Digital Signal Processing Reference
In-Depth Information
the analytic signal will reflect an accurate model of the real system with
dependable instantaneous frequency. Thus the signal should be as close as
possible to that of a sinusoid. A Matlab
function that implements (7.29) is given
in Figure 7.22. In the next section, we will focus briefly on the instantaneous
frequency and show how to extract the modulation frequency using the Hilbert
transform.
7.8.1
The Phase between Two Narrowband Signals
The phase between two narrowband signals could be retrieved using (7.7) or
(7.19). This was implied in our discussion on dc phase errors in Section 7.4. Of
course, if
ϕ
t
is constant then the signal is narrowband. Moreover, the frequency of
the input signals into the demodulator could be different. If this is the case, the
low-pass filters in both demodulation schemes must be capable of passing the
difference frequency of the resulting signals.
7.9 INSTANTANEOUS FREQUENCY
In this section, a brief review of the connection between the instantaneous phase
and frequency will be given. This will then be followed by straightforward
techniques to extract the instantaneous modulation frequency.
In the context of communications theory, the instantaneous frequency
F
inst
determines the message signal completely, and is given by
d
ϕ
1
t
F
=
(7.30)
inst
2
π
dt
where
t
is the time-dependent phase demodulated output. This is based on the
Carson-Fry model and has a specific interpretation [10,11], but is not strictly
analytic as discussed by Hahn [8]. In fact, such models approach analyticity under
narrowband conditions. As such, (7.30) is applicable to narrowband signals only,
and barring this, may lead to a misapplication of the derived signal. We must
emphasize though that in general, (7.30) gives the instantaneous frequency in
Hertz when the instantaneous phase is measured in radians, and its interpretation
is governed by the model used.
In some other applications, a physical meaning could be attached to the
instantaneous frequency. For example, in sensors involving optical interferometry,
where the optical path-length is designed to respond sympathetically to changes in
a physical property of interest, such as pressure or temperature, the instantaneous
frequency is often linked directly to certain dynamic properties of the material
forming the sensor. In optical Doppler flow measurements the instantaneous
frequency is linked to the instantaneous velocity of the flow particles.
ϕ
