# Frequency Modulation¶

Note

This tutorial is addressed to HF2LI lock-in amplifier users that have purchased both HF2-MF multi-frequency and HF2-MOD AM/FM modulation options.

## What is Frequency Modulation?¶

In frequency modulation (FM), the amplitude of the modulating signal is proportional to the instantaneous frequency deviation from a fixed frequency. In the simplest case shown in Figure 1(a), the modulated signal

is produced when a carrier signal of frequency \(f_{c} = \omega_{c}/2\pi\) is modulated by a sinusoidal modulation with frequency \(f_{m} = \omega_{m}/2\pi\). The maximum variation of the frequency around the carrier frequency, the peak frequency deviation, is \(f_{p} = \omega_{p}/2\pi\). The physical information is encoded in the parameters \(A\), \(f_{p}\) and \(\varphi\).

Because the frequency is the time derivative of the phase (divided by \(2\pi\)) and the phase is the argument of the cosine in equation Equation 1, we can define the instantaneous frequency as

The spectrum of the FM signal of Equation 1 is more complicated than in the case of amplitude modulation. It consists of the carrier and a series of pairs of sidebands at multiple integers of \(f_{m}\) around the carrier frequency, see Figure 1(d). The amplitudes of the carrier and sidebands are given by mathematical functions called Bessel functions usually indicated by \(J_{n}\)) evaluated at the modulation index \(h = f_{p}/f_{m}\). For instance, the n-th pair of sidebands is located symmetrically about \(f_{m}\) at \(f_{c} \pm n f_{m}\) and its amplitude is \(J_{n}\left( h \right)\).

A peculiarity of the Bessel functions is that they oscillate around zero: even for the carrier, as the modulation index is increased, its amplitude \(J_{0}\left( h \right)\) decreases, crossing zero at \(h \approx 2.41\) and then it increases in amplitude (in anti-phase) before reaching zero again at \(h \approx 5.52\).

At low modulation indexes, the amplitude of higher sidebands is very low and can thus be ignored: this is called the narrow-band approximation. In this limit (it is customary to assume \(h < 0.2\)), only the two sidebands at \(f_{c} \pm f_{m}\) have non-negligible amplitude and the signal s(t) can be approximated by

The first term is the carrier, the other two are the lower and upper sidebands. The problem of finding h (and the peak amplitude \(f_{p}\)) reduces now to comparing the amplitude of the first pair of sidebands and the carrier to the ratio \(J_{1}{\left( h \right)/J_{0}\left( h \right)}\). A plot of the ratio \(J_{1}{\left( h \right)/J_{0}\left( h \right)}\) and \(J_{2}{\left( h \right)/J_{0}\left( h \right)}\) is shown in Figure 1(e).

Even though \(\tilde{s}\left( t \right)\) looks very similar to an AM signal, there is a subtle but substantial difference: the phases of the sidebands are offset with respect to that of the carrier. This results in the sidebands being in quadrature with the carrier. For example, assume that \(\varphi\) = 0: demodulating \(\tilde{s}\left( t \right)\) with the carrier signal \(\sin\left( \omega_{c}t \right)\) gives the DC component (the carrier) but no sidebands; on the other hand, demodulating with the quadrature \(\cos\left( \omega_{c}t \right)\), only the two sidebands at \(f_{m}\) are observed and no carrier is present. Because of this, FM detection can be done in a similar way as AM detection scheme, using the tandem configuration described previously in Amplitude Modulation.

The HF2-MOD AM/FM Modulation option permits direct generation and demodulation of an FM signal. For demodulation, this option enables measurement of the parameters A, \(f_{p}\), and \(\varphi\).

Internally the HF2LI calculates the peak frequency \(f_{p}\) with the method described above, from the ratio \(J_{1}{\left( h \right)/J_{0}\left( h \right)}\), proportional to the carrier and first sideband amplitude. Since this method is valid only for narrow-band frequency modulation, users are advised to work at small values of the modulation index h < 1.

Another, intuitive way of demodulating an FM signal would be to use the PLL to track the frequency deviation \(\Delta f\) and to further demodulate this signal. However, using sideband demodulation with the HF2-MOD AM/FM Modulation option provides a better signal-to-noise ratio. This is because the signal can be averaged over several modulation cycles while keeping the bandwidth small.

## Generate the Test Signal¶

In this tutorial, you are going to generate an FM signal with a carrier frequency of 1 MHz, a modulation frequency of 100 kHz, and a modulation index of 0.1. The signal is generated at Signal Output 2 and is demodulated by the first lock-in unit by feeding it into Signal Input 1.

Start by enabling the Signal Output 2 in the Lock-in MF tab and disabling all demodulator Output Amplitudes. This will ensure that only the desired components of the frequency-modulated signal appear on the output.

Signal Output 2 Enable | ON |

Signal Output Amplitudes Demodulators 1-8 | OFF |

In the Modulation tab, in the MOD 2 section, select the following parameters:

MOD 2 Enable | ON |

Carrier Oscillator (Osc)/Frequency | Osc 1 / 1 MHz |

Sideband 1 Oscillator (Osc)/Frequency | Osc 2 / 100 kHz |

Carrier Mode/Enable | FM / ON |

Generation Carrier Amplitude/Enable | 100 mV / ON |

Generation Index | 0.1 |

This generates an FM signal consisting of a carrier and two sidebands at \(f_{c} \pm f_{m}\). To look at this signal, connect Signal Output 2 to Signal Input 1 of the HF2 Instrument. Select the correct input parameters in the Lock-in tab: for Signal Input 1, make sure Differential and 50 Ω are turned off. Then click the auto range button. In the Scope tab, select Signal Input 1 as Source, Signal Output 2 as Trigger, and click on Run/Stop to activate the Scope. Observe that the carrier amplitude is constant. The periodic frequency variation is hardly visible. In Frequency Domain FFT mode, the plot shows the carrier at 1 MHz and the two sidebands. You can increase the frequency resolution by selecting a smaller Sampling Rate and larger time scale in the Horizontal section of the Scope.

## Measure the Test Signal¶

In the Modulation tab, in the MOD 1 section, select the following parameters:

MOD 1 Enable | ON |

Carrier oscillator (Osc) | 1 |

Sideband 1 oscillator (Osc) | 2 |

Carrier Mode | FM |

Low-pass Filter BW (Carrier) | 10 Hz |

Low-pass Filter BW (Sideband 1) | 10 Hz |

Demod 1, 2, 3 Data Transfer Enable (Lock-in tab) | ON |

This sets the correct demodulation of the FM signal. In the Numerical
tab, look at the amplitude of the carrier, 71 mV_{RMS}. You can
also see that the two sidebands have an amplitude of 3.5
mV_{RMS}. This corresponds approximately to the carrier
amplitude multiplied by the ratio
\(J_{1}{\left( h \right)/J_{0}\left( h \right)}\), see Figure 1(e) for our modulation index of h=0.1. Note that
the phases of the two sidebands are 180° apart, which is typical for FM.