4. Multirate Systems and their Applications

We compute here … and throw away most of them here!!!! Inefficient Implementation of Downsampling

Most terms here are zero … and waste time to process them here!!!!$$ Inefficient Implementation of Upsampling

Same!!! Recall the Noble Identities

Example…

… continued

Example… Same

… continued

In general: Polyphase Decomposition. Take N=2: General Filter: Polyphase Decomposition with Polyphase Components

NOBLE IDENTITY POLYPHASE Downsampling using Polyphase Decomposition

This is a Serial to Parallel (a Buffer): S/P Serial to Parallel (Buffer)

NOBLE IDENTITY POLYPHASE NOBLE IDENTITY Upsampling using Polyphase Decomposition

This is a Parallel to Serial (an Unbuffer): P/S Parallel to Serial (Unbuffer or Interlacer)

Given any integer N: Example: take N=3 General Polyphase Decomposition

POLYPHASE Apply to Downsampling…

… apply Noble Identity

S/P Serial to Parallel (Buffer): Serial to Parallel (Buffer)

POLYPHASE Same for Upsampling…

NOBLE IDENTITY … apply Noble Identity

This is a Parallel to Serial (an Unbuffer): P/S Parallel to Serial (Unbuffer or Interlacer)

Processing Data by Blocks In most efficient implementations we process data by blocks, rather than one sample at a time. Real Time simulation and design software such as Simulink are designed to take advantages of block processing for two purposes: efficient computations, thus faster simulations; efficient design.

“Sample Based” and “Frame Based” Signals Sample Based: time They are MN distinct signals arranged in a matrix Particular Case: is like M distinct signals

“Sample Based” and “Frame Based” Signals Frame Based: They are N distinct signals, each one represented as a sequence of frames of length M Particular Case: one signal as a sequence of frames of length M

y0=Frame based y1=Sample based y2=y0=Frame based Convert “to Sample” and “to Frame” Example:

Serial to Parallel in Simulink Serial to Parallel is implemented by the “Buffer” Simulink block in Signal Processing Blcokset > Signal Management > Buffers: S/P Sample based Frame based

Example of Downsampling

Parallel to Serial in Simulink Parallel to Serial is implemented by the “UnBuffer” Simulink block in Signal Processing Blcokset > Signal Management > Buffers: Frame based Sample based P/S

Example of Upsampling