The discrete-time signal x (n) = (-1)n is periodic with fundamental period
Explanation: Period of the signal refers to the instant of time at which the signal repeats itself and for this Period =2 of the given discrete time signal.
The frequency of a continuous time signal x (t) changes on transformation from x (t) to x (α t), α > 0 by a factor
Explanation: x(t)->x(αt), α > 0
α > 1 compression in t, expansion in f by α.
α < 1 expansion in t, compression in f by α.
Two sequences x1 (n) and x2 (n) are related by x2 (n) = x1 (- n). In the z- domain, their ROC’s are
Explanation: x1(n) is the signal in the discrete domain and X1(z) is the signal in the z domain, RoC Rx
z Reciprocals x2(n) = x1(-n) X1(1/z), RoC 1/ Rx
The ROC of z-transform of the discrete time sequence x(n) =is:
Explanation: One part of the equation is the right sided signal and other part is the left sided signal hence the ROC of the system will be 1/3>|z|<1/2.
Which one of the following is the correct statement? The region of convergence of z-transform of x[n] consists of the values of z for which x[n] is:
Explanation: The region of convergence of z-transform of x[n] consists of the values of z for which x[n]r-n is absolutely summable.
The region of convergence of the z-transform of a unit step function is:
Explanation: h[n] =u[n] Hence, Region of Convergence is the region for which the values of the roots in z transform are lying in the function and is the range of values of z for which |z|>1.
If the region of convergence of x1[n]+x2[n] is 1/>|z|<2/3, the region of convergence of x1[n]-x2[n] includes:
Explanation: Region of Convergence is the region for which the values of the roots in z transform are lying in the function and ROC remains the same for addition and subtraction in z-domain.
A sequence x (n) with the z-transform X (z) = Z4 + Z2 – 2z + 2 – 3Z-4 is applied to an input to a linear time invariant system with the impulse response h (n) = 2δ (n-3). The output at n = 4 will be:
Explanation: H (z) = 2z-3
Then taking the inverse Laplace transform of the equation of Y (z) at n=4 y(n) =0.
H (z) is discrete rational transfer function. To ensure that both H(z) and its inverse are stable:
Explanation: For H(z) to be stable the poles must be inside the unit circle and for the inverse of H(z) to be stable the poles of it must be inside the unit circle.
Z and Laplace transform are related by:
Explanation: z = est
s =ln z/T.
Consider the following statements regarding a linear discrete-time system:
H (z) = z2+1/(z+0.5)(z-0.5)
1. The system is stable
2. The initial value of h(0) of the impulse response is -4
3. The steady-state output is zero for a sinusoidal discrete time input of frequency equal to one-fourth the sampling frequency
Which of these statements are correct?
Explanation: Characteristic equation is (z+0.5) (z-0.5) =0
Its root are z =0.5, -0.5
Since both roots are inside the unit circle, hence the system is stable.
The minimum number of delay elements required realizing a digital filter with transfer functionH (z) =
Explanation: H (z) =
Minimum number of delay elements= (Maximum power of z-minimum power of z)
Minimum number of delay elements = 3.
A system can be represented in the form of state equations as:
S (n+1) =A S (n) +B x (n)
Y (n) = C S (n) +D x (n)
Where, A, B, C, D are the matrices , S(n) is the state vector , x(n) is the input and y(n) is the output . The transfer function of the system.
H (z) =Y (z)/X (z) is given by:
Explanation: Solving both the equations and substituting the value of the output equation into the state equation we get the value of the transfer function as obtained.
Assertion (A): The signals anu(n) and anu(-n-1) have the same Z transform, z/(z-a)
Reason (R): the region of convergence of anu(n) is |z|>|a|, whereas the ROC for anu(-n-1) is |z|<|a|.
Explanation: Both have the ROC as given in the reason is true but the z transform for the second is with a minus sign.
What is the number of roots of the polynomial F(z) = 4z3-8z2-z+2, lying outside the unit circle?
Explanation: Factorizing F (z) and then the factors are the roots which here come out to be 3.
Assertion (A): The discrete time system described by y[n] =2x[n] +4x[n-1] is unstable
Reason (R): It has an impulse response with a finite number of non-zero samples
Explanation: For the system to be stable the value of the transfer function in the discrete time domain must be summable and H[n] calculated is summable hence the system is stable.
What is the z-transform of the signal x[n] = anu(n)?
Explanation: By definition this is the basic example of the z-transform and the Z-Transform of the equation is calculated is z/z-a.
Which one of the following rules determine the mapping of s-plane to z-plane?
Explanation: S- plane can be mapped into the z plane with certain rules than right side maps into the outside, left side maps into the inside and imaginary axis maps on the unit circle of the z plane.
Assertion (A): The z-transform of the output of the sampler is given by the series.
Reason (R): The relationship is the result of the application of z = e-sT, where T stands for the time gap between the samples.
Explanation: T is termed as the time of the sampling instant and z transform is always defined for the instant of the sampling event and this can be as desired by the user.
Convolution of two sequences X1[n] and X2[n] are represented by:
Explanation: Convolution of the two sequences is the combination of multiplication and addition of the two sequences at each instant and convolution in time domain is multiplication in the frequency domain.