Filters

This section explores applying (FIR) filters to time series.

Filters as Convolution

Applying a filter to a time series can be visualized as the convolution of a filter kernel with the data.

First, we can use a simplified example showing the application of a filter kernel (for a narrowband bandpass filter) to a burst signal:

Next, we can look at a applying a filter kernel to a ‘combined’ signal:

Applying a filter does not guarantee the extraction of a rhythmic component of the signal.

For example, we can see this in the ‘ringing’ in the output of applying a filter to a step function:

Notably, due to the shape of the filter kernel, outputs will tend to be smooth & rhythmic, even if the original data is neither.

For example, if we apply a bandpass filter to aperiodic (pink noise) data (non-rhythmic by definition) - the filter output still looks rhythmic:

Filter Outputs

Another way to look at filter outputs is to compare the outputs across different filter kernels.

First, we can see how the output varies across different n_cycles, when applied to a burst signal:

Next, we can see how the output varies across different bandwidths:

We can do the same for a step function signal, first with n_cycles:

And also with bandwidths:

Filter Properties

Finally, we can examine filter properties, explicitly examining the frequency response and filter kernel across different filter definitions.

First, we can look at the filter properties applied to a burst signal across different n_cycles:

We can also look at the filter properties applied to a burst signal across different bandwidths: