Virtual Labs

Study the statistical properties of the output response of a system when the input is wide sense stationary

In the context of a wide-sense stationary (WSS) process, what does ergodicity mean?

a: A WSS process is always ergodic, regardless of its autocorrelation properties. Explanation

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b: The statistical properties of the process can be obtained from a single realization over a long period of time. Explanation

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c: The process must have a zero mean and constant variance. Explanation

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d: The autocorrelation function must be non-negative at all time lags. Explanation

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What is the main effect of a moving average (MA) term in an ARMA model on the signal?

a: It introduces long-term memory into the signal. Explanation

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b: It smooths the signal by averaging past noise contributions. Explanation

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c: It makes the signal completely deterministic. Explanation

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d: It increases the signal frequency content. Explanation

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In the context of discrete-time systems, what is the implication of having poles outside the unit circle in the z-domain for the system's stability and behavior?

a: The system exhibits instability and potential unbounded output for certain input signals. Explanation

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b: The system's frequency response will exhibit unbounded growth for high frequencies. Explanation

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c: The system may exhibit non-linear behavior despite being linear in nature. Explanation

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d: The system can still be stable but may exhibit a non-minimum phase characteristic. Explanation

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Which of the following best describes a stochastic process?

a: A deterministic function that evolves over time according to a fixed rule. Explanation

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b: A collection of random variables indexed by time, representing the evolution of a system under uncertainty. Explanation

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c: A single random variable with a known probability distribution. Explanation

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d: A linear filter applied to a deterministic signal. Explanation

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Which of the following is a property of white noise in time series analysis?

a: It has a non-zero mean and time-dependent variance. Explanation

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b: Its values at different times are uncorrelated and have constant variance. Explanation

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c: It always follows a uniform distribution. Explanation

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d: It can only be used in deterministic models. Explanation

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When a wide-sense stationary (WSS) signal passes through a Linear Time-Invariant (LTI) system, what impact does the system's characteristics have on the stationarity and statistical properties of the output signal?

a: The output signal will remain WSS, but its autocorrelation function will be modified based on the system's impulse response. Explanation

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b: The output signal may become non-stationary depending on the system's impulse response and stability characteristics. Explanation

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c: The output signal will retain the same autocorrelation function as the input signal, as long as the system is causal. Explanation

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d: The stationarity of the output is guaranteed only if the system is both causal and stable. Explanation

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In an ARMA(1,1) model, what happens when the pole and zero are very close to each other in the z-domain?

a: The system exhibits oscillatory behavior. Explanation

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b: The system becomes unstable. Explanation

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c: The system's response becomes heavily damped. Explanation

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d: The pole-zero cancellation occurs, simplifying the system. Explanation

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For an AR(1) process X(n) = a·X(n−1) + W(n), where W(n) is zero-mean white noise, what happens to the power spectral density as |a| approaches 1?

a: The spectrum becomes more concentrated around low frequencies, and the process becomes nearly non-stationary. Explanation

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b: The process becomes more like white noise with a flat spectral density. Explanation

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c: The spectrum becomes uniform across all frequencies. Explanation

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d: The process loses all memory and becomes purely random. Explanation

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How does the inclusion of a high-order AR term in an ARMA model influence its autocorrelation properties?

a: It shortens the decay rate of the autocorrelation. Explanation

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b: It extends the memory of the process. Explanation

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c: It eliminates oscillatory behavior in the autocorrelation. Explanation

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d: It has no effect on the autocorrelation. Explanation

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Why is the ARMA model considered more parsimonious compared to using a pure AR or MA model for a stationary time series?

a: Because it eliminates the need for white noise in modeling. Explanation

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b: Because it combines both AR and MA terms, capturing dependencies efficiently with fewer parameters. Explanation

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c: Because it guarantees strict stationarity for all time series. Explanation

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d: Because it only uses past noise values and ignores past signal values. Explanation

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