4 Multirate Systems and their Applications Inefficient Implementation

4. Multirate Systems and their Applications

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

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

Recall the Noble Identities Same!!!

Example…

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Example… Same

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

Downsampling using Polyphase Decomposition POLYPHASE NOBLE IDENTITY

Serial to Parallel (Buffer) This is a Serial to Parallel (a Buffer): 0001234567 S/P 0 1 3 5 0 2 4 6

Upsampling using Polyphase Decomposition NOBLE IDENTITY POLYPHASE NOBLE IDENTITY

Parallel to Serial (Unbuffer or Interlacer) This is a Parallel to Serial (an Unbuffer): 0 1 3 5 0 2 4 6 P/S 0001234567

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

Apply to Downsampling… POLYPHASE

… apply Noble Identity

Serial to Parallel (Buffer): S/P 1 2 3 4 5 6 1 3 5 2 4 6

Same for Upsampling… POLYPHASE

… apply Noble Identity NOBLE IDENTITY

Parallel to Serial (Unbuffer or Interlacer) This is a Parallel to Serial (an Unbuffer): 1 3 5 2 4 6 P/S 1 2 3 4 5 6

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

Convert “to Sample” and “to Frame” Example: y 0= y 1= y 2=y 0= Frame based Sample based Frame based

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 “Un. Buffer” Simulink block in Signal Processing Blcokset > Signal Management > Buffers: P/S Frame based Sample based

Example of Upsampling