Transmitting an approximate CW with an FM modulator

Imagine we had an FM modulator with carrier frecuency $$f_c$$. My doubt is about the optimal approach to use it for transmitting a tone (CW) of a frequency $$f_c+\Delta f$$.

Let $$x(t)$$ a and $$y(t)$$ be the input and output of an FM modulator. Then,

$$y(t) = A \cos \left( 2\pi f_c t+2\pi \beta\int_0^t x(\tau)d\tau\right)$$

Where $$\beta$$ is the frequency deviation and $$|x(t)|\le1$$. Making this expression equal to the desired output:

$$A \cos \left( 2\pi f_c t+2\pi \beta\int_0^t x(\tau)d\tau\right) = A \cos \left( 2\pi f_c t+2\pi \Delta f t\right)$$

So that, for integer $$n$$,

$$\int_0^t x(\tau)d\tau = \Delta f t + 2\pi n$$

The solution to this equation is $$x(t)$$ beging a constant function. However, implementing this in real life leads to one big issue: modulators (and RF components in general) usually have DC blocks at their inputs in order to avoid transistor biasing and other problems.

This condition leads to the condition of $$x(t)$$ having a mean value of $$0$$ or, equivalently,

$$\int_0^\infty x(\tau)d\tau = 0$$

It is now clear that it is theoretically impossible to transmit a pure CW signal with such modulator (except for $$x(t)=0$$, which would lead to $$\Delta f=0$$). Nevertheless, what I am looking for is some way to transmit a similar (the cleaner, the better) signal to a CW.

For instance, let, for positive values of $$t$$,

$$x(t)=1 - 2 \pi N \sum_{k = 0}^\infty \delta(t-2 \pi Nk)$$

For integer $$N$$, this theoretical solution would have a mean value of $$0$$ while producing a good approximation to a CW at the output of the modulator. However, it seems to be far from easy to implement in a physical system.

My question then are:

· Is there any feasible way to implement an approximation to the above function in a physical system? If so, how clean would the CW at the output of the modulator be?

· What other funcions $$x(t)$$ can provide a relatively good tone at the output of an FM modulator?

Thank you very much in advance!

• Why do you think it's " theoretically impossible" to transmit a sine wave with FM? I don't follow your argument: sine waves do have zero mean, so it works just fine. Try $\lim_{T\to\infty} 1/T \int_{-T/2}^{+T/2} x(t) dt = 0$ Jun 4 '21 at 16:09
• I think the answer to your question very much hinges on your definition or even better, your way of measuring "clean" output, and how you really model your FM transmitter. For example, a square wave of alternatingly +your desired constant value, $-f_c-\text{const.}$ would work according to your formula for nearly all points in time – but will almost certainly be terrible in practice. But that's not covered by your transmitter model - you assume the only restriction is zero-mean $x$ (which certainly isn't the case). Jun 4 '21 at 16:46
• @MarcusMüller That's true. For example, I would have the same problem I describe with low frequencies by at high frequencies - it all depends on the frequency response of the modulator. Nevertheless, I find pretty strange that there is no "easy", "practical", "approximate" or "proof of concept" way to output a simple CW from an FM modulator Jun 4 '21 at 16:52
• seriously, though, you've found your solution that $x(\tau) \text{ const.}$ is the only solution, and that's correct. You argue that due to physical constraints of your transmitter, you can't do $x(t) \text{ const.}$, just, to, in the next paragraph suggest you can transmit Dirac $\delta$ distributions. Um. No, you can't, not in this universe with finite energy; that's why you really need to sit down and define how you measure the appropriateness of your $y$, and how you define what $x$ are admissible (you'll find that bandwidth becomes very important). Jun 4 '21 at 16:56
• so, define your modulator and its DC block! After you've don that, you can see whether that approach works. Before, we can postulate arbitrary things, none of which help you :) Jun 4 '21 at 18:33

One practical and traditionally used approach when the modulated waveform itself does not need to extend to $$f=0$$ (as in the case of the OP's wanting the transmit $$f+f_\Delta$$ is to use a PLL, or even an FLL as depicted in the block diagram below, such that the loop bandwidth of the loop is lower than the desired modulation frequency. The modulation signal is summed directly to the voltage controlled oscillator together with the frequency stabilization signal from the loop filter. The loop itself due to the tighter loop bandwidth will only correct relatively slow frequency changes (drift and close-in phase noise) while ignoring the higher frequency modulation signal. This is because the transfer function from the VCO input to the output is a high pass filter functionally, while the transfer function from the reference oscillator (used to stabilize the output) is a low pass filter. 