SC · LTI and convolution

The Intuition of Convolution

Convolution means flipping the impulse response, sliding it to one instant, multiplying it by the input and measuring the area where they overlap. See for yourself that this area is the output at that instant, and that sliding the instant along draws the whole output curve.

Flip, slide, and read the overlap

In the upper figure the gold rectangular pulse is the input x(τ), and the blue curve is h(t − τ), the response h flipped and slid to the instant t. The green area of their product is the output y(t), and the lower figure is that output curve. Move the slide position t and watch when the area is largest: at that instant the dot on the lower curve also reaches the peak.

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 and slid response barely overlaps the input. The product is nearly 0, so the area, and the output, are close to 0.

Little overlap

The output is the overlap area

To find the output at one instant t takes four steps: flip the response (h(−τ)), slide it to the instant t (h(t − τ)), multiply it by the input (x(τ)·h(t − τ)), and integrate the product. The stretch where the product is not 0 is the overlap, and its area is y(t), so flip, slide, multiply and integrate are all there is to convolution.

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, the 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 instant t is the sum of these contributions from every τ, so y(t) = ∫ x(τ) h(t − τ) dτ. With respect to the integration variable τ, the argument of h is t − τ, so its sign is reversed, and in the figure h appears flipped left to right. The flip comes naturally out of the delay.

Commutativity and discrete convolution

Convolution is commutative. Since x ∗ h = h ∗ x, the result is the same whichever of the input and the response is flipped. It is associative as well, so systems connected 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 the slide position moved, the green area changed, and that area matched the value of the lower output curve at that instant exactly. When there was hardly any overlap the output was close to 0, and near t = 1, where the overlap was largest, the output reached its peak. Convolution slides the flipped response across the input and gathers the overlapping area at each instant to build the output curve. With a single impulse response and this one operation, every output of an LTI system can be computed.

Convolution y(t) = (x ∗ h)(t) = ∫ x(τ) h(t − τ) dτ is made of four actions: flip the response, slide it to the instant t, multiply it by the input and integrate, and the overlapping area is the output y(t). Convolution is commutative and associative (x ∗ h = h ∗ x), and in discrete time it is y[n] = Σ x[k] h[n − k]. The output of an LTI system is the convolution of the input with the impulse response.

On to the next unit

With the picture of overlap in hand, it is time to turn it into numbers. In the next unit, computing convolution, you integrate the convolution of simple signals such as rectangular pulses piece by piece and check by hand which formulas produce the peak and the tail seen in the figure. For this unit's figure, the output rises as y = (1 − e^−1.5t)/1.5 for 0 ≤ t ≤ 1 and dies away exponentially for t > 1. The four actions of flip, slide, multiply and integrate set exactly the boundaries that divide the pieces.