Often sample rate conversion algorithms based on multiplying / dividing frequency and digital filtration.
Filtration used for elimination artifacts appears after each altering sample rate.
Exists two main types of filters: linear and minimal phase.
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Oversampling (increasing sample rate) How Convert Sample Rate. Oversampling
Downsampling (decreasing sample rate) How Convert Sample Rate. Divide. Downsampling
One of main troubles of digital filters is ringing audio. Artifacts appear after changing signal. Especially on fronts of musical notes.
Details here: What Is Ringing Audio