Further reading

I write resampling ratios in the form input samples:output samples. With my convention, a ratio less than unity is upsampling.

Consider resampling pseudocolour data. If you treat the data as continuous, the resulting image is liable to contain colours not in the source. If you use nearest-neighbour resampling to avoid generating “new” sample values, geometry will suffer.

HD to 1920× 1080 HD: 1280 samples in each input line must be converted to 1920 samples in the output, an upsampling ratio of 2:3.

One way to accomplish upsampling by an integer ratio of 1:n is to interpose n-1 zero samples between each pair of input samples. This causes the spectrum of the original signal to repeat at multiples of the original sampling rate. The repeated spectra are called “images.” (This is a historical term stemming from radio; it has nothing to do with pictures!) These “images” are then eliminated (or at least attenuated) by an anti-imaging lowpass filter. In some upsampling structures, such as the Lagrange interpolator that I will describe later in this chapter, filtering and upsampling are intertwined.

Downsampling produces fewer result samples than input samples. In audio, new samples can be created at a lower rate than the input. In video, downsampling is required when converting 4fSC NTSC digital video to BT.601 (”4:2:2“) digital video: 910 samples in each input line must be converted to 858 samples in the output, a downsampling ratio of 35:33; for each 35 input samples, 33 output samples are produced.

In an original sample sequence, signal content from DC to nearly 0.5fS can be represented. After downsampling, though, the new sample rate may be lower than that required by the signal bandwidth. After downsampling, meaningful signal content is limited by the Nyquist criterion at the new sampling rate – for example, after 4:1 downsampling, signal content is limited to 1⁄8 of the original sampling rate. To avoid the introduction of aliases, lowpass filtering is necessary prior to, or in conjunction with, downsampling. The corner frequency depends upon the downsampling ratio; for example, a 4:1 ratio requires a corner less than 0.125fS. Downsampling with an integer ratio of n:1 can be thought of as prefiltering (antialias filtering) for the new sampling rate, followed by the discarding of n-1 samples between original sample pairs.

Resampling produces new samples that assume that neighbouring input samples are related by a continuous function. If the underlying function is not continuous, problems can be expected. For example, pseudocolour images are not continuous: They cannot be meaningfully resampled without creating artifacts.

Figure 21.1 Two-times upsampling starts by interposing zero samples between original sample pairs. This would result in the folded spectral content of the original signal appearing in-band at the new rate. These “images” are removed by a resampling filter.

Figure 21.2 An original signal exhibits folding around half the sampling frequency. This is inconsequential providing that the signal is properly reconstructed. When the signal is upsampled or downsampled, the folded portion must be handled properly or aliasing will result.

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1:2-upsampled signal

Folded spectrum (“image”) prior to resampling (anti-imaging) filter

Folded spectrum following resampling filter

0.5 1.0 Frequency, 1:2-upsampled fs

UPSAMPLING

Original signal

Folding around half-sampling frequency

DOWNSAMPLING

Figure 21.3 Two-to-one downsampling requires a resampling filter to meet the Nyquist criterion at the new sampling rate. The solid green line shows the spectrum of the filtered signal; the shaded line shows its folded portion. Resampling without filtering would preserve the original baseband spectrum, but folding around the new sampling rate would cause alias products shown here in the crosshatched region.

Folded spectrum without resampling

Alias products

Signal spectrum

(antialiasing) filter

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00.5 1 Frequency, 2:1-downsampled fs

Figure 21.2, at the center above, sketches the spectrum of an original signal. Figure 21.1 shows the frequency domain considerations of upsampling; Figure 21.3 shows the frequency domain considerations of downsampling. These examples show ratios of 1:2 and 2:1; however, the concepts apply to resampling at any ratio.