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TRS 2023 | Spurs in Millimeter-Wave FMCW Radar System-on-Chip

毫米波雷达论文阅读: TRS 2023 | Spurs in Millimeter-Wave FMCW Radar System-on-Chip

论文链接：https://ieeexplore.ieee.org/document/10097751

0 Abstract

• 研究内容

  • present a nonlinear system model to evaluate the impact of circuit nonlinearities on spurs in millimeter-wave FMCW radar system-on-chips

  • The developed model includes:

    ✅ Harmonics of the frequency multiplier (频率乘法器的谐波)

    ✅ Nonlinearity of the power amplifier (PA) and low-noise amplifier (LNA) (功率放大器和低噪声放大器的非线性度)

    ✅ Switching operation of the receiver (RX) mixer (接收机混频器的切换操作)

    ✅ Limited bandwidth of the PA, LNA, transmitter (TX) and RX antennas (PA, LNA, 发射接收天线的有效带宽)

    ✅ TX-to-RX leakage (发射接收泄漏)

• 意义

  • The nonlinear model can be used to derive frequency and amplitude of spurs in the radar IF spectrum
  • without time-consuming simulations

• Insights : The major insights about the impact of different nonlinearities and their interactions

  • Harmonics of the frequency multiplier appear as spurs in the IF spectrum
  • PA can be driven in its nonlinear region to mitigate the harmonics generated by the frequency multiplier
  • Bandwidth of the TX and RX systems should be limited to attenuate the undesired harmonics(衰减不希望出现的谐波)
  • Interaction between the TX-to-RX leakage signal and the LNA nonlinearity can lead to spurs close to the target echo in the IF spectrum

1 Introduction

• 背景: Millimeter-wave high-resolution radars的重要性

  • Automotive sensors (ADAS, autonomous driving)
  • Medical imaging, vital signs monitoring, gesture recognition
  • IoT sensors, smart buildings, industrial transport, robotics

• 背景: mm-wave radar SoC 的最新进展

  • Implemented in CMOS, SOI, SiGe processes
  • Operating at 60 GHz, 77 GHz, 76-81 GHz, 140 GHz, 200-300 GHz
  • Achieving high range resolution by using higher mm-wave bands and wide bandwidth
  • Achieving high angular resolution using MIMO architecture

• 问题：广泛使用的FMCW雷达系统存在Spurs

  • Signal generation can be challenging due to requirements on:

    ✅ Bandwidth

    ✅ Phase noise

    ✅ Jitter

    ✅ Spurs

  • 前三种已经有解决方案：Nonlinear effects such as chirp nonlinearity can be mitigated using error correction or calibration techniques

  • Spurs: Spurs are undesired signals in the radar IF spectrum that can be generated by

    ❗ Harmonics of the frequency multiplier (因为low harmonic rejection ration(HRR) of the frequency multiplier)

    ❗ Nonlinearities of the power amplifier (PA) and low-noise amplifier (LNA) (通过 affect the harmonics level)

    ❗ TX-to-RX leakage (通过 affect the harmonics level)

  • 结论：一个全面的非线性模型对于推导 对 FMCW雷达电路组件的有效电路和系统设计 的要求 至关重要

• This paper ： presents a nonlinear system model for mm-wave FMCW radars

  • includes several important effects

    ✅ harmonics of the frequency multiplier

    ✅ nonlinearity of the PA, LNA

    ✅ the switching operation of the RX mixer

    ✅ the limited bandwidth of the TX and RX

    ✅ the stop-band rejections of the TX, RX, and IF filter

    ✅ the TX-to-RX leakage

  • To evaluate circuit nonlinearities and their interactions

  • 作用：

    ✅ The developed model can be used to estimate the frequency and amplitude of spurs in the radar IF spectrum

    ✅ without the need for time-consuming system and circuit simulations

2 Non-Linear System Model of FMCW Radar

2.1 Principles of FMCW Radar

• Architecture of a mm-wave FMCW radar system:

  • Reference chirp signal is generated at lower frequency (e.g. 10 GHz)

  • Then Frequency multiplier transforms it to mm-wave band

    ✅ 好处 ：useful to achieve lower phase noise in the reference signal

  • Multiplier output signal is amplified by PA (功率放大器) and transmitted

  • Echo signal is received, amplified by LNA (低噪声放大器) and mixed with replica of multiplier output

  • Mixer output signal passes through IF bandpass filter (BPF)

• Reference chirp signal

  • Linear time-dependent frequency:

    ✅ $f_{\mathrm{ref}}(t)=f_0+St0\le t\le T_c$

    ✅ $T_c$: chirp period

  • Instantaneous phase:

    ✅ ${\Phi }_{\mathrm{ref}}(t)=2\pi \int f_{\mathrm{ref}}(t)dt={\Phi }_0+2\pi f_{0}t+\pi S{t}^2$

• Frequency multiplier ideally generates

  • Chirp signal with frequency $N{f}_{ref}(t)$
  • Bandwidth of $NB$
  • Slope of $NS$
  • Period of $T_c$

• Received signal related to transmitted signal

  • $x_{\mathrm{RX}}(t)=G_R{x}_{\mathrm{TX}}\left(t-{\tau }_R\right)$,
  • GR: attenuation factor – $G_R={\left(\frac{G_{\mathrm{A},\mathrm{TX}}{G}_{\mathrm{A},\mathrm{RX}}{L}_{\mathrm{sys}}\sigma {\lambda }^2}{(4\pi {)}^3{R}^4}\right)}^{\frac{1}{2}}$
  • ${\tau }_R$: time-of-flight – ${\tau }_{R_0}=\frac{2R_0}{c}$

• Mixer + BPF output signal

  • instantaneous phase: ${\Phi }_{\mathrm{IF}}(t)=N{\Phi }_{\mathrm{ref}}(t)-N{\Phi }_{\mathrm{ref}}\left(t-{\tau }_R(t)\right)$

  • $\Rightarrow$ 大致表示一个正弦信号, frequency:

    ✅ $f_{\mathrm{IF}}=(NS){\tau }_{R_0}+f_D=\left(\frac{2NS}{c}\right)R_0+\left(\frac{2N{f}_0}{c}\right)v$

  • 2D FFT to get Range and Velocity

2.2 FMCW Radar Nonlinear Model

A nonlinear system model which properly captures the circuit nonlinearities and interactions of different nonlinear effects with each other can be very valuable for efficient design of a mm-wave FMCW radar SoC.

2.2.1 Important nonlinear effects in FMCW radar system model:

• Output harmonics of frequency multiplier

  • Neglect spurs from reference circuits like PLL or DDS

  • Harmonics level depends on:

    ✅ Multiplier circuit structure

    ✅ Frequency band

    ✅ Chirp bandwidth

    ✅ Quality factor of passive components

    ✅ IC process features

  • The output of the frequency multiplier

    ✅ $x_{\mathrm{out},\mathrm{MUL}}(t)={\sum }_{k=1}^{\infty }{a}_k\cos \left[k{\Phi }_{\mathrm{ref}}(t)\right]$

• Nonlinearity of PA

  • Can change harmonics’ relative amplitude
  • Especially at low supply voltages

• Bandwidth of PA and TX

  • Bandwidth of PA: how output harmonics of the frequency multiplier reach to the TX antenna

    ✅ 受负载阻抗(用来最大化输出功率/efficiency)的限制

  • Bandwidth of Tx

    ✅ three possible scenarios:

    ✅ 本文：假设为情况(b)

  • 谐波$k$的相对衰减系数：

    ✅ $A_{\mathrm{TX},\mathrm{k}}=\frac{|H_{\mathrm{TX}}\left(jk{\omega }_0\right)|}{|H_{\mathrm{TX}}\left(jN{\omega }_0\right)|}$,

    ✅ 决定因素：信号分量之间的频率差、PA电路中使用的片上无源元件的质量因子，以及TX天线带宽。

• TX-to-RX leakage signal

  • 产生原因 ：天线之间的耦合 + 芯片基板的耦合

  • 后果 ：desensitizes the RX circuits (特别是LNA) $\Rightarrow$ 掩盖目标

  • 解决 ：目前的leakage cancellation已经可以做到 30~50dB

    ✅ still limits system performance

  • The received signal: $x_{\mathrm{RX}}(t)=G_R{x}_{\mathrm{TX}}\left(t-{\tau }_R\right)+G_L{x}_{\mathrm{TX}}\left(t-{\tau }_L\right)$,

    ✅ leakage的后果can be different for short and long target ranges

• Bandwidth of LNA and RX

  • 意义：决定了能够到达混频器的谐波信号

  • LNA bandwidth:

    ✅ modeld by two transfer functions – $H_{\text{in, LNA }}(j\omega )$ and $H_{\text{out,LNA }}(j\omega )$

    ✅ 分别对应 输入和输出匹配网络的 frequency responses

    ✅ 带宽限制因素： input matching network (负责提供optimum noise matching)

  • RX bandwidth:

    ✅ 仍使用上图(b) (the same as for the TX)

  • The relative attenuation of the harmonic $k$:

    ✅ $A_{\mathrm{RX},\mathrm{k}}=\frac{|H_{\mathrm{RX}}\left(jk{\omega }_0\right)|}{|H_{\mathrm{RX}}\left(jN{\omega }_0\right)|}$.

• Switching operation of RX mixer

  • Generates intermodulation distortion (互调失真, IMD)
  • Mixes 参考信号和LNA输出中的不同谐波 $\Rightarrow$ 在IF上生成spurs

• Bandwidth and attenuation of IF filter

  • 通带纹波 的影响
  • Fig. 3(a)的情况： distort the chirp signal $\Rightarrow$ undesired amplitude and phase modulations

2.2.2 Model Limitations

存在Limitations的原因

• 模型考虑了 Most important effects for accurate radar performance evaluation
• Higher order imperfections excluded for clarity
• Apply approximations to keep model simple yet insightful

Limitations

• Frequency multiplier

  • Can have large amplitude variations of fundamentals/harmonics across bandwidth
  • 此时，不能用 constant HRR (harmonic rejection ratio) 进行描述
  • 可使用 average/worst-case HRR

• Largest harmonics of multiplier

  • May not be adjacent harmonics N±1
  • Attenuated by PA, LNA, TX/RX antennas

• PA/LNA nonlinearity

  • Can significantly change across bandwidth
  • Frequency-dependent $P_{1dB}$ makes analysis intractable
  • 解决：Use average P1dB for accuracy
  • 接下来的分析会假设 leakage does not saturate LNA

• Reference signal spurs neglected

  • 本文仅Focus on spurs generated by radar
  • Reference信号中的Spurs: PLL spurs via mechanisms like quantization noise

3 Analysis of FMCW Radar System

利用FMCW雷达的非线性系统模型,研究以下信号的spectral contents:

• TX输出
• RX输入
• IF

3.1 Analysis of Transmitter

• Frequency multiplier output signal

  • $x_{\mathrm{out},\mathrm{MUL}}(t)=a_{\mathrm{N}}\left[\cos (N\Phi )+k_{\mathrm{ref}}\cos [(N\pm 1)\Phi ]\right]$,
  • Desired harmonic N
  • Two adjacent harmonics N±1
  • $k_{\mathrm{ref}}$ related to HRR of multiplier

• PA nonlinearity model output signal:

  • 3rd order polynomial: $N\Phi ,3N\Phi ,(N\pm 1)\Phi ,(N\pm 2)\Phi ,(N\pm 3)\Phi ,(3N\pm 1)\Phi ,(3N\pm 2)\Phi ,(3N\pm 3)\Phi$,
  • Generates spectral regrowth
  • Output spectrum 3 times wider than input

• TX

  • bandwidth allows N and N±1 harmonics
  • Relative attenuation of N±1: ATX
  • 即 $A_{\mathrm{TX},\mathrm{N}\pm 1}=A_{\mathrm{TX}}$
  • (严格来说，TX system是一个动态非线性系统；本文，简化为一个 frequency-independent nonlinear model followed by a linear band-limited frequency response)

• TX output signal derived using PA model

  • $x_{\mathrm{TX}}(t)=a_{\mathrm{TX}}\left[\cos (N\Phi )+k_{\mathrm{TX}}\cos [(N\pm 1)\Phi ]\right]$,
  • 其中：$a_{\mathrm{TX}}={\alpha }_{1,\mathrm{PA}}{A}_{\mathrm{in}}+\frac{3}{4}\left(1+6k_{\mathrm{ref}}^2\right){\alpha }_{3,\mathrm{PA}}{A}_{\mathrm{in}}^3$, $k_{\mathrm{TX}}=\left(\frac{{\alpha }_{1,\mathrm{PA}}{A}_{\mathrm{in}}+\frac{9}{4}\left(1+k_{\mathrm{ref}}^2\right){\alpha }_{3,\mathrm{PA}}{A}_{\mathrm{in}}^3}{{\alpha }_{1,\mathrm{PA}}{A}_{\mathrm{in}}+\frac{3}{4}\left(1+6k_{\mathrm{ref}}^2\right){\alpha }_{3,\mathrm{PA}}{A}_{\mathrm{in}}^3}\right)A_{\mathrm{TX}}{k}_{\mathrm{ref}}$.
  • Includes fundamental and harmonic components
  • Harmonics level $k_{TX}<k_{ref}$ $\Rightarrow$ 意味着 TX输出时，谐波水平降低
  • 即：在非线性区域驱动PA，以减轻频率乘法器产生的谐波 + 提高PA效率 是有益的。

• Key insights:

  • PA nonlinearity decreases harmonics level

    🚩 Beneficial to drive PA in nonlinear region

  • TX bandwidth should limit harmonics

    🚩 Filter can further suppress harmonics

3.2 Analysis of Receiver

• Received signal :

  • Composed of target echo and TX-RX leakage

• Without leakage signal :

  • LNA output signal derived using LNA nonlinearity model: $\begin{array}{rl}& x_{\mathrm{out},\mathrm{LNA}}(t)\\ & =a_{\mathrm{out},\mathrm{LNA}}\left[\cos \left[N\Phi \left(t-{\tau }_R\right)\right]+k_{\mathrm{RX}}\cos \left[(N\pm 1)\Phi \left(t-{\tau }_R\right)\right]\right],\end{array}$

  • IF signal
    ✅ $x_{\mathrm{IF}}(t)=a_{\mathrm{IF}}\left[\cos \left[N{\Psi }_1(t)\right]+k_{\mathrm{ref}}{k}_{\mathrm{RX}}\cos \left[(N\pm 1){\Psi }_1(t)\right]\right]$

    ✅ ${\Psi }_1(t)=\Phi (t)-\Phi \left(t-{\tau }_R\right)$

    🚩 结论1：

• Key insights:

  • 结论1：中频处的激励电平低于参考信号的原始谐波电平 (IF spurs level < reference harmonics level)
    ✅ $k_{\mathrm{IF}}\approx \left(\frac{1-3\beta \frac{P_{\mathrm{in},\mathrm{LNA}}}{P_{\mathrm{ldB},\mathrm{LNA}}}}{1-\beta \frac{P_{\mathrm{in},\mathrm{LNA}}}{P_{\mathrm{ldB},\mathrm{LNA}}}}\right)\left(\frac{1-3\beta \frac{P_{\mathrm{in},\mathrm{PA}}}{P_{\mathrm{ldB},\mathrm{PA}}}}{1-\beta \frac{P_{\mathrm{in},\mathrm{PA}}}{P_{\mathrm{ldB},\mathrm{PA}}}}\right)A_{\mathrm{RX}}{A}_{\mathrm{TX}}{k}_{\mathrm{ref}}^2$.

  • 结论2：中频包含了三个频率分量：$f_{\mathrm{IF}1}=NS{\tau }_R$, $f_{\mathrm{IF}2}=(N+1)S{\tau }_R$, $f_{\mathrm{IF}3}=(N+2)S{\tau }_R$

    ✅ Spurs can cause overlap between IF spectra $\Rightarrow$ 降低雷达距离分辨率

---

• With leakage signal:

  • 相当于LNA有两个输入：$x_{\text{in,LNA }}(t)=x_{\text{echo }}(t)+x_{\text{leak }}(t)$,
  • Leakage and LNA nonlinearity interaction generates extra spurs
  • LNA output signal includes spectral components related to leakage： $\begin{array}{rl}& x_{\mathrm{out},\mathrm{LNA}}(t)\\ & G_R{a}_{\mathrm{TX}}[C_1\cos \left[N\Phi \left(t-{\tau }_R\right)\right]\\ & +C_2\cos \left[(N\pm 1)\Phi \left(t-{\tau }_R\right)\right]\\ & +C_3\cos \left[N\Phi \left(t-{\tau }_L\right)\right]+C_4\cos \left[(N\pm 1)\Phi \left(t-{\tau }_L\right)\right]\\ & +C_5\cos \left[2N\Phi \left(t-{\tau }_R\right)-N\Phi \left(t-{\tau }_L\right)\right]\\ & +C_6\cos \left[N\Phi \left(t-{\tau }_R\right)-2N\Phi \left(t-{\tau }_L\right)\right]],\end{array}$
  • IF signal ：$\begin{array}{rl}{x}_{\mathrm{IF}}(t)\approx & a_{\mathrm{IF}}[C_1\cos \left[N{\Psi }_1(t)\right]+C_2{k}_{\mathrm{ref}}\cos \left[(N\pm 1){\Psi }_1(t)\right]\\ & +C_3\cos \left[N{\Psi }_2(t)\right]+C_4{k}_{\mathrm{ref}}\cos \left[(N\pm 1){\Psi }_2(t)\right]\\ & +C_5\cos \left[N{\Psi }_3(t)\right]+C_6\cos \left[N{\Psi }_4(t)\right]]\end{array}$

• The Spurs level in the IF:

  • $\begin{array}{c}{k}_{\mathrm{IF}}\approx \left[\frac{1+r^2-\beta \left(3+2r^2\right)\frac{P_{\mathrm{in},\mathrm{LNA}}}{P_{1\mathrm{dB},\mathrm{LNA}}}}{1+r^2-\beta \left(1+2r^2\right)\frac{P_{\mathrm{in},\mathrm{LNA}}}{P_{1\mathrm{dB},\mathrm{LNA}}}}\right]\left[\frac{1-3\beta \frac{P_{\mathrm{in},\mathrm{PA}}}{P_{\mathrm{ddB},\mathrm{PA}}}}{1-\beta \frac{P_{\mathrm{in},\mathrm{PA}}}{P_{1\mathrm{dB},\mathrm{PA}}}}\right]\\ \times A_{\mathrm{RX}}{A}_{\mathrm{TX}}{k}_{\mathrm{ref}}^2.\end{array}$

  • 取决于：

    ✅ Reference harmonics

    ✅ Leakage signal power

    ✅ PA/LNA nonlinearity

    ✅ TX/RX bandwidth

    ✅ IF filter attenuation

  • 当 PA和LNA工作在线性功率范围时 （即$P_{\mathrm{in},\mathrm{PA}}\ll P_{1\mathrm{dB},\mathrm{PA}}$ and $P_{\mathrm{in},\mathrm{LNA}}\ll P_{1\mathrm{dB},\mathrm{LNA}}$）：

    ✅ $k_{\mathrm{IF}}\approx A_{\mathrm{RX}}{A}_{\mathrm{TX}}{k}_{\mathrm{ref}}^2$.

    🚩 $\Rightarrow$ PA和LNA的非线性有利于降低谐波 （因为上式中的前两项总是小于1）

• Leakage signal

  • Introduces new undesired components

  • 如下图Fig 6所示

  • 该理论能够指导：当给定相对泄露功率时，所需要的IF BPF阻带衰减

    ✅ $A_{IF,L/H}$ 需使得 $k_{\mathrm{IF},\mathrm{II}}<k_{\mathrm{IF}}$ & $k_{\mathrm{IF},\mathrm{IV}}<k_{\mathrm{IF}}$

3.3 Radar Dynamic Range

• Radar dynamic range (DR) :

  • $D{R}_{noise}$ : Difference between RX signal power and noise floor

  • $D{R}_{spur}$ (DR = $D{R}_{spur}$, 如果spur功率高于噪声功率): Difference between signal power and maximum spur level

• DR can be maximized by:

  • Limiting largest spur below noise floor

  • $D{R}_{spur}=D{R}_{noise}$

• $D{R}_{spur}$ can be derived as:

  • $-20{\log }_{10}\left(k_{\mathrm{IF},\max }\right)=P_{\mathrm{RX},\max }+174-NF-10{\log }_{10}\left(B_n\right)$,

  • $P_{RX,max}$: Maximum RX input power

  • $NF$: Noise figure

  • $B_n$: Noise bandwidth

  • $k_{\mathrm{IF},\max }$: Maximum spur level in IF

• 上述对DR的限制能够作为约束计算如下参数：

  • Frequency multiplier harmonics

  • TX-RX leakage power

  • PA/LNA nonlinearity

4 Simulations and Discussions

4.1 Simulation Setup

• 140 GHz chirp signal, 5 GHz bandwidth
• ×8 frequency multiplier (17.5G reference signal) with -20 dBc harmonics
• ADS and MATLAB used for TX/RX circuit modeling
• Target: at 10 m range

4.2 Radar IF Spectrum

• Compared theory and simulations for various nonlinearities

• Impact of circuit nonlinearities on radar IF spectrum

• Impact of leakage signal:
  • ≤0.5 dB difference up to 0 dB leakage power
  • Discrepancy increases to 2-3 dB at higher leakage
  • Extra spur observed besides two harmonic spurs
  • Amplitude matches theory
  • Location depends on leakage delay

Developed model provides accurate results

4.3 Radar Range-Doppler Diagram

• Shows probability of false alarm due to spurs
• Reduced with PA/LNA nonlinearity
• Leakage signal spur increases false alarm

• Developed theory provides:
  • Guidelines for IF filter attenuation
  • Required harmonic/leakage/nonlinearity levels

5 Conclusion

• 本文为FMCW毫米波系统的分析提供了一个非线性模型
• 模型考虑了收发机的多个组件带来的影响
• 能够预测雷达IF中Spurs的频率和幅度，为电路设计提供参考