# What's the difference between frequency shift, frequency offset, phase offset, and phase noise?

I'm confused about these terms: frequency shift, frequency offset, phase offset, and phase noise. My understanding that frequency shift and frequency offset are the same and caused by Doppler shift. Phase noise is caused by the instability of the local oscillator, and it changes per symbol. However, some books say this is nonrandom but unknown, I don't know what this means. I have no idea what phase offset is, and what causes it.

To complement Dan's answer, this is the way I usually use these terms.

• Frequency shift: a change in the frequency of a bandpass signal. It can be intentional, as when downconverting a signal in the receiver; or it can be unintentional, as when introduced by a Doppler effect.

• Frequency offset: the difference between two carriers that ought to (ideally) have the same frequency. For example, a receiver and a transceiver both set to 2.4 GHz will actually produce carriers with slightly different frequency (they have a frequency offset).

• Phase offset: similar to frequency offset, but regarding the oscillator's phase. In other words, the carriers $$c_1(t) = \textrm{exp}(2\pi f_0 t)$$ and $$c_2(t) = \textrm{exp}(2\pi (f_0+\Delta_f)t + \phi)$$ have a frequency offset of $$\Delta_f$$ and a phase offset of $$\phi$$. This is caused by the fact that two oscillators, even if matched perfectly in frequency, start operating at random times and in consequence the sine waves they produce have different phases.

Caution: some people define phase offset in a more general way that also includes frequency offset; that is, $$c_1(t) = \textrm{exp}(\varphi_1(t))$$ and $$c_2(t) = \textrm{exp}(\varphi_2(t))$$ would have phase offset $$\varphi_1(t) - \varphi_2(t)$$.

• Phase noise: your definition agrees with mine, and as Dan says, it is usually modeled as random, though it is neither uniform nor gaussian.

• Thanks, so both phase offset and phase noise are caused by the oscillator? How to include the phase noise in the formula? If it $c(t)=\exp\left(2\pi\left(f+\Delta_f\right)t+\phi+\varphi(t)\right)$, where $\phi$ is the phase offset, which is constant, while $\varphi(t)$ is the phase noise and it's random with time? Mar 10, 2023 at 11:33
• @Math_Novice Your interpretation is correct. I'd just add that the phase offset $\phi$ is the phase relative to a pre-defined reference and thus it is not really "caused" by the oscillator. Also, it can slowly drift over time, due to the chrystal aging, or to changes in the voltage or temperature.
– MBaz
Mar 10, 2023 at 18:35

Frequency shift and frequency offset could reasonably refer to the same thing, but a shift suggests something that has moved (from one frequency to the next as would occur in a changing Doppler) while a frequency offset suggests something more static such as a difference in frequency from a frequency reference after a shift has occurred.

Phase noise is clearly a random process, any book that says otherwise is wrong, or the statement as written is misinterpreted.

Modulations can change the phase, frequency or amplitude of a carrier frequency, and similarly unintentional noise and disturbances can modulate the carrier as well (in addition to additive noise sources that sum in power with the modulated signal). If we change the phase, that is "PM"; if we change the frequency, that is "FM"; and if we change the amplitude, that is "AM". AM and FM/PM are completely independent of each other but FM and PM are directly related. Instantaneous frequency is specifically the time derivative of phase (a change in phase versus a change in time):

$$f(t) = \frac{d\phi(t)}{dt}$$

If phase was in radians, the units of frequency would be radians/sec, or we can divide the result by $$2\pi$$ to have frequency in units of Hz. So if we change phase with time, we are also changing the frequency. A frequency offset therefore would be a phase ramp in time. Many confuse delay with phase given its explanation using sine waves, but that can be very misleading. A time delay will cause a phase offset for a specific frequency, and more specifically a phase that is directly proportional to carrier frequency (given a fixed time delay, a low frequency will have a much smaller phase shift than a higher frequency). Ultimately viewing waveforms as complex signals (a single tone at $$e^{j\omega t}$$ rather than sinusoids $$\cos(\omega t)$$) removes this confusion and provides much clarity of phase and frequency offsets and phase noise. Note as given by Euler's formula a sinusoid consists of two such tones:

$$\cos(\omega t) = e^{j\omega t} + e^{-j\omega t}$$

And if it wasn't clear the expression $$Ke^{j\phi}$$ is a complex phasor with magnitude $$K$$ and angle $$\phi$$ where $$K$$ and $$\phi$$ are both real.

$$e^{j\omega t}$$ is a spinning phasor and frequency is its rate of rotation (like a bicycle wheel, and we then have the notion of "positive" and "negative" frequencies since the rotation can now have direction). Phase is simply a rotation on the complex plane, and a Phase Offset is a static rotation. Phase noise is the random fluctuation of phase with time and is non-stationary.

Here are some graphical examples to clarify these different impairments (frequency and phase offsets, phase noise). The first graphic shows a waveform of phase versus time $$\phi(t)$$ as a noise free signal (the reference clean signal of what we actually want, which I chose arbitrarily as a fixed slope), and how it is received with the different impairments. The noise free signal here in this example is a fixed slope in phase versus time representing a constant frequency. This could for example be one symbol in a simple FSK (Frequency Shift Keying) transmission. The signal as received had a frequency shift which could have been due to Doppler if the transmitter or receiver were moving, or a frequency offset between the transmitter and receiver reference clocks (in practice these are both the primary contributors).

If our objective is to measure the frequency over an observation time (such as one symbol duration in FSK) then we would be interested in the average frequency which would be the best fit linear slope of phase over that duration of time. The difference of the two slopes (ideal vs actual) would be the frequency offset. The phase offset is the difference in phase at any point in time, typically given as a difference from the linear slopes: So if a constant frequency offset exists, the phase offset is linearly changing with time. The instantaneous deviations of phase from the noise free signal is phase noise. All oscillators have phase noise. Many of the sources of phase noise and its characteristics are captured in Leeson's equation.

The graphic below shows a QAM (Quadrature Amplitude Modulation) receiver with the effects of different noise impairments: AWGN, Phase Noise, Frequency and Phase Offset and how it would appear on the QAM constellation:

On the far left the noise free constellation is shown as symbols on a complex plane, here for 16-QAM. AWGN (Additive White Gaussian Noise) is added, and not shown but this would make each of those dots become circular clouds since the noise would be equally distributed on the real and imaginary axis. Phase noise adds noise in phase only, so stretches these circular clouds along the angular axis. A static phase offset (not shown by itself) would rotate the constellation by a fixed amount, while a frequency offset (which we do see) would cause the constellation to spin. Further of interest, the carrier tracking loop which is depicted would be able to track out the slower frequency variations of the random phase noise and thus reduce its impact (as we see in the constellation after carrier tracking). Finally a decision block converts the noise symbols to the decided (noise-free) result.

Other common impairments not shown are DC offsets which would move the origin, and amplitude and phase imbalance (also called IQ imbalance) as shown in the constellations below for 16-QAM with AWGN:

• Thank you for the explanation. I was talking about the above terms as impairments. Do I understand that phase offset is not an impairment, only the phase noise is? And if the phase noise is random, why some methods send a clock reference signal to compensate it? Mar 9, 2023 at 12:15
• You would need to carefully define “impairment” as all of these will effect the ability of the receiver to demodulate. It is very straightforward to compensate for phase and frequency offset. As far as phase noise if I randomly generate a phase offset in the transmitter, and then tell you what that offset is, it is still random. Phase noise is spread across frequency and the lowest frequency portions of phase noise (phase that fluctuates very slowly with time, randomly, is easy to compensate for even if we don’t send a reference (carrier tracking loops will do this— Mar 9, 2023 at 13:49
• similar to the feasibility of us to drive down a road even if it was being paved randomly ahead of us)), while the portions of phase noise that change faster cannot be easily corrected for. Mar 9, 2023 at 13:51