Convolution

Signal:   Impulse Impulse Train Single Step Square Pulse Triangular Step Triangular Pulse Sawtooth Pulse Random Step Sine (Low Frequency) Sine (High Frequency) Modulated Sine Sinc Random Low-Pass Filter High-Pass Filter Frequency:

System: Impulse Impulse Train Single Step Square Pulse Triangular Step Triangular Pulse Sawtooth Pulse Random Step Sine (Low Frequency) Sine (High Frequency) Modulated Sine Sinc Random Low-Pass Filter High-Pass Filter Frequency:

Swap Reverse Pause Forward Replay

Not Supported

Introduction: This application demonstrates the convolution of a signal by a system. Various signals and systems may be selected, and the output of the convolution sum is displayed.

Directions:

• Selecting Signal and System: Select the signal type and system type using the drop-down boxes. For periodic signals or systems, the frequency can be selected using the Frequency slider. Clicking "Swap" swaps the signal and the system.
• Play Speed: Clicking "Pause" toggles the convolution sum between paused and running. Clicking "Reverse" or "Forward" sets the convolution sum to play backward or forward. Clicking "Replay" causes the convolution sum to restart at the beginning.

This application demonstrates the convolution sum, as a system (green) that processes a signal (red). As the signal slides past the system, the system displays the current output (as the sum of the entire system multiplied by the overlapping portion of the signal) and the total output (as the sum over time).

This application demonstrates the effects of various systems on various signals. For example, a high-pass filter and a low-pass filter are included, and the filtering properties of these systems may be demonstrated by convolution with a high-frequency sinusoid signal and a low-frequency sinusoid signal. Another interesting convolution is the use of a cardinal sine ("sinc") system to convolve a square pulse; the output closely resembles the square pulse, with characteristically sharp discontinuities (the "Gibbs effect") apparent just before and after the sharp edges of the square pulse.

An interesting property of the convolution sum is that swapping the signal and the system produces the same output. This property may be demonstrated by selecting any signal and any system; noting the total output of the convolution; clicking the "Swap" button; and observing that the total output remains the same for the swapped signal and system.

Release history:

• v1.0: Initial release.
• v1.1: Bugfix release with small improvements (thanks to S. Titus for input).

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