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FIR Filters - Filter Fundamentals

Designed filter is excited with impulse signal. The first sample is assigned a value of one (impulse signal) and all other inputs from sample1 to sample7 are made zero. When ‘en’ signal is enabled filter starts the processing of accumulation of partial products. After N clock cycles, where N is the order of the filter | Chapter Filter Fundamentals Digital filters are often based upon common analog filter functions. Therefore a certain amount of background material concerning analog filters is a necessary foundation for the study of digital filters. This chapter reviews the essentials of analog system theory and filter characterization. Some common analog filter types Butterworth Chebyshev elliptical and Bessel are given more detailed treatment in subsequent chapters. 1.2 Systems Within the context of signal processing a system is something that accepts one or more input signals and operates upon them to produce one or more output signals. Filters amplifiers and digitizers are some of the systems used in various signal processing applications. When signals are represented as mathematical functions it is convenient to represent systems as operators that operate upon input functions to produce output functions. Two alternative notations for representing a system H with input x and output y are given in Eqs. 2.1 and 2.2 . Note that x and y can each be scalar valued or vector valued. y H X y Hx 2.1 2.2 This book uses the notation of Eq. 2.1 as this is less likely to be confused with multiplication of x by a value H. A system H can be represented pictorially in a flow diagram as shown in Fig. 2.1. For vector-valued x and y the individual components are sometimes explicitly shown as in Fig. 2.2a or lumped together as shown in Fig. 2.2b. Sometimes in order to emphasize their vector nature the input and output are drawn as in Fig. 2.2c. 35 36 Chapter Two Figure 2.1 Pictorial representation of a system. a Figure 2.2 Pictorial representation of a system with multiple inputs and outputs. In different presentations of system theory the notational schemes used exhibit some variation. The more precise treatments such as Chen 1984 use x or x - to denote a function of time defined over the interval 00 00 . A function defined over a more restricted interval such as f0 ti would be denoted as x to ti