# Chapter Two

# TRANSFORMATION METHODS

Switched-capacitor filters are *sampled-data* circuits, with *analog* signal representation. Hence, their analysis requires, in general, the mathematical tools of both analog signals (Laplace and Fourier transformations) and those of sampled signals (*z*-transformation). Furthermore, the relations between these two groups of transformations must be correctly formulated and used. For these reasons, this chapter gives a summary of the basic definitions of analog, digital, and sampled-analog systems, and then discusses the various transformation methods needed to analyze their time and frequency responses. Finally, design techniques will be described for obtaining the transfer function of a sampled-data system from that of a suitable analog “model” system.

## 2.1. ANALOG, DIGITAL, AND SAMPLED-ANALOG SIGNALS AND SYSTEMS^{1} – ^{3}

A *signal* is a function—its independent variable is, in our applications, time; the dependent variable is a physical quantity such as voltage, charge, or current. A *continuous-time signal* is a signal which has a well-defined value at every point in the time interval of interest (Fig. 2.1*a*). A *discrete-time signal* has values only at discrete (usually equally spaced) time instances (Fig. 2.1*b*); it is unspecified at any other time. Often, the discrete-time signal is obtained by sampling a continuous-time one. Thus, the signal of Fig. 2.1*b*