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Filtering Random Signals through LTI Systems

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LTI systems filtering random signals

Quick recall · 183 cards

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The autocorrelation of a wide-sense stationary random process is given by \( e^{-2|\tau|} \). The peak value of the power spectral density is:

A · \( \frac{1}{2} \)

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When plotted as a function of increasing frequency, noise phenomena are arranged in a specific order of dominance. Which of the following correctly represents this order?

B · Flicker noise, White noise, Transit time noise

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Which of the following best defines a random process?

B · A collection of random variables indexed by time or space

A random process is defined as a collection of random variables indexed by time or space, representing signals or phenomena that evolve randomly over time.

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Which property is NOT generally true for a random process?

B · Its future values can be exactly predicted from past values

Random processes are inherently unpredictable; their future values cannot be exactly predicted from past values, unlike deterministic signals.

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Which of the following statements correctly describes the mean function \( m_X(t) \) of a random process \( X(t) \)?

A · \( m_X(t) = E[X(t)] \), the expected value of \( X(t) \) at time \( t \)

The mean function of a random process is the expected value of the process at each time instant, representing its average behavior.

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The autocorrelation function \( R_X(\tau) \) of a wide-sense stationary (WSS) random process depends on:

C · Only the time lag \( \tau \)

For a WSS process, the autocorrelation function depends only on the time difference (lag) \( \tau \), not on the absolute time \( t \).

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Which of the following properties is true for the autocorrelation function \( R_X(\tau) \) of a real-valued random process?

B · \( R_X(\tau) = R_X(-\tau) \) (even function)

The autocorrelation function of a real-valued random process is an even function, i.e., symmetric about zero lag.

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If the autocorrelation function \( R_X(\tau) \) of a stationary random process satisfies \( R_X(0) = 5 \), what does this value represent?

A · The average power of the process

For a stationary random process, the autocorrelation at zero lag \( R_X(0) \) equals the average power of the process.

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Which of the following is NOT a valid property of the autocorrelation function \( R_X(\tau) \)?

B · It is always positive for all \( \tau \)

The autocorrelation function can take negative values for some lags \( \tau \); it is not always positive.

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Refer to the diagram below showing the autocorrelation function \( R_X(\tau) \) of a random process.
Which of the following statements is true about the process based on the diagram?

C · The process is wide-sense stationary with power \( R_X(0) \)

The symmetric autocorrelation function with a maximum at zero lag indicates a wide-sense stationary process with average power equal to \( R_X(0) \).

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The Power Spectral Density (PSD) \( S_X(f) \) of a random process is defined as:

B · The Fourier transform of the autocorrelation function \( R_X(\tau) \)

The PSD is defined as the Fourier transform of the autocorrelation function of the random process.

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Which of the following is a property of the Power Spectral Density (PSD) \( S_X(f) \) of a real-valued random process?

C · \( S_X(f) \) is real and even function of \( f \)

The PSD of a real-valued random process is a real and even function of frequency.

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The total average power of a stationary random process can be found by:

A · Integrating the PSD over all frequencies

The total average power is the integral of the PSD over all frequencies, which equals \( R_X(0) \).

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Refer to the PSD plot below of a random process. Which characteristic can be inferred from the plot?

B · The process is band-limited

A PSD plot that is zero outside a finite frequency range indicates the process is band-limited.

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Which theorem relates the autocorrelation function and the power spectral density of a wide-sense stationary random process?

B · Wiener-Khinchin theorem

The Wiener-Khinchin theorem states that the PSD is the Fourier transform of the autocorrelation function for WSS processes.

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According to the Wiener-Khinchin theorem, the power spectral density \( S_X(f) \) is the Fourier transform of \( R_X(\tau) \). What is the inverse transform of \( S_X(f) \)?

B · Autocorrelation function \( R_X(\tau) \)

The inverse Fourier transform of the PSD \( S_X(f) \) yields the autocorrelation function \( R_X(\tau) \).

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Which of the following statements is TRUE about the Wiener-Khinchin theorem?

C · It requires the random process to be wide-sense stationary

The Wiener-Khinchin theorem applies to wide-sense stationary random processes, relating their autocorrelation and PSD via Fourier transform.

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Refer to the diagram below showing the Fourier transform pair between \( R_X(\tau) \) and \( S_X(f) \). Which property of \( S_X(f) \) is illustrated?

B · PSD is real and even

The diagram shows that \( S_X(f) \) is symmetric about zero frequency and real-valued, consistent with PSD properties.

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A random process \( X(t) \) is said to be wide-sense stationary (WSS) if:

A · Its mean and autocorrelation are independent of time \( t \)

WSS requires the mean to be constant and the autocorrelation function to depend only on lag \( \tau \), not on absolute time \( t \).

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Which of the following is a necessary condition for strict-sense stationarity (SSS) of a random process?

A · All joint distributions are invariant under time shifts

SSS requires all finite-dimensional distributions to be invariant under time shifts, a stronger condition than WSS.

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Which statement is TRUE for a wide-sense stationary random process \( X(t) \)?

C · Autocorrelation depends only on time difference \( \tau \)

In WSS processes, autocorrelation depends only on the lag \( \tau \), not on absolute time.

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Refer to the diagram below showing the mean and autocorrelation of two processes. Which process is wide-sense stationary?

B · Process B with constant mean and autocorrelation depending on lag only

WSS requires constant mean and autocorrelation depending only on lag, as shown by Process B.

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The autocorrelation function of a random telegraph signal is given by \( R_X(\tau) = e^{-\lambda |\tau|} \). What is the corresponding PSD \( S_X(f) \)?

A · \( \frac{2\lambda}{\lambda^2 + (2\pi f)^2} \)

The Fourier transform of \( e^{-\lambda |\tau|} \) is \( \frac{2\lambda}{\lambda^2 + (2\pi f)^2} \).

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For a white noise process with power spectral density \( S_X(f) = N_0/2 \), the autocorrelation function \( R_X(\tau) \) is:

B · \( \frac{N_0}{2} \delta(\tau) \)

The autocorrelation of white noise is \( R_X(\tau) = \frac{N_0}{2} \delta(\tau) \), where \( \delta(\tau) \) is the Dirac delta function.

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Refer to the diagram below showing the PSD of two random processes. Which process has a wider bandwidth?

B · Process B with broad flat spectrum

A broad flat spectrum indicates a wider bandwidth compared to a narrow peak.

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The autocorrelation function of a stationary Gaussian random process is given by \( R_X(\tau) = \sigma^2 e^{-\alpha |\tau|} \). What is the bandwidth of the process?

A · \( \alpha / \pi \)

The bandwidth of an exponentially correlated Gaussian process is \( \alpha / \pi \).

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Which of the following is an application of autocorrelation in signal analysis?

A · Estimating signal power and detecting periodicities

Autocorrelation helps estimate signal power and detect periodic components in signals.

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Power spectral density (PSD) analysis is commonly used in noise characterization because it:

B · Identifies frequency components and power distribution of noise

PSD provides information about how noise power is distributed across frequencies, essential for noise analysis.

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Refer to the diagram below showing autocorrelation functions of noisy signals. Which signal likely has higher noise power?

A · Signal A with narrower autocorrelation peak

A narrower autocorrelation peak indicates less correlation and higher noise power.

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Which of the following is an analytical method to estimate the PSD from a finite-length signal?

A · Autocorrelation method using Fourier transform

The autocorrelation method estimates PSD by computing the autocorrelation function and applying Fourier transform.

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The PSD of a random process is given by \( S_X(f) = \frac{1}{1 + (f/f_c)^2} \). What type of process does this PSD correspond to?

C · First-order low-pass filtered white noise

This PSD corresponds to a first-order low-pass filtered white noise with cutoff frequency \( f_c \).

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Which of the following best describes a random process?

B · A collection of random variables indexed by time or space

A random process is defined as a collection of random variables indexed by time or space, representing signals whose values evolve randomly over time.

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Which of the following is NOT a classification of random processes based on time dependency?

C · Periodic

Periodic is a classification of deterministic signals, not random processes. Random processes are classified as stationary, ergodic, or non-stationary based on their statistical properties over time.

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Which of the following statements correctly describes the autocorrelation function \( R_X(\tau) \) of a random process \( X(t) \)?

A · \( R_X(\tau) = E[X(t)X(t+\tau)] \), where \( E[\cdot] \) denotes expectation

The autocorrelation function of a random process is defined as the expected value of the product of the process at two time instants separated by \( \tau \).

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Which property of the autocorrelation function \( R_X(\tau) \) is always true for a real-valued random process?

C · It is an even function of \( \tau \)

For a real-valued random process, the autocorrelation function is always an even function, i.e., \( R_X(\tau) = R_X(-\tau) \).

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If a random process \( X(t) \) is wide-sense stationary (WSS), which of the following statements about its autocorrelation function \( R_X(t_1, t_2) \) is true?

C · \( R_X(t_1, t_2) \) depends only on the time difference \( \tau = t_2 - t_1 \)

For a WSS process, the autocorrelation function depends only on the time difference \( \tau = t_2 - t_1 \), not on the absolute time instants.

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Refer to the diagram below showing the autocorrelation function \( R_X(\tau) \) of a WSS random process. What is the value of \( R_X(0) \) in terms of the process?

A · Average power of the process

The autocorrelation function at zero lag \( R_X(0) \) equals the average power of the random process.

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Which of the following defines the Power Spectral Density (PSD) \( S_X(f) \) of a WSS random process \( X(t) \)?

B · Fourier transform of the autocorrelation function \( R_X(\tau) \)

The PSD of a WSS random process is defined as the Fourier transform of its autocorrelation function.

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Which property is always true for the Power Spectral Density \( S_X(f) \) of a real-valued WSS random process?

C · \( S_X(f) \) is a real and even function of frequency \( f \)

For real-valued WSS processes, the PSD is real and even, i.e., \( S_X(f) = S_X(-f) \).

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Refer to the PSD plot below of a random process. What does the peak at frequency \( f_0 \) indicate about the process?

A · The process has maximum power concentrated at frequency \( f_0 \)

A peak in the PSD at frequency \( f_0 \) indicates that the process has maximum power concentrated at that frequency component.

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Which mathematical relationship correctly expresses the connection between the autocorrelation function \( R_X(\tau) \) and the power spectral density \( S_X(f) \) of a WSS random process?

C · Both A and B

The autocorrelation function and PSD form a Fourier transform pair, with \( S_X(f) \) being the Fourier transform of \( R_X(\tau) \) and vice versa.

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Which of the following statements about the Fourier transform relationship between autocorrelation and PSD is FALSE?

C · The autocorrelation function can be complex-valued

The autocorrelation function of a real-valued random process is always real and even, not complex-valued.

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Refer to the diagram below showing the autocorrelation function and its corresponding PSD. If the autocorrelation function is a rectangular pulse of width \( 2T \), what is the shape of the PSD?

A · A sinc squared function

The Fourier transform of a rectangular function is a sinc function, so the PSD will be a sinc squared function due to power spectral density being the magnitude squared.

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For a white noise random process with autocorrelation function \( R_X(\tau) = N_0/2 \delta(\tau) \), what is the corresponding PSD \( S_X(f) \)?

A · A constant \( N_0/2 \) for all frequencies

White noise has an autocorrelation function proportional to a delta function, resulting in a constant PSD across all frequencies.

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Given a random process with autocorrelation \( R_X(\tau) = e^{-\alpha |\tau|} \), where \( \alpha > 0 \), what is the shape of its PSD \( S_X(f) \)?

A · Lorentzian function \( \propto \frac{1}{\alpha^2 + (2\pi f)^2} \)

The Fourier transform of an exponential autocorrelation function is a Lorentzian function in the frequency domain.

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Refer to the diagram below showing the autocorrelation function of a random telegraph signal. Which property can be inferred from the shape of the autocorrelation?

A · The process is WSS with exponentially decaying correlation

The exponentially decaying autocorrelation indicates a WSS process with memory fading over time.

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Which of the following random processes has a PSD that is flat over all frequencies?

A · White noise

White noise has a flat PSD, meaning equal power at all frequencies.

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Given the PSD \( S_X(f) = \frac{1}{1 + (f/f_c)^2} \), where \( f_c \) is a cutoff frequency, what is the corresponding autocorrelation function \( R_X(\tau) \)?

A · \( R_X(\tau) = e^{-2\pi f_c |\tau|} \)

The inverse Fourier transform of the given Lorentzian PSD is an exponentially decaying autocorrelation function.

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Which application uses autocorrelation to detect periodicity in signals?

A · Radar signal processing

Autocorrelation is widely used in radar signal processing to detect periodicities and time delays.

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In communication systems, which of the following is a typical use of PSD analysis?

A · To determine the bandwidth of the transmitted signal

PSD analysis helps in determining the frequency content and bandwidth of signals in communication systems.

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Refer to the diagram below showing PSD plots of different noise types. Which noise type corresponds to the PSD decreasing as \( 1/f^2 \)?

C · Brownian noise

Brownian noise (or red noise) has a PSD that decreases proportionally to \( 1/f^2 \).

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Which noise type is characterized by a flat PSD across all frequencies?

A · White noise

White noise has a constant PSD, meaning equal power at all frequencies.

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Which of the following noise types has a PSD proportional to \( 1/f \)?

B · Pink noise

Pink noise has a PSD that decreases proportionally to \( 1/f \), also called 1/f noise.

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Which of the following is a key property of the autocorrelation function of a WSS random process?

C · It is a function of lag \( \tau \) only

For WSS processes, the autocorrelation depends only on the lag \( \tau \), not on absolute time.

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Which of the following random processes has an autocorrelation function that is a delta function?

A · White noise

White noise has an autocorrelation function proportional to a delta function, indicating no correlation at different times.

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Refer to the diagram below showing the PSD of a band-limited noise process. What is the bandwidth of the process?

A · The frequency range where PSD is non-zero

Bandwidth is defined as the frequency range over which the PSD is non-zero or significant.

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Which of the following statements is true about the relationship between autocorrelation and PSD for a WSS random process?

A · Autocorrelation is the inverse Fourier transform of PSD

The autocorrelation function is the inverse Fourier transform of the PSD, and vice versa.

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Which noise type is characterized by increasing power spectral density with frequency?

C · Blue noise

Blue noise has a PSD that increases with frequency, opposite to pink noise.

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Refer to the signal waveform below. Which method would you use to estimate its autocorrelation function?

A · Time averaging over multiple realizations

Autocorrelation estimation for random signals is typically done by time averaging over multiple realizations or long time intervals.

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Which of the following is a direct application of PSD analysis in telecommunications?

A · Determining channel noise characteristics

PSD analysis helps characterize the noise in communication channels, which is crucial for system design.

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Which of the following is NOT a property of the autocorrelation function of a WSS random process?

C · Always non-negative for all lags

The autocorrelation function can take negative values for some lags; it is not always non-negative.

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A random process X(t) has autocorrelation R_X(\tau) = \frac{1}{1 + (2\tau)^2}. Determine which of the following statements about its power spectral density S_X(f) is correct.

B · S_X(f) = \frac{\pi}{2} e^{-\pi |f|} and is not bandlimited

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A stationary random process X(t) has power spectral density S_X(f) = \frac{1}{1 + (f/0.7)^4}. Which of the following statements about its autocorrelation function R_X(\tau) is TRUE?

B · R_X(\tau) decays slower than exponential and has oscillatory behavior

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A WSS random process X(t) has autocorrelation R_X(\tau) = \frac{\sin(5\pi \tau)}{5\pi \tau} e^{-0.1|\tau|}. Which of the following statements about its power spectral density S_X(f) is TRUE?

A · S_X(f) is the convolution of a rectangular function of width 5 Hz and a Lorentzian with bandwidth 0.1 Hz

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A WSS random process X(t) has autocorrelation R_X(\tau) = e^{-\gamma \tau^2} cos(2\pi f_0 \tau). Which of the following best describes the nature of its power spectral density S_X(f)?

A · S_X(f) is the sum of two Gaussian functions centered at ±f_0

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Which of the following best defines white noise in the context of random signals?

B · A random signal with zero mean and constant power spectral density over all frequencies

White noise is defined as a random signal having zero mean and a constant power spectral density across all frequencies, implying equal power at all frequency components.

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Which characteristic is NOT true for an ideal white noise process?

C · It has a finite bandwidth

Ideal white noise has infinite bandwidth with a flat power spectral density. Having a finite bandwidth contradicts the ideal white noise definition.

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Which statement correctly describes the mean value of white noise?

B · Mean is zero

White noise is typically modeled as a zero-mean random process, meaning its expected value at any time instant is zero.

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Which of the following is a statistical property of white noise?

B · Its samples at different times are statistically independent

White noise samples at different time instants are statistically independent (or uncorrelated), which is a key statistical property.

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Which expression correctly represents the autocorrelation function \( R_{ww}(\tau) \) of an ideal white noise process with power \( N_0/2 \)?

A · \( R_{ww}(\tau) = \frac{N_0}{2} \delta(\tau) \)

The autocorrelation function of ideal white noise is an impulse function scaled by \( \frac{N_0}{2} \), indicating zero correlation at all non-zero time lags.

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Refer to the diagram below showing the autocorrelation function of a noise process.

Which noise type does this autocorrelation function represent?

B · White noise

The autocorrelation function shown is an impulse at zero lag, which is characteristic of ideal white noise, indicating no correlation at any other time lag.

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Which of the following correctly describes the power spectral density (PSD) of ideal white noise?

C · PSD is constant for all frequencies

Ideal white noise has a flat (constant) power spectral density across all frequencies, indicating equal power at every frequency component.

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Refer to the diagram below showing the power spectral density of a noise signal.

What type of noise does this PSD represent?

A · White noise

The PSD is flat across all frequencies, which is the defining characteristic of white noise.

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Which mathematical relationship correctly links the autocorrelation function \( R_{ww}(\tau) \) and power spectral density \( S_{ww}(f) \) of white noise?

A · \( S_{ww}(f) = \int_{-\infty}^{\infty} R_{ww}(\tau) e^{-j2\pi f \tau} d\tau \)

The power spectral density and autocorrelation function form a Fourier transform pair, with PSD being the Fourier transform of the autocorrelation function.

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Which statement best describes the relationship between white noise and random signals?

B · White noise is a special type of random signal with uncorrelated samples and flat PSD

White noise is a particular kind of random signal characterized by uncorrelated samples and a flat power spectral density.

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Which property distinguishes white noise from other random signals?

B · It has a flat power spectral density over all frequencies

White noise is distinguished by having a flat power spectral density, meaning equal power at all frequencies, unlike other random signals.

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A random signal has an autocorrelation function \( R_{xx}(\tau) = 5 \delta(\tau) \). What can be inferred about this signal?

B · It is white noise with power spectral density \( S_{xx}(f) = 5 \) for all frequencies

An autocorrelation function proportional to a delta function indicates white noise, with the constant factor representing the power spectral density.

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Which of the following is a practical application of white noise in signal processing?

B · Modeling thermal noise in electronic circuits

White noise is used to model thermal noise and other random noise sources in electronic circuits due to its statistical properties.

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Which implication of white noise properties is crucial for system identification and testing?

B · Its flat power spectral density excites all frequency components equally

White noise’s flat PSD means it contains all frequencies equally, making it ideal for exciting all modes of a system during testing.

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Which of the following is a limitation when using ideal white noise in practical applications?

A · It has infinite power

Ideal white noise has infinite bandwidth and thus infinite power, which is not physically realizable; practical white noise is band-limited.

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Refer to the signal waveform illustration below.

Which characteristic of the waveform indicates it is white noise?

B · Random fluctuations with no predictable pattern

White noise waveform shows random fluctuations with no predictable pattern or periodicity, which is characteristic of random noise signals.

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If the power spectral density of white noise is \( N_0/2 \), what is the total power in a band-limited white noise signal with bandwidth \( B \)?

A · \( N_0 B \)

The total power is the integral of the PSD over the bandwidth \( B \), which is \( N_0/2 \times 2B = N_0 B \).

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A white noise process \( w(t) \) passes through an ideal low-pass filter with bandwidth \( B \). What is the autocorrelation function of the output signal \( y(t) \)?

A · \( R_{yy}(\tau) = \frac{N_0}{2} \mathrm{sinc}(2B\tau) \)

Passing white noise through a low-pass filter results in band-limited noise with autocorrelation function proportional to a sinc function scaled by \( \frac{N_0}{2} \).

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Which of the following statements about the autocorrelation function of white noise is true?

B · It is non-zero only at \( \tau = 0 \)

The autocorrelation of white noise is an impulse function, non-zero only at zero lag \( \tau = 0 \), indicating no correlation at other time lags.

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Which of the following best explains why white noise is used as an input in system identification?

B · It excites all frequencies equally, allowing full system characterization

White noise contains all frequency components with equal power, making it ideal for exciting all modes of a system to analyze its response.

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Which of the following best defines white noise in the context of random signals?

B · A random signal with constant power spectral density over all frequencies

White noise is defined as a random signal whose power spectral density is constant across all frequencies, indicating equal power at every frequency component.

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White noise is characterized by which of the following statistical properties?

B · Zero mean and uncorrelated samples

White noise typically has zero mean and its samples are uncorrelated, which means the value at one time instant does not provide information about values at other times.

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Which statement correctly describes the autocorrelation function \( R_{ww}(\tau) \) of an ideal white noise process \( w(t) \)?

B · \( R_{ww}(\tau) = \delta(\tau) \) where \( \delta(\tau) \) is the Dirac delta function

The autocorrelation function of ideal white noise is an impulse (Dirac delta function) at zero lag, indicating zero correlation at any non-zero time difference.

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Refer to the diagram below showing the autocorrelation function of a white noise process. What does the spike at \( \tau=0 \) represent?

A · The average power of the noise signal

The value of the autocorrelation function at zero lag corresponds to the average power of the noise signal.

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Which of the following expressions correctly represents the power spectral density (PSD) \( S_{ww}(f) \) of an ideal white noise process?

A · \( S_{ww}(f) = N_0/2 \), a constant for all frequencies \( f \)

The PSD of ideal white noise is constant across all frequencies, often represented as \( N_0/2 \), indicating equal power distribution.

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If the power spectral density of a noise signal is flat over a certain bandwidth, which type of noise does it represent within that band?

B · White noise

A flat power spectral density over a bandwidth indicates white noise, which has equal power at all frequencies within that band.

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Which of the following statements correctly describes the relationship between white noise and random signals?

B · White noise is a special case of a random signal with uncorrelated samples and flat PSD

White noise is a specific type of random signal characterized by uncorrelated samples and constant power spectral density.

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In a communication system, white noise is often modeled as an additive noise source. Which property of white noise justifies this modeling?

B · Its constant power spectral density and uncorrelated samples

White noise's constant PSD and uncorrelated samples make it a good model for random disturbances added to signals in communication systems.

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Refer to the block diagram below of a system with white noise input \( w(t) \). If the system is linear and time-invariant with impulse response \( h(t) \), what is the power spectral density of the output \( y(t) \)?

A · \( S_{yy}(f) = |H(f)|^2 S_{ww}(f) \)

For an LTI system, the output PSD is the input PSD multiplied by the magnitude squared of the system's frequency response.

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Which of the following is a practical application of white noise in electronics and communication engineering?

B · Testing frequency response of systems

White noise is used to test the frequency response of systems because it contains all frequencies with equal power.

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Which property of white noise makes it suitable for dithering in analog-to-digital converters (ADCs)?

B · Its zero mean and uncorrelated samples

The zero mean and uncorrelated nature of white noise helps in dithering by reducing quantization errors without introducing bias.

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In the context of system identification, why is white noise often used as an input signal?

B · Because it excites all frequencies equally, allowing full system characterization

White noise excites all frequencies equally, making it ideal for identifying system characteristics across the entire frequency spectrum.

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Which mathematical model best represents an ideal white noise process \( w(t) \)?

A · A Gaussian process with zero mean and autocorrelation \( R_{ww}(\tau) = \sigma^2 \delta(\tau) \)

Ideal white noise is modeled as a Gaussian process with zero mean and autocorrelation equal to a scaled Dirac delta function.

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Consider a white noise process \( w(t) \) with power spectral density \( S_{ww}(f) = N_0/2 \). Which of the following is true about its autocorrelation function \( R_{ww}(\tau) \)?

A · \( R_{ww}(\tau) = N_0 \delta(\tau) \)

The autocorrelation function is the inverse Fourier transform of the PSD. For constant PSD \( N_0/2 \), the autocorrelation is \( N_0 \delta(\tau) \).

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Which of the following distinguishes white noise from colored noise?

A · White noise has a flat power spectral density, colored noise has frequency-dependent PSD

White noise has a flat PSD across frequencies, whereas colored noise has a PSD that varies with frequency.

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Refer to the power spectral density plots below. Which plot corresponds to white noise and which to colored noise?

A · Plot A is white noise (flat PSD), Plot B is colored noise (PSD varies with frequency)

White noise PSD is flat (constant across frequencies), while colored noise PSD shows variation with frequency.

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Which of the following is true about the autocorrelation function of colored noise compared to white noise?

B · Colored noise autocorrelation is spread out over time lags, white noise autocorrelation is an impulse at zero lag

Colored noise has correlated samples, so its autocorrelation function extends over time lags, unlike white noise which has an impulse autocorrelation.

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Given a white noise process \( w(t) \) with variance \( \sigma^2 \), what is the value of its autocorrelation function at zero lag \( R_{ww}(0) \)?

B · \( \sigma^2 \)

The autocorrelation at zero lag equals the variance of the process, which is \( \sigma^2 \).

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Which statistical property of white noise ensures that its samples at different times are independent?

C · Zero autocorrelation at non-zero lags

Zero autocorrelation at non-zero lags implies no linear dependence between samples at different times, indicating independence for Gaussian white noise.

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If a white noise process \( w(t) \) is passed through an ideal low-pass filter with bandwidth \( B \), what is the nature of the output noise?

B · Colored noise with bandwidth limited to \( B \)

Filtering white noise limits its bandwidth, resulting in colored noise with PSD shaped by the filter.

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Consider the autocorrelation function of a white noise process \( R_{ww}(\tau) = N_0 \delta(\tau) \). What is the Fourier transform of this function?

A · \( S_{ww}(f) = N_0 \), a constant for all frequencies

The Fourier transform of a delta function is a constant, so the PSD is constant across all frequencies.

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Refer to the power spectral density plot below of a noise signal. The PSD is constant up to frequency \( f_c \) and zero beyond. What type of noise does this represent?

B · Band-limited white noise

Noise with flat PSD up to a cutoff frequency and zero beyond is band-limited white noise.

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Which of the following is a key difference between mathematical modeling of ideal white noise and practical noise sources?

A · Ideal white noise has infinite power, practical noise has finite power

Ideal white noise has infinite power due to its infinite bandwidth, while practical noise sources have finite bandwidth and thus finite power.

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Which of the following best explains why white noise cannot exist physically as an ideal signal?

A · Because it requires infinite power due to infinite bandwidth

Ideal white noise has infinite bandwidth and constant power spectral density, which implies infinite total power, making it non-physical.

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Which of the following correctly describes the difference between white noise and pink noise?

A · White noise has constant PSD; pink noise PSD decreases with frequency

White noise has a flat PSD, while pink noise has a PSD that decreases inversely with frequency (1/f).

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Which of the following is NOT a property of a Linear Time-Invariant (LTI) system?

C · Memorylessness

LTI systems are defined by linearity and time invariance. They may or may not be memoryless. Memorylessness is not a required property of LTI systems.

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The impulse response of an LTI system completely characterizes the system because:

B · It allows output to be computed for any input via convolution

The impulse response characterizes an LTI system since the output for any input can be found by convolving the input with the impulse response.

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Which of the following statements about the frequency response \( H(\omega) \) of an LTI system is TRUE?

B · It is the Fourier transform of the system's impulse response

The frequency response \( H(\omega) \) is defined as the Fourier transform of the impulse response \( h(t) \) of the LTI system.

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Refer to the diagram below showing an LTI system block with input \( x(t) \) and output \( y(t) \). If the system is time-invariant, which of the following must hold true?

A · Output shifts by the same amount as input shift

Time invariance means that if the input is delayed by \( t_0 \), the output is also delayed by \( t_0 \) without any change in shape.

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Consider a random signal \( X(t) \) with mean zero and finite variance. Which of the following is a characteristic of such a random signal?

C · Mean value is constant and zero

A zero-mean random signal has a constant mean value of zero over time, though the signal values vary randomly.

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Which of the following best describes a wide-sense stationary (WSS) random process?

B · Mean is constant and autocorrelation depends only on time difference

A WSS process has a constant mean and an autocorrelation function that depends only on the time difference \( \tau \), not on the absolute time.

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A random signal \( X(t) \) has autocorrelation function \( R_X(\tau) = e^{-\alpha |\tau|} \). What is the nature of its power spectral density (PSD)?

B · Lorentzian spectrum

The Fourier transform of an exponential autocorrelation function is a Lorentzian-shaped PSD.

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Which of the following is TRUE about the mean of the output \( Y(t) \) when a random signal \( X(t) \) passes through an LTI system with impulse response \( h(t) \)?

C · Mean of \( Y(t) \) is the convolution of mean of \( X(t) \) with \( h(t) \)

The mean of the output is the convolution of the input mean with the system impulse response.

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If a stationary random process with power spectral density \( S_X(\omega) \) is passed through an LTI system with frequency response \( H(\omega) \), the output power spectral density \( S_Y(\omega) \) is given by:

B · \( S_Y(\omega) = |H(\omega)|^2 S_X(\omega) \)

The output PSD is the input PSD multiplied by the squared magnitude of the system frequency response.

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The autocorrelation function \( R_X(\tau) \) and power spectral density \( S_X(\omega) \) of a random process form a Fourier transform pair. Which of the following is TRUE?

A · Both \( R_X(\tau) \) and \( S_X(\omega) \) are always real and even functions

Autocorrelation and PSD of a real random process are real and even functions, and they form a Fourier transform pair.

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Refer to the diagram below showing the autocorrelation function \( R_X(\tau) \) of a random process. Which of the following statements about its Fourier transform \( S_X(\omega) \) is correct?

A · If \( R_X(\tau) \) is wider, \( S_X(\omega) \) is narrower

By the Fourier transform properties, wider autocorrelation corresponds to narrower PSD and vice versa.

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Which of the following describes the effect of an ideal low-pass LTI filter on the power spectral density of white noise?

C · Output PSD is flat up to cutoff frequency and zero beyond

An ideal low-pass filter passes frequencies up to cutoff unchanged and blocks higher frequencies, shaping the PSD accordingly.

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If a random process \( X(t) \) with autocorrelation \( R_X(\tau) \) passes through an LTI system with impulse response \( h(t) \), which expression gives the output autocorrelation \( R_Y(\tau) \)?

B · \( R_Y(\tau) = h(\tau) * R_X(\tau) * h(-\tau) \)

The output autocorrelation is the convolution of the input autocorrelation with the system autocorrelation \( h(\tau) * h(-\tau) \).

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Which of the following statements about the mean and variance of the output of an LTI system with input random signal \( X(t) \) is TRUE?

B · Output mean is zero if input mean is zero, variance is scaled by energy of impulse response

The output mean is zero if input mean is zero, and output variance is input variance scaled by the energy of the impulse response.

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Refer to the frequency response plot below of an LTI system. If white noise with PSD \( N_0/2 \) is input, what is the output noise power?

B · \( \frac{N_0}{2} \times \int |H(\omega)|^2 d\omega \)

Output noise power is the input noise PSD multiplied by the integral of the squared magnitude of the frequency response over all frequencies.

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The frequency domain representation of the output of an LTI system with input random signal \( X(t) \) is given by:

B · \( Y(\omega) = H(\omega) X(\omega) \)

In frequency domain, output is product of input spectrum and system frequency response.

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Which of the following is TRUE regarding the stationarity of a random process?

A · Strict-sense stationarity requires all joint distributions to be time-invariant

Strict-sense stationarity requires that all joint probability distributions are invariant to time shifts.

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Which noise type has a power spectral density inversely proportional to frequency (\( 1/f \))?

B · Pink noise

Pink noise has a PSD proportional to \( 1/f \), meaning power decreases with increasing frequency.

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Refer to the diagram below showing power spectral densities of different noise types. Which noise type corresponds to a flat PSD across all frequencies?

A · White noise

White noise has a constant (flat) power spectral density over all frequencies.

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Which of the following noise types is characterized by a power spectral density proportional to \( 1/f^2 \)?

C · Brownian noise

Brownian noise (or red noise) has a PSD proportional to \( 1/f^2 \).

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Which of the following is TRUE about the stationarity of the output \( Y(t) \) of an LTI system with WSS input \( X(t) \)?

B · Output is always WSS

An LTI system with WSS input produces an output that is also WSS.

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Which of the following statements about the autocorrelation function \( R_X(\tau) \) of a WSS random process is FALSE?

C · \( R_X(\tau) \) depends on absolute time \( t \)

For WSS processes, autocorrelation depends only on the time difference \( \tau \), not on absolute time \( t \).

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Which of the following noise types has a flat power spectral density and is often modeled as having equal power at all frequencies?

A · Thermal noise

Thermal noise is modeled as white noise with flat PSD across frequencies.

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A random signal \( X(t) \) passes through an LTI system with impulse response \( h(t) \). The output power spectral density \( S_Y(\omega) \) is related to the input PSD \( S_X(\omega) \) by:

A · \( S_Y(\omega) = |H(\omega)|^2 S_X(\omega) \)

The output PSD is the input PSD multiplied by the squared magnitude of the system frequency response.

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Refer to the diagram below showing the frequency response \( |H(\omega)| \) of an LTI system. Which frequency component of the input signal will be most attenuated?

B · Frequency at zero of \( |H(\omega)| \)

Frequency components where \( |H(\omega)| = 0 \) are completely attenuated by the system.

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Which of the following best describes the effect of filtering a WSS random process through an LTI system on its stationarity?

B · Output process is WSS if input is WSS

Filtering a WSS process through an LTI system produces an output that is also WSS.

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Which noise type is characterized by a power spectral density that increases with frequency?

C · Blue noise

Blue noise has a power spectral density that increases with frequency, proportional to \( f \).

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Which of the following statements about the power spectral density (PSD) of a random process is FALSE?

C · PSD can be negative for some frequencies

PSD is always non-negative since it represents power distribution over frequency.

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Which of the following noise types is typically modeled as Gaussian and white with flat PSD?

A · Thermal noise

Thermal noise is modeled as Gaussian white noise with flat PSD.

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Which of the following best describes the effect of an LTI system with frequency response \( H(\omega) \) on the autocorrelation function \( R_X(\tau) \) of a WSS input process?

C · Output autocorrelation is the convolution of input autocorrelation with \( h(t) * h(-t) \)

The output autocorrelation is the convolution of the input autocorrelation with the system autocorrelation \( h(t) * h(-t) \).

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Which of the following is TRUE for the power spectral density of a stationary random process?

B · It is always a real and even function

PSD of a stationary random process is real and even because it is the Fourier transform of a real and even autocorrelation function.

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Which of the following properties is NOT characteristic of a Linear Time-Invariant (LTI) system?

C · Memorylessness

LTI systems can have memory; memorylessness is not a necessary property of LTI systems. Linearity and time invariance define LTI systems, and causality is often assumed but not mandatory.

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The impulse response \( h(t) \) of an LTI system completely characterizes the system because:

B · It is the output when the input is a unit impulse

The impulse response \( h(t) \) is the output of an LTI system when the input is a unit impulse \( \delta(t) \), and it fully characterizes the system's behavior.

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Which of the following statements about the frequency response \( H(\omega) \) of an LTI system is TRUE?

B · It is the Fourier transform of the impulse response

The frequency response \( H(\omega) \) of an LTI system is the Fourier transform of its impulse response \( h(t) \).

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For an LTI system with impulse response \( h(t) = e^{-t}u(t) \), where \( u(t) \) is the unit step, the system is:

A · Causal and stable

Since \( h(t) = e^{-t}u(t) \) is zero for \( t < 0 \), the system is causal. The impulse response is absolutely integrable, so the system is stable.

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Refer to the block diagram below of an LTI system with input \( x(t) \) and output \( y(t) \). If the input is a sum of two signals \( x_1(t) + x_2(t) \), the output is:

A · \( y_1(t) + y_2(t) \), where \( y_i(t) \) is output due to \( x_i(t) \)

Due to linearity, the output of an LTI system to a sum of inputs is the sum of the outputs to each input separately.

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Which of the following is NOT a typical characteristic of a random signal?

A · Deterministic amplitude at every time instant

Random signals do not have deterministic amplitude values at every time instant; their values are uncertain and described statistically.

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The mean value of a wide-sense stationary (WSS) random process is:

B · Constant with respect to time

In a WSS process, the mean is constant over time, although it may not be zero.

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For a random process, the autocorrelation function \( R_X(\tau) \) satisfies which of the following properties?

A · It is an even function of \( \tau \)

The autocorrelation function of any real-valued random process is an even function, i.e., \( R_X(\tau) = R_X(-\tau) \).

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If a random signal has zero mean and its autocorrelation function is \( R_X(\tau) = \sigma^2 e^{-\alpha |\tau|} \), the parameter \( \sigma^2 \) represents:

B · Variance of the signal

The value of the autocorrelation function at \( \tau=0 \) is the variance \( \sigma^2 \) of the zero-mean random signal.

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Which of the following best describes the output of an LTI system when a stationary random signal is input?

B · The output is also a stationary random signal

An LTI system preserves stationarity; thus, the output of an LTI system to a stationary input is also stationary.

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Refer to the block diagram below of an LTI system with impulse response \( h(t) \). If the input is a random signal \( X(t) \), the output \( Y(t) \) is given by:

A · \( Y(t) = X(t) * h(t) \) (convolution)

The output of an LTI system is the convolution of the input signal with the system's impulse response.

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The autocorrelation function of the output \( R_Y(\tau) \) of an LTI system with impulse response \( h(t) \) and input autocorrelation \( R_X(\tau) \) is given by:

C · \( R_Y(\tau) = R_X(\tau) * (h(\tau) * h(-\tau)) \)

The output autocorrelation is the convolution of the input autocorrelation with the autocorrelation of the impulse response \( h(\tau) * h(-\tau) \).

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Refer to the diagram below showing the power spectral density (PSD) \( S_X(\omega) \) of an input random process and the magnitude response \( |H(\omega)| \) of an LTI system. The PSD of the output \( S_Y(\omega) \) is:

A · \( S_Y(\omega) = S_X(\omega) \cdot |H(\omega)|^2 \)

The output PSD is the input PSD multiplied by the squared magnitude of the system's frequency response.

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If the input to an LTI system is a stationary random process with autocorrelation \( R_X(\tau) \), and the system has impulse response \( h(t) \), the output autocorrelation \( R_Y(\tau) \) can be expressed as:

A · \( R_Y(\tau) = R_X(\tau) * (h(\tau) * h^*(-\tau)) \)

The output autocorrelation is the convolution of the input autocorrelation with the autocorrelation of the impulse response, which involves the conjugate flipped impulse response.

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Which LTI system parameter primarily affects the bandwidth of the output random signal when a random input is filtered?

A · Impulse response duration

The duration of the impulse response relates inversely to the system bandwidth, thus affecting the bandwidth of the output signal.

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Increasing the cutoff frequency of an LTI low-pass filter applied to a stationary random input signal will generally:

A · Increase the output signal power

A higher cutoff frequency allows more frequency components to pass, increasing the output power of the filtered random signal.

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Refer to the diagram below showing the impulse response \( h(t) \) of an LTI system. If the system's energy increases, the variance of the output random signal when white noise is input will:

A · Increase proportionally

The output variance is proportional to the energy of the impulse response when the input is white noise.

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Which of the following statements about stationarity of a random process is TRUE?

A · Strict-sense stationarity requires all joint distributions to be time-invariant

Strict-sense stationarity requires that all joint probability distributions do not change with time shifts.

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A random process is ergodic if:

A · Time averages equal ensemble averages

Ergodicity means that statistical properties can be obtained from a single realization over time, i.e., time averages equal ensemble averages.

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Which of the following is a necessary condition for a random process to be wide-sense stationary (WSS)?

A · Mean is constant and autocorrelation depends only on time difference

WSS requires the mean to be constant and the autocorrelation function to depend only on the time difference \( \tau \), not on absolute time.

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In practical applications, filtering a noisy random signal through an LTI system is primarily done to:

B · Extract desired frequency components

Filtering is used to extract or enhance desired frequency components and reduce noise outside the passband.

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Which of the following is TRUE about the effect of an LTI system's phase response on a filtered random signal?

C · Phase response affects the time-domain waveform but not the power spectral density

Phase response affects the time-domain shape of the output but does not affect the power spectral density, which depends on magnitude squared of frequency response.

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A stationary random process with autocorrelation \( R_X(\tau) = e^{-\beta |\tau|} \) is passed through an LTI system with impulse response \( h(t) = e^{-\alpha t}u(t) \). The output autocorrelation \( R_Y(\tau) \) will:

A · Decay faster if \( \alpha > \beta \)

The output autocorrelation is influenced by both input and system parameters; a larger \( \alpha \) causes faster decay of the impulse response, thus faster decay of \( R_Y(\tau) \).

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Which of the following random processes is ergodic in mean but not necessarily in autocorrelation?

A · Wide-sense stationary process

Wide-sense stationary processes can be ergodic in mean but may not be ergodic in autocorrelation.

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Refer to the diagram below showing the autocorrelation function \( R_X(\tau) \) of a random process. Which of the following indicates that the process is wide-sense stationary?

A · Symmetry about \( \tau = 0 \) and dependence only on \( \tau \)

WSS processes have autocorrelation functions symmetric about zero and dependent only on the lag \( \tau \), not on absolute time.

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In a practical scenario, filtering a random signal through an LTI system is used to:

B · Modify the statistical properties such as variance and bandwidth

Filtering modifies the statistical properties of the input random signal, including variance and bandwidth, but does not convert non-stationary to stationary or vice versa.

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Refer to the block diagram below of a random signal passing through a filter with transfer function \( H(s) \). If the input is a Gaussian white noise, the output random signal is:

A · Gaussian with modified mean and variance

An LTI system preserves Gaussianity; the output remains Gaussian with mean and variance modified by the system.

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Which of the following is an analytical method to determine the output power spectral density of a filtered random signal?

A · Fourier transform of the input autocorrelation multiplied by \( |H(\omega)|^2 \)

The output PSD is the Fourier transform of the output autocorrelation, which equals the input PSD multiplied by the squared magnitude of the system frequency response.

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A random process \( X(t) \) is passed through an LTI system with frequency response \( H(\omega) \). The output power is given by:

A · \( P_Y = \int_{-\infty}^{\infty} S_X(\omega) |H(\omega)|^2 d\omega \)

The output power is the integral over frequency of the input PSD multiplied by the squared magnitude of the system frequency response.

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Which of the following statements about ergodicity is FALSE?

B · All stationary processes are ergodic

Not all stationary processes are ergodic; ergodicity is a stronger condition.

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Refer to the diagram below showing the power spectral density of a filtered random signal. The notch in the PSD corresponds to:

A · Frequency components attenuated by the filter

A notch in the PSD indicates frequencies where the filter attenuates the signal components.

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Which of the following best describes the effect of an LTI system with a narrow bandwidth on a wideband random input signal?

A · Output signal bandwidth is reduced

A narrow bandwidth filter passes only a limited frequency range, reducing the output signal bandwidth.

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A random signal \( X(t) \) with autocorrelation \( R_X(\tau) \) is filtered by an LTI system with impulse response \( h(t) \). The output autocorrelation \( R_Y(\tau) \) is:

A · \( R_Y(\tau) = R_X(\tau) * h(\tau) * h(-\tau) \)

The output autocorrelation is the convolution of the input autocorrelation with the autocorrelation of the impulse response.

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Which of the following is an example of a practical application of filtering random signals through LTI systems?

A · Noise reduction in communication receivers

Filtering random signals is widely used for noise reduction in communication systems by removing unwanted frequency components.

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Refer to the diagram below of an LTI system with input \( X(t) \) and output \( Y(t) \). If the input is a stationary random process with PSD \( S_X(\omega) \), the output PSD \( S_Y(\omega) \) is:

A · \( S_Y(\omega) = |H(\omega)|^2 S_X(\omega) \)

The output PSD is the input PSD multiplied by the squared magnitude of the system frequency response.

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Which of the following is a key assumption when analyzing random signals through LTI systems?

B · Input is stationary or wide-sense stationary

Analysis typically assumes the input random signal is stationary or wide-sense stationary for meaningful statistical characterization.

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A WSS random process x(t) with PSD S_x(\omega) = \frac{4}{1 + (\frac{\omega}{1.5})^2} is passed through an LTI system with impulse response h(t) = \delta(t) - 0.3 \delta(t-2). The output PSD S_y(\omega) is:

D · All of the above

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A WSS random process x(t) with autocorrelation R_x(\tau) = e^{-0.4|\tau|} is filtered by an LTI system with frequency response H(\omega) = \frac{1}{1 + j\omega/2}. The output autocorrelation R_y(\tau) is:

A · R_y(\tau) = R_x(\tau) * h(\tau) * h(-\tau), where h(t) = 2 e^{-2t} u(t)

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A WSS random process x(t) with PSD S_x(\omega) = \frac{6}{1 + (\frac{\omega}{2})^2} is passed through an LTI system with frequency response H(\omega) = \frac{j\omega}{1 + j\omega/3}. The output variance \sigma_y^2 is:

A · \int_{-\infty}^{\infty} \frac{6 \omega^2}{(1 + (\frac{\omega}{2})^2)(1 + (\frac{\omega}{3})^2)} d\omega

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A WSS random process x(t) with autocorrelation R_x(\tau) = e^{-0.5|\tau|} is filtered by an LTI system with impulse response h(t) = e^{-t}u(t). The output autocorrelation R_y(\tau) is:

B · R_y(\tau) = (h * h_{-})(\tau) * R_x(\tau), where h_{-}(t) = h(-t)

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A WSS random process x(t) with PSD S_x(\omega) = \frac{8}{1 + (\frac{\omega}{3})^4} is passed through an LTI system with frequency response H(\omega) = \frac{1}{1 + j\frac{\omega}{6}}. The output variance \sigma_y^2 is:

D · Both A and C

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A WSS random process x(t) with autocorrelation R_x(\tau) = e^{-0.3|\tau|} is filtered by an LTI system with impulse response h(t) = \delta(t) - 0.4 \delta(t-1) + 0.16 \delta(t-2). The output autocorrelation R_y(\tau) is:

D · Both B and C

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Filtering Random Signals through LTI Systems

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