If x_k = 2^k for k ≤ 0 and = 0 for k > 0 X(z) = 2/(2 - z).

If p is the probability of success and q is the probability of failure, then the probability that there are r successes in n trails is

Consider the following statements

1. If ensemble and time averages of a random process are identical, the process is ergodic.
2. If ensemble and time average of a random process are not identical, the process is ergodic.
3. An ergodic process is stationary.
4. A stationary process is necessarily ergodic.

In a oscillator, a sine wave generator starts at t =It consist of

The theoretical power of white noise is

If f(t) ↔ F(jω), f(- t) ↔

Assertion (A): When a function f(t) is represented as exponential Fourier series, the set of complex coefficients F_n is called frequency spectrum of f(t)

Reason (R): Frequency spectrum is also called line spectrum.

The area of an impulse is

In plotting line spectrum

To plot frequency spectrum we draw two graphs viz |F_n| versus frequency and φ_n versus frequency.

If f(t) is an even function, the coefficients F_n in the exponential form of Fourier series

(sI - A)^-1 is equal to

For a periodic waveform, the power spectral density and correlation function form

Assertion (A): Laplace transform can be used to evalute integrals.

Reason (R): Laplace transform can be used to solve differential equations.

The peak factor is

Assertion (A): The function ke^st with both σ and ω positive and s > 0 depicts a sinusoid whose amplitude increases with time.

Reason (R): If σ =ke^st becomes a sinusoid.

If F(jω) is the Fourier transform of f(t), then

Assertion (A): Transient analysis uses Laplace transformation.

Reason (R): Laplace transform method is suitable for solution of integro differential equations.

The minor M_ij of an n x n matrix is the determinant of (n -x (n -matrix formed by deleting the ith row and j the column of n x n matrix.

For matrix A, A^-1 A = 1

e^-3t is a

For a Gaussian process, auto-correlation sequence also implies that

Assertion (A): A half cycle of sine wave can be expressed as sum of sine wave and another sine wave shifted by T/2 where T is the time period.

Reason (R): The function u(t - t₀) - u(t - t₀ - T) is a gate function occuring at t = t₀ and of duration T.

Assertion (A): Heaviside partial expansion gives a simple procedure to find inverse Laplace transform of the terms having a complex conjugate pair of roots.

Reason (R): If I(s) = P(s)/Q(s) and all roots of Q(s) = 0 are simple, i(t) will have terms with exponentials having real exponents only.

The energy of highest value of Autocorrelation of a functioncospt is

If p is the probability of success and q is the probability of failure, then the probability that there are r successes in n trails is

Which one of the following is correct? Energy of a power signal is

If x^k = a^k and k ≥ 0
X(z) = (1 - az^-1)^-1 with |a| < |z|

The function δ' (t-b) is

An impulse train is

For formation of state equations, the inductors and current sources

If u is input, y is output and a, b are constants, the system y = au + b

In RLC circuits the state variables generally selected are

Assertion (A): Transient periods are of short duration but can result in dangerously high voltages and currents.

Reason (R): Circuit equations in transient analysis are integral differential equations.

If f(t) =F(jω) =

Auto correlation function

If the value of

For the signal in the given figure the Fourier transform is Then the Fourier transform of the signal in the given figure

Assertion (A): If

Reason (R): Displacement in time domain by T becomes multiplication by e^-st in s domain.

If the value of

The minor M_ij of an n x n matrix is the determinant of (n -x (n -matrix formed by deleting the ith row and j the column of n x n matrix.

The following table gives some time functions and the Laplace transforms

Of these correctly matched pairs are

Auto correlation function

If the value of

The following table gives some time functions and the Laplace transforms

Of these correctly matched pairs are

Assertion (A): Transient periods are of short duration but can result in dangerously high voltages and currents.

Reason (R): Circuit equations in transient analysis are integral differential equations.

The following table gives some time functions and the Laplace transforms

Of these correctly matched pairs are

Assertion (A): If

Reason (R): Displacement in time domain by T becomes multiplication by e^-st in s domain.

Assertion (A): If

Reason (R): Displacement in time domain by T becomes multiplication by e^-st in s domain.

If f(t) is an odd function, F(jω) =