This chapter addresses the most fundamental question of digital signal processing: What is a discrete-time frequency and how is it related to continuous-time frequencies? If you think you know the answers, read this chapter anyway. It is short, and it presents a viewpoint somewhat different from that found in other expositions. A thorough understanding of these basic issues goes a long way toward clearing up all kinds of muddled thinking about digital signal processing.
As is often the case in a complicated subject, the fundamentals of digital signal processing are actually quite simple once they are grasped. These underlying foundation stones are easily lost in the complicated superstructure built on top of them. These basic issues can be understood through everyday experience, such as watching wagon wheels in old Western movies turning backwards when you know that that cannot really be happening (see Section 1.3).
What is a continuous-time frequency? You may think you know the answer, and you may just write down a single number or letter, ω or f, to represent a “frequency”–but think again. The answer is really a bit more complicated.
The most elementary notion of a frequency is a real number, ω, that in turn specifies a waveform, cos(ωt + ), for some initial phase angle, . In signal ...