Thus far we have looked at transforms from a somewhat abstract viewpoint. In contrast, here we look at an application where transforms take center stage.
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One of the worst problems encountered in terrestrial radio communication, including broadcasting, is multipath reception, in which a receiver gets a direct signal along with a signal that has reflected from something. The reflected signal must have taken a longer path and so cannot be time aligned with the direct signal.
The adoption of digital techniques made matters worse, because one of the tradeoffs of digitizing signals is that the bandwidth required increases. This is simply because as the frequency rises, the wavelength falls and the time delay of multipath reception causes a bigger phase shift. Reflections can be reduced using directional antennas, but these need skill in setting up and are not really appropriate for portable devices.
Similar problems are found in cable data transmission, especially over existing telephone wires, where there is little control of impedance changes that cause reflections.
In both cases, attention turns to modulation schemes that can combat reflections. If the frequency or bit rate is low enough, the length of time needed to transmit one bit becomes long compared to the multipath delay and the result is that the direct signal and the reflection have considerable overlap and can reinforce one another. Such a low bit rate transmission would have very narrow bandwidth.
Obviously the provision of an adequate bit rate requires many such narrow carriers to operate in parallel. Using conventional radio receiver technology, designing a receiver that could simultaneously tune to and receive large numbers of carriers would make it unwieldy, expensive and probably subject to drift. With such narrow carriers, the slightest instability would cause them to be lost or mixed up.
This suggests a digital solution, which would offer the necessary stability and which could support the necessary complexity. A Digital Fourier Transform allows us to look at a large number of frequencies at once and to keep them separate.
Going back to the principles of sampling, we know that it is necessary to have filters that are appropriate for the sampling rate. Shannon tells us the sampling rate must be twice the bandwidth of the filters. When we reconstruct a waveform from samples, each sample produces an impulse or spike, whose height is proportional to the binary code. Each spike is then filtered. The impulse response of an ideal rectangular low-pass filter is a sinx/x curve, which, as can be seen in Fig.1, contains periodic zero crossings.
When the filter frequency is correct, the zero crossings are spaced apart by the sample period, which means that at the center of any one sample, the filtered result of all other samples is zero. In other words at the center of a sample, the voltage output waveform is determined by that sample alone. Mathematically we say the samples are orthogonal; they live in different dimensions and so do not interfere with one another.
Fig. 1 - Transform duality in action. In the time domain, the impulses from individual samples are orthogonal so that perfect reconstruction of the waveform is possible. In the frequency domain, the same orthogonality eliminates crosstalk between the frequency bands. The DFT allows those frequency bands to be separated.
However, transform duality allows an interesting twist. In sampling, we have a filter whose response in the frequency domain is rectangular and which transforms in the time domain to an impulse response that is a sinx/x function. If we take the dual of that situation, we find that a binary digit becomes a rectangular pulse in the time domain and transforms to a spectrum in the frequency domain that is a sinx/x function.
It follows that if the correct parameters are used, multiple sinx/x spectra can be made orthogonal because the center frequency of a given channel coincides with the zero crossings of all other channels. That, in a nutshell, is how OFDM (orthogonal frequency division multiplexing) works. It enables a large number of narrow channels to be packed together in such a way that they do not interfere with one another. The packing is also very efficient according to information theory.
The total bandwidth required by the multiple carriers of an OFDM system is the same as that needed by a single carrier modulated by the same data.
If such a signal is sampled at the correct sampling rate and a block of samples are subject to a DFT, the DFT will analyze each channel individually so that if the channels are modulated in some way the modulation will be seen in the coefficients emerging from the DFT. As the DFT is complex, meaning that it analyses the phase of the signal as well as the amplitude, various different types of coding can be used in conjunction with OFDM, including quadrature amplitude modulation (QUAM).
The slow speed of the individual channels in OFDM give some resistance to multipath reception, but this can be increased by the adoption of guard intervals. The guard interval is shown in Fig.2. It essentially causes the transmitted symbol to be extended in time. If the receiver sees two or more signals having a relative time shift, this will be ignored provided the useful time, Tu, which is essentially a window, is positioned at a time within the symbol time Ts when the signals are the same.
Fig.2 - When guard intervals are used, transmission rate is slowed down so that every symbol is longer. That means the receiver useful time, Tu, can better avoid reflections of an earlier symbol encroaching on the beginning of the symbol time Ts.
The OFDM receiver is essentially an analog to digital convertor that produces from the received signal a set of samples of the waveform equal to the number of orthogonal channels. As the system performs a DFT on the data, it is the number of samples in the set and the sampling rate that determines the frequencies to be analyzed, and the reciprocal of the frequency spacing determines the effective time period Tu over which the signal is considered.
The effect of introducing guard intervals is that the symbol rate and thus the data rate goes down slightly. The receiver takes a set of samples in the same way and at the same rate, meaning that orthogonality is maintained, but the sample sets are taken slightly further apart, causing the data rate to go down.
The advantage of guard intervals are that large delays in multipath signals can be tolerated and this allows one of the most powerful features of OFDM, which is that multiple transmitters can be used on the same frequency. The signals from additional transmitters are simply treated as if they were reflections from one transmitter, which of course means that the multiple transmitters must radiate identical signals with very accurate synchronization.
If the samples of the received waveform are taken with the slightest timing error, the result will be a phase shift in all of them that will be revealed by the phase rotation of the complex coefficients. One solution is to use differential QUAM, in which the data are encoded in phase differences rather than as absolute phases. The transmitted signal can also have periodic and distributed symbols known as pilots that are un-modulated and have a known phase. The receiver can use these to perform fine-tuning.
One of the weaknesses of OFDM is that when there is relative movement between the transmitter and the receiver the resultant Doppler shift changes all of the frequencies in the received signal and the receiver is required to re-tune. That in itself is not impossible, but the real difficulty is when a moving receiver encounters multipath reception. The direct and received signals arrive from different directions and the amount of Doppler shift in each will be different, causing a potential loss of orthogonality.
The intricacies of OFDM in the frequency domain require that transmitters should be reasonably linear, because intermodulation has a damaging effect on the OFDM spectrum.
OFDM is purely a frequency division scheme. In practice OFDM needs a suitable modulation scheme for the carriers and needs an error correcting strategy outside that. The result is then known as coded OFDM, or just COFDM. The FFT can identify channels in which the signal is very weak and replace the data with error flags. The worst case is probably where a reflection is received at the same strength as the direct signal, as this can cause perfect cancellation. The direct and delayed signals form a comb filter having periodic nulls in the frequency domain whose spacing is a function of the geometry.
Small delays cause many narrow but close-spaced nulls, whereas long delays cause large nulls that are far apart. The small nulls are best dealt with using error correction. Dealing with large nulls requires the error correction to be assisted by interleaving.
In the case of digital radio, several different audio channels are multiplexed into the set of orthogonal carriers. The multiplexing is arranged so that adjacent carriers are used for different channels. Thus in an N-channel multiplex, N symbols can be corrupted yet each audio channel sees one error.
When such interleaving is used, the DFT only needs to compute coefficients for the carriers that are actually used in the channel to be decoded. The basis functions for other channels are not generated and the multiplications do not take place. This gives a useful saving in processing power and energy consumption.
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