Upsampling
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Upsampling is the process of increasing the sampling rate of a signal.
The upsampling factor (commonly denoted by L) is usually an integer or a rational fraction greater than unity. This factor multiplies the sampling rate or, equivalently, divides the sampling period. For example, if compact disc audio is upsampled by a factor of 5/4 then the resulting sampling rate goes from 44,100 Hz to 55,125 Hz, which increases the bit rate from 1,411,200 bit/s to 1,764,000 bit/s. The range of valid frequencies (i.e., those that satisfy the Nyquist-Shannon sampling theorem) has gone from 22,050 Hz to 27,562.5 (an increase in 5,512.5 Hz).
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[edit] Sampling theorem satisfaction
The upsampled signal satisfies the Nyquist-Shannon sampling theorem if the original signal does.
Unlike in downsampling which uses a low-pass filter as an anti-aliasing filter, upsampling uses an interpolation filter, which also is a low-pass filter.
[edit] Upsampling process
Consider a discrete signal <math>f(k)</math> on a radian frequency digital frequency range.
[edit] Upsampling by integer factor
Let L denote the upsampling factor.
- Add L-1 zeros between each sample in <math>f(k)</math>. Or, equivalently define <math>g(k) = \left \{ \begin{matrix} f\left(\frac{k}{L}\right) & \mbox{if } \frac{k}{L} \mbox{ is an integer} \\ 0 & \mbox{otherwise} \end{matrix} \right.</math>
- Filter with a low-pass filter which, theoretically, should be the sinc filter with frequency cut off at <math>\frac{\pi}{L}</math>
The second step calls for the use of a perfect low-pass filter, which is not implementable. When choosing a realizable low-pass filter this will have to be considered and it will have aliasing effects.
[edit] Upsampling by rational fraction
Let L/M denote the upsampling factor.
- Upsample by a factor of L
- Downsample by a factor of M
Note that upsampling requires an interpolation filter after increasing the data rate and that downsampling requires a filter before decimation. These two filters can be combined into a single filter. Since both interpolation and anti-aliasing filters are low-pass filters, the filter with the smallest bandwidth is more restrictive and, thus, can be used in place of both filters. Since the rational fraction L/M is greater than unity when <math>M < L</math> and the single low-pass filter should have cutoff at <math>\frac{\pi}{L}</math>.
[edit] See also
[edit] References
- Oppenheim, Alan V., Ronald W. Schafer, John R. Buck (1999). Discrete-Time Signal Processing, 2nd Edition, Prentice Hall. ISBN 0-13-754920-2.
| Digital Signal Processing |
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| Theory — Nyquist–Shannon sampling theorem, estimation theory, detection theory |
| Sub-fields — audio signal processing | control engineering | digital image processing | speech processing | statistical signal processing |
| Techniques — Discrete Fourier transform (DFT) | Discrete-time Fourier transform (DTFT) | bilinear transform | Z-transform, advanced Z-transform |
| Sampling — oversampling | undersampling | downsampling | upsampling | aliasing | anti-aliasing filter | sampling rate | Nyquist rate/frequency |
