainment and consumer electronics, and much more.

Other non standard solutions are becoming popular, such as one by Flarion (now owned by Qualcomm). The proposal was the basis of work to another IEEE group to be created: 802.20. The proposal initially used 113 subcarriers, 17 of which are used for pilots. (The next four questions refer to this solution)
(For simplicity, ignore guard bands, cyclic prefix, etc, and assume that the entire symbol duration is for user data).