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Digital Measuring Instruments
 Consider the two clocks shown in the diagram below. To use clock A we observe the position of the "hands" as they move past the markings on a circular scale. The face of this clock is an analogue display*. Clock B tells us the time by displaying numbers (digits) which change at regular intervals. This is a digital display. Notice that the analogue display gives us the possibility of estimating the number of seconds whereas the digital display tells us only hours and minutes. Similarly, we might use a voltmeter having a pointer which moves across a scale (an analogue meter) or one having numbers which change according to the voltage applied to its terminals (a digital meter), as shown below. Again, the analogue display gives us the possibility of making an estimate when the pointer lies between two marks on the scale but, in this case, a suitable choice of calibration of the digital meter has given us greater precision. * It is worth remembering that an ordinary "ticking" clock shares some properties of the digital instrument. If you look closely you might see the minute hand moving in small steps... Quantization Error When using an analogue voltmeter, there is no theoretical limit to how small a displacement the pointer can experience. There is, of course, a practical limit, for example, with the voltmeter shown above, a change from 2.45V to 2.46V might just be visible but any smaller change would be totally undetectable. With a voltmeter having a digital display, the changes in the reading only occur in discrete "steps". A variable which can only change in this way is said to be quantized and for this reason, the precision of a digital instrument is said to be limited by its quantization error which is simply the smallest amount by which the display of the instrument can change. So, when recording a result measured by a digital instrument the indeterminacy is ±1 in the least significant digit (in other words, the last decimal place) of the display. The reading of the meter in the diagram is therefore 2.4 ±0.1V. Sampling Frequency When a voltage is applied to a digital voltmeter, a variable voltage source inside the meter is increased from zero until it reaches the same value as the applied voltage. The display then indicates the value of the internal voltage source. This process of comparing the external voltage with an internal voltage is called sampling the external voltage and, of course, takes a certain time. Suppose the sampling process occurs 4 times a second. Any change in voltage which lasts for less than ¼ of a second will (probably) be missed. For this reason, a digital instrument should operate with a sampling frequency much higher than the expected frequency of variation of the quantity being measured. The speed of response of an analogue meter is, of course, also limited by the inertia of its moving parts. The importance of having a (relatively) high sampling rate can be seen if we consider the process of digital sound recording. A varying voltage from, for example, a microphone must be converted into a series of numbers representing the magnitude of the voltage at regular intervals. In other words, what we need is a fast response digital voltmeter (the device used is actually called an analogue to digital converter). These numbers are then recorded and when "replayed" can be used to reproduce the original sound. The first two diagrams below represent the voltage to be recorded (left) and its digital equivalent (right) produced using a sampling frequency which is much higher than the frequency of the original sound. The marks on the horizontal axis represent the instants at which the sampling occurred. At this sampling frequency the original wave is immediately recognizable and could be reproduced accurately with the aid of filters. Now consider using a sampling rate which is about the same as the frequency of the sound to be recorded. This situation is represented by the diagrams below. The digital signal is nothing like the original sound but at least has a similar frequency. Finally, consider using a sampling rate which is lower than the frequency of the sound to be recorded. This situation is represented by the diagrams below. Here, we get something which has very little in common with the original sound. In this case it would be impossible to extract the original sound. It has been shown (by C. Shannon in 1948) that, to be able to obtain a good "copy" of the original sound, the sampling frequency must be at least twice as high as the highest frequency of sound to be recorded.
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