EXAMPLE APPLICATIONS
At this point let's take a look at where we have been and where we are going. So far, we've been concerned with the mechanics of getting a signal into and out of our DSP system, and with reviewing some general math principles we will use later on. We have seen that we can sample a waveform, optionally store it, and then send it back out to the world. This is, in and of itself, a very useful ability. However, it represents only a small fraction of the things we can do with a DSP system.

Understanding how a DSP system is designed and used basically requires two types of knowledge. The first is an understanding of the applications that lend themselves best to DSP. The second type is an understanding of the tools necessary to design the system to accommodate these applications.

With this in mind, let's now turn our attention to the subject of filtering, beginning with a simple filter that is easily understood intuitively. We will then move on to developing the tools and techniques that will allow us to create more sophisticated, higher-performance filters of professional quality.

FILTERS
One of the most common DSP operations is filtering. As with analog filters, DSP filters can provide low-pass, bandpass, and high-pass filtering. (Specialized functions, such as notch filters, are also possible, though we will not be covering them in this book.) The basic idea behind filtering in general is this: An input signal, generally a function of time, is input to a transfer function. Normally, the transfer function is a differential equation expressed as a function of frequency. The output of the transfer function is some subset of the input signal.

A block diagram of a low-pass filter is shown in Figure 4-8. In the figure, the input signal is a sum of two sine waves: one of them at a fundamental frequency, the other at the third harmonic. After passing through the transfer function H(ω) only the fundamental frequency remains; the first harmonic has been blocked. The top portion of Figure 4-8 depicts the low-pass filter as a function of time. The bottom portion of Figure 4-8 shows the filter as a function of frequency. We will be revisiting these concepts in greater detail in later chapters.

Figure 4-8. The basic low-pass filter.

Key Concept
In the world of analog electronics, the transfer function H(ω) is realized by arranging a combination of resistors, capacitors, inductors, and possibly operational amplifiers. In DSP applications, a computer is substituted for the resistors, capacitors, and inductors. The computer then computes the output using the input and H(ω).

The question for the DSP applications developer then becomes: How do we define H(ω) to give us the desired transfer function? This chapter shows, in an intuitive way, how simple digital filters operate. After that, several key concepts are introduced that lay the groundwork for developing more sophisticated filters. In the next chapters, we will see how to apply these tools to develop some practical working filters.