# Extracting instantaneous amplitude,phase,frequency – application of Analytic signal/Hilbert transform

(2 votes, average: 4.00 out of 5)

## Introduction

The concept of instantaneous amplitude/phase/frequency are fundamental to information communication and appears in many signal processing application. We know that a monochromatic signal of form $$x(t) = a cos(\omega t + \phi)$$ cannot carry any information. To carry information, the signal need to be modulated. Take for example the case of amplitude modulation, in which a positive real-valued signal $$m(t)$$ modulates a carrier $$cos(\omega_c t)$$. That is, the amplitude modulation is effected by multiplying the information bearing signal $$m(t)$$ with the carrier signal $$cos(\omega_c t)$$.
$$x(t) = m(t) cos\left(\omega_c t\right)$$
Here, $$\omega_c$$ is the angular frequency of the signal measured in radians/sec and is related to the temporal frequency $$f_c$$ as $$\omega_c = 2 \pi f_c$$. The term $$m(t)$$ is also called instantaneous amplitude.

Similarly, in the case of phase or frequency modulations, the concept of instantaneous phase or instantaneous frequency is required for describing the modulated signal.
$$x(t) = a cos \left(\phi(t) \right)$$
Here, $$a$$ is the constant amplitude factor (no change in the envelope of the signal) and $$\phi(t)$$ is the instantaneous phase which varies according to the information. The instantaneous angular frequency is expressed as the derivative of instantaneous phase.
$$\omega(t) = \frac{d}{dt} \phi(t)$$
$$f(t) = \frac{1}{2 \pi} \frac{d}{dt} \phi(t)$$

## Definition

Generalizing these concepts, if a signal is expressed as
$$x(t) = a(t) cos\left(\phi(t) \right)$$

• The instantaneous amplitude or the envelope of the signal is given by $$a(t)$$
• The instantaneous phase is given by  $$\phi(t)$$
• The instantaneous angular frequency is derived as $$\omega(t) = \frac{d}{dt} \phi(t)$$
• The instantaneous temporal frequency is derived as $$f(t) = \frac{1}{2 \pi} \frac{d}{dt} \phi(t)$$

### Problem statement

An amplitude modulated signal is formed by multiplying a sinusoidal information and a linear frequency chirp. The information content is expressed as $$a(t) = 1 + 0.7 \; sin \left( 2 \pi 3 t\right)$$ and the linear frequency chirp is made to vary from $$20 \; Hz$$ to $$80 \; Hz$$. Given the modulated signal, extract the instantaneous amplitude (envelope), instantaneous phase and the instantaneous frequency.

### Solution

We note that the modulated signal is a real-valued signal. We also take note of the fact that amplitude/phase and frequency can be easily computed if the signal is expressed in complex form. Which transform should we use such that the we can convert a real signal to the complex plane without altering the required properties ?? Answer: Apply Hilbert transform and form the analytic signal on the complex plane.  Figure 1 illustrates this concept.

If we express the real-valued modulated signal $$x(t)$$ as an analytic signal, it is expressed in complex plane as
$$z(t) = z_r(t) + j z_i(t) = x(t) + j HT \left[ x(t) \right]$$
where, $$HT[]$$ represents the Hilbert Transform operation. Now, the required parameters are very easy to obtain.

The instantaneous amplitude (envelope extraction) is computed in the complex plane as
$$a(t) = |z(t)| = \sqrt{z_r^2(t) + z_i^2(t)}$$

The instantaneous phase is computed in the complex plane as
$$\phi(t) = \angle z(t) = arctan \left[ \frac{z_i(t)}{z_r(t)} \right]$$

The instantaneous temporal frequency is computed in the complex plane as
$$f(t) = \frac{1}{2 \pi} \frac{d}{dt} \phi(t)$$

Once we know the instantaneous phase, the carrier can be regenerated as $$cos[\phi(t)]$$. The regenerated carrier is often referred as Temporal Fine Structure (TFS) in Acoustic signal processing [1].

## Results

(2 votes, average: 4.00 out of 5)