Measurement noise and signal processing

5.6.4 Digital-to-analogue (D/A) conversion

Digital-to-analogue conversion is much simpler to achieve than analogue-to-digital conversion and the cost of building the necessary hardware circuit is considerably less. It is required wherever a digitally processed signal has to be presented to an analogue control actuator or an analogue signal display device. A common form of digital-to-analogue converter is illustrated in Figure 5.24. This is shown with 8 bits for simplicity of explanation, although in practice 10 and 12 bit D/A converters are used more frequently. This form of D/A converter consists of a resistor-ladder network on the input to an operational amplifier. The analogue output voltage from the amplifier is given by:

                   V_A = V₇ + V₆/2 + V₅/4 + V₄/8 + V₃/16 + V₂/32 + V₁/64 + V₀/128

V₀ … V₇ are set at either the reference voltage level Vref or at zero volts according to whether an associated switch is open or closed. Each switch is controlled by the logic level of one of the bits 0–7 of the 8 bit binary signal being converted. A particular switch is open if the relevant binary bit has a value of 0 and closed if the value is 1. Consider for example a digital signal with binary value of 11010100. The values of V₇ … V₀ are therefore:

                                     V₇ = V₆ = V₄ = V₂ = V_ref; V₅ = V₃ + V₁ = V₀ = 0

The analogue output from the converter is then given by:

                                           V_A = V_ref + V_ref/2 + V_ref/8 + V_ref/32

5.6.5 Digital filtering

Digital signal processing can perform all of the filtering functions mentioned earlier in respect of analogue filters, i.e. low pass, high pass, band pass and band stop. However, the detailed design of digital filters requires a level of theoretical knowledge, including the use of z-transform theory, which is outside the scope of this book. The reader interested in digital filter design is therefore referred elsewhere (Lynn, 1989; Huelsman, 1993).

5.6.6 Autocorrelation

Autocorrelation is a special digital signal processing technique that has the ability to extract a measurement signal when it is completely swamped by noise, i.e. when the noise amplitude is larger than the signal amplitude. Unfortunately, phase information in the measurement signal is lost during the autocorrelation process, but the amplitude and frequency can be extracted accurately. For a measurement signal s (t), the autocorrelation coefficient ­0 is the average value of the product of s (t) and s (t - #), where s (t - #) is the value of the measurement signal delayed by a time #. 0­s can be derived by the scheme shown in Figure 5.25, and mathematically it is given by: 

The autocorrelation function ­0s (#)  describes the relationship between ­0s and # as # varies:

If the measurement signal is corrupted by a noise signal n(t) (such that the total signal y (t) at the output of the measurement system is given by y (t) = s (t) + n (t), the noise can be represented by an autocorrelation function of the form ­0n (#) where:

If n (t) only consists of random noise, ­0n (#) has a large value close to # = 0, but, away from # = 0, ­0n (#) decreases to a very small value. The autocorrelation function for the combined signal plus noise is given by ­0s (#) + 0­n (#). For # >> 0, ­0n (#) à 0 and thus ­0s (#) + 0­n (#)à ­0s (#). Thus, at large time delays, the amplitude and period of the signal can be found from the amplitude and period of the autocorrelation function

of the signal at the output of the measurement system. Further details can be found in Healey, (1975).

5.6.7 Other digital signal processing operations

Once a satisfactory digital representation in discrete form of an analogue signal has been obtained, many signal processing operations become trivial. For signal amplification and attenuation, all samples have to be multiplied or divided by a fixed constant. Bias removal involves simply adding or subtracting a fixed constant from each sample of the signal. Signal linearization requires a priori knowledge of the type of non-linearity involved, in the form of a mathematical equation that expresses the relationship between the output measurements from an instrument and the value of the physical quantity being measured. This can be obtained either theoretically through knowledge of the physical laws governing the system or empirically using input–output data obtained from the measurement system under controlled conditions. Once this relationship has been obtained, it is used to calculate the value of the measured physical quantity corresponding to each discrete sample of the measurement signal. Whilst the amount of computation involved in this is greater than for the trivial cases of signal amplifica[1]tion etc. already mentioned, the computational burden is still relatively small in most measurement situations.