# Skew-Circulant-Matrix-Based Harmonic-Canceling Synthesizer for BIST Applications

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## Abstract

**:**

## 1. Introduction

## 2. Harmonic-Canceling Filter

#### 2.1. Constant-Amplitude HCF

#### 2.2. Constant-Delay HCF

## 3. Proposed SCM-Based HCF

#### 3.1. Matrix Representation of the HCF

#### 3.2. HCF with Multi-Stage Open-Loop SCM-Based Coefficient Generator

#### 3.3. HCF with Single-Stage Closed-Loop SCM-Based Coefficient Generator

#### 3.4. High-Order HCF

#### 3.5. Band-Pass HCF

## 4. Circuit Implementation

#### 4.1. System Architecture

#### 4.2. Frequency Divider

#### 4.3. Phase Scrambler

#### 4.4. Retimer and Buffer

#### 4.5. 24-Tap HCF Core

## 5. Measurement Results

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BIST | Built-In Self-Test |

HC | Harmonic Canceling |

HCF | Harmonic-Canceling Filter |

SCM | Skew-Circulant Matrix |

SFDR | Spurious-Free Dynamic Range |

DUT | Device-Under-Test |

ADC | Analog-to-Digital Converter |

THD | Total Harmonic Distortion |

BPF | Band-Pass Filter |

DDFS | Direct Digital Frequency Synthesizer |

P2AM | Phase-to-Amplitude Mapping |

CG | Coefficent Generator |

SW | Square-Wave |

CMOS | Complementary Metal-Oxide Semiconductor |

CM | Current-Mirror |

FD | Frequency Divider |

PS | Phase Scrambler |

R&B | Retimer and Buffer |

NR | Not Reported |

VCCS | Voltage-Controlled Current Source |

DEM | Dynamic Element Matching |

## Appendix A. Eigenvalues of the Even-Order SCM with Ideal (Irrational) HCF Coefficients

## Appendix B. Eigenvalues of the Even-Order SCM with Non-Ideal (Integer) HCF Coefficients

## Appendix C. Equivalence between a Cascade of Lower Order HCFs and a Higher Order HCF

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**Figure 2.**Different single-tone generators: (

**a**) BPF-based oscillator, (

**b**) DDFS, and (

**c**) Harmonic-Canceling synthesizer.

**Figure 4.**Constant–amplitude HCF: (

**a**) Block diagram and (

**b**) transfer function of most basic implementation, and (

**c**) block diagram and (

**d**) transfer function of HCF with rejection of 3rd and 5th harmonic and their odd multiples.

**Figure 5.**Constant–delay HCF: (

**a**) Block diagram and (

**b**) transfer function of the half-sine HCF; (

**c**) block diagram, (

**d**) transfer function, and (

**e**) implementation of the 4-tap sampled half-sine HCF.

**Figure 6.**(

**a**) Implementation of ideal HCF based on cascade of non-ideal SCMs, and (

**b**) normalized eigenvalues of M SCMs $\left[{\mathit{A}}_{\mathbf{6}}\right]$ in cascade, and (

**c**) improved implementation.

**Figure 7.**(

**a**) Block diagram of the closed-loop CG, and (

**b**) THD versus CG’s order n, for different M-stages open-loop CGs and single-stage closed-loop CG, with ${\parallel A\parallel}^{-1}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}8/5n$.

**Figure 9.**(

**a**) Comparison between impulse responses of the basic and band-pass HCFs, and (

**b**) impulse response of several band-pass HCFs for $n\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}8$.

**Figure 17.**(

**a**) Measured total power consumption of HCFs, (

**b**) power consumption per block of 24-tap HCF for $H\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$, and simulated (s) and measured (m) SFDR for (

**c**) $H\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$ and (

**d**) $H\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1$.

**Figure 18.**(

**a**) Transient waveform and (

**b**) PSD of HCF’s output for $H\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$, and (

**c**) Transient waveform and (

**d**) PSD of HCF’s output for $H\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1$.

Year | Tech. | ${\mathit{V}}_{\mathbf{DD}}$ (V) | Area (mm${}^{2}$) | Coefficient Generation | HCF Order | Bypassed Harmonic | ${\mathit{f}}_{\mathit{o}}$ (MHz) | SFDR/-THD * (dBc) | Power (mW) | FoM | |
---|---|---|---|---|---|---|---|---|---|---|---|

@ ${\mathit{f}}_{\mathit{o}}$ (MHz) | @ ${\mathit{f}}_{\mathit{o}}$ (MHz) | ||||||||||

This Work | 2022 | 180 nm CMOS | 1.8 | 0.505 | SCM-based | 6-tap | 1st | 0.8–60 | 66.4 @ 0.8 | 6.8 @ 0.8 | 1797 |

52.9 @ 60 | 19.1 @ 60 | ||||||||||

5th | 33–100 | 46.5 @ 33 | 6.1 @ 33 | ||||||||

38.4 @ 100 | 8.7 @ 100 | ||||||||||

12-tap | 1st | 0.8–32 | 64 @ 0.8 | 6.8 @ 0.8 | |||||||

53 @ 32 | 15.3 @ 32 | ||||||||||

5th | 8.3–75 | 43.7 @ 8.3 | 5.3 @ 8.3 | ||||||||

38.8 @ 75 | 8.7 @ 75 | ||||||||||

24-tap | 1st | 0.8–12.5 | 63.7 @ 0.8 | 6.9 @ 0.8 | |||||||

54.6 @ 12.5 | 13.3 @ 12.5 | ||||||||||

5th | 2–50 | 53.6 @ 2 | 5.1 @ 2 | ||||||||

46.2 @ 50 | 10.2 @ 50 | ||||||||||

[15] | 2019 | 28 nm FDSOI | NR | 0.011 | VCCS + calib. + LPF | 6-tap | 1st | 1–333 | 41.5 ${}^{\u2020}$ @ 166.67 | NR | - |

52 ${}^{\u2021}$ @ 166.67 | |||||||||||

[16] | 2017 | 130 nm CMOS | 1.2–1.5 | 0.056 | CM ratios | 12-tap | 1st | 0.01–1 | NR | 4 (single-tone) | - |

[14] | 2017 | 130 nm CMOS | 1.2–1.5 | 0.066 | Unit-current switches + DEM | 4-tap | 1st | 2 | 69 ${}^{\u2021}$ | 0.94 | 840 |

[13] | 2015 | 180 nm CMOS | 1.0–1.8 | 0.08 | Resistor-ratios + calibration + LPF | 6-tap | 1st | 150–850 | 50.5 ${}^{\u2020}$ @ 150 | ||

60.3 ${}^{\u2021}$ @ 150 | 9.1 @ 150 | 698 ${}^{\u2020}$ | |||||||||

47 ${}^{\u2020}$ @ 750 | 57.2 @ 850 | 6642 ${}^{\u2021}$ | |||||||||

70 ${}^{\u2021}$ @ 750 | |||||||||||

[12] | 2015 | 180 nm CMOS | 1.8 | 0.04 | Capacitor ratios + LPF | 8-tap | 1st | 1.11 | 77 * | 3.24 | 938 |

[10] | 2010 | 130 nm CMOS | 1.2 | 0.186 | N/A | N/A | 1st | 10 | 72 * | 4 | 716 |

^{†}: without calibration or DEM,

^{‡}: with calibration or DEM, *: -THD. VCCS: Voltage-controlled current source, DEM: Dynamic element matching.

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**MDPI and ACS Style**

Garayar-Leyva, G.G.; Osman, H.; Estrada-López, J.J.; Moreira-Tamayo, O.
Skew-Circulant-Matrix-Based Harmonic-Canceling Synthesizer for BIST Applications. *Sensors* **2022**, *22*, 2884.
https://doi.org/10.3390/s22082884

**AMA Style**

Garayar-Leyva GG, Osman H, Estrada-López JJ, Moreira-Tamayo O.
Skew-Circulant-Matrix-Based Harmonic-Canceling Synthesizer for BIST Applications. *Sensors*. 2022; 22(8):2884.
https://doi.org/10.3390/s22082884

**Chicago/Turabian Style**

Garayar-Leyva, Guillermo G., Hatem Osman, Johan J. Estrada-López, and Oscar Moreira-Tamayo.
2022. "Skew-Circulant-Matrix-Based Harmonic-Canceling Synthesizer for BIST Applications" *Sensors* 22, no. 8: 2884.
https://doi.org/10.3390/s22082884