1.4 Wireless communications background

Basics of digital communications

Communication theory aims to explore and develop methods that suppress (as far as possible) the effect of noise and to simultaneously transmit as many discrete signals as possible through a channel. Spectral analysis is a tool that connects time-domain signals to the frequency domain, allowing insight into the characteristics of broadband and narrowband signals in a communication bandwidth.

Spectral analysis

Frequency has a ubiquitous role in the process of communication. It is used as a carrier and bandwidths are specified in terms of it. It is therefore important to have tools with which you can easily determine the frequency content in a signal. This can be achieved using the Fourier transform (FT) and its discrete counterpart, the DFT. The FT of a signal $\mathrm{w(t)}$ is defined in Equation 1 as:

$\mathrm{W(f)}={{\int }_{\mathrm{-\infty }}}^{}\mathrm{w(t)exp(-j2\pi ft)dt}$

Its inverse is calculated with Equation 2:

$\mathrm{w(t)}={{\int }_{\mathrm{-\infty }}}^{}\mathrm{W(f)exp(j2\pi ft)df}$

The above equations show that $\mathrm{w(t)}$ is a weighted sum of sinusoids in the interval $\mathrm{-\infty \; to\; +\infty }$ . The weights are complex numbers $\mathrm{W(f)}$ . If at any particular frequency the magnitude spectrum is strictly positive, then that frequency is said to be present in $\mathrm{w(t)}$ . The set of all frequencies present in a signal is its frequency content. If this content consists of frequencies in a certain range, then $\mathrm{w(t)}$ is said to be bandlimited with a certain bandwidth.

Digital modulation

It is the process of converting digital symbols into waveforms that are compatible with the characteristics of the transmission medium. In the case of baseband modulation, these waveforms take the shape of pulses designed to reduce intersymbol interference (ISI). In the case of bandpass modulation, these shaped pulses modulate a sinusoidal carrier wave that is converted to an electromagnetic (EM) field for propagation over distances. In free space, antennas radiate and receive EM signals. Antennas operate effectively only when their dimensions are of the order of magnitude of quarter wavelength $(\frac{\Lambda }{4})$ of the transmitted signal. If the signal frequency is very high, antenna dimension becomes practical; however, high frequencies get attenuated by the atmosphere and therefore cannot travel great distances.

Basic modulation techniques

Any message can be converted into binary digits called bits. For transmission, these bits are grouped together and encoded into sequences whose elements are the symbols of an alphabet set. To utilize bandwidth more efficiently, these alphabets are encoded in waveforms called pulses, which are then combined to form a baseband signal. For example, bitstream 01001001010010111010101 can be paired as 01 00 10 01 01 11 and so on. Then the pairs can be encoded as -1, -3, 1, -1, +3 and so on to produce the symbol sequence. There are many ways to map from bits to symbols. Bitstreams can be mapped to eight-level, 16-level, 256-level, etc. After the original message is grouped into alphabets, it must be turned into analog waveforms by choosing a pulse shape $\mathrm{p(t)}$ and then transmitting $\mathrm{–p(t\; -\; kT),\; -3p(t\; –\; kT),\; p(t\; –\; kT),\; -p(t\; –\; kT),\; 3p(t\; –kT)}$ . In general, this four-level signal takes the form shown in Equation 3: