An audio engineer fires a starter pistol in a concert hall and records the resulting decaying echoes. She then convolves this recording with a dry piano track. Why does the result sound like the piano played in that hall?
AThe impulse response captures the hall's frequency spectrum, which is then added to the piano's power
BBy linearity, each scaled impulse in the piano signal produces a scaled copy of h(t); by time-invariance, each delayed impulse produces a delayed copy of h(t) — the sum of all these copies is the convolution y = x * h
CThe impulse response averages the room reflections, and convolution applies this average uniformly to the piano signal
DConvolution blends the two signals by computing a running average of their amplitudes
This is the direct derivation of why convolution works. The piano signal x(t) decomposes into scaled, shifted impulses. Linearity says the response to a scaled impulse is a scaled output; time-invariance says the response to a shifted impulse δ(t−τ) is a shifted h(t−τ). Summing all these responses gives y(t) = ∫x(τ)h(t−τ)dτ — convolution. Options A and C misrepresent the mechanism; D confuses convolution with averaging (only true for a box filter, not in general).
Question 2 Multiple Choice
Two different LTI systems produce identical outputs when given a 440 Hz sinusoidal test signal. What can you conclude about the two systems?
AThe systems are identical — matching outputs for any single input means they have the same impulse response
BThe systems have the same gain and phase shift at 440 Hz, but may differ for every other frequency
CThe systems have identical impulse responses except at a single point in time
DThe systems are time-invariant but may not be linear
The impulse response encodes a system's behavior across ALL frequencies and ALL possible inputs. Matching output for a single sinusoidal input tells you only about behavior at that one frequency. The two systems could have entirely different impulse responses — and thus entirely different outputs — for any other input. This illustrates why the impulse response is a complete characterization: you need to test with the impulse (which contains all frequencies) to determine the full system.
Question 3 True / False
For an LTI system, knowing the impulse response h(t) is sufficient to determine the system's output for any possible input.
TTrue
FFalse
Answer: True
This is the central theorem of LTI system analysis. The output is always y(t) = x(t) * h(t) — the convolution of the input with the impulse response. This single function h(t) encodes everything the system can do to any input. The key caveat is that the system must truly be LTI; if the system is nonlinear or time-varying, the impulse response does not fully characterize it.
Question 4 True / False
The impulse response h(t) of an LTI system is defined as the input signal that produces the most useful output from the system.
TTrue
FFalse
Answer: False
This reverses the definition. The impulse response is the system's OUTPUT when the INPUT is an impulse δ(t). It is not a special input — it is what the system produces in response to an idealized spike. The confusion between 'input to the system' and 'output from the system' is the most common error in understanding this definition.
Question 5 Short Answer
Explain why time-invariance is just as essential as linearity for the impulse response to fully characterize an LTI system.
Think about your answer, then reveal below.
Model answer: Linearity alone allows the output to be written as a sum of responses to scaled, shifted impulses: ∫x(τ)·(response to δ(t−τ)) dτ. But without time-invariance, the system's response to δ(t−τ) could be an entirely different function depending on when τ occurs — not simply a shifted version of h(t). Time-invariance guarantees that the response to δ(t−τ) is exactly h(t−τ). Without this, no single function can summarize the system's behavior for all possible inputs and times.
The convolution formula y(t) = ∫x(τ)h(t−τ)dτ requires both properties simultaneously. Linearity enables superposition (building the output from individual impulse responses); time-invariance ensures those responses are all just shifts of the same h(t). Remove either property and the formula breaks down.