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 and transition bandwidth specifications range from 0 to 1; 1 corresponds to , where is the sampling frequency.
- Calculate each channel stopband frequency, such that .
- Calculate the modal bandwidth such that = GCD/2. This corresponds to the modal stopband frequency.
- Calculate the decimation factor M of the masking filter using the formula M=. The interpolated factor is calculated using the formula , where is the stopband frequency for each channel. Thus, the fractional rate for the masking filter can be calculated as .
- Calculate the decimator factor of the complementary filter using the formula . The interpolated factor is calculated using the formula . Thus, the fractional rate for complementary filter can be calculated as .
- Determine the transition bandwidth for the masking and complementary filters, , such that = , where is the fractional rate for masking or complementary filter.
- Determine the base modal or complementary modal TBW as= min. This corresponds to the modal transition width.
- Calculate the modal, masking, and complementary passband widths using = .
- Determine the passband edge and stopband edge of the modal or prototype filter, masking, and complementary filter, using Table 2, where m = for the masking filter and m = for the complementary filter.
- Find the stopband ripple using = .
- The modal passband peak ripple is calculated as: = min().
- 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 = [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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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 |
---|---|---|
= | = | |
= | = | |
= | = | |
= | = |
Filter Bank |
Stopband Frequency () |
Passband Frequency () |
Stopband Attenuation () |
Passband Ripples () |
Filter Length | |
---|---|---|---|---|---|---|
Modal filter, | 0.025 | 0.022625 | 0.1 | −50 | 196 | |
Bluetooth, | 0.025 | 0.0224 | 0.0975 | −39 | 159 | |
Zigbee, | 0.1 | 0.089 | 0.09 | −39 | 34 | |
WCDMA, | 0.2 | 0.125 | 0.0875 | −48.25 | 13 |
Filter Specification | Sampling Frequency (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 |
Filter Bank | Passband Frequency () | Stopband Frequency () | Stopband Attenuation () | Passband Ripples () | Filter Length | |
---|---|---|---|---|---|---|
Modal filter, | 0.027307 | 0.02269 | 0.1 | −50 | 209 | |
Bluetooth, | 0.027307 | 0.02269 | 0.092 | −36.92 | 150 | |
Zigbee, | 0.1080 | 0.0911 | 0.088 | −35.5 | 37 | |
WCDMA, | 0.2 | 0.125 | 0.0875 | −48.25 | 13 |
Filter Bank | Filter Order | Total Number of Multiplication | ||
---|---|---|---|---|
Modal filter | 405 | - | - | 320 |
BT | - | 159 | 150 | 156 |
Zigbee | - | 34 | 37 | 37 |
WCDMA | - | 13 | 13 | 13 |
Filter Bank | Filter Order | Total Number of Multiplication | ||
---|---|---|---|---|
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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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
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 StyleOtunniyi, 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