Signals and Systems
LTI Systems
Practice questions from LTI Systems.
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IncorrectConsider a discrete-time linear time-invariant (LTI) system , where
Let
where is the discrete-time unit impulse function. For an input signal , the output is
Given a discrete-time LTI system ,
In other way,
Hence, option (a) is correct.
Selected data points of the step response of a stable first-order linear time-invariant (LTI) system are given below. The closest value of the time-constant, in sec, of the system is
For standard order system,
Output,
where and Time Constant
For
Hence, option (b) is correct.
Let continuous-time signals and be
Consider the convolution . Then is
from area property,
Option (a) is correct.
The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant system . The output is observed to be the continuous-time unit step signal . Which one of the following statements is true?
Every bounded input signal applied to results in a bounded output signal.
It is possible to find a bounded input signal which when applied to results in an unbounded output signal.
On applying any input signal to , the output signal is always bounded.
On applying any input signal to the output signal is always unbounded.
Given
Since the unit impulse input produces output as u(t). Hence u(t) is the impulse response of the system.
For input (bounded signal),
Any signal convolved with , gives its running integral,
Thus, every bounded signal does not produces every bounded o/p signal
h(t) = u(t) is not an absolutely integrable signal
Hence, the system is not BIBO stable so it can produce unbounded output for bounded input
Hence, option (b) is correct.
Suppose signal is obtained by the time-reversal of signal , i.e., . Which one of the following options is always true for the convolution of and ?
Since convolution is commutative in nature
is even.
Which of the following statement(s) is/are true?
If an LTI system is causal, it is stable.
A discrete time LTI system is causal if and only if its response to a step input is 0 for .
If a discrete time system has an impulse response of finite duration the system is stable.
If the impulse response for all , then the LTI system is stable.
(a) for a causal LTI system, h[n] = 0 ∀ n < 0 & for stability
∴ Causal LTI system may or may not be stable.
(b) for causal LTI system, h[n] = 0 ∀ n < 0
Step response, s[n] = u[n] * h[n]
∴ both u[n] & h[n] = 0 ∀ n < 0
∴ S[n] = 0 ∀ n < 0
(c) if h[n] has finite duration, then system may or may not be stable
Has finite duration but h[n] is not absolutely summable ∴ system is unstable.
(d) if 0 < |h[n]| < 1 then system may be unstable because if signal has infinite duration then h[n] is not absolutely summable.
For the signals shown in the figure, is maximum at . Then in seconds is __________ (Round off to the nearest integer).
For maximum value of convolution we must have maximum area under product of x(t - τ) & y(τ).
∴ (t – 1) must coincide with t = 3 so that entire positive area under y(t) coincide with x(t - τ)
∴ t – 1 = 3
t = 4
∴ z(t) = max occurs at t = 4