# Improving Generalized Discrete Fourier Transform (GDFT) Filter Banks with Low-Complexity and Reconfigurable Hybrid Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Proposed Hybrid Generalized Discrete Fourier Transform (HGDFT-FB)

#### 2.1.1. Proposed Design Steps

- Normalise all the channel bandwidths (BWs), such that the $B{W}_{i}$ and transition bandwidth $TB{W}_{i}$ specifications range from 0 to 1; 1 corresponds to $\frac{{f}_{s}}{2}$, where ${f}_{s}$ is the sampling frequency.
- Calculate each channel stopband frequency, such that ${\omega}_{si}=\frac{B{W}_{i}}{2}$.
- Calculate the modal bandwidth such that $B{W}_{Modal}$ = GCD$(B{W}_{1}^{\prime},B{W}_{2}^{\prime},B{W}_{3}^{\prime})$/2. This corresponds to the modal stopband frequency.
- Calculate the decimation factor M of the masking filter using the formula M=$\frac{\pi}{{\omega}_{ms}}$. The interpolated factor is calculated using the formula $L=\lfloor \frac{\pi}{{\omega}_{ms}}\rfloor $, where ${s}_{i}$ is the stopband frequency for each channel. Thus, the fractional rate for the masking filter can be calculated as $\frac{{L}_{ma}}{{M}_{ma}}$.
- Calculate the decimator factor of the complementary filter using the formula $M=\frac{\pi}{\pi +{\omega}_{mcs}}$. The interpolated factor is calculated using the formula $L=\lfloor \frac{\pi}{\pi +{\omega}_{mcs}}\rfloor $. Thus, the fractional rate for complementary filter can be calculated as $\frac{{L}_{mc}}{{M}_{mc}}$.
- Determine the transition bandwidth for the masking and complementary filters, $tbwi$, such that $tb{w}_{k}^{\prime}$ = $tb{w}_{k}\times \frac{{L}_{k}}{{M}_{k}}$, where $\frac{{L}_{k}}{{M}_{k}}$ is the fractional rate for masking or complementary filter.
- Determine the base modal or complementary modal TBW as$tb{w}_{modal}$ = min$(tb{w}_{1}^{\prime},tb{w}_{2}^{\prime},.....,tb{w}_{n}^{\prime})$. This corresponds to the modal transition width.
- Calculate the modal, masking, and complementary passband widths using ${\omega}_{p}$ = ${\omega}_{s}-TB{W}_{modal}$.
- Determine the passband edge and stopband edge of the modal or prototype filter, masking, and complementary filter, using Table 2, where m = $\lfloor \frac{{\omega}_{mp}\frac{{L}_{ma}}{{M}_{ma}}}{2pi}\rfloor $ for the masking filter and m = $\lceil \frac{{\omega}_{ms}\frac{{L}_{mc}}{{M}_{mc}}}{2pi}\rceil $ for the complementary filter.
- Find the stopband ripple using ${\delta}_{s1}^{\prime}$ = ${\delta}_{s1}\frac{{L}_{i}}{{M}_{i}}$.
- The modal passband peak ripple is calculated as: ${\delta}_{pmodal}$ = min(${\delta}_{p1}^{\prime},{\delta}_{p2}^{\prime},...,{\delta}_{pn}^{\prime}$).
- The cutoff frequency of the prototype and masking filter are calculated using Table 2.
- Determine the prototype filter order and the individual channel filter order using the Bellanger formula N = $\frac{-2lo{g}_{10}\left({\delta}_{p}^{\prime}{\delta}_{s}^{\prime}\right)}{3{\Delta}_{TBW}}}-1$ [58].

#### 2.2. Improving HGDFT with Parallel Distributed Arithmetic-Based Residual Number System (PDA-RNS)

#### 2.2.1. Application of HGDFT with PDA-RNS Filter Bank to Non-Uniform Channels: BT, ZIGBEE, WCDMA

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Diagram depicting hybrid generalized discrete Fourier transform (HGDFT) Channelisation Algorithm.

Channelisation Technique | Computational Load | Reconfigurability |
---|---|---|

Per Channel [1,2,3] | Very High | Bad |

Binary Approach [4] | High | Bad |

DFT [5] | Very High | Bad |

DFT (poly-phase) [6,7,8,9] | High | Bad |

EMFB [10,11] | Low | Bad |

FPCC [12,13,14,15] | Low | Bad |

QMFB [16,17] | High | Bad |

TQMB [18] | High | Good |

GDFT [5] | High | Good |

HTQMB [19,20] | High | Good |

PGDFT [10,21] | High | Good |

RGDFT [10,22] | High | Good |

FRM [22,53] | High | Good |

CDM 1 and 11 [23,24,25,26] | Low | Bad |

MCD 1 and 11 [23,24,25,26] | Low | Bad |

ICDM 1 and 11 | Low | Bad |

Parameter | Case 1 | Case 2 |
---|---|---|

${\mathrm{H}}_{a}\left(z\right)$ | ${\theta}_{a}=2m\pi -{\omega}_{s}L/M$ | ${\theta}_{a}={\omega}_{s}L/M-2\pi m$ |

${\varphi}_{a}=2m\pi -{\omega}_{p}L/M$ | ${\varphi}_{a}={\omega}_{p}L/M-2m\pi $ | |

${\mathrm{H}}_{ma}\left(z\right)$ | ${\omega}_{mp}$=$\frac{2\pi (m+1)-{\varphi}_{a}}{L/M}$ | ${\omega}_{mp}$=$\frac{2\pi m-{\varphi}_{a}}{L/M}$ |

${\omega}_{ms}$=$\frac{2\pi m+{\varphi}_{a}}{L/M}$ | ${\omega}_{ms}$=$\frac{2\pi m-{\varphi}_{a}}{L/M}$ | |

${\mathrm{H}}_{mc}\left(z\right)$ | ${\omega}_{mcp}$=$\frac{\left(2\pi \right)(m+1)+{\varphi}_{a}}{L/M}$ | ${\omega}_{mcp}$=$\frac{2\pi m-{\varphi}_{a}}{L/M}$ |

${\omega}_{mcs}$=$\frac{2\pi m-{\varphi}_{a}}{L/M}$ | ${\omega}_{mcs}$=$\frac{2\pi m-{\varphi}_{a}}{L/M}$ |

Filter Bank | $\frac{{L}}{{M}}$ |
Stopband Frequency (${\mathit{\omega}}_{\mathit{ms}}$) |
Passband Frequency (${\mathit{\omega}}_{\mathit{mp}}$) |
Stopband Attenuation (${\mathit{\delta}}_{\mathit{ms}}$) |
Passband Ripples (${\mathit{\delta}}_{\mathit{mp}}$) |
Filter Length |
---|---|---|---|---|---|---|

Modal filter, ${H}_{a}$ | $\frac{39}{40}$ | 0.025 | 0.022625 | 0.1 | −50 | 196 |

Bluetooth, ${H}_{ma}$ | $\frac{39}{40}$ | 0.025 | 0.0224 | 0.0975 | −39 | 159 |

Zigbee, ${H}_{ma}$ | $\frac{9}{10}$ | 0.1 | 0.089 | 0.09 | −39 | 34 |

WCDMA, ${H}_{ma}$ | $\frac{7}{8}$ | 0.2 | 0.125 | 0.0875 | −48.25 | 13 |

Filter Specification | Sampling Frequency ${\mathit{F}}_{\mathit{s}}$ (MHz) | Channel Bandwidth (MHz) | Transition Bandwidth (kHz) | Passband Ripples (dB) | Stopband Ripples (dB) |
---|---|---|---|---|---|

Bluetooth | 40 | 1 | 50 | 0.1 | −40 |

Zigbee | 40 | 4 | 200 | 0.1 | −40 |

WCDMA | 40 | 5 | 500 | 0.1 | −55 |

**Table 5.**Frequency characteristics of complementary masking filter implemented using the HGDFT filter bank.

Filter Bank | $\frac{\mathit{L}}{\mathit{M}}$ | Passband Frequency (${\mathit{\omega}}_{\mathit{mcp}}$) | Stopband Frequency (${\mathit{\omega}}_{\mathit{mcs}}$) | Stopband Attenuation (${\mathit{\delta}}_{\mathit{mcs}}$) | Passband Ripples (${\mathit{\delta}}_{\mathit{mcp}}$) | Filter Length |
---|---|---|---|---|---|---|

Modal filter, ${H}_{a}$ | $\frac{8}{9}$ | 0.027307 | 0.02269 | 0.1 | −50 | 209 |

Bluetooth, ${H}_{mc}$ | $\frac{8}{9}$ | 0.027307 | 0.02269 | 0.092 | −36.92 | 150 |

Zigbee, ${H}_{mc}$ | $\frac{8}{9}$ | 0.1080 | 0.0911 | 0.088 | −35.5 | 37 |

WCDMA, ${H}_{mc}$ | $\frac{7}{8}$ | 0.2 | 0.125 | 0.0875 | −48.25 | 13 |

Filter Bank | Filter Order | Total Number of Multiplication | ||
---|---|---|---|---|

${\mathit{H}}_{\mathit{a}}$ | ${\mathit{H}}_{\mathit{ma}}$ | ${\mathit{H}}_{\mathit{mc}}$ | ||

Modal filter | 405 | - | - | 320 |

BT | - | 159 | 150 | 156 |

Zigbee | - | 34 | 37 | 37 |

WCDMA | - | 13 | 13 | 13 |

Filter Bank | Filter Order | Total Number of Multiplication | ||
---|---|---|---|---|

${\mathit{f}}_{\mathit{a}}$ | ${\mathit{f}}_{\mathit{ma}}$ | ${\mathit{f}}_{\mathit{mc}}$ | ||

CDFB [54] | 3089 | 400 | - | 1745 |

ICDM FB [53] | 2929 | 160 | - | 1545 |

NU-MDFT FB [55] | 187 | 430 | 469 | 1090 |

Proposed HGDFT filter bank Bank | 320 | 104 | 102 | 526 |

Filter | Input Bits | Passband Filter PB Ripples(dB) | Stopband Attenuation | No of Adders | Amp-Litude Distor-Tion |
---|---|---|---|---|---|

Prototype filter | 8-bits | 3.360 | 11.204 | 0.20 | |

12-bits | 0.087 | 48.146 | 391 | 0.205 | |

16-bits | 0.09 | 48.146 | 0.19 | ||

Bluetooth | 8-bits | 3.457 | 10.532 | 0.05 | |

12-bits | 0.094 | 42.818 | 150 | 0.054 | |

16-bits | 0.022 | 43.062 | 0.047 | ||

Zigbee filter | 8-bits | 0.089 | 39.095 | 0.164 | |

12-bits | 0.088 | 39.094 | 34 | 0.164 | |

16-bits | 0.089 | 39.094 | 0.164 | ||

WCDMA | 8-bits | 0.146 | 43.638 | 0.28 | |

12-bits | 0.068 | 50.314 | 13 | 0.276 | |

16-bits | 0.068 | 50.34 | 0.27 |

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**MDPI and ACS Style**

Otunniyi, T.O.; Myburgh, H.C.
Improving Generalized Discrete Fourier Transform (GDFT) Filter Banks with Low-Complexity and Reconfigurable Hybrid Algorithm. *Digital* **2021**, *1*, 1-17.
https://doi.org/10.3390/digital1010001

**AMA Style**

Otunniyi TO, Myburgh HC.
Improving Generalized Discrete Fourier Transform (GDFT) Filter Banks with Low-Complexity and Reconfigurable Hybrid Algorithm. *Digital*. 2021; 1(1):1-17.
https://doi.org/10.3390/digital1010001

**Chicago/Turabian Style**

Otunniyi, Temidayo O., and Hermanus C. Myburgh.
2021. "Improving Generalized Discrete Fourier Transform (GDFT) Filter Banks with Low-Complexity and Reconfigurable Hybrid Algorithm" *Digital* 1, no. 1: 1-17.
https://doi.org/10.3390/digital1010001