downsampling and upsampling【转】

orig: http://www.eetimes.com/document.asp?doc_id=1275556

downsampling

The process of reducing a sampling rate by an integer factor is referred to as downsampling of a data sequence.We also refer to downsampling as ''decimation'' (not taking one of ten). The term ''decimation'' used for the downsampling process has been accepted and used in many textbooks and fields. To downsample a data sequence x(n) by an integer factor of M, we use the following notation:

y(m) = x(mM), (12.1)

where y(m) is the downsampled sequence, obtained by taking a sample from the data sequence x(n) for every M samples (discarding M – 1 samples for every M samples). As an example, if the original sequence with a sampling period T = 0.1 second (sampling rate = 10 samples per sec) is given by

x(n):8 7 4 8 9 6 4 2 –2 –5 –7 –7 –6 –4 …

and we downsample the data sequence by a factor of 3, we obtain the downsampled sequence as

y(m):8   8   4   –5   –6 … ,

with the resultant sampling period T = 3 × 0.1 = 0.3 second (the sampling rate now is 3.33 samples per second). Although the example is straightforward, there is a requirement to avoid aliasing noise.

upsampling

Increasing a sampling rate is a process of upsampling by an integer factor of L. This process is described as follows:

y(m) = { x(m/L)       m = nL,
             0              otherwise, (12.9)

where n = 0, 1, 2, … , x(n) is the sequence to be upsampled by a factor of L, and y(m) is the upsampled sequence. As an example, suppose that the data sequence is given as follows:

x(n):8   8   4   –5   –6 …

After upsampling the data sequence x(n) by a factor of 3 (adding L – 1 zeros for each sample), we have the upsampled data sequence w(m) as:

w(m): 8 0 0  8 0 0  4 0 0  –5 0 0  –6 0 0 …

The next step is to smooth the upsampled data sequence via an interpolation filter. The process is illustrated in Figure 12-5a.