Equation 1 is a modification to the standard exponential filter calculation. The first term uses the projected value of x based only on previous information, as in the exponential filter. But instead of just taking the last filtered value, the best estimate of the trend is also added to account for the straight line assumption. The second term uses the correction for the newly-observed input at time k. The exponential filter factor a weights the past vs the new information.
Equation 2 is an estimate of the trend, exponentially filtered with its own filter constant g. The first term uses the best estimate from past information, and the second term uses the correction for the most recently observed trend, y[k] - y[k-1] . The exponential filter factor g weights the past vs the new information.
If the input signal becomes a ramp (constant slope), then this filter output will track the ramp, approaching it asymptotically. If the slope of the ramp is changed, the filter output will again approach the new ramp. For that reason, this filter is competitive with a linear least squares filter, which also makes a straight line assumption. This is an IIR approximation of the FIR linear least squares filter.
Alpha-beta filters are based on the same assumptions as double exponential smoothing. There are estimates of two variables, one the derivative of the other. The derivative is assumed constant for prediction. Upon receipt of new data, both the variable and its derivative estimates are updated. There are two parameters for the weighting of prediction vs correction: alpha and beta. It should have the same general characteristics as double exponential smoothing, although there may be no published comparison of this. See the Wikipedia article.
Other signal models can be used. For instance, the predictions can include factors based on a sine wave. This is common practice in fields such as economics where seasonal adjustements are made. That is beyond the scope of this guide.
A comb filter adds a delayed version of a signal to itself. This is used mainly in some specific electrical engineering cases where a signal at a particular known frequency is of interest. That is beyond the scope of this guide.
Robust filtering attempts to reduce the impact of outliers. Most filters have some sensitivity to outliers. That is beyond the scope of this guide.