Resolution increase

Increasing resolution is the process of increasing the sampling rate or increasing the number of pixels per unit length. The sampling rate is measured in Hz , and the resolution is in pixels per centimeter or dots per inch.

Images, such as high-quality photographs, are an example of high-resolution source data, but often you need to see the details of a small fragment of the image. In this case, methods for increasing resolution may be used.

If you want to play back sampled sound at a slower speed or to dub sound at a higher sampling rate, then an increase in resolution is also required.

The resolution increase factor (usually denoted by L) is an integer or rational number usually greater than 1. The sampling frequency is multiplied by this coefficient or, equivalently, the sampling period is divided. For example, if for sound from an audio CD, the resolution is increased with a factor of 5/4, then the final resolution changes from 44.100 Hz to 55.125 Hz.

Content

Fulfillment of the conditions of Kotelnikov's theorem

A signal with an increased resolution satisfies the Kotelnikov theorem if the original signal satisfies it.

Indeed, with increasing resolution, either the sampling rate increases, or the cutoff frequency of the signal decreases. In any of these cases, the ratio 2F _max <F _{d is} maintained.

To eliminate the effect of aliasing (aliasing) when changing the resolution, an interpolation filter is required, both with increasing and decreasing resolution. This is usually a high-quality low-pass filter.

Resolution Process

Resolution increase with coefficient L = 3

In the formulas below, we will consider the circular sampling frequency, measured in radians / seconds .

Let L be the coefficient of increase in resolution.

Add L-1 zeros between each pair of neighboring samples f (k) f (k + 1), which can be formally written as $g(k)=\left\{{\begin{matrix}f\left({\frac {k}{L}}\right)&{\mbox{if }}{\frac {k}{L}}{\mbox{ is an integer}}\\0&{\mbox{otherwise}}\end{matrix}}\right.$ ${\ displaystyle g (k) = \ left \ {{\ begin {matrix} f \ left ({\ frac {k} {L}} \ right) & {\ mbox {if}} {\ frac {k} { L}} {\ mbox {is an integer}} \\ 0 & {\ mbox {otherwise}} \ end {matrix}} \ right.}$
We filter the resulting sequence using a good low-pass filter. The filter should theoretically be a sinc filter (ideal filter) with a suppression frequency ${\frac {\pi }{L}}$ ${\ displaystyle {\ frac {\ pi} {L}}}$ .

The second stage involves the use of an ideal low-pass filter, which is an impossible requirement. When choosing a realizable low-pass filter, overlapping effects (aliasing) will occur. These effects can be greatly reduced by properly designing the FIR filter. The presence of zeros in the sequence passing through the filter can be used to reduce the complexity of filter implementation. The source filter can be divided into L subfilters, each of which is used sequentially to obtain a filtered output sequence.

Increase Resolution with a Rational Ratio

Let L / M be a rational resolution increase factor. The algorithm for increasing resolution in this case is as follows:

Resolution increase with coefficient L.
Decrease resolution with M.

Note that increasing the resolution requires the use of an interpolation filter after increasing the sampling frequency. A decrease in resolution requires the use of a filter before thinning. These two filters can be combined into one filter. Since the interpolation and smoothing filters are low-pass, the filter with the lowest passband can be used in both filters. Since the rational coefficient L / M is greater than unity, it means that M <L. This must be taken into account when determining the parameters of the low-pass filter.

Notes

Oppenheim, Alan V. Discrete-Time Signal Processing. - 2nd Edition. - Prentice Hall, 1999 .-- ISBN 0-13-754920-2 .
Digital Audio Resampling Home Page (discusses a technique for bandlimited interpolation ) (Retrieved May 18, 2009)
Matlab example of using polyphase filters for interpolation (Retrieved May 18, 2009)