CHAPTER 1

THE DIGITAL SIGNAL PROCESSING

An introductory overview

The digital signal processing finds its origins, as a well defined subject, during the seventeenth and eighteenth centuries, thanks also to the work of the two great mathematicians Newton and Gauss. At that time its applications were typically oriented to the numerical calculus, since this subject was mainly concerned with the representation of the mathematical functions by means of sequences of numbers, or symbols, and the elaboration of such sequences. In this respect, the classical numerical techniques, such as those used for the interpolation, integration and derivation of functions can be fully considered as Digital Signal Processing (DSP) techniques.

Only recently, approximately starting from the second half of the 70s, in the 20th

century, the evolution of the computing devices and the Analog-to-Digital (AD) converters has greatly extended the application field of this subject.

One of the fields which most benefit of this evolution is that of measurement, and in particular the electric and electronic measurement. It is well known that the measurement activity typically consist in observing a physical phenomenon, both spontaneous (as is the case, for instance, of measurements of natural phenomena, such as those performed in astronomy or seismology), or artificially originated in order to perform the measurements (as is the case, for instance, of the test measurements on devices or machines, where the device under test is properly stimulated).

It is known as well that a physical phenomenon can be observed through the variation of physical quantities to which the information describing that phenomenon is associated. For instance, the presence of an electric field can be detected by analysing the distribution of the electric potential in the region of space where the electric field is present. Once a proper reference potential is taken, the observation of the electric field can be attained by means of measurements of potential differences. It is then possible to associate the information related to the observed physical phenomenon (the electric field) to the values taken by these potential differences and their variation in time.

In this example, the potential differences play the role of measurement signals, whose processing allows to extract the required information. The possible ways the measurement signals can be processed may be, in this case, for instance, the evaluation of the peak value, from which the maximum intensity of the field can be obtained, the evaluation of the rms value, to which the capability of the field to do an electric work can be associated, or the analysis of the spectral components, to which the field behaviour in the frequency domain can be associated.

The key point in a measurement process is therefore processing one or more measurement signals in order to extract the required information from the signals

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themselves. This processing is generally described by a more or less complex mathematical relationship. Still considering the above example, the rms value of a potential difference u(t), periodic with period T over time t, is expressed by the following relationship:

( ) 02 ,1 0

0tdttu

TU

t

Tt∀=

− (1.1)

This result is not easy to obtain with the traditional analog techniques. In the past, electromechanical devices have been adopted, such as the electrodynamic instruments, where, due to the mechanical action caused by the interaction between currents, a driving torque, proportional to the square of the applied voltage, is exerted on a pivoted moving coil. The mechanical inertia of the moving coil acts as a low-pass filter, so that, in the presence of a spring restraining torque, the displacement angle of the moving coil is proportional to the mean value of the squared voltage. If a pointer is fixed on the moving coil rotation axes, its displacement on a square-law scale provides the desired rms value.

It can be easily understood that, if a good accuracy is required, the mechanical structure becomes critical and very expensive. When the analog electronic components became available, instruments based on electronic multipliers and low-pass filters could be realized, with a far better accuracy and wider bandwidth than the electromechanical instruments. However, the number of components required to process the input signal is still large, and therefore there is a large number of uncertainty sources to control in order to feature a good accuracy, which results in a high cost for the whole system. The problem becomes even more difficult to solve with the analog techniques when the measurement of quantities defined by more complex mathematical relationships are considered, such as, for instance, the spectral components of a signal.

On the other hand, the numerical calculus provides a relatively simple and direct solution to (1.1). In fact, if a device is available, able to sample u(t) with a constant sampling period Tc, convert the obtained samples into a digital code, store the obtained sequence of codes into a digital memory and process them digitally, it is possible to prove that, under a number of given conditions that will be discussed in the following chapters, (1.1) can be written as:

( )=k

TkTuT

U cc21 (1.2)

The block diagram of an instrument able to perform the above-listed operations is schematically drawn in Figure 1.1.

In such an instrument the result of the measurement is obtained by computing (1.2), with a proper program stored in the memory of the processor; this means that it is possible to perform different measurements, always starting from the same samples of u(t), by simply changing the program, thus attaining also a dramatic cost reduction.

The digital signal processing 3

The above examples show that the operating principle of a digital instrument that employs DSP techniques is based on the instrument capability of sampling the input signals over time, converting the samples into digital values and processing them with a suitable algorithm in order to attain the result of the desired measurement.

+ Memory

Measurement result

U

Input signal

u(t)

processing techniques.

When an analog signal is sampled and the samples are converted into digital values, the mathematical object sequence is obtained, defined in the discrete-time domain. This

variable n, whilst a function f(t) of time t is defined for real values of the independent variable t, and is therefore defined in the continuous-time domain.

Since the obtained values are coded into a finite number of digits, generally representing a binary code, the input signal is allowed to assume only a finite number of values also in amplitude, thus giving rise to the quantization phenomenon.

A digital instrument, based on DSP techniques is hence characterized by a double transformation of the input signal from the continuous to the discrete domain, both in time and amplitude. This double transformation is the characterizing element of this kind of instruments, and differentiates them from another class of so-called digital instruments that process the input signals in the continuous time and amplitude domains and provide only the measurement result in a digital format.

The double domain transformation represents also the most critical part of the theoretical analysis of the DSP-based instruments, since the very first question to be answered is whether this double transformation modifies the information associated to the input signals or not.

This book is therefore aimed to analyse the mathematical theory that describes the two domain transformations, and in particular the transformation from the continuous to the discrete time, since this transformation is responsible for the most dangerous changes in the information associated with the input samples when nowadays sampling and converting devices are employed.

The devices in the block diagram in Fig. 1.1 are then considered, in order to evaluate their metrological characteristics and the way they contribute to the metrological performance of the whole DSP-based instrument.

Sampling device

A/D converter

Samples Processor

means that any sequence s(n) is defined only for integer values of the independent

At last the discrete systems are considered and analyzed in order to study a very imporant family of systems in the DSP technique: the digital filters.

Figure 1.1. Block diagram for a measuring instrument based on the digital signal

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Some of the considered items can be analysed only by means of advanced mathematical tools. Since this book is concerned with the digital processing of measurement signals rather than the mathematics of the discrete-time signals, the mathematical derivations will be limited to the amount that is strictly required to the correct comprehension of the presented theory. The readers are invited to refer to mathematics books if they wish to investigate the mathematical approach more in depth.