Getting Started with Signal Processing
This chapter introduces basic signal processing theory. We examine using basic network equations how passive components such as inductors and capacitors behave like filters in analog circuits. We introduce the Laplace Transform and examine the effect it h
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Getting Started with Signal Processing
3.1 Introduction In the previous chapter, we introduced DSP using a few non-engineering and some well-known electrical engineering examples. We had also compared DSP with analog signal processing with advantages and disadvantages of both. With an overview of DSP, in this chapter, let us dive into signal processing theory. In this chapter, we will start with circuits made of passive components such as inductors and capacitors and look at their properties as filters. We will analyze basic filters such as the LC filter using network laws to be able to lay the foundation of the frequency dependent behaviour of circuits. We will then present the Laplace Transform and examine its properties in transforming common electrical signals as well as common circuit laws. We will examine the impact of Laplace Transform and the significance of the resultant function. We will then examine how to convert a system from the continuous domain to the digital domain. This chapter will present theory with basic examples but will not get started with the actual filtering or the performance of filters. The focus of this chapter is to understand how starting from a time domain representation of the system, we can arrive at a frequency domain representation of a system. This frequency domain representation of a system is the fundamental basis of signal processing as it allows us to analyze the behaviour of the system for different frequencies.
3.2 Reviewing Capacitors and Inductors We saw in Chap. 2, a few basic filter examples of how we used capacitors and inductors as filters. Let us continue with analog filters such as these but try to take a step back and look at the process mathematically instead. Take for instance the © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 S. V. Iyer, Digital Filter Design using Python for Power Engineering Applications, https://doi.org/10.1007/978-3-030-61860-5_3
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3 Getting Started with Signal Processing
capacitor filter. The primary equations of a capacitor are [2]: dv dt 1 v= idt C i=C
(3.1) (3.2)
Let us try to map the physical properties of the capacitor to these equations. The current through the capacitor branch being a derivative with respect to time of the voltage across the capacitor implies that the current will increase as the voltage changes rapidly. As an extreme case if the voltage is constant—a dc voltage—the current will be zero, as the voltage does not change at all, and so the rate of change is zero. Physically, if you apply a dc voltage across a capacitor, it will charge up to a voltage equal to the applied voltage after which the current will be zero. Take a very high frequency noise signal as the applied voltage as the other extreme case. Now, the rate of change is very high simply because it has a high frequency to begin with. The current through the capacitor will be high and will be a high frequency component as well. The capacitor will therefore block a dc current and
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