LTI Systems
The machines that process signals
You know what signals are. Now meet the things that process them.
A system is anything that takes an input signal and produces an output signal. An amplifier, a filter, a reverb, a noise canceler, a moving average, even a simple delay: all are systems. The input goes in, something happens, and the output comes out.
Most systems are complicated. But one special class of systems, linear time-invariant (LTI) systems, has a remarkable property: a single measurement tells you everything the system will ever do. Feed in one spike, record the output, and you can predict the response to any input.
Input, System, Output
A discrete-time system is a rule that maps an input sequence x[n] to an output sequence y[n]. We write:
x[n] → System → y[n]
Some examples:
- Amplifier: y[n] = 3 · x[n] — scales every sample by 3
- Delay: y[n] = x[n - 2] — shifts the signal 2 samples later
- Moving average: y[n] = (x[n] + x[n-1] + x[n-2]) / 3 — smooths by averaging 3 neighbors
- Squarer: y[n] = x[n]² — squares each sample (nonlinear!)
The first three are LTI. The last one is not. Let's see why.
Linearity
A system is linear if it satisfies two conditions:
- Scaling: If input x produces output y, then input a·x produces output a·y (for any constant a)
- Additivity: If x1 produces y1 and x2 produces y2, then x1 + x2 produces y1 + y2
Together: the response to a weighted sum of inputs is the same weighted sum of individual responses. This is superposition, and it is what makes analysis possible.
The squarer y[n] = x[n]² fails: (x1 + x2)² ≠ x1² + x2². The cross-term 2x1x2 breaks additivity.
Time-Invariance
A system is time-invariant if delaying the input simply delays the output by the same amount:
If x[n] → y[n], then x[n - k] → y[n - k] for any k
The system doesn't care when the signal arrives. It behaves the same today, tomorrow, and next year. A moving average is time-invariant: the formula y[n] = (x[n] + x[n-1] + x[n-2])/3 depends only on relative positions, not on absolute time.
A counterexample: y[n] = n · x[n] is not time-invariant. The output depends on n itself, so the same input produces different outputs at different times.
The Impulse Response
Here's where LTI becomes powerful. Feed the unit impulse δ[n] into an LTI system and record the output. Call it h[n]:
δ[n] → LTI System → h[n]
This is the impulse response. For the moving average, h[n] = [1/3, 1/3, 1/3]. For a pure delay by 2, h[n] = δ[n - 2]. For an echo system, h[n] might be [1, 0, 0, 0.5] (original plus a quieter copy 3 samples later).
Frequently Asked Questions
Are real-world systems actually LTI?
Strictly, no. Real amplifiers saturate (nonlinear at high levels), and component aging makes them time-varying. But many systems behave approximately as LTI over useful operating ranges. The LTI model is enormously useful as a first approximation, and understanding it is a prerequisite for analyzing more complex nonlinear and time-varying systems.
What is a moving average and why is it LTI?
A 3-point moving average computes y[n] = (x[n] + x[n-1] + x[n-2]) / 3. It is linear because it only uses addition and scaling. It is time-invariant because the formula does not depend on the absolute value of n, only on relative positions of input samples. Its impulse response is h[n] = [1/3, 1/3, 1/3].
Why can't you characterize a nonlinear system with an impulse response?
The impulse response trick relies on superposition: the response to a sum of signals equals the sum of individual responses. Nonlinear systems violate this. For example, a squaring system's response to δ[n] is δ[n], but its response to 2·δ[n] is 4·δ[n], not 2·δ[n]. Knowing the impulse response does not predict the output for arbitrary inputs.
Quick Check
Test your understanding of the key concepts from this lesson.
