You can instateously realize that the Phase Noise is "Noise in terms of phase". Then what is Phase. Again, the mathematical definition of phase just come from high school math as shown below.

This page first builds the idea of phase noise from a single phase shift, and then shows the problems it causes in a receiver. After that it shows how phase noise is specified as L(f) in dBc/Hz, and how that curve turns into an rms phase error and an EVM figure.

• What is phase noise ?
• Why phase noise is a problem ?
• How is phase noise specified ?
• How does phase noise turn into phase error and EVM ?

What is phase noise ?

Let's start from the sine wave itself, because phase noise is a property of its phase term and nothing else. The amplitude and the nominal frequency can be perfect while the phase still moves. So we first look at what a phase change does to a signal in time and in frequency.

How would a signal changes when we changes the phase of the signal. Following plots would give you the answer. The red plot and blue plot has the same frequency and amplitude and the only difference is the phase. In plot (c), you would clearly see the difference. If you see a signal in time domain, the phase difference would result in the delay or advance of a signal.

Now let's think about how a signal would vary with phase in frequency domain. Look at the following three plots. The red plot (a) and blue plot (b) has the same frequency and amplitude and the only difference is the phase. I already mentioned the time domain difference (left column), now let's look at the frequency domain difference (the center column). If you just focus on the peak point you would not find any obvious difference between the blue signal and red signal. But if you look closely into the region right next to the peak point, you would notice pretty outstanding difference between the blue signal and red signal.

Phase noise is a kind of noise of unexpected (unwanted) phase shift of a signal. In the description above, I compare only two signals but in reality the phase of the noised signal keep changing within a ceratain range. That's why we call it a noise. (If the phase get shifted to a certain value and remain the same all the time as in the example shown above, we would not call it a noise. it is just a phase shift which can easily corrected. I used a case with only two phase just for easy explanation).

One detail of the plots above needs care. For an ideal sine wave of infinite length, a constant phase offset does not change the magnitude spectrum at all. It changes only the phase of the spectrum, as the right hand column shows. The difference next to the peak in the center column comes from the finite length of the plotted records. What creates real sidebands is a phase that changes with time. A phase that moves randomly spreads the power of the carrier into the frequencies next to it, and that spread is the skirt we call phase noise.

In mathematical form, the noisy signal is sin(2π f t + φ(t)), where φ(t) is a small random process. For a small φ(t), its spectrum is the carrier line plus sidebands whose shape follows the spectrum of φ(t). A slowly moving φ(t) puts power close to the carrier, and a fast moving φ(t) puts power further away.

• Phase noise is a random, time varying phase : the signal is sin(2π f t + φ(t)) with a small random φ(t).
• A constant phase offset is not phase noise : it is a fixed shift that a receiver corrects once.
• The random phase shows up as a skirt around the carrier : the shape of the skirt follows the spectrum of φ(t).

Why phase noise is a problem ?

Why phase noise become a problem ?  Let's look at a couple of problems which is caused by the phase noise and you will get some idea on why we want to avoid the phase noise.

The top row in the following plots shows a kind of idea signal which does not have any noise (not even phase noise). Look at the domain and frequency domain and I just assume that the signal is QAM modulated (If it is really QAM modulated, the time domain and frequency domain should be a little bit different, but let's just assume that everything is right).

Then suppose that the signal is suffering from phase noise. If you plot the noised signal in time domain, you would get a plot shown in the second row. One thing you need to notice is that the frequency response. If a signal has phase noise, you would see the spreadind skirts at the bottom of the signal peak as shown below. And that phase noise would appear in the constellation as shown on the right side. You would see as if the constellation gets rotated.

Let's look at another problemetic situation caused by phase noise. Let's assume that your local oscillator in the reciever chain has the ideal frequency response as shown in (a) and gets under phase noise as shown in (d). Now this reciever chain is receiving a signal (wanted signal) and an interferer (unwanted signal) as shown in (b), (e). The signal and interferer goes through a down converter and problem starts here. When the local oscialltor signal does not have any phase noise, you can remove the most of interferer after the downconverter (e.g, by using filter) since they are separated from each other as shown in (c). But when your local oscillator has phase noise and the (signal + interferer) is downconverted by the noised local oscillator, the result would become as shown in (f). Since both signal and interferer frequency response got spreaded by the phase noise of local oscillator and the inteferer frequency response is overlapping the signal frequency response. In this case, it would be almost impossible to completely remove the interferer in analog stage (e.g, RF or IF stage). This is why the phase noise property of a local oscillator is so important.

• The left column uses an ideal local oscillator at f_c, drawn as a single line in (a). After the Down Converter, the wanted signal and the interferer stay separate at f_IF in (c).
• The right column uses a local oscillator with phase noise, drawn with a skirt in (d). The same pair of signals in (e) comes out in (f) with both signals spread, and the red interferer covers the green wanted signal.

This effect is called reciprocal mixing, and it can be estimated with one line of arithmetic. The interferer mixes with the phase noise skirt of the local oscillator. So the noise it lands in the wanted channel is the interferer power, plus L(f) at the frequency offset between the two signals, plus 10 log of the channel bandwidth. Let's take an interferer 40 dB above the wanted signal, an L(f) of -120 dBc/Hz at that offset, and a 1 MHz channel. The skirt then contributes -120 + 60 = -60 dB relative to the interferer, which is 40 - 60 = -20 dB relative to the wanted signal. So the SNR of the wanted signal cannot exceed 20 dB, however clean the wanted signal is.

There is another case where the phase noise become an issue. Let's suppose following situation. In this case, a very clean signal with a weak power is coming into the reciever and at the same time a strong signal with interferer is coming into the reciever at the same time, in that case even the signal (wanted signal) is clean it may be burried under the phase noise of nearby signal and will not be properly decoded.

The case above is the transmit side of the same problem. The strong signal carries its own phase noise skirt, and a weak signal next to it falls under that skirt before it ever reaches the receiver. No receiver design can remove this noise, because it is already inside the wanted channel. That is why transmitters also carry limits on the noise they put into adjacent channels.

• Phase noise rotates and smears the constellation : every symbol gets a random phase error, which raises the EVM.
• Reciprocal mixing moves an interferer into the wanted channel : the leaked power is the interferer power + L(f) + 10 log(BW).
• A strong neighbor can bury a weak signal under its own skirt : this noise is inside the wanted channel, so no filter removes it.

How is phase noise specified ?

To compare oscillators we need one number, or one curve, that describes the skirt. The industry uses the single sideband phase noise L(f). It is the noise power in a 1 Hz bandwidth at an offset f from the carrier, relative to the carrier power, and its unit is dBc/Hz.

L(f) is always given together with its offset. A datasheet line such as -100 dBc/Hz at 10 kHz means that, 10 kHz away from the carrier, the noise in each 1 Hz is 100 dB below the carrier. A single value without an offset says nothing, because the skirt falls steeply as the offset grows.

We can read L(f) from a spectrum analyzer in three steps. The first step measures the carrier power. The second step measures the noise power at the chosen offset in the resolution bandwidth, RBW. The third step normalizes the noise to 1 Hz by subtracting 10 log(RBW). For example, take a carrier at 0 dBm and a noise reading of -70 dBm at 10 kHz offset with RBW = 1 kHz. Then L(10 kHz) = -70 - 0 - 30 = -100 dBc/Hz. A noise marker on the analyzer does the normalization directly, and a dedicated phase noise analyzer measures the whole curve.

A typical synthesizer curve has two main parts. Inside the loop bandwidth of the PLL, the reference and the loop set the phase noise, and the curve is often nearly flat. Outside the loop bandwidth, the free running VCO sets the phase noise, and the curve falls at about 20 dB per decade. Figure 1 in the next section uses a curve of this shape.

• L(f) is noise in 1 Hz at an offset f, relative to the carrier : its unit is dBc/Hz, and it is meaningless without the offset.
• From a spectrum analyzer, L(f) = noise - carrier - 10 log(RBW) : -70 dBm noise in 1 kHz RBW next to a 0 dBm carrier is -100 dBc/Hz.
• The PLL shapes the curve : flat inside the loop bandwidth, and falling about 20 dB per decade outside it.

How does phase noise turn into phase error and EVM ?

A receiver does not see L(f) at one offset. It sees the total phase jitter of the local oscillator over the offsets its demodulator is sensitive to. So we integrate the curve over that range and convert the result into an rms phase error, which maps directly to EVM.

The integration works on linear values. Convert L(f) from dBc/Hz to a ratio, integrate it over the offset range from f₁ to f₂, and double the result, because L(f) describes one sideband and the phase error comes from both. The rms phase error in radians is the square root of that sum: σ_φ = sqrt(2 x integral of 10^L(f)/10 df).

Figure 1. Integrated phase noise of an example synthesizer. The flat part inside the loop bandwidth contributes about half of the total, so lowering the in-band floor pays off more than improving the far out skirt.

• The curve is an example profile, not a measurement. It is flat at -95 dBc/Hz from 1 kHz to 100 kHz, and it falls 20 dB per decade to -135 dBc/Hz at 10 MHz.
• The flat part from 1 kHz to 100 kHz integrates to 3.13 x 10^-5. The part from 100 kHz to 1 MHz adds 2.85 x 10^-5, and the part from 1 MHz to 10 MHz adds only 2.85 x 10^-6.
• The total is 6.26 x 10^-5, or -42.0 dBc. Doubling it and taking the square root gives σ_φ = 0.0112 rad, which is 0.64 deg rms.

For small errors, the EVM caused by phase noise alone is close to σ_φ in radians. So 0.0112 rad gives an EVM of about 1.1 %. This is the reason RF engineers quote integrated phase noise in deg rms next to the L(f) curve. Higher order modulation has closer constellation points and so needs a smaller σ_φ.

In OFDM the phase error splits into two parts. The slow part of φ(t), below the subcarrier spacing, rotates all subcarriers of one symbol by the same angle. This is the common phase error, and the receiver can estimate and remove it. The fast part leaks power from each subcarrier into its neighbors as inter-carrier interference, which acts like noise. A PLL that multiplies its reference by N also raises the phase noise by 20 log(N) dB. So the phase noise grows with the carrier frequency, and the problem is largest in the mmWave bands. This is one reason 5G NR defines the phase tracking reference signal, PT-RS, which lets the UE and the gNB track the common phase error.

• Integrate L(f) in linear units and double it : σ_φ = sqrt(2 x integral of 10^L(f)/10 df) in radians.
• EVM from phase noise is about σ_φ in radians : 0.64 deg rms, or 0.0112 rad, gives about 1.1 % EVM.
• The flat in-band floor usually dominates the total : in the example it contributes about half of the integrated value.
• In OFDM, slow phase noise is common phase error and fast phase noise is ICI : PT-RS in 5G NR tracks the common part.