ly represent the analogue signal. As demonstrated above in the temperature example (Table 5.1 and Table 5.2) sampling must also be efficient and use a minimum number of bits in its representation. Figure 5.5(a) shows a section of an analogue signal with a length of 1 S. The signal has three peaks and two troughs, and therefore can be estimated to have a frequency of ~3 Hz, although one can note that the rate of peaks and troughs is increasing over time. The signal is sampled at regular intervals at a sampling rate of 40 Hz in Figure 5.5(b), and the resulting discrete rep- resentation is shown in Figure 5.5(c). The representation appears to be a close replica of the analogue waveform. Another representation is shown in Figure 5.5(d) where the sampling frequency is 20 Hz or half as much compared with Figure 5.5(c). Figure 5.4 A one second voice clip as a time domain sound waveform Broadband Telecommunications Technologies and Management 1 s Time 1 S Time 1 S Time 1 S Time Figure 5.5 An analogue signal and its discrete representations with 40 and 20 Hz sampling rate It is clear that Figure 5.5(d) is more efficient as it needs half as many samples to represent the analogue signal. However, it does not represent the analogue signal as closely as Figure 5.5(c). Is it therefore an equally accurate representation? This leads to the question whether there exists a minimum sampling frequency at which all information of an analogue signal can be preserved in its digital representation. Nyquist Theorem ² The necessary sampling rate is given by the Nyquist theorem. As described in the Historical Note, Harry Nyquist, a scientist at Bell Laboratories, researched on topic of transmission of telegraph signals over telephone lines. His aim was to determine the fastest possible rate at which telegrams could be sent over band-limited telephone lines. In order to determine this he 2 This theorem is also known as the Nyquist-Shannon theorem. Several other scientists are considered to have arrived at the same result independently. From