What is FM modulation?

Although less intuitive of amplitude modulation, frequency modulation is still a fairly simple method for transmitting wireless data.

We are all at least vaguely familiar with frequency modulation, It is the origin of the term “radio FM“. If we think of frequency as something that has an instantaneous value, rather than something that consists of several cycles each for a corresponding period of time, we can continuously vary the frequency according to the instantaneous value of a baseband signal.

Frequency Modulation (FM) and phase modulation (PM)

FM and PM are closely related because frequency and phase are closely related. This is not so obvious when you consider the rate as the number of complete cycles per second: what you have to do the cycles per second with the position of the sine wave at a given time during its cycle? But it makes more sense when you consider the instantaneous frequency, namely the frequency of a signal at a given time. (It is certainly paradoxical to describe a frequency as instantaneous, ma, in the context of the signal practice, we can safely ignore the complicated theoretical details associated with this concept).

In a basic sine wave, the value of the instantaneous frequency is equal to the frequency “normal”. The analytical value of the instantaneous frequency rises when it comes to signals which have a variable frequency over time, ie the frequency is not a constant value but rather a function of time, written as ω (t). Anyhow, the important point for our present discussion on the close relationship between frequency and phase are as follows: the instantaneous angular frequency is the derivative, with respect to time, stage. So if you have an expression φ (t) variable that describes the behavior over time of the phase of the signal, the rate of change (with respect to time) of φ (t) It gives you the expression for the instantaneous angular frequency:

In mathematics

Above we discussed the paradox quantity called instantaneous frequency. If you feel that this term is unfamiliar or confused, back to that page and read the section “Frequency Modulation (FM) and phase modulation (PM)”. It may still be a little confused, and this is understandable – the idea of ​​an instantaneous frequency violates the basic principle that “frequency” indicated How often a complete signal a complete cycle: ten times per second, one million times per second or whatever it may be.

We will not attempt any kind of comprehensive or thorough treatment of the instantaneous frequency as a mathematical concept. In the context of FM, the important thing is to realize that the instantaneous frequency naturally follows from the fact that the frequency of the carrier varies continually in response modulating wave (that is, the base-band signal). The instantaneous value of the baseband signal influences the frequency at a given time, not the frequency of one or more complete cycles. In reality, But, this is only true for analog FM; digital FM, a bit corresponds to a discrete number of cycles. This leads to the interesting situation in which older technology (analog FM) It is less intuitive than the latest technology (FM digital, detta anche shifting keying, o FSK).

It is not necessary to reflect on the instantaneous frequency to understand the digital frequency modulation.

As for AM, we contact the carrier as sin (ωCt). It already has a frequency (that is, ωC), so we will use the term excess frequency to refer to the frequency component supplied from the modulation process. This term is somewhat misleading, however, as “excess” It implies a higher frequency, while the modulation can determine a carrier frequency higher or lower than the nominal carrier frequency. In fact, this is the reason why the frequency modulation (unlike the amplitude modulation) It does not require a shifted base band signal: the positive baseband values ​​increase the carrier frequency and baseband negative values ​​decrease the carrier frequency. In these conditions, demodulation is not a problem, because all of the baseband values ​​are mapped on a single frequency.

Anyhow, back to our carrier signal: without (ωCt). If we add the baseband signal (xBB) the quantity within the brackets, we are making phase excess linearly proportional to the base-band signal. But we're looking for frequency modulation, no phase modulation, so we want frequency excess is linearly proportional to the base-band signal. We know that we can get the frequency taking the derivative, with respect to time, stage. Then, if we want that the frequency is proportional to xbb, we should not add the baseband signal but rather the integral of the baseband signal (why take the derivative cancels the integral, and we remain with xbb as excess frequency).

The only thing we have to add here is the modulation index, m. Previously we have seen that the modulation index can be used to make changes in the amplitude of the carrier more or less sensitive to changes in the value baseband. Its FM function is equivalent: the modulation index allows us to fine-tune the intensity of the change in frequency produced by a change in the value in base band.

The time domain

Let's look at some forms of onda.Ecco our carrier from 10 MHz:

The baseband signal is a sine wave of 1 MHz, as follows:

The FM waveform is generated by applying the above formula. The integral of sin (x) is -cos (x) + C. Here the constant C is not relevant, so we can use the following equation to calculate the FM signal:

Here is the result (the signal of the baseband is shown in red):

It almost seems that the carrier has not changed, but if you look closely, the peaks are slightly closer when the signal of the base band is close to its maximum value. So we have here the frequency modulation; the problem is that the variations of the base band does not produce a variation of the carrier frequency sufficient. We can easily remedy this situation by increasing the modulation index. We use m = 4:

Now we can see more clearly how the frequency of the modulated carrier continuously follows the instantaneous value of the baseband.

The frequency domain

The waveforms in the domain of the AM and FM time for the same baseband and carrier signals seem very different. It's interesting, then, find that AM and narrow band FM produce similar changes in the frequency domain. (The narrow band FM involves a bandwidth limited modulating and allows easier analysis.) In both cases, a low-frequency spectrum (including negative frequencies) It is translated into a band that extends above and below the carrier frequency. Con AM, the spectrum in base band moves upwards. with FM, is the spectrum integral of the baseband signal that appears in the band that surrounds the carrier frequency.

For the modulation baseband frequency, m = 1 shown above, we have the following:

The next spectrum is with m = 4:

This clearly demonstrates that the modulation index influence the frequency content of the modulated waveform. The spectral analysis with frequency modulation is more complicated with respect to the amplitude modulation; It is difficult to predict the bandwidth of the frequency modulated signals.

For the moment I finish here but, I will publish as soon as the last item on the phase modulation, the physical time to stilarlo and create fornule and images.

And if the topic interested I can also talk about the digital modulation amplitude components of the three phase and frequency

Here's the direct link to the WIDTH MODULATION

Here's the direct link to the MODULATION PHASE

Direct link to them here DIGITAL MODULATION

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