# Communication Systems/FM and PM Generalization

## Concept

We can see from our initial overviews that FM and PM modulation schemes
have a lot in common. Both of them are altering the angle of the carrier
sinusoid according to some function. It turns out that we can go so far
as to generalize the two together into a single modulation scheme known
as **angle modulation**. Note that we will never abbreviate \"angle
modulation\" with the letters \"AM\", because Amplitude modulation is
completely different from angle modulation.

## Instantaneous Phase

Let us now look at some things that FM and PM have of common:

$$s_{FM} = A\cos (2 \pi [f_c + ks(t)]t + \phi)$$

$$s_{PM} = A\cos (2 \pi f_c t + \alpha s(t))$$

What we want to analyze is the *argument of the sinusoid*, and we will
call it Psi. Let us show the Psi for the bare carrier, the FM case, and
the PM case:

$$\Psi_{carrier}(t) = 2 \pi f_c t + \phi$$

$$\Psi_{FM}(t) = 2 \pi[f_c + ks(t)] t + \phi$$

$$\Psi_{PM}(t) = 2 \pi f_c t + \alpha s(t)$$

$$s(t) = A\cos(\Psi(t))$$

This Psi value is called the *Instantaneous phase* of the sinusoid.

## Instantaneous Frequency

Using the Instantaneous phase value, we can find the *Instantaneous
frequency* of the wave with the following formula:

$$f(t) = \frac{d\Psi(t)}{dt}$$

We can also express the instantaneous phase in terms of the
instantaneous frequency:

$$\Psi(t) = \int_{-\infty}^t f(\lambda)d\lambda$$

Where the Greek letter \"lambda\" is simply a dummy variable used for
integration. Using these relationships, we can begin to study FM and PM
signals further.

## Determining FM or PM

If we are given the equation for the instantaneous phase of a particular
angle modulated transmission, is it possible to determine if the
transmission is using FM or PM? it turns out that it is possible to
determine which is which, by following 2 simple rules:

1.  In PM, instantaneous phase is a linear function.
2.  In FM, instantaneous frequency minus carrier frequency is a linear
    function.

For a refresher course on Linearity, there is a chapter on the subject
in the Signals and Systems
book
worth re-reading.

## Bandwidth

In a PM system, we know that the value $\alpha s(t)$ can never go
outside the bounds of $(-\pi, \pi]$. Since sinusoidal functions
oscillate between \[-1, 1\], we can use them as a general PM generating
function. Now, we can combine FM and PM signals into a general equation,
called *angle modulation*:

$$v(t) = A \sin ( 2 \pi f_c t + \beta \sin (2 \pi f_m t))$$

If we want to analyze the spectral components of this equation, we will
need to take the Fourier transform of this. But, we can\'t integrate a
sinusoid of a sinusoid, much less find the transform of it. So, what do
we do?

It turns out (and the derivation will be omitted here, for now) that we
can express this equation as an infinite sum, as such:

$$v(t) = A \sum_{n = -\infty}^{\infty}J_n(\beta) \sin [2 \pi(nf_m + f_c)t]$$

But, what is the term $J_n(\beta)$? J is the *Bessel function*, which is
a function that exists only as an open integral (it is impossible to
write it in closed form). Fortunately for us, there are extensive tables
tabulating Bessle function values.

## The Bessel Function

The definition of the Bessel function is the following equation:

$$J_n(\beta) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{j[\beta sin\theta - n\theta]} d\theta$$

The bessel function is a function of 2 variables, N and $\beta$.

Bessel Functions have the following properties:

-   If n is even:

$$J_{-n}(\beta) = J_n(\beta)$$

-   If n is odd:

$$J_{-n}(\beta) = -J_n(\beta)$$

-   $J_{n-1} + J_{n+1} = \frac{2n}{\beta}J_n(\beta)$.

The bessel function is a relatively advanced mathematical tool, and we
will not analyze it further in this book.

## Carson\'s Rule

If we have our generalized function:

$$v(t) = A\sin ( 2 \pi f_c t + \beta \sin (2 \pi f_m t))$$

We can find the bandwidth BW of the signal using the following formula:

$$BW=2(\beta + 1)f_m = 2(\Delta f + f_m)$$

where $\Delta f$ is the maximum frequency deviation, of the transmitted
signal, from the carrier frequency. It is important to note that
Carson\'s rule is only an approximation (albeit one that is used in
industry frequently).

## Demodulation: First Step

Now, it is important to note that FM and PM signals both do the same
first 2 steps during demodulation:

1.  Pass the signal through a **limiter**, to remove amplitude peaks
2.  Pass the signal through a bandpass filter to remove low and high
    frequency noise (as much as possible, without filtering out the
    signal).

Once we perform these two steps, we no longer have white noise, because
we\'ve passed the noise through a filter. Now, we say the noise is
**colored**.

here is a basic diagram of our demodulator, so far:

`      channel`\
`s(t) ---------> r(t) --->|Limiter|--->|Bandpass Filter|---->z(t)`

Where z(t) is the output of the bandpass filter.

## Filtered Noise

To denote the new, filtered noise, and new filtered signal, we have the
following equation:

$$z(t) = \gamma A \cos (\Psi(t)) + n_0(t)$$

Where we call the additive noise $n_0(t)$ because it has been filtered,
and is not white noise anymore. $n_0(t)$ is known as **narrow band
noise**, and can be denoted as such:

$$n_0(t) = \mathbf{W}(t) \cos(2 \pi f_c t) + \mathbf{Z}(t) \sin(2 \pi f_c t)$$

Now, once we have it in this form, we can use a trigonometric identity
to make this equation more simple:

$$n_0(t) = \mathbf{R}(t) \cos(2 \pi f_c t + \mathbf{\Theta}(t))$$

Where

$$\mathbf{R}(t) = \sqrt{\mathbf{W}(t)^2 + \mathbf{Z}(t)^2}$$

$$\mathbf{\Theta}(t) = tan^{-1} (\mathbf{Z}(t)/\mathbf{W(t)})$$

Here, the new noise parameter R(t) is a **rayleigh** random variable,
and is discussed in the next chapter.

## Noise Analysis

R(t) is a noise function that affects the amplitude of our received
signal. However, our receiver passes the signal through a limiter, which
will remove amplitude fluctuations from our signal. For this reason,
R(t) doesnt affect our signal, and can be safely ignored for now. This
means that the only random variable that is affecting our signal is the
variable $\mathbf{\Theta}(t)$, \"Theta\". Theta is a uniform random
variable, with values between pi and -pi. Values outside this range
\"Wrap around\" because phase is circular.
