Multi-rate Signal Processing - Nobel Identity

Sep 23, 2024#数字通信78

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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 order of extraction relative to multiplication and addition is interchangeable. 2. Delaying a signal by M samples before or after M extractions yields the same result. 3. Filtering a signal after M extractions is equivalent to extracting the signal after filtering. 4. The position of interpolation relative to multiplication and addition does not affect the outcome. 5. A delay followed by L interpolation is equivalent to L interpolation followed by L delays. 6. Filtering a signal before L interpolation is equivalent to performing L interpolation before filtering with a modified filter. These identities simplify the design of systems by allowing flexibility in the arrangement of operations.

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 in front, simplifying the system design.

First Identity#

Indicates that the extraction operation after the multiply-accumulate operation is equivalent to the extraction operation before the multiply-accumulate operation.

Second Identity#

Performing M extractions after M delays is equivalent to performing M delays after M extractions.

Third Identity#

The signal passing through the filter H(zM) and then M extractions is equivalent to the signal passing through M extractions and then 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 going through 1 delay before L interpolation is equivalent to going through L interpolation before L delays.

Sixth Identity#

The signal passing through the filter $H(z)$ before L interpolation is equivalent to the signal going through L interpolation before passing through the filter $H(z^L)$.