Windowing the Data
Sometimes an experiment yields a very limited amount of data, which must be used carefully to get the maximum amount of information. For example, a very short impulse response might be recorded with a high-speed A/D converter and still yield only a few data points. In other situations, the problem is just the opposite — there is so much data that it cannot, or should not, be used all at once. For example, in speech processing, the digitized speech waveform is analyzed in short segments, so that the time evolution of the waveform can be observed. In both cases, the data must be handled with care. It turns out that the data should seldom be used as originally obtained. Usually a better result is obtained if the data points are tapered off toward the edges of the data window. This process is called windowing the data. The window shape has a significant impact on the final results. The effects of various windowing options are explored in this chapter. This chapter is essential reading for Part III.
10.1 RESTRICTIONS TO FINITE TIME
Theorem 11.8 of Chapter 11 covers all cases of the Fourier transform simultaneously. Of course, in digital signal processing, the only cases that can be used directly are those that have finitely many group elements. Only a finite number of data points can be used, or even obtained, in any finite duration of time. Often, however, the theory has been worked out using the group G = or maybe G = S1, and the question arises: What ...