convolve_pwc

Graph.convolve_pwc(pwc, kernel)

Create the convolution of a piecewise-constant function with a kernel.

Parameters:

pwc (Pwc) – The piecewise-constant function \(\alpha(t)\) to convolve. You can provide a batch of functions, in which case the convolution is applied to each element of the batch.
kernel (ConvolutionKernel) – The node representing the kernel \(K(t)\).

Returns:

The sampleable function representing the signal \((\alpha * K)(t)\) (or batch of signals, if you provide a batch of functions).

Return type:

Stf

See also

Graph.discretize_stf: Discretize an Stf into a Pwc.
Graph.filter_and_resample_pwc: Filter a Pwc with a sinc filter and resample it.
Graph.gaussian_convolution_kernel: Create a convolution kernel representing a normalized Gaussian.
Graph.pwc: Create piecewise-constant functions.
Graph.sample_stf: Sample an Stf at given times.
Graph.sinc_convolution_kernel: Create a convolution kernel representing the sinc function.

Notes

The convolution is

\[(\alpha * K)(t) \equiv \int_{-\infty}^\infty \alpha(\tau) K(t-\tau) d\tau.\]

Convolution in the time domain is equivalent to multiplication in the frequency domain, so this function can be viewed as applying a linear time-invariant filter (specified via its time domain kernel \(K(t)\)) to \(\alpha(t)\).

For more information on Stf nodes see the Working with time-dependent functions in Boulder Opal topic.

Examples

Filter a piecewise-constant signal using a Gaussian convolution kernel.

>>> gaussian_kernel = graph.gaussian_convolution_kernel(std=1.0, offset=3.0)
>>> gaussian_kernel
<ConvolutionKernel: operation_name="gaussian_convolution_kernel">
>>> pwc_signal
<Pwc: name="alpha", operation_name="pwc_signal", value_shape=(), batch_shape=()>
>>> filtered_signal = graph.convolve_pwc(pwc=pwc_signal, kernel=gaussian_kernel)
>>> filtered_signal
<Stf: operation_name="convolve_pwc", value_shape=(), batch_shape=()>

Refer to the How to add smoothing and band-limits to optimized controls user guide to find the example in context.