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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:

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.