Instantaneous Frequency

definition:
The differential of oscillation phase over time divided by 2 π.

When describing non monochromatic signals, instantaneous frequency is required, and its definition is:

 

That is to say, the differential of oscillation phase φ with respect to time. Excluding the factor 1/2 π, the instantaneous angular frequency is obtained. Unlike Fourier frequency, instantaneous frequency is usually related to time. The instantaneous frequency of a sine signal is a constant equal to the frequency of oscillation.  

 

Figure 1: Electric field of strongly chirped pulses, where the instantaneous frequency increases with time.  
When considering frequency noise and phase noise, instantaneous frequency is very important and is often used for chirping optical pulses (Figure 1), whose instantaneous frequency is time-dependent. The basic idea is more intuitive than Fourier frequency. There is also a similar concept in music: in sheet music, notes are represented as time intervals, and the instantaneous frequency within each interval is a constant value (corresponding to pitch). However, this concept poses problems for complex signals such as white noise. In lasers, it is easy to define the instantaneous frequency for single frequency lasers, while for multimode lasers, it is necessary to first separate the different frequency components (using filtering techniques) before obtaining the instantaneous frequency. The concept of instantaneous frequency is also very useful in chirped light pulses, where the instantaneous frequency of different pulses is different.  
The Fourier spectrum of an oscillating signal does not represent the probability distribution of instantaneous frequency, so the linewidth measured using this spectrum is not the root mean square value of instantaneous frequency. The relationship between Fourier frequencies in the instantaneous frequency domain is more complex and microsecond.  
The variation of instantaneous frequency over time can be obtained from the spectrum. However, a curve of instantaneous frequency variation over time alone cannot provide all the information about the changes.  

Measure instantaneous frequency
The instantaneous frequency of electronic signals (such as beats) can be obtained by a phase-locked loop (PLL), which includes a voltage controlled oscillator (VCO) and a phase detector in the feedback system to synchronize the VCO with the incident signal. The incident signal of VCO can measure instantaneous frequency. The principle of this scheme can also be applied to phase tracker software to measure the instantaneous frequency of the recorded signal. This solution is simple but also has some drawbacks, especially its limited bandwidth and delayed response. The fast Fourier transform method is more powerful, but it can also be more complex.