SC · LTI and convolution

The Intuition of Convolution

Convolution flips the impulse response, slides it to one instant, multiplies it against the input, and measures the overlapping area. That area is the output at that instant; slide it and watch the whole output curve appear.

Flip, slide, and read the overlap

The top shows the fixed input x(τ) and the response h(t−τ) flipped and slid to instant t. Their green overlap area is the output y(t). Below is that output curve. Move the slide position t and find when the area is largest.

input x(τ)flipped h(t−τ)overlap = y(t)

output y(t)

slide position t-0.50

Output at this instant

y(-0.50) = 0.000

The flipped response barely overlaps the input. The product is almost 0, so the area and the output are near 0.

Little overlap

The output is the overlap area

To get the output at one instant t, take four steps: flip the response (h(−τ)), slide it to t (h(t−τ)), multiply by the input (x(τ)·h(t−τ)), and integrate the product. Where the product is nonzero is the overlap, and its area is y(t). Flip, slide, multiply, integrate — that is all convolution is.

Observey(t) = ∫ x(τ) h(t − τ) dτ

The output is the integral of the product, the overlap area.

Choosey(t) = ∫ x(τ) ? dτ

The integrand is input times the flipped, shifted response.

Fill iny[n] = ?

For discrete signals the integral becomes a sum.

On your owny = x ?

Convolution is written with a star.

Why the flip

In the previous unit, an input impulse x(τ) at instant τ left x(τ)·h(t−τ) in the output, because its delay was t−τ. The output at t adds up these contributions from every τ, so y(t) = ∫ x(τ) h(t−τ) dτ. In the integration variable τ, h’s argument t−τ carries a minus, so h appears flipped left-to-right in the picture. The flip is the natural consequence of delay.

Commutativity and discrete convolution

Convolution is commutative: x ∗ h = h ∗ x, so flipping either the input or the response gives the same result. It is also associative, so systems in series can be merged into one. For discrete signals the integral becomes a sum, y[n] = Σ x[k] h[n−k], and a digital filter computes exactly this sum at every sample.

Back to the first screen

Every time you moved the slide position the green area changed, and that area exactly equalled the value of the output curve below at that instant. Where overlap was scarce the output was near 0; where overlap was maximal the output touched its peak. Convolution slides the flipped response across the input, collecting the overlap area at each instant to build the output curve. One impulse response and this single operation compute every output of an LTI system.

The convolution y(t) = (x ∗ h)(t) = ∫ x(τ) h(t − τ) dτ is four moves: flip the response, slide it to t, multiply by the input, integrate. That overlap area is the output y(t). Convolution is commutative and associative (x ∗ h = h ∗ x), and discretely y[n] = Σ x[k] h[n − k]. An LTI system’s output is the convolution of input with impulse response.

On to the next unit

Now that you hold the picture of overlap, it is time to turn it into numbers. The next unit, computing convolution, integrates the convolution of simple signals like a rectangular pulse piece by piece, so you confirm by hand how the peak and tail you saw arise as formulas. The four moves — flip, slide, multiply, integrate — set the very breakpoints that split the intervals.