Interpolation, Decimation, and Rate Changing by Integer Fractions 1.1 2005/01/14 15:25:31 US/Central 2005/05/24 14:35:04 GMT-5 Douglas L. Jones dl-jones@uiuc.edu Douglas L. Jones dl-jones@uiuc.edu Charlet Reedstrom charlet@rice.edu Kyle Clarkson kclarks@rice.edu Multirate signal processing

Interpolation: by an integer factor L Interpolation means increasing the sampling rate, or filling in in-between samples. Equivalent to sampling a bandlimited analog signal L times faster. For the ideal interpolator, X 1 ω X 0 L ω ω L 0 L ω We wish to accomplish this digitally. Consider and . y m X 0 m L m 0 ± L ± 2 L … 0

The DTFT of y m is Y ω m ω y m ω m n x 0 n ω L n n x n ω L n X 0 ω L Since X 0 ω ′ is periodic with a period of 2 , X 0 L ω Y ω is periodic with a period of 2 L (see ).

By inserting zero samples between the samples of x 0 n , we obtain a signal with a scaled frequency response that simply replicates X 0 ω ′ L times over a 2 interval! Obviously, the desired x 1 m can be obtained simply by lowpass filtering y m to remove the replicas. x 1 m y m h L m Given H L m 1 ω L 0 L ω In practice, a finite-length lowpass filter is designed using any of the methods studied so far ().

Interpolator Block Diagram

Decimation: sampling rate reduction (by an integer factor M) Let y m x 0 L m ()

That is, keep only every Lth sample ()

In frequency (DTFT): Y ω m y m ω m m x 0 M m ω m n M m n x 0 n k δ n M k ω n M ω ′ ω M n x 0 n k δ n M k ω ′ n DTFT x 0 n DTFT δ n M k Now DTFT δ n M k 2 k 0 M 1 X k δ ω ′ 2 k M for ω as shown in homework #1 , where X k is the DFT of one period of the periodic sequence. In this case, X k 1 for k 0 1 … M 1 and DTFT δ n M k 2 k 0 M 1 δ ω ′ 2 k M . DTFT x 0 n DTFT δ n M k X 0 ω ′ 2 k 0 M 1 δ ω ′ 2 k M 1 2 μ ′ X 0 μ ′ 2 k 0 M 1 δ ω ′ μ ′ 2 k M k 0 M 1 X 0 ω ′ 2 k M so Y ω k 0 M 1 X 0 ω M 2 k M i.e., we get digital aliasing.()

Usually, we prefer not to have aliasing, so the downsampler is preceded by a lowpass filter to remove all frequency components above ω M ().

Rate-Changing by a Rational Fraction L/M This is easily accomplished by interpolating by a factor of L, then decimating by a factor of M ().

The two lowpass filters can be combined into one LP filter with the lower cutoff, H ω 1 ω L M 0 L M ω Obviously, the computational complexity and simplicity of implementation will depend on L M : 23 will be easier to implement than 10611060 !