Signal and systems miscellaneous

Signal and systems miscellaneous

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Signals and Systems

Signal and systems miscellaneous
  1. If f(t) = – f(– t) and f(t) satisfy the Dirichlet’s conditions, then f (t) can be expanded in a Fourier series containing—








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    Since
    f(t) = – f(– t)
    Represents the condition of an odd function, which contains only sine terms.

    Correct Option: A

    Since
    f(t) = – f(– t)
    Represents the condition of an odd function, which contains only sine terms.

  1. The autocorrelation of a wide-sense stationary random process is given by e– 2|τ| The peak value of the spectral density is—








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    Given that
    Rxx(τ) = e– 2|t|
    then spectral density will be

    Sxx(ω) =
    2a
    		where, a = 2
    a2 + ω2

    Spectral density will be maximum at zero frequency
    i.e., ω = 0
    So, Sxx (0)max =
    2a
    2 × 2
    		= 1
    a222

    Hence, alternative (B) is the correct choice.

    Correct Option: B

    Given that
    Rxx(τ) = e– 2|t|
    then spectral density will be

    Sxx(ω) =
    2a
    		where, a = 2
    a2 + ω2

    Spectral density will be maximum at zero frequency
    i.e., ω = 0
    So, Sxx (0)max =
    2a
    2 × 2
    		= 1
    a222

    Hence, alternative (B) is the correct choice.

  1. The co-variance function, Cx(τ) of a stationary stochastic process, x(t), is said to be positive definite. This means that—








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    NA

    Correct Option: C

    NA

  1. Which of the following represent a stable system?
    1. Impulse response of the system decreases exponentially.
    2. Area within the impulse response is finite.
    3. Eigen values of the system are positive and real.
    4. Roots of the characteristic equation of the system are real and negative. Select the correct answer using the codes given below—








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    For the system to be stable—
    ● Impulse response of the system decreases exponentially.
    ● Root of the characteristic equation should be left half of the s-plane or we can say real and negative.
    ● Area within the impulse response is finite.
    Hence, alternative (D) is the correct choice.

    Correct Option: D

    For the system to be stable—
    ● Impulse response of the system decreases exponentially.
    ● Root of the characteristic equation should be left half of the s-plane or we can say real and negative.
    ● Area within the impulse response is finite.
    Hence, alternative (D) is the correct choice.

  1. Which one of the following is the correct Fourier transform of the unit step signal?










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    We know that

    u(t) =
    1
    		[1 + sgn(t)]
    2

    The Fourier transform is given by
    F{u(t)} =
    1
    		[F{1} + F{sgn(t)}]
    2

    =
    1
    		πδ(ω) +
    1
    2jω

    = πδ(ω) +
    1
    jω

    Hence, alternative (C) is the correct choice.

    Correct Option: C

    We know that

    u(t) =
    1
    		[1 + sgn(t)]
    2

    The Fourier transform is given by
    F{u(t)} =
    1
    		[F{1} + F{sgn(t)}]
    2

    =
    1
    		πδ(ω) +
    1
    2jω

    = πδ(ω) +
    1
    jω

    Hence, alternative (C) is the correct choice.