# Bandwidth

A generally accepted definition of the bandwidth of an analogue transmission channel is the difference between the highest and lowest frequencies that it can support. Bandwidth is typically measured in *hertz*. In the case of a baseband channel, the bandwidth is generally considered to be the highest frequency supported. The bandwidth of a channel that is made up of a number of distinct physical transmission links is limited by the range of frequencies supported by *all* of the links. In data communication networks, the term bandwidth often refers to the nominal maximum *data rate* measured in bits per second (bps). The maximum data rate (or channel capacity) of a physical communication link is related to its bandwidth in hertz, sometimes referred to as its *analogue bandwidth*.

An analogue telephone line in Europe or North America typically has a bandwidth of 3 kHz, and can carry frequencies of between 400 Hz and 3.4 Khz. The frequency response of the channel is artificially limited by filters in the telephone transmission system (the type of twisted pair cable employed in the subscriber loop can actually carry a much wider range of frequencies). By comparison analogue TV signals, which comprise both video and audio components, require a 6MHz bandwidth RF channel. The graphic below provides a comparison of the typical bandwidths achievable using current or proposed Internet access technologies.

Comparative bandwidth of current and proposed Internet access technologies

Since digital signals are often represented by discrete voltage levels, the signal elements that make up a digital transmission can essentially be considered to be *square wave pulses*. Such waveforms do not occur naturally, and the French scientist *Jean Baptiste Joseph Fourier* (1768 - 1830) was able to demonstrate that such a signal can only be generated by combining a number of sine waves, each having a different frequency and amplitude, to create a more complex waveform.

The frequency of the square wave itself is said to be the *fundamental frequency*. It can be shown that by taking a sine wave with same frequency as the required square wave, and adding successive odd-numbered *harmonics* to it, a square wave can be approximated. A harmonic is a sine wave with a frequency that is an integer multiple of the fundamental frequency. By adding together the fundamental, third harmonic and fifth harmonic, we can achieve a waveform that is an approximation of a square wave. The fundamental, 3^{rd} and 5^{th} harmonics are shown below, and are labelled **A**, **B** and **C** respectively. Notice that the amplitude of each harmonic relative to that of the fundamental is approximately the inverse of its harmonic number.

Fundamental sine wave with third and fifth harmonics

The image below illustrates the effect of *adding* these sine waves together. The resulting waveform is starting to resemble our ideal square wave, although in practice it would require an infinite number of harmonics to produce a "perfect" square wave. Since no transmission medium is capable of supporting an infinite range of frequencies, the best that can ever be achieved will be an approximation of a square wave. It is the properties of the *receiver* in a commununications channel that will determine how good an approximation is required, and therefore the bandwidth that must be supported by the channel.

Adding the fundamental, third and fifth harmonics produces an approximation of a square wave

We so far have looked at the waveform of a complex wave (in this case a square wave) as it might appear on an *oscilloscope*, which displays the *amplitude* of a waveform as a function of *time*. In otherwords, we have looked at these waveforms in the *time domain*. We could also look at the waveform using a *spectrum analyser*, which displays the amplitude and frequency of each sine wave used to generate the complex waveform. Looking at the same square wave illustrated above in the *frequency domain*, therefore, we would see something like the image below.

A time-domain view of a squarewave comprising the fundamental, third and fifth harmonics