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An important part of digital or computing systems that interface to the “real/world” of sensors and analog interfaces is the ability to process sampled data in the digital domain. This is often called Sampled Data Systems (SDS) or defined as operating in the Z-domain. Most engineers are familiar with the operation of filters in the Laplace or S-domain where a continuous function defines the characteristics of the filter and this is the digital domain equivalent to that.
For example, consider a simple RC circuit in the analog domain, which is designed to be a low pass filter, as shown in Figure 9.1. This has a low pass filter behavior and can be represented mathematically using the continuous Laplace (or S-domain) notation:
L(s) = 1 1 + sRC (9.1)
This function is a low pass filter because the Laplace operator s is equivalent to jω, where ω = 2πf (with f being the frequency). If f is zero (the d.c. condition), then the gain will be 1, but if the value of sRC is equal to 1, then the gain will be 0.5. This in dB is −3 dB and is the classical low pass filter cut-off frequency.
In the digital domain, the s operation is replaced by Z. Z−1 is equivalent in a practical sense to a delay operator, and similar functions to the Laplace filter equations can be constructed for the digital, or Z domain, equivalent.
There are a number of design techniques, many beyond the scope of this book (if the reader requires a more detailed introduction to the realm of digital filters, Cunningham’s Digital Filtering: An Introduction is a useful starting point); however, it is useful to introduce some of the basic techniques used in practice and illustrate them with examples.
The remainder of this chapter will cover the introduction to the basic techniques and then demonstrate how these can be implemented using VHDL and Verilog on FPGAs.
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