Multi-rate Signal Processing - Nobel's Identity

Sep 23, 2024#数字通信139

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The document discusses various equivalences in signal processing related to interpolation and filtering operations. It presents six key identities: 1. The extraction operation can be placed either after or before the multiplication and addition operations without affecting the outcome. 2. Delaying a signal by M samples before extracting M samples is equivalent to extracting M samples and then delaying by one sample. 3. Passing a signal through a filter $ H(z^M) $ after M extraction is equivalent to extracting M samples first and then filtering with $ H(z) $. 4. The position of interpolation relative to multiplication and addition (before or after) does not change the result. 5. Delaying a signal by one sample before L interpolation is equivalent to performing L interpolation followed by L delays. 6. Filtering a signal with $ H(z) $ before L interpolation is equivalent to performing L interpolation first and then filtering with $ H(z^L) $.

The structure of a typical interpolator, with the filter located after the interpolation operation, means that the filter operates at a higher sampling rate, putting pressure on the design of the filter. The interpolation operation can be moved to the back through an identity transformation, placing the filter at the front, simplifying the system design.

First Identity#

Indicates that the extraction operation after the multiply-accumulate operation and the extraction operation before the multiply-accumulate operation are equivalent.

Second Identity#

Performing M delays followed by M extractions and performing M extractions followed by 1 delay are equivalent.

Third Identity#

The signal passing through the filter $H(z^M)$ and then undergoing M extraction is equivalent to the signal undergoing M extraction and then passing through the filter $H(z)$ .

Fourth Identity#

The interpolation operation before the multiply-accumulate operation is equivalent to the interpolation operation after the multiply-accumulate operation.

Fifth Identity#

The signal undergoing 1 delay followed by L interpolation is equivalent to undergoing L interpolation followed by L delays.

Sixth Identity#

The signal passing through the filter $H(z)$ followed by L interpolation is equivalent to the signal undergoing L interpolation followed by the filter $H(z^L)$ .