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Categories: Digital signal processing | Signal processing

Upsampling

Upsampling is the process of increasing the sampling rate of a signal. This is usually done to increase the bandwidth 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 was 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

Sampling theorem satisfaction

Since upsampling increases the bandwidth of the signal then the upsampled signal also 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.

Upsampling process

Consider a discrete signal $f (k)$ on a radian frequency digital frequency range.

Upsampling by integer factor

Let L denote the upsampling factor.

Add L-1 zeros between each sample in $f (k)$ . Or, equivalently define $g(k) = \left \{ \begin{matrix} f\left(\frac{k}{L}\right) & \mbox{if } \frac{k}{L} \mbox{ is an integer} \\ 0 & \mbox{otherwise} \end{matrix} \right.$
Filter with a low-pass filter which, theoretically, should be the sinc filter with frequency cut off at $\frac{\pi}{2L}$

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 aliasing effects it will have.

Upsampling by rational fraction

Let L/M denote the downsampling 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 then $M < L$ and the single low-pass filter should have cutoff at $\frac{\pi}{2L}$ .

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