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When we talk about the Mathematical description of OFDM then we cannot neglect the following mathematical treatments.
The Fourier transform
The use of the Fast Fourier Transform in OFDM
The guard interval and its implementation
As we have discussed above that a large number of narrowband carriers which are spaced close to each other in frequency domain are transmitted by OFDM. The modern digital technique that is used in the OFDM is FFT i-e Fast Fourier transform (FFT) and due to the use of FFT it reduces the number of modulators and demodulators both at the receiver and transmitter side.
Fig. 4 Examples of OFDM spectrum (a) a single subchannel, (b) 5 carriers
At the central frequency of each subchannel, there is no crosstalk from other subchannels.
Mathematically, each carrier can be described as a complex wave:
sc(t) = the real part of original signal.
Ac(t) = the Amplitude
f c(t) = Phase of carrier
(t)= symbol duration period
Ac(t) and f c(t) use to fluctuate on symbol by symbol basis. Parameter values are constant over (t).
As we know that OFDM posses many carriers. So the complex signals ss(t) is represented as:
This is of course a continuous signal. If we consider the waveforms of each component of the signal over one symbol period, then the variables Ac(t) and f c(t) take on fixed values, which depend on the frequency of that particular carrier, and so can be rewritten:
If the signal is sampled using a sampling frequency of 1/T, then the resulting signal is represented by:
At this point, we have restricted the time over which we analyse the signal to N samples. It is convenient to sample over the period of one data symbol. Thus we have a relationship:
If we now simplify eqn. 3, without a loss of generality by letting w 0=0, then the signal becomes:
Now Eq. 4 can be compared with the general form of the inverse Fourier transform:
In eq. 4, the function is no more than a definition of the signal in the sampled frequency domain, and s(kT) is the time domain representation. Eqns. 4 and 5 are equivalent if:
This is the same condition that was required for orthogonality (see Importance of orthogonality). Thus, one consequence of maintaining orthogonality is that the OFDM signal can be defined by using Fourier transform procedures.
The Fourier transform
Fourier transform actually relate events in time domain to events in frequency domain. There are different version of FFT which are used according to requirement of different sort of work
The conventional transform provide the relation of continuous signals. Note that Continuous signals are not limited in both time and frequency domain. Though, it is better to sample the signal so that the signal processing becomes simpler. But it lead to an aliasing when we sample the signals with infinite spectrum and the processing of signals which are not time limited can lead to another problem that is referred to as space storage.
DFT (discrete Fourier transforms) is use to overcome the above problem of signal processing. The original definition of DFT reveals that the time waves have to repeat frequently and similarly frequency spectrum repeat frequently in frequency domain. Basically in DFT the signals can be sampled in time domain as well as in frequency domain.
The Fourier transform is the process in which the signal represented in the time domain transformed in frequency domain, while the reverse process uses IFT which is the inverse Fourier transform.
The use of the Fast Fourier Transform in OFDM
The main reason that the OFDM technique has taken a long time to become a prominence has been practical. It has been difficult to generate such a signal, and even harder to receive and demodulate the signal. The hardware solution, which makes use of multiple modulators and demodulators, was somewhat impractical for use in the civil systems.
The ability to define the signal in the frequency domain, in software on VLSI processors, and to generate the signal using the inverse Fourier transform is the key to its current popularity. The use of the reverse process in the receiver is essential if cheap and reliable receivers are to be readily available. Although the original proposals were made a long time ago [Weinstein and Ebert], it has taken some time for technology to catch up.
At the transmitter, the signal is defined in the frequency domain. It is a sampled digital signal, and it is defined such that the discrete Fourier spectrum exists only at discrete frequencies. Each OFDM carrier corresponds to one element of this discrete Fourier spectrum. The amplitudes and phases of the carriers depend on the data to be transmitted. The data transitions are synchronised at the carriers, and can be processed together, symbol by symbol (Fig. 5).
Fig. 5 Block diagram of an OFDM system using FFT, pilot PN sequence and a guard bit insertion [Zou and Wu]
The definition of the (N-point) discrete Fourier transform (DFT) is:
and the (N-point) inverse discrete Fourier transform (IDFT):
A natural consequence of this method is that it allows us to generate carriers that are orthogonal. The members of an orthogonal set are linearly independent.
Consider a data sequence (d0, d1, d2, …, dN-1), where each dn is a complex number dn=an+jbn. (an, bn=± 1 for QPSK, an, bn=± 1, ± 3 for 16QAM, … )
k=0,1,2, …, N-1 (9)
where fn=n/(ND T), tk=kD t and D t is an arbitrarily chosen symbol duration of the serial data sequence dn. The real part of the vector D has components
If these components are applied to a low-pass filter at time intervals D t, a signal is obtained that closely approximates the frequency division multiplexed signal
Fig. 5 illustrates the process of a typical FFT-based OFDM system. The incoming serial data is first converted form serial to parallel and grouped into x bits each to form a complex number. The number x determines the signal constellation of the corresponding subcarrier, such as 16 QAM or 32QAM. The complex numbers are modulated in a baseband fashion by the inverse FFT (IFFT) and converted back to serial data for transmission. A guard interval is inserted between symbols to avoid intersymbol interference (ISI) caused by multipath distortion. The discrete symbols are converted to analog and low-pass filtered for RF upconversion. The receiver performs the inverse process of the transmitter. One-tap equalizer is used to correct channel distortion. The tap-coefficients of the filter are calculated based on the channel information.
Fig. 6 Example of the power spectral density of the OFDM signal with a guard interval D = TS/4 (number of carriers N=32) [Alard and Lassalle]
Fig 4a shows the spectrum of an OFDM subchannel and Fig. 4b and Fig. 6 present composite OFDM spectrum. By carefully selecting the carrier spacing, the OFDM signal spectrum can be made flat and the orthogonality among the subchannels can be guaranteed.
The guard interval and its implementation
The orthogonality of subchannels in OFDM can be maintained and individual subchannels can be completely separated by the FFT at the receiver when there are no intersymbol interference (ISI) and intercarrier interference (ICI) introduced by transmission channel distortion. In practice these conditions can not be obtained. Since the spectra of an OFDM signal is not strictly band limited (sinc(f) function), linear distortion such as multipath cause each subchannel to spread energy into the adjacent channels and consequently cause ISI. A simple solution is to increase symbol duration or the number of carriers so that distortion becomes insignificant. However, this method may be difficult to implement in terms of carrier stability, Doppler shift, FFT size and latency.
Fig. 7 The effect on the timing tolerance of adding a guard interval. With a guard interval included in the signal, the tolerance on timing the samples is considerably more relaxed.
Fig. 8 Example of the guard interval. Each symbol is made up of two parts. The whole signal is contained in the active symbol (shown highlighted for the symbol M) The last part of which (shown in bold) is also repeated at the start of the symbol and is called the guard interval
One way to prevent ISI is to create a cyclically extended guard interval (Fig. 7, 8), where each OFDM symbol is preceded by a periodic extension of the signal itself. The total symbol duration is Ttotal=Tg+T, where Tg is the guard interval and T is the useful symbol duration. When the guard interval is longer than the channel impulse response (Fig. 3), or the multipath delay, the ISI can be eliminated. However, the ICI, or in-band fading, still exists. The ratio of the guard interval to useful symbol duration is application-dependent. Since the insertion of guard interval will reduce data throughput, Tg is usually less than T/4.
The reasons to use a cyclic prefix for the guard interval are:
to maintain the receiver carrier synchronization ; some signals instead of a long silence must always be transmitted;
cyclic convolution can still be applied between the OFDM signal and the channel response to model the transmission system.
In an OFDM-based WLAN architecture, as well as many other wireless systems, multipath distortion is a key challenge. This distortion occurs at a receiver when objects in the environment reflect a part of the transmitted signal energy. Figure 2 illustrates one such multipath scenario from a WLAN environment.
Figure 2: Multipath reflections, such as those shown here, create ISI problems in OFDM receiver designs.
Click here for larger version of Figure 1b
Multipath reflected signals arrive at the receiver with different amplitudes, different phases, and different time delays. Depending on the relative phase change between reflected paths, individual frequency components will add constructively and destructively. Consequently, a filter representing the multipath channel shapes the frequency domain of the received signal. In other words, the receiver may see some frequencies in the transmitted signal that are attenuated and others that have a relative gain.
In the time domain, the receiver sees multiple copies of the signal with different time delays. The time difference between two paths often means that different symbols will overlap or smear into each other and create inter-symbol interference (ISI). Thus, designers building WLAN architectures must deal with distortion in the demodulator.
Recall that OFDM relies on multiple narrowband subcarriers. In multipath environments, the subcarriers located at frequencies attenuated by multipath will be received with lower signal strength. The lower signal strength leads to an increased error rate for the bits transmitted on these weakened subcarriers.
Fortunately for most multipath environments, this only affects a small number of subcarriers and therefore only increases the error rate on a portion of the transmitted data stream. Furthermore, the robustness of OFDM in multipath can be dramatically improved with interleaving and error correction coding. Let's look at error correction and interleaving in more detail.
Error Correction and Interleaving
Error correcting coding builds redundancy into the transmitted data stream. This redundancy allows bits that are in error or even missing to be corrected.
The simplest example would be to simply repeat the information bits. This is known as a repetition code and, while the repetition code is simple in structure, more sophisticated forms of redundancy are typically used since they can achieve a higher level of error correction. For OFDM, error correction coding means that a portion of each information bit is carried on a number of subcarriers; thus, if any of these subcarriers has been weakened, the information bit can still arrive intact.
Interleaving is the other mechanism used in OFDM system to combat the increased error rate on the weakened subcarriers. Interleaving is a deterministic process that changes the order of transmitted bits. For OFDM systems, this means that bits that were adjacent in time are transmitted on subcarriers that are spaced out in frequency. Thus errors generated on weakened subcarriers are spread out in time, i.e. a few long bursts of errors are converted into many short bursts. Error correcting codes then correct the resulting short bursts of errors.
OR for guard interval
The time-domain counter part of the multipath is the ISI or smearing of one symbol into the next. OFDM gracefully handles this type of multipath distortion by adding a "guard interval" to each symbol. This guard interval is typically a cyclic or periodic extension of the basic OFDM symbol. In other words, it looks like the rest of the symbol, but conveys no 'new' information.
Since no new information is conveyed, the receiver can ignore the guard interval and still be able to separate and decode the subcarriers. When the guard interval is designed to be longer than any smearing due to the multipath channel, the receiver is able to eliminate ISI distortion by discarding the unneeded guard interval. Hence, ISI is removed with virtually no added receiver complexity.
It is important to note that discarding the guard interval does have an impact on the noise performance since it reduces the amount of energy available at the receiver for channel symbol decoding. In addition, it reduces the data rate since no new information is contained in the added guard interval. Thus a good system design will make the guard interval as short as possible while maintaining sufficient multipath protection.
Why don't single carrier systems also use a guard interval? Single carrier systems could remove ISI by adding a guard interval between each symbol. However, this has a much more severe impact on the data rate for single carrier systems than it does for OFDM. Since OFDM uses a bundle of narrowband subcarriers, it obtains high data rates with a relatively long symbol period because the frequency width of the subcarrier is inversely proportional to the symbol duration. Consequently, adding a short guard interval has little impact on the data rate.
Single carrier systems with bandwidths equivalent to OFDM must use much shorter duration symbols. Hence adding a guard interval equal to the channel smearing has a much greater impact on data rate.
As we know that cyclic prefix is used to restore the orthogonality and preserve ISI, but the question that arises is that how the orthogonality destroyed between the subcarriers and how cyclic prefix restore the orthogonality.  
The orthogonality between subcarriers is destroyed due to the channel dispersion whenever the signal is transmitted over a channel and this cause ICI and due to the longer delay ISI occur among the OFDM symbols which are in sequence.  Further more there is no any interference in uncorrupted OFDM signal when they are demodulated but when we talk about the time dispersive channel the OFDM subcarriers lost there orthogonality. The main cause behind this is that the demodulator correlation interval for one path will overlap with the symbol boundary of a different path as show in the figure [ ] [ 2]
Fig. 4.11 16QAM constellation
We will see that this makes equalization
in the receiver very simple. If multipath exceeds the CP, then constellation points
in the modulation is distorted. As can be seen from Fig. 4.11, when multipath delay
exceeds the CP, the subcarriers are not guaranteed to be orthogonal anymore, since
modulation points may fall into anywhere in the respective contour. As delay spread
gets more severe, the radius of the contour enlarges and crosses the other contours.
Hence, this causes error.
The CP is utilized in the guard period between successive blocks and constructed
by the cyclic extension of the OFDM symbol over a period τ :
The required criteria is that τ is chosen bigger than channel length τh so as not to
experience an ISI. The CP requires more transmit energy and reduces the bit rate to
(Nb/NT +τ ), where b is the bits that a subcarrier can transmit.
The CP converts a discrete time linear convolution into a discrete time circular
convolution. Thus, transmitted data can be modeled as a circular convolution between
the channel impulse response and the transmitted data block, which in the
frequency domain is a pointwise multiplication of DFT samples. Then received signal
Hence, kth subcarrier now has a channel component Hk, which is the fourier transform
of h(t) at the frequency fk.
The OFDM symbol is sampled (t = nT and fk = k/NT) in the receiver and demodulated
with an FFT. Consequently, the received data has the following form
yk = Hk xk, k = 0, . . . ,N −1. (4.6)
The received actual data can be retrieved with N parallel one-tap equalizers. One-tap
equalizer simply uses the estimated channel ( ˆHk) components and use it to retrieve
estimated ˆ xk as follows
Also note that the spectrum of OFDM decays slowly. This causes spectrum leakage
to neighboring bands. Pulse shaping is used to change the spectral shape by
either commonly used raised cosine time window or passing through a filter.
An OFDM system design considers setting the guard interval (τ ) as well as the
symbol time (T) and FFT size with respect to desired bit rate B and given tolerable
delay spread. The guard interval is selected according to delay spread, and typically
it is 2–4 times the root-mean-squared delay spread with respect to chosen coding
Symbol time is set with respect to guard time and it is desirable to select much
larger than the guard time since the loss in SNR in the guard time is compensated.
Symbol time as we know determines the subcarrier spacing ( fb = 1/T). Number
of subcarriers N is found with respect to desired bit rate, since total number of
bits (bT ) to carry in one symbol is found with B/(T +τ ) and selected coding and
modulation determines the number of bits (b) in one subcarrier. Hence, the number
of subcarriers is N = bT /b. For instance, b is two for 16QAM with rate 1/2. The
required bandwidth (W) is then N ∗ fb. Alternatively, this method is reversed to find
out the symbol time starting from the given bandwidth.
As I m talking about the time dispersive channel I want to include that in time dispersive channel the subcarrier not only have inter symbol interference within them but they also posses interference between them. As we know that in case of time dispersive channel the frequency-selective channel frequency response is equivalent to time dispersion on the radio channel.
There are two reasons of orthogonality between OFDM subcarriers.
Due to frequency-domain separation.
The specific frequency-domain structure of each subcarrier.
Even if the frequency-domain channel is constant over a bandwidth corresponding to the main lobe of an OFDM subcarrier and only the subcarrier side lobes are corrupted due to the radio-channel frequency selectivity, the orthogonality between subcarriers will be lost with inter-subcarrier interference as a consequence. Due to the relatively large side lobes of each OFDM subcarrier, already a relatively limited amount of time dispersion or, equivalently, a relatively modest radio-channel frequency selectivity may cause non-negligible interference between subcarriers.
Time dispersion and corresponding received-signal timing
Figure 9 Time dispersion and corresponding received-signal timing.
To deal with this problem and to make an OFDM signal truly insensitive to time dispersion on the radio channel, so-called cyclic-prefix insertion is typically used in case of OFDM transmission. As illustrated in Figure 10, cyclic-prefix insertion implies that the last part of the OFDM symbol is copied and inserted at the beginning of the OFDM symbol. Cyclic-prefix insertion thus increases the length of the OFDM symbol from Tu to Tu +TCP, where TCP is the length of the cyclic prefix, with a corresponding reduction in the OFDM symbol rate as a consequence. As illustrated in the lower part of Figure 10, if the correlation at the receiver side is still only carried out over a time interval Tu =1/∆f , subcarrier orthogonality will then be preserved also in case of a time-dispersive channel, as long as the span of the time dispersion is shorter than the cyclic-prefix length.
Figure 10. Cyclic-prefix insertion
Cyclic-prefix insertion is beneficial in the sense that it makes an OFDM signal insensitive to time dispersion as long as the span of the time dispersion does not exceed the length of the cyclic prefix. The drawback of cyclic-prefix insertion is that only a fraction Tu /( Tu +TCP) of the received signal power is actually utilized by the OFDM demodulator, implying a corresponding power loss in the demodulation. In addition to this power loss, cyclic-prefix insertion also implies a corresponding loss in terms of bandwidth as the OFDM symbol rate is reduced without a corresponding reduction in the overall signal bandwidth. One way to reduce the relative overhead due to cyclic-prefix insertion is to reduce the subcarrier spacing ∆f , with a corresponding increase in the symbol time Tu as a consequence.