Multi-rate Signal Processing - Nobel Identity

Sep 23, 2024#数字通信78

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The document discusses the structure of typical interpolators, emphasizing that placing the filter after the interpolation operation requires the filter to operate at a higher sampling rate, which complicates its design. It presents six key equivalences (identities) related to signal processing operations: 1. The extraction operation can be positioned either after or before the multiplication-addition operation without affecting the outcome. 2. Delaying a signal by M samples before extraction is equivalent to extracting first and then delaying. 3. Passing a signal through a filter after M extraction is equivalent to extracting first and then filtering. 4. The position of the interpolation operation relative to the multiplication-addition operation does not change the result. 5. Delaying a signal before L interpolation is equivalent to interpolating first and then delaying. 6. Filtering a signal before L interpolation is equivalent to interpolating first and then applying the filter to the adjusted signal. These identities simplify the design and analysis of signal processing systems.

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

Second Identity#

Performing M extractions after M delays is equivalent to performing 1 delay 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 located before the multiply-accumulate is equivalent to the interpolation operation located after the multiply-accumulate.

Fifth Identity#

The signal first undergoes 1 delay and then L interpolation is equivalent to first undergoing L interpolation and then L delays.

Sixth Identity#

The signal first passes through the filter $H(z)$ and then undergoes L interpolation is equivalent to the signal first undergoing L interpolation and then passing through the filter $H(z^L)$.