7.1 INTRODUCTION

The Hilbert transform is a relatively well-known technique to convert a real signal into its analytic form ^[1], ^[2]. An analytic signal is a signal whose imaginary part is the Hilbert transform of the real part. Thus given a real signal s _t sampled at time t, the analytic signal S _t is

where H{ x} is the Hilbert transform of x, and j is the complex term ? ?1. The Hilbert transform therefore imparts a ?/2 phase shift to a real signal to make it analytic. The fact that we can create an orthogonal signal from an existing set of measurements means that it is possible to extract the frequency and phase information carried by the original signal. It is this property that is exploited in many phase and frequency demodulation applications in communications. To see this, we convert (7.1) into polar coordinates, assuming the signal s _t has angular frequency ? and phase ? _t at time t, and write

where

Thus, the amplitude and phase are readily recovered from a working knowledge of the Hilbert transform of a signal. Note that ? _t may itself be a function of time producing a wide range of spectral components. It is also useful to treat the rate of change of phase as an instantaneous frequency; this aspect will be discussed later.

The goal of such demodulation schemes is to determine...