Questions: System Causality and Realizability Constraints

Question 1

A signal processing system produces its output at time t = 5 s using input values sampled at t = 3 s, t = 5 s, and t = 8 s. What can be said about this system?

• A. It is causal — it uses two past values and only one future value, so it is mostly causal
• B. It is non-causal — using input at t = 8 s to produce output at t = 5 s requires knowledge of a future input
• C. It is stable — averaging multiple samples prevents unbounded output
• D. It is realizable — it uses a finite number of input samples, so it can be implemented

Answer: (B)

Causality is a strict requirement: the output at any time t may only depend on inputs at times ≤ t. Using even one future input value (t = 8 > t = 5) makes the system non-causal. There is no such thing as 'mostly causal.' Realizability (option D) is related but distinct — a system can use a finite number of inputs and still be non-causal if any input is in the future. Stability (option C) is a separate property entirely and cannot be inferred from the number of input samples.

Question 2

An ideal low-pass filter has a perfectly flat passband magnitude and perfectly linear phase. What does causality theory imply about implementing it in real time?

• A. It is realizable in real time because it has finite passband gain
• B. It is realizable in real time because linear phase is easier to implement than nonlinear phase
• C. It cannot be implemented causally in real time — it requires access to future input samples and must introduce delay
• D. It violates stability constraints, making any implementation impossible

Answer: (C)

An ideal low-pass filter has a symmetric impulse response centered at t = 0, meaning it requires samples from both before and after the current moment — it is non-causal. To run it in real time, you must introduce a delay equal to half the filter length, buffering 'future' samples so they are available when needed. This is exactly why audio and communication processing systems have latency: they are buffering future input to enable non-causal filtering. The filter is not unstable (option D) — it is perfectly stable; it is simply non-causal.

Question 3

A stable system is typically causal, because any system that responds to inputs indefinitely in time is expected to eventually incorporate future input knowledge.

• T. True
• F. False

Answer: False

Stability and causality are independent properties. An ideal low-pass filter is stable (bounded input produces bounded output) but non-causal (its symmetric impulse response extends into negative time). Conversely, a system with a right-half-plane pole is causal (h(t) = 0 for t < 0) but unstable (its output grows without bound). You can have any combination of the two properties: stable-causal, stable-non-causal, unstable-causal, and unstable-non-causal all exist.

Question 4

For a causal system, the magnitude and phase of the frequency response are not independently choosable — specifying the magnitude constrains the achievable phase, and vice versa.

• T. True
• F. False

Answer: True

This follows from the Kramers-Kronig relations: the real and imaginary parts of the frequency response of a causal system are Hilbert transform pairs of each other. In filter design terms, you cannot have both a perfectly flat passband and perfectly linear phase in a causal filter — the two goals are in tension. Minimum-phase filters achieve the sharpest magnitude roll-off but with highly nonlinear phase. Linear-phase FIR filters have excellent phase but require a delay to be made causal. Engineers must choose which degradation to accept.

Question 5

Why does audio processing software often add latency, and what does this have to do with causality?

Model answer: Many desirable audio filters (such as linear-phase FIR filters) are non-causal — their impulse response is symmetric around t = 0, meaning they need to 'see' both past and future samples to compute the output. To run a non-causal filter on a live audio stream, the software buffers incoming audio, introducing a delay equal to the 'look-ahead' needed (half the filter length for a symmetric FIR). This delay is the latency. It is the engineering price of using non-causal filter designs that would otherwise require future input knowledge.

This trade-off appears throughout real-time signal processing: more filter sophistication (flatter magnitude, more linear phase, sharper cutoffs) generally requires more look-ahead, which means more buffer and more latency. Low-latency applications like phone calls tolerate less processing; offline applications like music mastering can use arbitrarily non-causal filters because the entire recording is already available.