Minimal group delay filter to remove harmonics but preserve fundamental

Question

I have a signal that contains energy at a fundamental frequency (60 Hz) plus significant energy at harmonics (2nd, 3rd etc up to 15th). The signal is sampled by an ADC at 1920 Hz. I want to construct a digital filter that removes / reduces the harmonics but allows the fundamental and non-harmonic content to pass through. In my application it is important to minimize the overall time delay introduced by the filter as well as minimize over / undershoot on steep leading edges. Such edges occur when a relay closes while the signal is not at a zero crossing. Since the signal source is 3-phase it is not possible to close the relay at a zero crossing for all three phases. I am particularly concerned with removing the so-called triplen harmonics (3rd, 9th & 15th) and could sacrifice performance of the filter at other harmonics if necessary.

I am currently using a series of IIR notch filters, one at each harmonic frequency starting at the 3rd harmonic. I wonder if there might be more efficient / elegant solutions that I should consider. I considered a comb filter, but don't know how to make one that allows the fundamental to pass through. I also considered a simple LPF but found it difficult to create one that has good performance at the 3rd harmonic without introducing significant group delay at the fundamental. I'd like to hold the fundamental delay to 500uS or less, which is approximately 1 sample period in my case.

I am a novice on digital filter design. Perhaps others on this forum have faced similar design challenges? Any suggestions or advice is appreciated.

Answer 1

The following is useful in applications where we want to null out interfering tones that are within are spectrum of interest where our signal occupies, so with that we want to minimize how much spectrum we remove. In cases where we are really only interested in the signal at 60 Hz (as the OP may be) then a PLL or 2nd order resonator would be most applicable and simply done, and I have already detailed in this link.

The comb filter described in this link shows the approach to notch out the harmonics while preserving the fundamental, but has a longer delay equal to half the number of filter coefficients and much roll-off at all the frequencies in between.

An approach where you can minimize group delay and narrow the width of the notch is with a zero-stuffed 2nd order notch filter cascaded with pole-zero cancellation filter for the pass frequency of interest. This is accomplished by combining the details from both of the posts linked below.

This first post shows how to make a harmonic notch filter by simply zero stuffing the 2nd order notch filter:

how to create a harmonic mask from fundamental frequency?

Similar to a decomposition of a moving average to a cascade-integrator-comb structure, we can cascade the above structure of the interpolated notch with the inverse of the single tone notch filter that is described in this post:

Transfer function of second order notch filter

This will be a very easy solution since the OP has a sampling rate that is a multiple of the tone of interest.

The filter is created by cascading the harmonic nulling filter which is given by the following transfer function:

$$H_1(z) = K_1 \frac{1 - z^{-N}}{1-\alpha^N z^{-1}}$$

With the pole / zero canceller at the desired Harmonic which is given by the following transfer function:

$$H_2(z) = K_2 \frac{1-2\alpha \cos(2\pi/N)z^{-1}+\alpha^2}{1-2\cos(2\pi/N)z^{-1}+1}$$

Where:

$N$ is the number of total harmonics up to the sampling frequency (in the OP's case $N=32$.

$\alpha$ is a filter Q-factor with the closer $\alpha$ is to 1, the tighter the notches will be (and the greater the required total number of bits will be in the extended precision accumulators for fixed point implementations!).

$K_1$ and $K_2$ are gain normalization constants, and computed as follows:

$$K_1 = \frac{1+\alpha^N}{2}$$

$$K_2 = \frac{2+2\cos(2\pi/N)}{1+2\alpha\cos(2\pi/N)+\alpha^2}$$

Below shows the resulting filter and group delay using $\alpha=0.99$ with $N=32$. (Note that we can optionally select any particular Harmonic to pass by changing $2\pi/N$ where used in both numerator and denominator in $H_2(z)$ and $K_2$ to be $2\pi k/N$ where $k$ is the selected harmonic):

Notice that the Group Delay is minimized at the passband of interest. (1st Harmonic).

Interesting side-note: A moving average filter similarly has a frequency response given by the Dirichlet Kernel (basically an aliased Sinc) which is useful when we want to pass DC but null every harmonic of a given frequency. When this same technique is used to sharpen the frequency response nulls in the moving average filter, the resulting structure is the moving average in the forward path, and a "leaky" moving average fed back and subtracted as an IIR structure. The "leaky" moving average is given by coefficients $c_n = \alpha^n$ for $n = 0,1,2 \ldots N-1$.

Answer 2

My apologies if this is more of a comment then an answer, given my low reputation.

As the other comments have mentioned, a sinusoidal impulse response at 60 Hz would be adequate for satisfying the removal of harmonics. If you are concerned with preserving group-delay (and thus phase-delay), consider using a zero-phase digital filter. The MATLAB documentation on the filtfilt function that has some good examples to see if it would suite your needs. Essentially, phase-delay is maintained by performing two filtering operations: one in the forward direction, and one in reverse. The filter's impulse response should be selected such that it is even with respect to the filter's center, i.e. $$h(n)=h(-n)$$ These two filtering operations will have phase delays that will cancel each other out. Keep in mind, this filter is non-causal.