In order to be able to communicate many bits per symbol interval we can decide to use many carrier frequencies ‘in parallel’. In effect, using many simultaneous transmissions, each providing a modest information rate, which combine to deliver a higher data rate. A group of modulated carriers used together in this way is called an Ensemble. If we do this there is a neat trick which can make the approach particularly efficient. To understand this, lets consider the frequency spectrum of a typical symbol. For the sake of a convenient example, we can choose one with 32 carrier cycles during the chosen symbol interval. If we record the signal over just the duration of the symbol interval and work out its power-frequency spectrum the results will be as shown in Figure 19.8 by the solid (blue) line.
Now if we’d asked for the spectrum of a continuous sinusoid which extends over a very long (i.e. essentially infinite) time interval, then the spectrum would have only had one non-zero component, at the appropriate frequency. From Information Theory we can expect that limiting the observed waveform to a finite interval, 
, causes the spectrum to broaden. The result is a sinc-squared pattern as shown in Figure 19.8. The spectrum has a peak at the expected (carrier) frequency of 32 cycles in the interval. However note that it also has zeros or null at other frequencies corresponding to integer numbers of cycles in the interval which are 
32. This behaviour arises from the properties of Orthogonal Functions. We won’t bother with the details of this mathematical topic here beyond noting the specific result which is useful in this case.
Consider two possible sinusoidal waveforms which have different carrier frequencies,
Where 
and 
are integers. It can be mathematically shown that if 
then
and this is so regardless of the values we may choose for the phases, 
, and 
.
The practical result is as illustrated in Figure 19.8 which superimposes the power spectra for two different choices of possible carrier frequency, each choice having an integer number of cycles in the chosen symbol interval. The broken (red) line shows the spectrum for a symbol whose carrier frequency is 34 cycles per symbol duration. This can be seen to have zeros at all the frequencies which have an integer number of cycles during the symbol period except 34.
If we now use both carriers simultaneously as parallel QAM streams we find in our receiver/demodulator that a circuit designed to measure the signal amplitude and phase at one frequency, 
, will be unaffected by the presence of another carrier at 
since this produces no contribution at 
. This result is a very important one. It means that we can choose to simultaneously transmit a whole series of carriers and modulate each of them with their own bit (information) patterns. By choosing an Orthogonal set of carrier frequencies we can prevent them from interfering with one another, and hence recover all the information they carry. The requirement we must obey for this to be possible is that we choose a finite symbol duration, 
, and then select all the carrier frequencies, so that they have different integer number of cycles in this time period.
This ability means we can transmit, say, 100 carriers, and modulate them to send 100 times more bits in a given time than we could using just one modulated carrier. Techniques of this kind are called Orthogonal Frequency Division Multiplexing (OFDM). More commonly you will see this referred to as COFDM or Coded Orthogonal Frequency Division Multiplexing which refers to the way the system is modified and used in practice. COFDM is the basic modulation and communication method now used for ‘digital’ TV and sound radio in the UK.
Multipath
A common problem in radio communications is Multipath. As the name indicates, this is the situation where signals may travel from the transmitter to the receiver by more than one path. For example, we may get one signal path in a straight line from the transmitter to receiver, and another via a reflection from a hill or tall building. When these paths have different lengths their contributions to the received results arrive with different propagation delays.. This situation is illustrated in Figure 19.9 below. Note that in practice, there may be many such paths, giving contributions arriving with various time delays and amplitudes. Here we’ll just consider one delayed contribution for the sake of simplicity.
Looking at Fig 19.9 we can see that the addition of the delayed contribution via the ‘reflected’ path has two consequences.
- The symbols have been extended and now have a ‘tail’ due to the reflected contribution continuing to arrive for a short period after the direct symbol has ended.
- The received result has its effective amplitude and phase altered by an amount that depends on the time difference between the paths.
The two paths are of lengths we can call 
(direct) and 
(indirect). This allows us to define the time delay between their relative arrivals to be
Where 
is the speed of light.
The above shows why it is useful to have a Guard Interval, 
, in between the Useful Symbol Periods, 
. Provided that we have arranged for 
then the delayed version of each symbol will have finished arriving before the next symbol in the sequence starts to arrive. This prevents the pattern transmitted for a given symbol from affecting what we see during the next symbol period. Where one symbol pattern interferes with, or unintentionally alters, the pattern we wish to send for another then this is called Inter-Symbol Interference (ISI). In general we wish to avoid ISI because it would may make it harder for the receiver to demodulate the symbols correctly.
As a result, the chosen value for the Guard Interval is important for avoiding data loss or errors due to multipath. We wish to choose a value for 
that is large enough that no reflections will be arriving with delays greater than 
– or at least, we wish to ensure that any that do arrive later than this will have such low amplitudes as to have almost no effect on the amplitude and phase of the following symbol pattern.
A long Guard Interval can avoid one symbol significantly affecting the next, but the multipath may still alter the amplitudes and phases of the received symbols. To help deal with this we can insert a regular pattern of what are called Pilot Symbols or Reference Symbols into the transmitted stream. These also allow the transmitter to inform the receiver what phase/amplirtude corresponds to a given symbol. In effect, these Pilot or Reference symbols make it possible to ‘calibrate’ the receiver and help correct for multipath effect.
Summary
You should now understand how Digital Modulation can be used to convey information for a stream of ‘bits’ in terms of a set of predefined Symbols. That the number of bits which each symbol can convey depends on the number of different Symbols available in the chosen modulation scheme, in terms of their distinct amplitudes and phases. It should also be clear that these have to differ by enough to be distinguishable over the noise which may be present, and that the power required increases as we want more bits per symbol. You should also now understand that we can use Orthogonal sets of carriers to provide ‘parallel’ streams which can be modulated and detected without interfering with one another, thus increasing the total rate at which information can be conveyed.
Content and pages maintained by: Jim Lesurf (jcgl@st-and.ac.uk)
using
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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.
