Polyphase FIR filter for sample rate converter

Polyphase filter is an efficient structure for applying resampling and filtering to the signal. It is also useful in filter banks processing. A typical polyphase can be showed as below:

Assume that a N-1 th FIR filter is $H(z)$ , if we want to split the filter into M bands, then $H(z)$ can be rewrote as: $$ H(z) = \Sigma_{n=0}^{n=N-1}{h(n)*z^{-n}}=\Sigma_{k=0}^{k=M-1}z^{-k}E_k(z^M)\E_k(z)=\Sigma_{l=0}^{l=N/M-1}h(Ml+k)*z^{-n},k=[0,M-1] $$ Now the FIR filter $H(z)$ has split into M bands, the system function of sub-band filter is $E_k$, that is the polyphase structure equivalence transformation.

As we know that for a sample rate conversion, if we want to upsample the signal from F to I*F, we need to interpolate the input sample Itimes. If a down-samplerate from F to F/M is required , we should decimated every M samples. Moreover, a low-pass filter is needed to avoid the frequency aliasing in the processing.

For example:

As the noble identity,

So look at the formula (1), we here replace the $H(z)$ with a polyphase structure ,

Applying decimation before filter to decrease the operators:

For a interpolation, it can be showed as below:

Turn into the code, you can find the polyphase filter in the file fir.h, here is a simple unit test for the polyphase which convert 96 kHz signal to 48 kHz for decimation, and interpolate the signal from 48 kHz to 96 kHz.

...sineGenerator()...
... output2File()...
void polyphaseFilterUnittest() {
	size_t fs = 96000;
	float* x96 = sineGenerator(1000, 1, 1, fs);
	PolyphaseFilter* p = newPolyphaseFilter(FIR_96_48, 2, FIR_96_48_24);
	float* y96 = (float*)malloc(sizeof(float) * fs
	// test case 1: 96k->48k	|	2:1
	// Use the polyphase decimation,M		|| test pass
	for (size_t n = 0; n < fs; n+=2)
	{
		runPolyphaseDecimation(p, x96+n, y96+n/2, 2);
	}
	output2File("96 to 48.txt", y96, fs / 2);

	// test case 2: 48k->96k	|	1:2
	// Use the polyphase interpolation,L	|| test pass
	memset(y96, 0, sizeof(float) * fs);

	float* x48 = sineGenerator(2000, 1, 1, 48e3);
	for (size_t n = 0; n < 48000; n++)
	{
		runPolyphaseInterpolation(p, x48 + n, y96 + n*2, 2);

	}
	output2File("48 to 96.txt", y96, fs);

	free(x96);
	free(y96);
	freePolyphaseFilter(p);
}