FM Frequency Modulation - moerjielovecookie

1 Modulation Algorithm#

1.1 FM Signal Description#

Using a baseband modulation signal to control the instantaneous frequency of the carrier signal, causing it to vary according to the modulation signal's pattern. When the modulation signal is an analog signal, this process is called frequency modulation. The time-domain expression of the frequency modulation signal is as follows:

S_{FM}(t)=A_m\times \cos[\omega_ct+K_f \int^{t}_{0}m(\tau)d\tau]\tag{1}

Where $A_m$ is the carrier amplitude, $K_f$ is the frequency modulation sensitivity (in units of $rad/(s\cdot V)$), $m(t)$ is the modulation signal, $cos\omega_c t$ is the carrier, and $\omega_c$ is the carrier angular frequency.
From equation (1), the instantaneous frequency deviation of the FM signal relative to the carrier frequency can be calculated as:

\frac{d\left[K_f\int_0^tm(\tau)d\tau\right]}{dt}=K_fm(t)\tag{2}

From equation (1), it can be seen that the instantaneous phase shift of the FM signal relative to the carrier phase varies linearly with the integral of $m(t)$. From equation (2), it can be seen that the instantaneous frequency shift of the FM signal relative to the carrier frequency varies linearly with $m(t)$, with a proportionality constant of $K_f$. Let $k_f$ represent the frequency modulation sensitivity (in units of Hz/V), the relationship is $K_f=2\pi k_f$.
The frequency modulation index $\beta$ of FM is:

\beta=\frac{k_f |m(t)|_{max}}{W}\tag{3}

Where $W$ is the bandwidth or maximum frequency of the baseband signal $m(t)$.

1.1.1 Narrowband Frequency Modulation (NBFM)#

The situation where the maximum instantaneous phase shift caused by $m(t)$ is much less than 30° is called narrowband frequency modulation. Narrowband frequency modulation has a narrow bandwidth and limited data transmission capacity, mainly used for wireless voice transmission.

|K_f\int_0^tm(\tau)d\tau|\ll\frac\pi6 \tag{4}

At this time, equation (1) can be approximated as:

\begin{aligned}s_{NBFM}(t)&=Acos\left[\omega_ct+K_f\int_0^tm(\tau)d\tau\right]\\\\&=Acos(\omega_ct)cos{\left[K_f\int_0^tm(\tau)d\tau\right]}-Asin(\omega_ct)sin{\left[K_f\int_0^tm(\tau)d\tau\right]}\\\\&\approx Acos(\omega_ct)\cdot1-Asin(\omega_ct)\cdot K_f\int_0^tm(\tau)d\tau\\\\&=Acos(\omega_ct)-\left[AK_f\int_0^tm(\tau)d\tau\right]sin(\omega_ct)\end{aligned}\tag{5}

Performing an FFT transformation on it, the frequency spectrum of the narrowband frequency modulation signal is obtained as:

S_{NBFM}(\omega)=\pi A\left[\delta(\omega+\omega_c)+\delta(\omega-\omega_c)\right]+\frac{AK_f}{2}\left[\frac{M(\omega-\omega_c)}{\omega-\omega_c}-\frac{M(\omega+\omega_c)}{\omega+\omega_c}\right]\tag{6}

Where $M(\omega)$ is the frequency spectrum of the modulation signal $m(t)$. Unlike AM signals, the two sidebands of the NBFM signal are multiplied by the factors $1/(\omega-\omega_c)$ and $1/(\omega+\omega_c)$, respectively. Since these factors are functions of frequency, their weighting is frequency-weighted, resulting in distortion of the modulation signal's frequency spectrum, and one sideband of NBFM is opposite to AM.

1.1.2 Wideband Frequency Modulation (WBFM)#

When the conditions of equation (4) are not satisfied, it is called wideband frequency modulation. Wideband frequency modulation occupies a wider frequency band, allowing for a larger data transmission capacity, mainly used for FM stereo broadcasting. The time-domain expression of WBFM cannot be simplified. When $m(t)=A_m \cos(\omega_m t)$, substituting into equation (1) gives:

\begin{aligned}s_{WBFM}(t)&=Acos\left[\omega_ct+K_f\int_0^tA_mcos(\omega_m\tau)d\tau\right]\\\\& =Acos\left[\omega_ct+\frac{K_fA_m}{\omega_m}sin(\omega_mt)\right]\\\\& =Acos[\omega_ct+\beta sin(\omega_mt)]\\\\& =A\sum_{n=-\infty}^\infty J_n(\beta)cos[(\omega_c+n\omega_m)t]\end{aligned}\tag{7}

Where $J_n(\beta)$ is the Bessel function of the first kind of order n, which is a function of the frequency modulation index $\beta$.
The frequency spectrum is:

S_{WBFM}(\omega)=\pi A\sum_{n=-\infty}^\infty J_n(\beta)\left[\delta(\omega-\omega_c-n\omega_m)+\delta(\omega+\omega_c+n\omega_m)\right]\tag{8}

1.2 Bandwidth of FM Signals#

The frequency spectrum of wideband frequency modulation signals contains an infinite number of frequency components; therefore, theoretically, the bandwidth of frequency modulation signals is infinitely wide. However, in practice, the amplitude of the side frequencies $J_n(\beta)$ decreases as n increases, so an appropriate n value can be chosen to make the side frequency components small enough to be negligible, thus approximating that FM signals have a finite bandwidth. The general principle is that the bandwidth of the signal should include side frequency components with amplitudes greater than 10% of the unmodulated carrier. Empirically, when $\beta \geq 1$, selecting the side frequency number $n=\beta +1$ is sufficient, as the amplitudes of side frequencies greater than $\beta+1$ are all less than 0.1.
Based on this experience, the effective bandwidth of FM signals is:

BW_{FM}=2(\beta+1)W\tag{9}

This formula is known as Carson's formula.

2 FM Modulation Methods#

2.1 Direct Frequency Modulation#

Direct frequency modulation is the method of directly controlling a high-frequency oscillator with the modulation signal $m(t)$, causing the parameters of the loop components to change, resulting in an output frequency that varies linearly according to the modulation signal. Common components used are varactor diodes. The main advantage of direct frequency modulation is that it can achieve a large frequency deviation while meeting the requirements for linear frequency modulation, and the implementation circuit is simple; the main disadvantage is that the frequency stability is not high, often requiring an automatic frequency control system to stabilize the center frequency.

2.2 Indirect Frequency Modulation#

Indirect frequency modulation is also known as the frequency multiplication method. First, the modulation signal is integrated, then phase modulation is applied to the carrier to produce an NBFM signal, which is then processed through $n$ frequency multipliers to obtain a WBFM signal. The advantage is high frequency stability, while the disadvantage is the need for multiple frequency multiplications and mixing, making the circuit complex.

2.3 Quadrature Modulation#

Expanding equation (1) using trigonometric identities gives:

\begin{aligned}s_{FM}(t)&=Acos\left[\omega_ct+K_f\int_0^tm(\tau)d\tau\right]\\\\&=Acos(\omega_ct)cos\left[K_f\int_0^tm(\tau)d\tau\right]-Asin(\omega_ct)sin{\left[K_f\int_0^tm(\tau)d\tau\right]}\end{aligned}\tag{10}

The process is as follows:

Integrate the modulation signal $m(t)$ to obtain $\Phi=K_f\int^{t}_{0}m(\tau)d\tau$
Take the cosine and sine of the integrated signal to obtain I-channel data $I(t)=\cos(\Phi)$ and Q-channel data $Q(t)=\sin(\Phi)$
Multiply by the carrier $A\cos(\omega_c t)$ and $-A\sin(\omega_ct)$ respectively and add them together to obtain the FM signal $s_{FM}(t)=I(t)Acos(\omega_ct)-Q(t)Asin(\omega_ct)$

3 FM Signal Format#

The modulation frequency deviation of FM systems in China is 150kHz, distributed in the frequency band of 87MHz to 107.9MHz.
The baseband signal is the MPX signal, which includes mono (0~15kHz), pilot (19kHz), stereo subcarrier (center frequency 38kHz), and RDS data (center frequency 58kHz). Its spectral components are as follows:
Spectrum components of MPX signal