Measurement noise and signal processing

 5.6 Digital signal processing

Digital techniques achieve much greater levels of accuracy in signal processing than equivalent analogue methods. However, the time taken to process a signal digitally is longer than that required to carry out the same operation by analogue techniques, and the equipment required is more expensive. Therefore, some care is needed in making the correct choice between digital and analogue methods.

 Whilst digital signal processing elements in a measurement system can exist as sep[1]arate units, it is more usual to find them as an integral part of an intelligent instrument (see Chapter 9). However, the construction and mode of operation of such processing elements are the same irrespective of whether they are part of an intelligent instrument of not. The hardware aspect of a digital signal-processing element consists of a digital computer and analogue interface boards. The actual form that signal processing takes depends on the software program executed by the processor. However, before consider[1]ation is given to this, some theoretical aspects of signal sampling need to be discussed.

5.6.1 Signal sampling

Digital computers require signals to be in digital form whereas most instrumentation transducers have an output signal in analogue form. Analogue-to-digital conversion is therefore required at the interface between analogue transducers and the digital computer, and digital-to-analogue conversion is often required at a later stage to convert the processed signals back into analogue form. The process of analogue-to-digital conversion consists of sampling the analogue signal at regular intervals of time. Each sample of the analogue voltage is then converted into an equivalent digital value. This conversion takes a certain finite time, during which the analogue signal can be changing in value. The next sample of the analogue signal cannot be taken until the conversion of the last sample to digital form is completed. The representation within a digital computer of a continuous analogue signal is therefore a sequence of samples whose pattern only approximately follows the shape of the original signal. This pattern of samples taken at successive, equal intervals of time is known as a discrete signal. The process of conversion between a continuous analogue signal and a discrete digital one is illustrated for a sine wave in Figure 5.21.

The raw analogue signal in Figure 5.21 has a frequency of approximately 0.75 cycles per second. With the rate of sampling shown, which is approximately 11 samples per second, reconstruction of the samples matches the original analogue signal very well. If the rate of sampling was decreased, the fit between the reconstructed samples and the original signal would be less good. If the rate of sampling was very much less than the frequency of the raw analogue signal, such as 1 sample per second, only the samples marked ‘X’ in Figure 5.21 would be obtained. Fitting a line through these ‘X’s incorrectly estimates a signal whose frequency is approximately 0.25 cycles per second. This phenomenon, whereby the process of sampling transmutes a high-frequency signal into a lower frequency one, is known as aliasing. To avoid aliasing, it is necessary theoretically for the sampling rate to be at least twice the highest frequency in the analogue signal sampled. In practice, sampling rates of between 5 and 10 times the highest frequency signal are normally chosen so that the discrete sampled signal is a close approximation to the original analogue signal in amplitude as well as frequency.

Problems can arise in sampling when the raw analogue signal is corrupted by high[1]frequency noise of unknown characteristics. It would be normal practice to choose the sampling interval as, say, a ten-times multiple of the frequency of the measurement

component in the raw signal. If such a sampling interval is chosen, aliasing can in certain circumstances transmute high-frequency noise components into the same frequency range as the measurement component in the signal, thus giving erroneous results. This is one of the circumstances mentioned earlier, where prior analogue signal conditioning in the form of a low-pass filter must be carried out before processing the signal digitally.

One further factor that affects the quality of a signal when it is converted from analogue to digital form is quantization. Quantization describes the procedure whereby the continuous analogue signal is converted into a number of discrete levels. At any particular value of the analogue signal, the digital representation is either the discrete level immediately above this value or the discrete level immediately below this value. If the difference between two successive discrete levels is represented by the parameter Q, then the maximum error in each digital sample of the raw analogue signal is ±Q/2. This error is known as the quantization error and is clearly proportional to the resolution of the analogue-to-digital converter, i.e. to the number of bits used to represent the samples in digital form.