# Instantaneous Frequency¶

Analytical approach to continuous Instantaneous Frequency and Frequency Modulation.

## 1. Instantaneous frequency¶

A modulated signal can be expressed as: $$\large x(t)=a(t)\cos(\phi(t))$$ where:

• The instantaneous amplitude or envelope is given by $\large a(t)$;
• The instantaneous phase is given by $\large \phi(t)$;
• The instantaneous angular frequency is given by $\large \omega(t)=\frac{d}{dt}\phi(t)$;
• The instantaneous ordinary frequency is given by $\large f(t)=\frac{1}{2\pi}\frac{d}{dt}\phi(t)$.

## 2. Examples¶

Modulation using instantaneous frequency

### 2.1. Linear frequency modulation¶

Given a modulation frequency, defined by:

$$\large f(t)=f_a+\frac{(f_b-f_a)t}{T}$$

where $f(t)$ is linearly interpolated from $f_a$ to $f_b$.

The modulated signal whithout instantaneous frequency is:

$$\large x(t)=sin(2\pi f(t)t)$$

In this example the carrier frequency is zero ($\phi_c=0$).

If we considere:

$$\large \phi(t)=\int\omega(t)dt=2\pi\int f(t)dt$$

the modulated signal by the instantaneous frequency could be expressed by:

$$\large x_m(t)=sin(\phi(t))=sin\left(2\pi\left[f_at+\frac{(f_b-f_a)t^2}{2T}\right]\right)$$

### 2.2. Exponencial frequency modulation¶

$$\large f(t)=f_a+\frac{f_b-f_a}{1+e^{-k\left(\frac{t}{T}-\frac{1}{2}\right)}}$$

$f(t)$ is a mudulation frequency which interpoles from $f_a$ to $f_b$ exponentially. It's inspired by the sigmoid function where $k$ is the slope of the middle point.