Basic Wireless Communication for Microcontrollers
Chapter 1 - Electricity and Magnetism
Propagation Speed in General
We have already noted how the speed of light can be less in some materials than in free space. There are situations, however, where the speed of light can actually be a function of frequency. Two examples of situations where this happens are in ionized gasses (such as in the ionosphere) and in microwave waveguides. Such media are called "dispersive" media.
For discussing the velocity of propagation, it is useful to define a quantity called the wavenumber, usually denoted by "k". K tells you the number of cycles of an EM wave which fit in a meter length. It is measured in radians, though, so one cycle per meter would be K=2*pi, for example. K is related directly to the wavelength: K=2*pi/lambda.
If the signal being sent through a medium is a pure sinusoid, then the velocity of propagation is measured by observing how a particular phase point on the wave moves in time (for example, the maximum of the wave is a point of constant phase, so would be a zero crossing point, etc.). The velocity with which such a point moves is called the phase velocity, and is equal to the ratio w/k, which is the same as f*lambda.
Group Velocity
If the signal is more complex, so that a range of sinusoidal components are propagating together, then the situation can be more complicated. For non-dispersive media, the ratio w/k will remain a constant over frequency and the signal propagates without changing shape. In dispersive media, each component will travel at a different phase velocity, and the wave shape will get distorted as it travels. The shape will get broader and lower in peak magnitude.
Assuming the signal is relatively narrowband, there will only be a small difference in speed among the different components, and dispersion or spreading of the waveform will occur at a much slower rate than the average forward speed of the whole group or packet of frequency components. We can then define the speed at which the main peak (or peaks) of the waveform will move, and call that the group velocity. It turns out that group velocity is equal to dw/dk. If phase velocity is independent of frequency, dw/dk=w/k.
The group velocity concept is especially useful when pulses are being transmitted through a dispersive medium because we are usually concerned with two quantities: how fast is the peak traveling and how much wider does the pulse get as it travels.
For signals where it makes sense to talk about group velocity (narrowband signals), group velocity is the speed at which information is being conveyed by the wave. As such, it is possible for phase velocity to be higher, in some media, than the speed of light is in a vacuum. However, those media are dispersive and the group velocity will be lower than C, the vacuum speed of light.
Energy Velocity
Finally, there are situations where the signal is so broadband that the concept of group velocity breaks down. In these cases, the only type of speed which makes sense (for the waveform as a whole, not individual components) is the speed at which energy is being transferred by the wave. We will not discuss computing the energy velocity because it should not be necessary for the scope of this tutorial.
Relation to Filters
Although people who are not specialists in RF rarely have to worry about dispersive media, we have discussed this topic mainly because it is very relevant to something we will discuss later: designing filters. Filters, whether made out of resistors, capacitors, and inductors, or implemented using digital signal processing techniques (DSP), emulate a medium in which an EM wave is propagating. They do this in the sense that there is a delay between when the signal (or some change in the signal) enters the filter and when it comes out the other side, just as it takes some time for an EM wave to propagate from one point to another in space.
While most common media for EM waves are not very dispersive, most filters are. The delay from input to output varies depending on frequency, so in addition to causing the desired filtering operation, the filter may distort the signal. For many analog signals (such as speech), this isn't a problem. For digital signals, though, this means that pulses which enter the filter with sharply defined edges may come out with wider, more rounded shapes. This can severly affect the ability to decode the information contained in the signal. A concept called group delay is defined for filters, and is similar to group velocity in EM wave propagation.
In DSP, there are a whole class of common filters (linear-phase finite impulse response (FIR) filters) which are non-dispersive. All analog filters made using resistors, capacitors, and inductors, though, are dispersive to some degree, and it is something that will have to be taken into account in designing radio equipment for digital signals.
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