# 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

### 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.