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Polyphase filtering is a powerful technique in digital signal processing (DSP) that improves the efficiency of multirate operations like decimation (downsampling) and interpolation (upsampling). It's especially useful when working with large sample rate changes or designing efficient filter banks.

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What Is Polyphase Filtering?
At its core, polyphase filtering involves breaking a filter into multiple smaller sub-filters, called polyphase components, each responsible for a subset of the output. This allows for:

  • Efficient implementation
  • Reduced computation
  • Better performance in real-time systems

Use Cases

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Key Concepts
1. Decimation (Downsampling)

  • Reduces the sampling rate by a factor M
  • Naively: filter → keep every Mth sample
  • Polyphase version: Split the filter into M sub-filters; compute only what's needed for each output sample

2. Interpolation (Upsampling)

  • Increases the sampling rate by a factor L
  • Naively: insert zeros → filter
  • Polyphase version: Use L sub-filters, each computing part of the upsampled output without wasting effort on zero samples

Mathematical View
Let h[n] be your FIR filter and 𝑀 your decimation factor. The polyphase components are:
hk​[n]=h[nM+k],for k=0,1,,M−1
The output is then computed using only the necessary phase components.

Benefits

  • Computational Efficiency: Avoids unnecessary multiplies with zero (e.g., in upsampling)
  • Lower Memory Footprint: Works with smaller sub-filters
  • Pipeline-Friendly: Well-suited for hardware/DSP chips

Applications

  • Audio compression (e.g., MP3, AAC)
  • Sample rate converters
  • Digital communications (e.g., multicarrier systems)
  • Software-defined radios

Summary

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