Signal Processing in C

CHAPTER 13

From Waveforms to Filters

13.1 INTRODUCTION

Part I of this book was primarily concerned with waveform synthesis using simple harmonic oscillators. We now turn to the task of waveform analysis: obtaining information from a waveform. A receiver, for example, must separate the desired signal from the noise. In other cases, the desired signal is not known a priori as it is for receiver designs, and the goal is to learn something about the physics of the object producing the waveform. There is always noise mixed with the desired signal, and the noise must somehow be rejected even when the signal is not well specified.

For example, a pulsed radar sends discrete pulses of microwave radiation toward a target and waits for a return. The radiated signal is known, but the environment scatters the signal; moving targets cause doppler shifts; birds, buildings, and the ground bounce part of the energy back to the receiver; and some targets do their best to hide from or jam the radar. In the midst of all that clutter, the radar receiver is supposed to extract useful information about the significant targets.

In this part of the book, we will analyze waveforms from various musical instruments and attempt to construct digital filters that could have produced these waveforms when properly driven. The equation to be solved in this process is deceptively simple. Every linear filter can, as discussed below, be expressed as a convolution. If y is the filter output. x is the filter input. ...