EC-7.3 Impulse Response and Convolution
You can compute the output of a linear time-invariant system by convolution, in both continuous and discrete time, and read causality and stability off the impulse response.
Before:EC-4. Transforms, Complex Variables and Numerical MethodsUnlocks:S4-1. Analog CircuitsS4-3. Control SystemsS4-5. Probability Theory & Stochastic ProcessesS5-3. Digital Signal ProcessingPE2-3. Biomedical Instrumentation and Signal ProcessingEC-9. Microcontrollers and Embedded Systems
This is the single most reused result in the area. A matched filter, a channel model, a moving average, an echo, the blur in an image and the response of a circuit to a switch closing are all the same operation with different inputs. It is also the result that makes the transform methods worth learning, because the reason to move to a transform domain is that this operation becomes a multiplication there.
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The impulse response, and why it describes such a system completely
Any input can be written as a sum of shifted and scaled impulses, so linearity and time invariance together mean the response to the impulse determines the response to everything. Nothing else about the system needs to be known.
MIT RES.6-007: Signals and Systems · CourseThe convolution sum, built from shifted and scaled impulses
In discrete time the output at each position is a weighted total of input values, with the impulse response supplying the weights. Working two or three of these by hand is what makes the operation stop feeling like a formula.
NPTEL: Principles of Signals and Systems · CourseThe convolution integral, and the graphical method for it
In continuous time the same operation becomes an integral, and the standard method is to reflect one signal, slide it across the other, and integrate the overlap at each position. The limits change as the overlap changes, which is where most errors live.
NPTEL: Signals and Systems (IIT Kanpur) · CourseProperties of convolution, and the systems in series or parallel
The operation is commutative, associative and distributive, which is exactly what lets two systems in series be replaced by one and two in parallel be added. Those two rules are used constantly in block diagram work.
Reading causality and stability off the impulse response
A causal system has an impulse response that is zero before the origin, and a stable one has an impulse response whose total absolute size is finite. Both properties become inspections rather than arguments.
NPTEL: Signals and Systems (IISER Bhopal) · CourseThe step response, and moving between it and the impulse response
The step response is the running total of the impulse response, and the impulse response is the rate of change of the step response. The step response is often the one you can actually measure on a real system.
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