This page describes the concepts behind the fractional delay compensation option in the waveguide elements. Proper use of this option can help to improve the accuracy of time domain simulations and ease the process of tuning global simulation properties such as the sampling rate and the simulation time window.

## Digital filter

For time domain simulations, the frequency dependent elements rely on either infinite impulse response (IIR) or finite impulse response (FIR) digital filters. FIR filters can provide much better frequency responses provided the signal is composed of tones that fall exactly on the frequencies corresponding to the FIR taps. The following is an example of a Gaussian filter that relies on FIR digital filters.

\(x(n)\) \(y(n)= \sum_{k=0}^{M-1} b_k x(n-k)\)

where \(b_k\) are the filter tap coefficients, the filter transfer function in z-domain is:

$$ H(z)=\sum_{k=0}^{M-1} b_{k} \cdot z^{-k} $$

From the transfer function above we can see, FIR filters introduce extra M delays, which adds a constant group delay to the signal path.

To compensate for the artificial group delay, fractional delay and delay compensation options are available in related waveguide elements.

## Fractional delay

Waveguide elements including Straight Waveguide, MODE Waveguide, Multimode Waveguide and Waveguide Bends adopt an option to enable a fractional delay filter, where time delays don’t have to be an integer multiple of the sampling period. It offers better accuracy when running transient simulations, where waveguides are typically represented as delay lines. This option is especially useful in ring structures, since the delay adds up by the resonance effect and become more crucial in these structures.

The default setting for "fractional delay" is "true" for all newly added waveguide elements. The "delay compensation" number depends on the number of connections (or number of elements) in the structure. Please note that, every time when signal passes the waveguide element, the designated number of samples of delay compensation will be performed.

The following example wg_delay_compensation_comparison.icp has a simple ring structure built by a pair of optical coupler and two pieces of straight waveguide. The result is the comparison of the modulation gain curves with scattering data analysis, impulse response without and with delay compensation. The schematic layout and result are shown below.

The results shown above proves that, if time delay is not an integer multiple of the sampling period and no delay compensation is performed, the impulse response of the modulation gain curve only matches the scattering data analysis at around the center frequency (green curve and blue curve). As the frequency deviates from the center, the FSR goes more and more off. When including fractional delay compensation, the impulse response result matches the scattering data analysis result very well for most of the frequency ranges (red curve and blue curve), but when frequency is far off from the center (~ 8 THz on each side), the impulse response suffers from some distortion. Please make sure to operate in the valid frequency range when using the fractional delay compensation option.

### See also

Straight Waveguide (reference guide)

MODE Waveguide (reference guide)

Multimode Waveguide (reference guide)