EC-7.3 Impulse Response and Convolution

What this is and why it exists

This is the most reused result in the area, by a wide margin.

Consider a matched filter in a receiver, a channel model, a moving average, or an echo added to a recording. Consider the blur in an out-of-focus photograph, or the response of a circuit to a switch closing. All of these are the same operation, with different inputs. Learn it once properly and six later subjects become recognisable rather than new.

It is also the result that makes the transform methods worth learning at all. The operation here is awkward to compute and awkward to picture. In the frequency domain it becomes an ordinary multiplication. That trade is the reason anyone bothers with transforms. It cannot be appreciated until the awkward version has been done by hand a few times.

The vocabulary

• Impulse response — the output of a system when the input is a unit impulse.
• Convolution — the operation that produces the output from the input and the impulse response.
• Convolution sum — the discrete-time form, a weighted total.
• Convolution integral — the continuous-time form.
• Graphical convolution — the method of reflecting one signal, sliding it, and measuring the overlap at each position.
• Absolutely summable — a sequence whose total absolute size is finite.
• Step response — the output when the input is a unit step.
• Cascade — two systems in series, the output of one feeding the other.

The mental model

Here is the argument, and it is short enough to hold in your head.

Any input can be written as a sum of impulses, each shifted to a different position and scaled by the value of the input there. That is not a trick; in discrete time it is a restatement of what the values are.

Now use the two properties from the last topic. Linearity says the response to that sum is the sum of the responses to each piece. Time invariance says the response to an impulse at some position is the impulse response, shifted to that position.

Put the two together. The output is a sum of shifted impulse responses, each scaled by the corresponding input value. Written out, that sum is convolution. Nothing else about the system was needed. One measured response describes it completely.

That is worth pausing on. A system might contain a thousand components. Measure its response to one sharp tap and you can predict its response to anything at all.

In discrete time the computation is concrete. The output at each position is a weighted total of input values, and the impulse response supplies the weights. Work two or three of these by hand. The formula stops being a formula somewhere in the second one.

In continuous time the sum becomes an integral and the standard method is graphical. Reflect one of the two signals about the origin. Slide it across the other. At each position, multiply the two where they overlap and find the area of the product. That area is the output at that position.

The difficulty is never the multiplication; it is the limits. As the sliding signal moves, the region of overlap changes, and each distinct region needs its own integral with its own limits. Sketch the two signals at three or four representative positions, and mark where the overlap begins and ends. That is what turns this from guesswork into bookkeeping.

Convolution has three properties that get used constantly. It is commutative, so which signal you reflect is your choice; reflect whichever is simpler. It is associative, which means two systems in series can be replaced by a single system whose impulse response is the convolution of the two. It is distributive over addition, which means two systems in parallel can be replaced by one whose impulse response is the sum. Those last two rules are the whole of block diagram simplification, and they are why a long signal chain can be reasoned about at all.

Two properties from the last topic now become inspections rather than arguments.

A system is causal exactly when its impulse response is zero before the origin. The reason is immediate: a nonzero value before the origin would mean the response began before the tap arrived.

A system is stable exactly when the total absolute size of its impulse response is finite. In discrete time that is a sum; in continuous time an integral. The picture behind it is a bounded input arriving forever, with the response to every past instant still adding up. If those contributions have a finite total, the output stays bounded. If not, it does not.

One practical note to end on. The impulse response is the natural object in theory and an awkward one in the laboratory, because a true impulse cannot be produced. The step response can be measured, by switching something on and recording what happens. The step response is the running total of the impulse response, and the impulse response is the rate of change of the step response. So the measurable one gives you the theoretical one, which is how this is done in practice.

What you should now be able to explain or do

• Explain why the impulse response describes a linear time-invariant system completely, using both properties in the argument.
• Compute a discrete convolution by hand and say what the impulse response supplies.
• Carry out a graphical convolution, identifying the distinct regions of overlap and the limits for each.
• Use the associative and distributive properties to combine systems in series and in parallel.
• Decide causality and stability by inspecting an impulse response.
• Move between the step response and the impulse response in either direction.

Check yourself