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A function f(t) is an even function, if for all values of (t)
( T is the timeperiod of the function)
For even function, f(t) = f(t)
For odd function, f(t) = f(t)
A control system transfer function is H(s) = 1/s^{3}. Express its impulse response in terms of unit step signal
Convolution in the time domain implies multiplication in S(or frequency) domain
The Laplace transform of any signal h(t) is given by
So we observe that the H(s) = 1/s^{3}, corresponds to a unit step signal convoluted with itself thrice.
Therefore the correct answer is option 1
The unit impulse response of a linear time invariant system is the unit step function u(t) for t > 0, the response of the system to an excitation e^{at} u(t), a > 0 will be
Given: h(t) = u(t)
x(t) = e^{–at} u(t)
∴ Y(s) = X(s) H(s) =
y(t) = 1/a (1 – e^{– at}) u(t)
A system with an input x(t) and output y(t) is described by the relation: y(t) = tx(t). This system is
y(t) = tx(t)
y_{1}(t) = t.x_{1}(t) = r_{1}(t)
y_{2}(t) = tx_{2}(t) = r_{2}(t)
y_{1}(t) + y_{2}(t) = t(x_{1}(t) + x_{2}(t))
= r_{1}(t) + r_{2}(t) ∴ linear
y(t) = t.x(t)
y( t  t_{o}) = (t  t_{o}) x ( t  t_{o})
and for delayed input signal,
y(t) = t x (t  t_{o})
y(t) ≠ y( tt_{o})
∴ Time varying signal
If a function f(t) u(t) is shifted to the right side by t_{0}, then the function can be expressed as
Since f(t) u(t) = f(t) for t > 0 also we know
u ( t  t_{o}) = 1, for t > t_{o}
Here in right side shifting that means t_{o} > 0
by property on shifting right side,
The impulse response of a causal, linear, time invariant, continuous time system is h(t). The output y(t) of the same system to an input x(t). Where x(t) = 0 for t < 2 is
Since, causal system
h(t) = 0 for t < 0
Input y(t) = x(t) * h(t)
h(τ) = 0 for τ < 0
x ( t  τ) = 0, for t  τ <  2
∴ τ > f + 2
The unit step response of a system is given by (1  e^{αt}) u(t), the impulse response is given by
Figure I and Figure II, shows the input x(t) to a linear time invariant system and the impulse response h(t) of the system
the output of the system is zero everywhere except for the time interval.
Given x(t) and h(t) y(t) = x(t) * h(t)
y(t) =^{∞}x(τ) h(t – τ)dτ_{– ∞}
Thus, we conclude that the output of the system is zero everywhere except for the time interval.
1 < t < 5
A signal f(t) = cos8πt + 0.5cos4πt is instantaneously sampled. The maximum allowable value of sampling interval T_{s} in sec is
The impulse response. h[n] of a linear time invariant system is given by h[n] = u[n + 3] + u[n  3]  2u[n  7], the above system is
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