The concept of a linewidth is borrowed from
laser spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
. The linewidth of a laser is a measure of its
phase noise
In signal processing, phase noise is the frequency-domain representation of random fluctuations in the phase of a waveform, corresponding to time-domain deviations from perfect periodicity (jitter). Generally speaking, radio-frequency engineers ...
. The spectrogram of a laser is produced by passing its light through a prism. The spectrogram of the output of a pure noise-free laser will consist of a single infinitely thin line. If the laser exhibits phase noise, the line will have non-zero width. The greater the phase noise, the wider the line. The same will be true with oscillators. The spectrum of the output of a noise-free oscillator has energy at each of the harmonics of the output signal, but the bandwidth of each harmonic will be zero. If the oscillator exhibits phase noise, the harmonics will not have zero bandwidth. The more phase noise the oscillator exhibits, the wider the bandwidth of each harmonic.
Phase noise
In signal processing, phase noise is the frequency-domain representation of random fluctuations in the phase of a waveform, corresponding to time-domain deviations from perfect periodicity (jitter). Generally speaking, radio-frequency engineers ...
is a noise in the phase of the signal. Consider the following noise free signal:
:''v''(''t'') = ''A''cos(2π''f''
0''t'').
Phase noise is added to this signal by adding a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
represented by φ to the signal as follows:
:''v''(''t'') = ''A''cos(2π''f''
0''t'' + φ(''t'')).
If the phase noise in an oscillator stems from
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
sources, then the power spectral density (PSD) of the phase noise produced by an oscillator will be ''S''
φ(''f'') = ''n''/''f''
2, where ''n'' specifies the amount of noise. The PSD of the output signal would then be
:
where ''n'' = 2''cf''
02.
Define the corner frequency ''f''
Δ = ''c''π ''f''
02 as the linewidth of the oscillator. Then
:
It is more common to report oscillator phase noise as ''L'', the ratio of the single-sideband
(SSB) phase noise power to the power in the fundamental (in dBc/Hz). In this case
:
Adding phase noise neither increases nor decreases the power of the signal. It simply redistributes the power by increasing the bandwidth over which the signal is present while decreasing the amplitude of the signal that occurs at the nominal oscillation frequency. The total noise power, as found by integrating the power spectral density over all frequencies, remains constant regardless of the amount of phase noise. This is illustrated in the figures on the right. It can be proven by integrating ''L'' over all frequencies to compute the total power of the signal.
:
See also
*
Laser linewidth
*
Spectral linewidth
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
''Introduction to RF Simulation and its Application'' by Ken Kundert
Oscillators
Stochastic processes
Statistical signal processing