Using the Dirichlet kernel as a lowpass filter

Question

I have a question about using the Dirichlet kernel as a filter. Let us suppose that I have samples of a continuous function sampled with frequency $F_s=10 \,\texttt{Hz}$. The function is band-limited and the sampling frequency is well above the Nyquist frequency, but there is noise added to the data. I need a lowpass filter to remove all the high frequency components introduced by the noise above a certain frequency $B$.

My understanding is that $\text{sinc}(2Bt)$ would represent the ideal lowpass filter because computing its Fourier transform (FT) gives a rectangular frequency response of single-sided width $B$. However, this filter has an infinite duration in time and we need to truncate it. The abrupt truncation with a rectangular window causes unwanted Gibbs oscillations in the frequency domain and, consequently, we smooth the $\text{sinc}$ with an appropriate window instead to mitigate this problem.

However, because we implement all our calculations on machines, continuous functions are sampled and FT become Discrete Fourier transforms (DFT). Then, why don't we just use the Dirichlet kernel as the lowpass filter? Let's say that my filter has $N$ coefficients, with $N$ odd. The DFT will have spacing ${\Delta}f=F_s/N$. If I take $N_B$ as the nearest integer to $B/{\Delta}f$, then the Dirichlet kernel $D_n=\sin[{\pi}(2N_B+1)n/N]/\sin({\pi}n/N)$ will have a rectangular DFT with single-sided bandwidth given by $N_B{\Delta}f$. Of course, if I have $M$ samples of the input function, the output will have $M-(N-1)$ samples. It seems to me that $D_n$ is the ideal filter that we need. Am I wrong about this?

I know that $D_n$ tends to $\text{sinc}(2Bt)$ as $N$ goes to $\infty$. So I guess an issue related to my question is, for what value of $N$ is the discrete approximation to the continuous case good enough that we don't have to worry about it anymore? On the other hand, when we are limited to relatively small values of $N$ (for instance, if we don't want to exacerbate the data loss $M-(N-1)$), should we just do things as if everything were discrete or should we carry over results from the continuous case (like filtering with $\text{sinc}$ for example)?

Thank you in advance for your answers!

Answer 1

The Dirichlet Kernel is a "Time Aliased Sinc Function", and using it as the coefficients for the low pass filter avoids truncation issues we would see as the OP describes correctly with the truncated Sinc function, but leads to greater distortion in the frequency domain for the frequency response corresponding to all frequencies that are in between the samples given by the DFT (The frequency response is the DTFT of the time domain samples, as even though we are sampled in time, we can have samples of any frequency on the continuous frequency domain). The Dirichlet Kernel would result in the exact frequency values sampled for a rectangular low pass (such as 1,1,1,0,0,0,0 .....0, 0, 1,1] in the frequency domain, but the continuous frequency response would have much more ripple (error) at all the frequencies in between those "bin centers" than what we would get with using samples of the truncated Sinc function as the filter.

Below shows the two functions; the truncated Sinc labeled "IFT", and the Dirichlet Kernel labeled "IDFT"

The Sinc in orange has it's distortion due to being truncated from the ideal Sinc that extends to $\pm \infty$, while the Dirichlet Kernel has it's distortion as deviating from the ideal Sinc, which can be shown to be mathematically equivalent to aliasing from the tails of the Sinc that extend beyond the boundary shown (hence the elevated values). This distortion appears as an error in the frequency response.

I detail the difference between the two approaches in DSP #31905, with some bottom-line plots from that post copied below.

Below is the comparative frequency response for a 99-Tap FIR using coefficients of the Dirichlet Kernel (and labeled "Frequency Sampling") in Blue, compared to coefficients of a truncated Sinc (and labeled "Window" as it has been windowed with a rectangular window) in red. In the scale of the first plot we primarily see the difference in the stop-band, in that it has greater error which is attributed to the time domain aliasing of an ideal Sinc response in the time domain.

Below is a zoom in of the passband, again showing the greater error that results.

The lesson from above is that both the truncated Sinc and the Dirichlet Kernel are non-ideal (neither is a brick-wall response), with the deviation due to either truncation error for the rectangular windowed Sinc, or time-domain aliasing for the Dirichlet Kernel. But importantly, the truncated Sinc out-performs the Dirichlet Kernel. We can continue to improve the truncated Sinc with higher performing windows, or we can continue to improve the Dirichlet Kernel by oversampling first in the frequency domain to reduce the time domain aliasing (effectively push the time duration out further), and then windowing that result. Oversampling the frequency domain to reduce aliasing for a rectangular frequency response just makes the Dirichlet Kernel approach the ideal Sinc!