 # Sample-rate conversion

The process of changing the sampling rate of a discrete signal to obtain a new discrete representation of the underlying continuous signal.

Sample-rate conversion is the process of changing the sampling rate of a discrete signal to obtain a new discrete representation of the underlying continuous signal. Application areas include image scaling and audio/visual systems, where different sampling rates may be used for engineering, economic, or historical reasons.

For example, Compact Disc Digital Audio and Digital Audio Tape systems use different sampling rates, and American television, European television, and movies all use different frame rates. Sample-rate conversion prevents changes in speed and pitch that would otherwise occur when transferring recorded material between such systems.

Within specific domains or for specific conversions, the following alternative terms for sample-rate conversion are also used: sampling-frequency conversion, resampling, upsampling, downsampling, interpolation, decimation, upscaling, downscaling. The term multi-rate digital signal processing is sometimes used to refer to systems that incorporate sample-rate conversion.

Conceptual approaches to sample-rate conversion include: converting to an analog continuous signal, then re-sampling at the new rate, or calculating the values of the new samples directly from the old samples. The latter approach is more satisfactory since it introduces less noise and distortion. Two possible implementation methods are as follows:

• If the ratio of the two sample rates is (or can be approximated by) a fixed rational number L/M: generate an intermediate signal by inserting L − 1 0s between each of the original samples. Low-pass filter this signal at half of the lower of the two rates. Select every M-th sample from the filtered output, to obtain the result.
• Treat the samples as geometric points and create any needed new points by interpolation. Choosing an interpolation method is a trade-off between implementation complexity and conversion quality (according to application requirements). Commonly used are ZOH (for film/video frames), cubic (for image processing) and windowed sync function (for audio).

The two methods are mathematically identical: picking an interpolation function in the second scheme is equivalent to picking the impulse response of the filter in the first scheme. Linear interpolation is equivalent to a triangular impulse response; windowed sync approximates a brick-wall filter (it approaches the desirable "brick wall" filter as the number of points increases). The length of the impulse response of the filter in method 1 corresponds to the number of points used in interpolation in method 2.

In method 1, a slow pre-computation (such as the Remez algorithm) can be used to obtain an optimal (per application requirements) filter design. Method 2 will work in more general cases, e.g. where the ratio of sample rates is not rational, or two real-time streams must be accommodated, or the sample rates are time-varying ### Key Terms

application requirements
filter
impulse response
interpolation method
method
process
rate
rates
sample rate conversion
sample rates

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### Synonymns

Sample-rate conversion
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