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.