FFT-Based Simultaneous Calculations of Very Long Signal Multi-Resolution Spectra for Ultra-Wideband Digital Radio Frequency Receiver and Other Digital Sensor Applications
The discrete Fourier transform (DFT) is the most commonly used signal processing method in modern digital sensor design for signal study and analysis. It is often implemented in hardware, such as a field programmable gate array (FPGA), using the fast Fourier transform (FFT) algorithm. The frequency...
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| Format: | Article |
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MDPI AG
2024-02-01
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| Series: | Sensors |
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| Online Access: | https://www.mdpi.com/1424-8220/24/4/1207 |
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| author | Chen Wu Michael Low |
| author_facet | Chen Wu Michael Low |
| author_sort | Chen Wu |
| collection | DOAJ |
| description | The discrete Fourier transform (DFT) is the most commonly used signal processing method in modern digital sensor design for signal study and analysis. It is often implemented in hardware, such as a field programmable gate array (FPGA), using the fast Fourier transform (FFT) algorithm. The frequency resolution (i.e., frequency bin size) is determined by the number of time samples used in the DFT, when the digital sensor’s bandwidth is fixed. One can vary the sensitivity of a radio frequency receiver by changing the number of time samples used in the DFT. As the number of samples increases, the frequency bin width decreases, and the digital receiver sensitivity increases. In some applications, it is useful to compute an ensemble of FFT lengths; e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mi>P</mi><mo>−</mo><mi>j</mi></mrow></msup></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><mi>J</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>j</mi></semantics></math></inline-formula> is defined as the spectrum level with frequency resolution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo> </mo><msup><mn>2</mn><mi>j</mi></msup><mo>·</mo><mo>Δ</mo><mi>f</mi></mrow></semantics></math></inline-formula>. Here <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mi>f</mi></mrow></semantics></math></inline-formula> is the frequency resolution at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. However, calculating all of these spectra one by one using the conventional FFT method would be prohibitively time-consuming, even on a modern FPGA. This is especially true for large values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>P</mi></semantics></math></inline-formula>; e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>≥</mo><mn>20</mn></mrow></semantics></math></inline-formula>. The goal of this communication is to introduce a new method that can produce multi-resolution spectrum lines corresponding to sample lengths <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo> </mo><msup><mn>2</mn><mrow><mi>P</mi><mo>−</mo><mi>j</mi></mrow></msup></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> levels, concurrently, while one long <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mi>P</mi></msup></mrow></semantics></math></inline-formula>-length FFT is being calculated. That is, the lower resolution spectra are generated naturally as by-products during the computation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mi>P</mi></msup></mrow></semantics></math></inline-formula>-length FFT, so there is no need to perform additional calculations in order to obtain them. |
| first_indexed | 2024-03-07T22:15:07Z |
| format | Article |
| id | doaj.art-941f85426f40477abdf833a1cf4b4331 |
| institution | Directory Open Access Journal |
| issn | 1424-8220 |
| language | English |
| last_indexed | 2024-03-07T22:15:07Z |
| publishDate | 2024-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Sensors |
| spelling | doaj.art-941f85426f40477abdf833a1cf4b43312024-02-23T15:33:50ZengMDPI AGSensors1424-82202024-02-01244120710.3390/s24041207FFT-Based Simultaneous Calculations of Very Long Signal Multi-Resolution Spectra for Ultra-Wideband Digital Radio Frequency Receiver and Other Digital Sensor ApplicationsChen Wu0Michael Low1Defence Research and Development Canada—Ottawa Research Centre, Ottawa, ON K1A 0Z4, CanadaDefence Research and Development Canada—Ottawa Research Centre, Ottawa, ON K1A 0Z4, CanadaThe discrete Fourier transform (DFT) is the most commonly used signal processing method in modern digital sensor design for signal study and analysis. It is often implemented in hardware, such as a field programmable gate array (FPGA), using the fast Fourier transform (FFT) algorithm. The frequency resolution (i.e., frequency bin size) is determined by the number of time samples used in the DFT, when the digital sensor’s bandwidth is fixed. One can vary the sensitivity of a radio frequency receiver by changing the number of time samples used in the DFT. As the number of samples increases, the frequency bin width decreases, and the digital receiver sensitivity increases. In some applications, it is useful to compute an ensemble of FFT lengths; e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mi>P</mi><mo>−</mo><mi>j</mi></mrow></msup></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><mi>J</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>j</mi></semantics></math></inline-formula> is defined as the spectrum level with frequency resolution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo> </mo><msup><mn>2</mn><mi>j</mi></msup><mo>·</mo><mo>Δ</mo><mi>f</mi></mrow></semantics></math></inline-formula>. Here <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mi>f</mi></mrow></semantics></math></inline-formula> is the frequency resolution at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. However, calculating all of these spectra one by one using the conventional FFT method would be prohibitively time-consuming, even on a modern FPGA. This is especially true for large values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>P</mi></semantics></math></inline-formula>; e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>≥</mo><mn>20</mn></mrow></semantics></math></inline-formula>. The goal of this communication is to introduce a new method that can produce multi-resolution spectrum lines corresponding to sample lengths <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo> </mo><msup><mn>2</mn><mrow><mi>P</mi><mo>−</mo><mi>j</mi></mrow></msup></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> levels, concurrently, while one long <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mi>P</mi></msup></mrow></semantics></math></inline-formula>-length FFT is being calculated. That is, the lower resolution spectra are generated naturally as by-products during the computation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mi>P</mi></msup></mrow></semantics></math></inline-formula>-length FFT, so there is no need to perform additional calculations in order to obtain them.https://www.mdpi.com/1424-8220/24/4/1207digital receiverdiscrete Fourier transformfast Fourier transformmulti-resolution spectrumblocking fast Fourier transformsignal processing |
| spellingShingle | Chen Wu Michael Low FFT-Based Simultaneous Calculations of Very Long Signal Multi-Resolution Spectra for Ultra-Wideband Digital Radio Frequency Receiver and Other Digital Sensor Applications Sensors digital receiver discrete Fourier transform fast Fourier transform multi-resolution spectrum blocking fast Fourier transform signal processing |
| title | FFT-Based Simultaneous Calculations of Very Long Signal Multi-Resolution Spectra for Ultra-Wideband Digital Radio Frequency Receiver and Other Digital Sensor Applications |
| title_full | FFT-Based Simultaneous Calculations of Very Long Signal Multi-Resolution Spectra for Ultra-Wideband Digital Radio Frequency Receiver and Other Digital Sensor Applications |
| title_fullStr | FFT-Based Simultaneous Calculations of Very Long Signal Multi-Resolution Spectra for Ultra-Wideband Digital Radio Frequency Receiver and Other Digital Sensor Applications |
| title_full_unstemmed | FFT-Based Simultaneous Calculations of Very Long Signal Multi-Resolution Spectra for Ultra-Wideband Digital Radio Frequency Receiver and Other Digital Sensor Applications |
| title_short | FFT-Based Simultaneous Calculations of Very Long Signal Multi-Resolution Spectra for Ultra-Wideband Digital Radio Frequency Receiver and Other Digital Sensor Applications |
| title_sort | fft based simultaneous calculations of very long signal multi resolution spectra for ultra wideband digital radio frequency receiver and other digital sensor applications |
| topic | digital receiver discrete Fourier transform fast Fourier transform multi-resolution spectrum blocking fast Fourier transform signal processing |
| url | https://www.mdpi.com/1424-8220/24/4/1207 |
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