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Unit 5 - Practice Quiz

ECE180 50 Questions
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1 A stochastic process is best described as:

A. A family of random variables indexed by a parameter (usually time).
B. A sequence of constant values.
C. A single random variable defined at a specific time.
D. A deterministic function of time.

2 If the time index is a countable set and the random variable takes continuous values, the process is classified as:

A. Discrete Random Sequence
B. Continuous Random Sequence
C. Continuous Random Process
D. Discrete Random Process

3 A stochastic process is called deterministic if:

A. Its future values can be predicted exactly from its past values.
B. Its power spectral density is flat.
C. Its values are purely random and unpredictable.
D. It has a constant mean.

4 A specific waveform or function of time observed from a stochastic process is called a:

A. Sample function or realization
B. Probability density function
C. Correlation coefficient
D. Random variable

5 Which function fully characterizes the statistical properties of a stochastic process at a single time instant ?

A. The autocorrelation function
B. The second-order probability density function
C. The first-order probability density function
D. The power spectral density

6 Two stochastic processes and are said to be statistically independent if their joint density function satisfies:

A.
B.
C.
D. for all

7 A stochastic process is Strict-Sense Stationary (SSS) if:

A. Its power spectral density is constant.
B. Its statistical properties are invariant to a shift in the time origin.
C. Its autocorrelation depends only on the time difference.
D. Its mean is constant.

8 For a first-order stationary process, which of the following must be true?

A.
B.
C.
D. All of the above

9 A process is Wide-Sense Stationary (WSS) if:

A. The process is strictly stationary.
B. The variance is zero.
C. The mean is constant and the autocorrelation depends only on time difference .
D. All higher-order moments are constant.

10 Which of the following statements is true regarding SSS and WSS?

A. WSS is a stricter condition than SSS.
B. All SSS processes are WSS (assuming moments exist).
C. All WSS processes are SSS.
D. There is no relationship between SSS and WSS.

11 For a Gaussian random process, Wide-Sense Stationarity (WSS) implies:

A. The process is non-deterministic.
B. The process is white noise.
C. Nothing about SSS.
D. Strict-Sense Stationarity (SSS).

12 The autocorrelation function of a WSS process is defined as:

A.
B.
C.
D.

13 Which of the following is a property of the autocorrelation function of a real WSS process?

A. is always 0 for .
B.
C. (Even symmetry)
D. is an odd function.

14 The maximum value of the autocorrelation function occurs at:

A. It has no maximum.
B.
C.
D.

15 The value represents the:

A. Total average power (mean square value) of the process.
B. Variance of the process.
C. Mean value of the process.
D. DC power of the process.

16 A stochastic process is said to be ergodic if:

A. Its ensemble averages are equal to its corresponding time averages.
B. Its power spectral density is infinite.
C. All time averages are zero.
D. It is strictly stationary.

17 For a process to be mean ergodic, which condition must hold as ?

A.
B.
C. The process must be Gaussian.
D.

18 The autocovariance function is related to the autocorrelation and mean by:

A.
B.
C.
D.

19 The cross-correlation function is defined as:

A.
B.
C.
D.

20 For jointly WSS processes and , the property indicates:

A. Orthogonality.
B. Odd symmetry.
C. Symmetry with respect to the process indices and time reversal.
D. Even symmetry.

21 Two processes and are said to be orthogonal if:

A. for all
B.
C.
D. for all

22 Two processes and are uncorrelated if:

A. They have different power spectral densities.
B.
C. and are orthogonal.
D. (or )

23 The Power Spectral Density (PSD), , of a WSS process is the Fourier Transform of:

A. The autocorrelation function .
B. The cross-correlation function.
C. The probability density function.
D. The mean function.

24 Which of the following is a fundamental property of the Power Spectral Density ?

A. is purely imaginary.
B. is always real and non-negative ().
C. decreases exponentially.
D. can be negative.

25 The total average power of a process can be obtained from the PSD by:

A.
B.
C.
D.

26 If is the PSD of a real random process, it is an:

A. Odd function of frequency.
B. Undefined function.
C. Exponential function.
D. Even function of frequency.

27 White noise is defined as a process having:

A. A constant Power Spectral Density for all frequencies.
B. Zero variance.
C. A Gaussian PDF only.
D. A constant Autocorrelation function.

28 The autocorrelation function of ideal white noise is:

A. A sinc function.
B. A triangular function.
C. A Dirac delta function .
D. A constant.

29 The Cross-Power Spectral Density is the Fourier Transform of:

A. Cross-correlation
B. Covariance
C. Joint PDF
D. Autocorrelation

30 Which relationship holds for the Cross-Power Spectral Density?

A.
B.
C.
D.

31 If two processes and are orthogonal, their Cross-PSD is:

A. Zero.
B. A constant.
C. Equal to .
D. Infinite.

32 If is the input to a linear time-invariant (LTI) system with transfer function , and is the output, the output PSD is given by:

A.
B.
C.
D.

33 For the same LTI system where is the output and is the input, the Cross-PSD is:

A.
B.
C.
D.

34 The DC component of a random process contributes to the PSD as:

A. A constant value across all frequencies.
B. A Gaussian pulse.
C. Zero value.
D. An impulse (delta function) at .

35 Real, wide-sense stationary noise with PSD is passed through an ideal low-pass filter with cutoff frequency . The output noise power is:

A. (or if is in Hz)
B. $0$
C.
D.

36 If , the process is:

A. Periodic.
B. Not periodic (aperiodic).
C. Unbounded.
D. White noise.

37 The inequality is a form of:

A. Cauchy-Schwarz inequality.
B. Markov inequality.
C. Chebyshev inequality.
D. Wiener-Khinchin theorem.

38 For a periodic WSS process with period , the autocorrelation function is:

A. Decaying exponentially.
B. Periodic with period .
C. A delta function.
D. Zero everywhere.

39 The spectral components of two different uncorrelated random processes are:

A. Summed.
B. Identical.
C. Uncorrelated.
D. Orthogonal.

40 What is the physical unit of Power Spectral Density if is a voltage signal?

A. Volts/Hz (or Watts/Hz into 1 Ohm)
B. Watts
C. Volts/Hz
D. Volts

41 If , where is a random variable with , the process is:

A. Ergodic.
B. Time-varying.
C. Neither Stationary nor Ergodic.
D. Stationary but not Ergodic.

42 The cross-correlation of input and output of a system can be used to determine:

A. The mean of the output only.
B. The probability density of the input.
C. The total power.
D. The impulse response of the system (if input is white noise).

43 If , the mean square value is:

A. 26
B. 1
C. Infinite (theoretically, for delta function) or undefined without bandwidth limits.
D. 25

44 If the PSD is a constant , the process is called:

A. Ideal white noise.
B. Pink noise.
C. Band-limited white noise.
D. Colored noise.

45 Which of the following functions cannot be a valid autocorrelation function?

A.
B.
C.
D. $1$

46 For a WSS process, if , the autocorrelation is:

A.
B.
C.
D.

47 If and are independent WSS processes, the autocorrelation of their sum is:

A.
B.
C.
D.

48 A random process where is uniformly distributed in is:

A. WSS.
B. Non-ergodic.
C. WSS but not SSS.
D. Not stationary.

49 The Power Spectral Density of the derivative of a process, , is related to by:

A.
B.
C.
D.

50 Which of the following is true for the Real part of the Cross-PSD, ?

A. It is an even function of .
B. It is always zero.
C. It is always negative.
D. It is an odd function of .
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