File:Spectral views of zero-fill and interpolation by lowpass filtering.svg
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Captions
Summary edit
DescriptionSpectral views of zero-fill and interpolation by lowpass filtering.svg |
English: The first triangle of the first graph represents the Fourier transform X(f) of a continuous function x(t). The entirety of the first graph depicts the discrete-time Fourier transform of a sequence x[n] formed by sampling the continuous function x(t) at a low-rate of 1/T. The second graph depicts the application of a lowpass filter at a higher data-rate, implemented by inserting zero-valued samples between the original ones. And the third graph is the DTFT of the filter output. The bottom table expresses the maximum filter bandwidth in various frequency units used by filter design tools. |
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Date | ||||
Source | Own work. The first and second graphs correspond to the first and third graphs of Harris[1], Figure 2.12, except the upsample rates are different (3 and 5) | |||
Author | Bob K | |||
Permission (Reusing this file) |
I, the copyright holder of this work, hereby publish it under the following license:
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Other versions |
This file was derived from: Spectral views of zero-fill and interpolation by lowpass filtering.pdf |
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SVG development InfoField | This W3C-invalid vector image was created with LibreOffice. |
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References InfoField |
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Interpolation filter design edit
Let be the Fourier transform of any function, whose samples at some interval, equal the sequence. Then the discrete-time Fourier transform (DTFT) of the sequence is the Fourier series representation of a periodic summation of
When has units of seconds, has units of hertz (Hz). Sampling times faster (at interval ) increases the periodicity by a factor of
which is also the desired result of interpolation. An example of both these distributions is depicted in the first and third graphs.
When the additional samples are inserted zeros, they decrease the sample-interval to Omitting the zero-valued terms of the Fourier series, it can be written as:
which is equivalent to the first formula above, regardless of the value of What determines is the DTFT periodicity of a digital filter implemented at the higher data-rate. The second graph depicts a lowpass filter and resulting in the desired spectral distribution (third graph). The filter's bandwidth is the Nyquist frequency of the original sequence. In units of Hz that value is but filter design applications usually require normalized units.
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 22:36, 24 March 2024 | 1,047 × 922 (99 KB) | Bob K (talk | contribs) | Modified a label and a table heading. But I might have lost the dashed-lines representing the lowpass filter and its repetitions. | |
15:04, 5 January 2020 | 1,047 × 922 (104 KB) | Bob K (talk | contribs) | same frequency scale for all 3 graphs | ||
18:11, 3 January 2020 | 1,047 × 922 (105 KB) | Bob K (talk | contribs) | replacing ω character with an svg-friendly version | ||
17:08, 3 January 2020 | 1,047 × 922 (105 KB) | Bob K (talk | contribs) | trying to fix a π character that renders correctly in svg | ||
15:55, 3 January 2020 | 1,047 × 922 (105 KB) | Bob K (talk | contribs) | User created page with UploadWizard |
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Width | 295.4mm |
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Height | 260.35mm |