Virtual Labs

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

1. In the context of a wide-sense stationary (WSS) process, which of the following is true regarding the concept of **ergodicity**?

a: A WSS process is always ergodic in both time and ensemble. Explanation

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b: For a WSS process to be ergodic, the time average of the process must equal the ensemble average for all values of \( t \). Explanation

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c: Ergodicity implies that the statistical properties of the process can be determined from a single realization of the process over time. Explanation

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d: A WSS process is guaranteed to be ergodic if its autocorrelation function is non-negative. Explanation

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2. An LTI system has an impulse response that is zero for all values of time greater than a certain constant. What does this imply about the system?

a: The system is time-invariant but not necessarily causal. Explanation

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b: The system is causal and has finite memory. Explanation

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c: The system is unstable. Explanation

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d: The system is invertible. Explanation

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3. 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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4. How does the inclusion of a moving average (MA) term in an ARMA model affect its frequency response?

a: It sharpens the peak of the response. Explanation

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b: It smoothens the response by adding zeros. Explanation

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c: It introduces poles into the system. Explanation

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d: It does not affect the frequency response. Explanation

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5. In the context of stochastic processes, what distinguishes a strict-sense stationary (SSS) process from a wide-sense stationary (WSS) process in terms of their moment invariance properties?

a: WSS requires the invariance of all higher-order moments, while SSS only requires the invariance of the second-order moments. Explanation

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b: SSS guarantees the invariance of all statistical moments, including higher-order moments, whereas WSS only ensures invariance of the first and second moments. Explanation

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c: SSS demands invariance of all moments, including higher-order moments, while WSS only imposes conditions on the mean, variance, and autocovariance (second-order moments). Explanation

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d: There is no significant difference in the moment invariance requirements between WSS and SSS processes. Explanation

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6. 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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7. 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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8. For an AR(2) process defined by X(n) = a1·X(n−1) + a2·X(n−2) + W(n), where W(n) is white noise, what does it imply about the spectral characteristics of the process if the characteristic equation has roots lying exactly on the unit circle in the z-plane?

a: The process exhibits infinite power at specific frequencies, indicating a non-stationary behavior. Explanation

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b: The process is stationary with a flat power spectral density. Explanation

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c: The process is marginally stable and exhibits purely deterministic sinusoidal oscillations. Explanation

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d: The process is unstable and its power spectrum is undefined. Explanation

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9. 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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10. What is the effect of adding a long-memory MA term to an ARMA model in the frequency domain?

a: It increases the bandwidth of the process. Explanation

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b: It adds sharp resonances at specific frequencies. Explanation

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c: It suppresses high-frequency components. Explanation

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d: It enhances low-frequency components. Explanation

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