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The behavior of a linear, time-invariant discrete-time system
with input signal x[n] and output signal y[n]
is described by the convolution sum:
The signal h[n], assumed known, is the response of the system to a unit-pulse input.

The convolution summation has a simple graphical interpretation. First, plot h[k] and the
"flipped and shifted" x[n - k] on the k axis, where n is fixed.
Second, multiply the two signals to obtain a plot of the summand sequence indexed by k.
Summing the values of this sequence with respect to k yields y[n]. These operations can be
repeated for every value of n of interest.

The behavior of a linear, continuous-time, time-invariant system with input signal x(t)
and output signal y(t) is described by the convolution integral
The signal h(t), assumed known, is the response of the system to a unit impulse input.
To compute the output y(t) at a specified t, first the integrand h(v)x(t - v) is computed as
a function of v.Then integration with respect to v is performed, resulting in y(t).

These mathematical operations have simple graphical interpretations.First, plot h(v) and
the "flipped and shifted" x(t - v) on the v axis, where t is fixed. Second, multiply the
two signals and compute the signed area of the resulting function of v to obtain y(t).
These operations can be repeated for every value of t of interest.

To explore graphical convolution, select signals
x[n] or h[n]
from the provided examples below, or use the mouse to create your own or modify a signal.
Furthmore, by using a right mouse click, one can start drawing polygons between right mouse click events and end them with a single left mouse click.

Click and drag on the the plot displaying x[n-k] and h[k] to modify the current k-value.
During dragging the plot of x[n-k] * h[k] as well as the plot of y[n]will be updated live.