Low-pass vs. windowing function in front of FFT

Question

I am sampling sound at 48 kHz because that's what the DAC for the headset needs. I take every 6th of these. So I have 8 kHz and process 1024 at a time. My goal is to analyse the spectrum in the final end.

My array of data, I have found out, I have presented in three different ways, instead of pointer casts:

typedef struct {
    union {
        // ================================================
        // Reuse of buffer, avoids ram usage and memcpy
        // with union instead of complex casts to pointers
        // ================================================
        mic_sample_t  mic_samples          [NUM_MIC_SAMPLES_REAL];                   // 1 * 1024
        dsp_complex_t complex_array        [NUM_FFT_POINTS_COMPLEX];                 // 2 * 512
        dsp_complex_t complex_half_spectra [NUM_FREQ_SPECTRA][NUM_FREQ_POINTS_REAL]; // 2 * 2 * 256
    } u;
} mic_samples_buff_t;

My data goes through this:

// collect as mic_samples
dsp_fft_bit_reverse (
        buffer.u.complex_array,  // 1024 signed seen as 512 complex
        NUM_FFT_POINTS_COMPLEX); // 512
dsp_fft_forward (
        buffer.u.complex_array,
        NUM_FFT_POINTS_COMPLEX, // 512 complex
        FFT_SINE(NUM_FFT_POINTS_COMPLEX)); // dsp_sine_512 is dim (512/4) + 1 = 129
dsp_fft_split_spectrum (
        buffer.u.complex_array,
        NUM_FFT_POINTS_COMPLEX);
// Handle as two complex_half_spectra

I end up with two spectra, each of 256 values, which I take the magnitude and square root of. I end up with 7.81 Hz/bin - which takes me to about 2 kHz.

I see that as my clean signal passes 2 kHz the mirror of it comes down again and moves downwards in the spectrum.

Should I put a low-pass in front of the bit reverse only? Should it have a cut-off of 2 kHz or somewhat lower? I reckon I'd have a high Q.

But then, I think I perhaps need a windowing function also, in addition. I assume that even if I end up with two spectra, I would need one window across all of the 1024 samples? I think it's the values at both ends that deliver most of the convoluted mirror frequencies, so we'd need to keep them down - in addition to low-pass? If I need both, would Hanning or Hamming be preferred?

(Aside: this is a hobby project, as a retired engineer. I plan to make one of the box, and I have no income or ads from my blog notes. I publish "everything" there. So I have described this at My Beep-BRRR notes)

Answer 1

There are two questions here: when do we need an "anti-alias" filter and when do we need to window the signal?

ANTI-ALIAS FILTER

The need for an anti-alias filter is associated with the sampling process. Once any signal is sampled, the spectrum that exists from $-f_s/2$ to $+f_s/2$ where $f_s$ is the sampling rate, will periodically repeat (exactly) if we were to extend the frequency axis to $\pm \infty$. This is mathematically valid to do so, but because the results are an exact repeat, we only need to show the spectrum from $-f_s/2$ to $+f_s/2$. Further, under the narrower condition that the signal is real, the negative frequency axis will be Hermitian symmetric (same magnitude, opposite phase), so for those cases the information is also redundant and we only need to show the spectrum from $0$ to $f_s/2$.

These points are demonstrated in the graphic below showing the frequency spectrums when sampling an analog 3 Hz cosine wave with a 20 Hz sampling clock. The top row is the spectrum of the analog input, the middle row is the spectrum for the sampling process itself (which is impulses in time that we multiply with our signal to sample; the Fourier Transform of a train of impulses in time is a train of impulses in frequency). We multiplied in time, so the first two rows would convolve in frequency, resulting in the bottom row as the spectrum of our digital signal, when we show the frequency axis extending to $\pm \infty$.

With observation of the above, it is clear that any signal at the input that has a frequency between $-f_s/2$ to $+f_s/2$ would be represented accurately in the digital spectrum at the same frequency as the input. This frequency range $-f_s/2$ to $+f_s/2$ at the input is referred to as the "First Nyquist Zone".

What gets really interesting is with further observation we can see how the sampling process is a multi-band down-converter. We see from above how the block from the first Nyquist Zone $-f_s/2$ to $+f_s/2$ in the analog was converted directly to this block in the digital spectrum. What we are about to see is that all higher blocks (such as the "Second Nyquist Zone" from $-f_s$ to $-f_s/2$ and $+f_s/2$ to $f_s$ are also translated to $-f_s/2$ to $+f_s/2$ in the digital spectrum!

Following the same description of the process for sampling the 3 Hz cosine, if we instead sampled a 23 Hz cosine, with the same 20 Hz sampling clock, we would get the spectrums as shown in the graphic below. The convolution process when dealing with impulses is simply adding the frequencies for all the impulses shown: convolving the input with the impulse at 0 in the sampling process creates the tones at +23 Hz and -23 Hz in the digital spectrum. Convolving the input with the impulse at -20 Hz in the sampling process creates the tones at -43 Hz and -3 Hz in the digital spectrum. The main point is that whatever results in any of the bands that extend from $\pm f_s/2$ centered around each multiple of the sampling rate MUST repeat in the digital spectrum. So we get the result as shown and notably, from observation of the digital result alone- we don't know if the input signal was actually as 23 Hz, or 17 Hz, or 3 Hz, or 37 Hz, or 43 Hz, or 63 Hz,...

If we were however to use a filter prior to sampling that selected any one of these Nyquist Zones (and ultimately the A/D converter itself is a low pass filter as it does not have infinite bandwidth, and finite aperture width of the sampling process also limits the bandwidth), then under this condition we can really know what the sampled signal represents in the analog domain.

Everything described applies to resampling as well: when we resample a signal all the same aliasing results can apply with respect to the final output rate. Here we have the opportunity to apply the anti-alias filter for the resampled conditions in either the digital domain (prior to re-sampling!) or in the analog domain. So in the OP's case we have two options:

A more relaxed anti-alias filter with respect to the original 48 KHz sampling rate-- this filter should be concerned with passing the signal of interest, for example DC to 3 KHz, and rejecting the alias frequency bands, which would be at N x 48 KHz +/- 3 KHz. So here we could use a very simple and cheap low pass filter given the huge transition band passing up to 3KHz and rejection starting at 45 KHz. This would then be followed by a digital anti-alias filter prior to resampling to 8 KHz which would need to pass the 0 to 3 KHz signal of interest and reject the alias band from N x 8 KHz +/- 3KHz. Multiband filter techniques can easily be used in the digital domain to do this filter very efficiently.

OR

All anti-aliasing can be done in the analog domain prior to filtering. This would be a much more challenging analog filter as it would need to pass DC to 3KHz and reject 5KHz and above (it could have multiband rejection at all the alias bands at N x 8 KHz +/- 3KHz, but that isn't as simple to implement in the analog domain).

Finally, note that even if we don't expect to have signals in any of the frequency bands that would alias, there is inevitably a noise floor and not providing the proper filtering can lead to noise enhancement.

WINDOWING

Windowing a signal prior to taking an FFT of the signal is a completely different process for different considerations. Windowing is the process of multiplying the signal in the time domain with a tapering shape. This serves to reduce severe spectral smearing in the frequency domain. Without further windowing, a "rectangular window" is implied just by having selected a finite duration of our signal in time. The Fourier Transform of a rectangular window is a Sinc function in frequency; multiplying in time is convolving in frequency so by having multiplied our time domain signal with the rectangular function, we convolved it's frequency response with a Sinc function. Sinc functions have side-lobes that go down relatively slowly (the peaks go down by $1/f$) so this results in a significant smearing of what would otherwise be narrow impulses in frequency (for the case of single tones). Windowing with more advanced windows serves to reduce this, as detailed further in these other posts:

Why would one use a Hann or Bartlett window?

Filtering sidelobes