reconstruct the low resolution signal before upsampling

Question

When we upsampling a discrete 1d signal by 2x, we first interleave the signal by 0 and add zero padding, then pass through a low pass filter.

low resolution signal [x1, x2, x3, x4] -> interleave 0 and pad 0 (padding size is filter_size-1) -> [0, 0, x1, 0, x2, 0, x3, 0, x4, 0, 0] -> low pass filter (kaiser filter, convolution) -> high resolution signal

Given the high resolution signal and the low pass filter (kaiser filter), is there a way to reconstruct the low resolution signal? If there is not a way to perform precise reconstruction, is there a way to perform roughly reconstruction?

Answer 1

If the interpolation is done correctly and with good image rejection, selecting every other sample in this case will reconstruct the original signal (although with a possible timing offset).

The interpolation is simply zero stuff and low pass filter. The low pass is designed to pass the original signal with minimum distortion and reject the image which will be at fs/2 +/- b where fs is the original sampling rate and b is the single sided bandwidth of the original signal.

Downsampling is the reverse process where typically we would first filter (to eliminate alias frequencies) and then select every other sample. Since your interpolation has already removed the image (which would otherwise be the alias) you can simply select every other sample in this case with no additional filtering first.