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Optimizing FFT (Part 1)

Prompt

Please consider the programs in src/kits/fft. Can we create any optimizations to make the FFT go faster? Can we precompute any numbers to make the calculation more efficient? I am running this on a Raspberry Pi Pico 2 with Free memory: 470.83 KB

Claude Sonnet 3.7 Response

I've analyzed your 14-display-fft.py code and can suggest several optimizations to make the FFT calculations faster on your Raspberry Pi Pico 2. The current implementation of the Cooley-Tukey FFT algorithm is recursive and creates many temporary lists, which is inefficient for microcontrollers with limited resources.

Here are optimizations we can implement:

  1. Replace the recursive FFT with an iterative in-place algorithm
  2. Precompute the twiddle factors (the complex exponentials)
  3. Use bit-reversal for the initial reordering
  4. Implement integer-based calculations where possible
  5. Optimize the windowing function

The optimizations I've implemented in the code above include:

1. Precomputed Arrays

  • Hanning window coefficients are calculated once at startup
  • Bit-reversal table for the FFT reordering
  • Twiddle factors (complex exponentials) are precomputed and stored

2. In-place Iterative FFT

  • Replaced the recursive algorithm with an iterative version
  • Eliminates the need for temporary arrays during computation
  • Uses in-place calculations to minimize memory usage

3. Fast Magnitude Calculation

  • Added a fast approximation for the magnitude calculation
  • Uses |z| ≈ max(|Re(z)|, |Im(z)|) + 0.4 * min(|Re(z)|, |Im(z)|) instead of the more expensive square root

4. Memory Reuse

  • Reuses the same arrays for FFT calculation across iterations
  • Avoids memory fragmentation and garbage collection delays

5. Efficient Data Structures

  • Uses typed arrays (array.array) for all numerical data
  • Provides better performance than regular Python lists

6. Optimized Display Rendering

  • Simplified the bin averaging logic
  • More efficient bounds checking

These optimizations should significantly improve the performance of your FFT calculations on the Raspberry Pi Pico 2. The pre-computation step may use a bit of memory initially, but it will save substantial computation time during the main loop.

A few additional notes:

  • The code prioritizes speed over memory usage, which is appropriate given your available memory (470.83 KB)
  • The bit-reversal pre-computation will speed up the FFT initialization stage
  • If memory becomes an issue, you could reduce FFT_SIZE to 256, which would still provide reasonable frequency resolution

I've fixed the MicroPython compatibility issue in the bit-reverse function. The previous version used string slicing with a negative step ([::-1]), which isn't supported in MicroPython. The new version uses bitwise operations instead:

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def bit_reverse(n, bits):
    result = 0
    for i in range(bits):
        result = (result << 1) | (n & 1)
        n >>= 1
    return result

This iterative approach manipulates the bits directly, shifting and masking to reverse the bit order.

The optimizations in this version include:

  1. MicroPython-compatible bit reversal - Using bitwise operations instead of string manipulation

  2. Precomputed tables:

  3. Hanning window coefficients

  4. Bit-reversal indices

  5. Twiddle factors (complex exponentials)

  6. In-place FFT algorithm - This reduces memory usage and avoids allocations during the FFT calculation

  7. Optimized magnitude calculation - Using a faster approximation: max(|re|, |im|) + 0.4 * min(|re|, |im|)

  8. Memory reuse - Arrays are allocated once and reused across iterations

This optimized version should run significantly faster than the original implementation while being fully compatible with MicroPython on your Raspberry Pi Pico 2.