Ferkans — Interactive Telecom Tutor

Phase Noise — The Silent Killer of OFDM Performance

Every local oscillator (LO) exhibits random phase fluctuations known as phase noise. In single-carrier systems, phase noise causes a slow constellation rotation that is easily tracked by a PLL. In OFDM, however, phase noise has a far more insidious effect: it destroys the orthogonality between subcarriers, creating inter-carrier interference (ICI) that cannot be removed by simple phase tracking. As 5G NR pushes to mmWave frequencies with wider bandwidths and higher-order modulation, phase noise requirements become extremely stringent. This section develops the analytical framework for understanding phase noise in OFDM systems and deriving the oscillator specifications needed for each numerology.

Definition:
Lorentzian Phase Noise Model

A free-running oscillator at carrier frequency $f_0$ produces phase noise $\phi(t)$ modelled as a Wiener process with two-sided Lorentzian power spectral density:

$S_\phi(f) = \frac{\beta_{3\text{dB}}/\pi}{f^2 + \beta_{3\text{dB}}^2}$

where $\beta_{3\text{dB}}$ is the 3 dB linewidth of the oscillator. The single-sideband phase noise at offset $\Delta f$ from the carrier (in dBc/Hz) is:

$\mathcal{L}(\Delta f) = 10\log_{10}\!\left( \frac{\beta_{3\text{dB}}/\pi}{\Delta f^2 + \beta_{3\text{dB}}^2}\right)$

For offsets $\Delta f \gg \beta_{3\text{dB}}$ (the typical operating region), this simplifies to the $1/f^2$ rolloff:

$\mathcal{L}(\Delta f) \approx 10\log_{10}\!\left( \frac{\beta_{3\text{dB}}}{\pi\,\Delta f^2}\right) \;\text{dBc/Hz}$

The Lorentzian model is the simplest physically motivated phase noise model, arising from white frequency noise in the oscillator. Real oscillators also exhibit $1/f^3$ noise (flicker FM) at small offsets, but the $1/f^2$ region dominates at the offsets relevant to OFDM subcarrier spacing.

Definition:
Common Phase Error and Inter-Carrier Interference

When an OFDM signal with $N$ subcarriers and subcarrier spacing $\Delta f_{\text{SCS}}$ is affected by phase noise $\phi(t)$ , the demodulated symbol on subcarrier $k$ becomes:

$Y_k = H_k X_k \cdot I_0 + \sum_{l \neq k} H_l X_l \cdot I_{k-l} + W_k$

where $I_m = \frac{1}{N}\sum_{n=0}^{N-1} e^{j\phi(nT_s)}\,e^{-j2\pi mn/N}$ are the DFT coefficients of $e^{j\phi(t)}$ sampled at the OFDM symbol rate.

Common Phase Error (CPE): $I_0 = \frac{1}{N}\sum_n e^{j\phi(nT_s)}$ is the DC component — a single complex scalar that rotates all subcarriers identically. CPE can be estimated and corrected using pilot subcarriers.
Inter-Carrier Interference (ICI): $\{I_m\}_{m \neq 0}$ spread energy from each subcarrier to its neighbours. ICI is a random, data-dependent interference that is much harder to mitigate.

The CPE/ICI partition depends on the ratio of phase noise bandwidth to subcarrier spacing. Wider SCS pushes more of the phase noise energy into the CPE term (easier to correct), which is why 5G NR uses wider SCS at higher frequencies.

Theorem: ICI Power from Phase Noise in OFDM

For an OFDM system with $N$ subcarriers and subcarrier spacing $\Delta f_{\text{SCS}}$ , operating with a Lorentzian phase noise source of 3 dB linewidth $\beta_{3\text{dB}}$ , the ICI power (normalised to signal power) is:

$\sigma_{\text{ICI}}^2 = 1 - e^{-2\pi\beta_{3\text{dB}} T_{\text{sym}}} \cdot \left(\frac{\sin(\pi\beta_{3\text{dB}} T_{\text{sym}})} {\pi\beta_{3\text{dB}} T_{\text{sym}}}\right)^2 \approx 2\pi\beta_{3\text{dB}} T_{\text{sym}}$

where $T_{\text{sym}} = 1/\Delta f_{\text{SCS}}$ is the useful OFDM symbol duration (excluding CP). The approximation holds when $\beta_{3\text{dB}} T_{\text{sym}} \ll 1$ .

The effective signal-to-ICI ratio is:

$\text{SIR}_{\text{ICI}} = \frac{1}{\sigma_{\text{ICI}}^2} \approx \frac{1}{2\pi\beta_{3\text{dB}} T_{\text{sym}}} = \frac{\Delta f_{\text{SCS}}}{2\pi\beta_{3\text{dB}}}$

Key insight: SIR improves linearly with subcarrier spacing. Doubling SCS doubles SIR (3 dB improvement).

Phase noise accumulates over the OFDM symbol duration. Longer symbols (smaller SCS) experience more phase drift within one symbol, causing greater ICI. This is why mmWave systems, which have worse phase noise, benefit from the wider SCS options (120, 240 kHz) in 5G NR.

Proof

Phase noise variance over one OFDM symbol

For a Wiener process with linewidth $\beta_{3\text{dB}}$ :

$\mathbb{E}[|\phi(t) - \phi(0)|^2] = 2\pi\beta_{3\text{dB}}\,|t|$

Over the symbol duration $T_{\text{sym}}$ :

$\sigma_{\phi}^2 = 2\pi\beta_{3\text{dB}}\, T_{\text{sym}}$

CPE and ICI decomposition

The total phase noise energy is partitioned:

$\mathbb{E}[|I_0|^2] + \sum_{m \neq 0}\mathbb{E}[|I_m|^2] = 1$

The CPE power is $|I_0|^2 \approx e^{-\sigma_\phi^2}$ (for small $\sigma_\phi^2$ ). Therefore:

$\sigma_{\text{ICI}}^2 = 1 - |I_0|^2 \approx 1 - e^{-2\pi\beta_{3\text{dB}} T_{\text{sym}}} \approx 2\pi\beta_{3\text{dB}} T_{\text{sym}}$

SIR scaling with SCS

$\text{SIR}_{\text{ICI}} = \frac{1}{\sigma_{\text{ICI}}^2} \approx \frac{1}{2\pi\beta_{3\text{dB}} T_{\text{sym}}} = \frac{\Delta f_{\text{SCS}}}{2\pi\beta_{3\text{dB}}}$ $Since$ T_{\text{sym}} = 1/\Delta f_{\text{SCS}} $, doubling the SCS halves the symbol duration and thus halves the ICI power.$ \blacksquare$

Phase Noise Decomposition — CPE vs ICI

Visualise how phase noise energy partitions into correctable common phase error (CPE) and residual inter-carrier interference (ICI) as the oscillator linewidth increases relative to the subcarrier spacing.

Green bars show CPE power (correctable via pilots); red bars show ICI power (residual distortion). As

\beta_{3\text{dB}} T_{\text{sym}}

increases, ICI dominates and system performance degrades.

Phase Noise Impact on OFDM

Investigate how phase noise degrades OFDM performance. The plot shows the received constellation and the effective SIR as a function of subcarrier index, decomposing the total degradation into CPE (correctable) and ICI (residual) components. Adjust the phase noise level, number of subcarriers, and subcarrier spacing to see how wider SCS mitigates the ICI component while the CPE remains easily trackable.

Parameters

Phase noise (dBc/Hz)-80

N subcarriers64

SCS (kHz)15

Example: Phase Noise Requirements for 5G NR Numerologies

A 5G NR system must support 256-QAM, requiring $\text{SIR}_{\text{ICI}} > 30$ dB from phase noise alone.

(a) For SCS = 15 kHz (sub-6 GHz), find the maximum tolerable oscillator linewidth $\beta_{3\text{dB}}$ . (b) Repeat for SCS = 120 kHz (FR2 mmWave). (c) A typical sub-6 GHz PLL achieves $\beta_{3\text{dB}} = 1$ Hz. What is the resulting SIR margin?

Solution

SIR requirement

(a) From $\text{SIR}_{\text{ICI}} = \Delta f_{\text{SCS}}/(2\pi\beta_{3\text{dB}})$ :

$10^{30/10} = 1000 = \frac{15000}{2\pi\beta_{3\text{dB}}}$

$\beta_{3\text{dB}} \leq \frac{15000}{2\pi \times 1000} = 2.39 \;\text{Hz}$

mmWave SCS

(b) For SCS = 120 kHz:

$\beta_{3\text{dB}} \leq \frac{120000}{2\pi \times 1000} = 19.1 \;\text{Hz}$

The 8 $\times$ wider SCS relaxes the phase noise requirement by a factor of 8 (9 dB), which is critical because mmWave oscillators inherently have worse phase noise.

Margin calculation

(c) With $\beta_{3\text{dB}} = 1$ Hz and SCS = 15 kHz:

$\text{SIR} = \frac{15000}{2\pi \times 1} = 2387 = 33.8 \;\text{dB}$

Margin above requirement: $33.8 - 30 = 3.8$ dB. This leaves room for other impairments (I/Q imbalance, quantisation noise) in the total EVM budget. $\blacksquare$

Quick Check

Why does 5G NR use wider subcarrier spacing (e.g., 120 kHz) at mmWave frequencies compared to sub-6 GHz bands (15 kHz)?

To increase the number of subcarriers and thus improve frequency diversity

To reduce the OFDM symbol duration, which mitigates phase noise ICI and suits shorter channel delay spreads at mmWave

To make the system backward-compatible with 4G LTE

To increase the peak data rate by using more bandwidth

Correction:

To reduce the OFDM symbol duration, which mitigates phase noise ICI and suits shorter channel delay spreads at mmWave

mmWave oscillators have worse phase noise, and ICI power scales as $2\pi\beta_{3\text{dB}}/\Delta f_{\text{SCS}}$ . Wider SCS reduces the symbol duration, limiting phase accumulation and ICI. Additionally, mmWave channels have shorter delay spreads, so the CP can be shorter (enabled by the shorter symbol).

Phase Noise

Random fluctuations in the phase of a local oscillator output, characterised by the single-sideband power spectral density $\mathcal{L}(\Delta f)$ in dBc/Hz. In OFDM, phase noise causes common phase error (CPE) and inter-carrier interference (ICI).

Lorentzian Spectrum

The power spectral density of a free-running oscillator: $S_\phi(f) = (\beta_{3\text{dB}}/\pi)/(f^2 + \beta_{3\text{dB}}^2)$ , where $\beta_{3\text{dB}}$ is the 3 dB linewidth. Exhibits $1/f^2$ rolloff at offsets beyond the linewidth.

Common Phase Error (CPE)

The DC component of the phase noise DFT coefficients in OFDM, representing a uniform phase rotation of all subcarriers within one symbol. CPE can be estimated and corrected using pilot subcarriers, unlike the residual ICI component.

Oscillators and Phase Noise