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).

Contents 1 Sampling theorem satisfaction 2 Upsampling process 2.1 Upsampling by integer factor 2.2 Upsampling by rational fraction 3 See also 4 References

[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 f(k) on a radian frequency digital frequency range.

[edit] Upsampling by integer factor

Let L denote the upsampling factor.

1. Add L-1 zeros between each sample in f(k). Or, equivalently define
2. Filter with a low-pass filter which, theoretically, should be the sinc filter with frequency cut off at

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.

1. Upsample by a factor of L
2. 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 M < L and the single low-pass filter should have cutoff at .

[edit] See also

• Downsampling
• Oversampling
• Sampling (information theory)
• Signal (information theory)
• Data conversion

[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
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
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