NOTES¶

Convolution — Foundation of Signal Processing

Flip the kernel, slide it across the signal, multiply and accumulate at each position. FIR filtering is convolution by definition.

Algorithm¶

Full Convolution¶

\[ y[n] = \sum_{k=0}^{M-1} h[k] \cdot x[n - k], \quad n = 0, \dots, L+M-2 \]

\(L\) = signal length, \(M\) = kernel length.

Output Modes¶

Mode	Output length	Description
`CONV_FULL`	\(L + M - 1\)	Full, including transients
`CONV_CENTER`	\(L\)	Same length as input
`CONV_HEAD`	\(L\)	First \(L\) points (real-time)
`CONV_TAIL`	\(L\)	Last \(L\) points (trailing)

Padding¶

Mode	Behavior	Use case
`PAD_ZERO`	Zero outside	Default
`PAD_SYMMETRIC`	Mirror at edges	DWT, reduce artifacts
`PAD_PERIODIC`	Periodic extension	Periodic signals

Convolution vs Correlation

Convolution flips the kernel; correlation does not. This flip is what makes convolution the natural operation for system response (FIR filtering).

FFT acceleration

Direct convolution \(O(LM)\). For long signals, use FFT: \(\text{conv}(x, h) = \text{IFFT}(\text{FFT}(x) \cdot \text{FFT}(h))\), \(O(N \log N)\).

API Reference¶

// Basic convolution (full mode + zero padding)
tiny_error_t tiny_conv_f32(const float *Signal, int siglen,
                           const float *Kernel, int kernlen,
                           float *convout);

// Extended convolution (configurable padding + output mode)
tiny_error_t tiny_conv_ex_f32(const float *Signal, int siglen,
                              const float *Kernel, int kernlen,
                              float *convout,
                              tiny_padding_mode_t padding_mode,
                              tiny_conv_mode_t conv_mode);

Notes¶

Output buffer size

Full mode: at least \(L+M-1\). Buffer overflow is a silent bug.

ESP32: longer signal should come first

The implementation assumes Signal is the longer array. Order still works if reversed but performance degrades.