## Signals & Systems Questions and Answers Part-9

1. For the system, y (t) = cos 2πt x (t), which of the following holds true?
a) System is Linear, time-invariant, causal and stable
b) System is time-invariant, causal and stable
c) System is Linear, causal and stable
d) System is Linear, time-invariant and stable

Explanation: y2 (t) k cos 2πt v (t) = k y1 (t)
Again, x3 (t) = v (t) + w (t)
So, y3 (t) = cos 2πt [v (t) + w (t)] = y1 (t) + y2 (t)
Since, the system is both homogeneous and additive, therefore it is linear.
Again, y1 (t) = cos 2πt v (t)
And y2 (t) = cos 2πt (t-t0) ≠ y (t-t0)
= cos [2π (t-t0)] v (t-t0)
∴ The system is time variant.
The response at any time t=t0, depends only on the excitation at that time and not on the excitation at any later time, so Causal system.
If x (t) is bounded then y (t) is also bounded, so a stable system.

2. For the system, y (t) = |x (t)|, which of the following holds true?
a) System is Linear, time-invariant, causal and stable
b) System is Linear, time-invariant and causal
c) System is Linear, time-invariant and stable
d) System is Linear, causal and stable

Explanation: y1 (t) = |v (t)|, y2 (t) = |k v (t)|= |k|y1 (t)
If k is negative, |k| y1 (t) ≠k y1 (t)
Since it is not homogeneous, so non-linear system.
Again, y1 (t) = |v (t)|, y2 (t) = |y (t-t0)| = y1 (t-t0)
∴ System is time invariant.
The response at any time t=t0, depends only on the excitation at that time and not on the excitation at any later time, so causal system.
If x (t) is bounded then y (t) is also bounded, so stable system.

3. For the system, $$t\frac{dy (t)}{dt}$$ – 8 y (t) = x (t), which of the following holds true?
a) System is Linear, time-invariant, causal and stable
b) System is Linear, time-invariant and causal
c) System is time-invariant, causal and stable
d) System is Linear, causal and stable

Explanation: All options are linear. Hence linearity is not required to be checked.
Let x1 (t) = v (t), then $$t\frac{dy_1 (t)}{dt}$$ – 8 y1 (t) = v (t)
Let x2 (t) = v (t-t0)
Then, $$t\frac{dy_2 (t)}{dt}$$ – 8 y2 (t) = v (t-t0)
The first equation can be written as (t-t0) $$t\frac{dy (t-t_0)}{dt}$$ – 8 y (t-t0) = x (t-t0)
This equation is not satisfied if y2 (t) = y1 (t-t0). Therefore y2 (t) ≠ y1 (t-t0)
∴ System is time variant.
The response at any time t=t0, depends only on the excitation at that time and not on the excitation at any later time, so Causal system.
The response will increase without bound as time increases, so unstable system.

4. For the system, $$y (t) = \int_{-∞}^{t+3} x(t) \,dt$$, which of the following holds true?
a) System is Linear, time-invariant and causal
b) System is time-invariant and causal
c) System is Linear and time-invariant
d) System is Linear and stable

Explanation: $$y_1 (t) = \int_{-∞}^{t+3} v(t) \,dt$$
And, $$y_2 (t) = \int_{-∞}^{t+3} kv(t) \,dt$$
= $$k \int_{-∞}^{t+3} x(t) \,dt$$ = k y1 (t)
Now, x3 (t) = v (t) + w (t)
And, y3 (t) = $$\int_{-∞}^{t+3}$$ [v(t) + w(t)]dt
= $$\int_{-∞}^{t+3}$$v(t) dt + $$\int_{-∞}^{t+3}$$w(t) dt
= y1 (t) + y2 (t)
Since, it is both homogeneous and additive, so linear system.
Again, y1 (t) = $$\int_{-∞}^{t+3}$$ v(t) dt
And, y2 (t) = $$\int_{-∞}^{t+3}$$ v(t-t0) dt
= y1 (t-t0)
∴ System is time invariant.
The response at any time t=t0, depends partially on the excitation at time t0 < t < (t0 + 3), which are in future, so non-causal system.
The response will increase without bound as time increases, so unstable system.

5. The impulse response of a continuous time LTI system is $$H (t) = (2e^{-2t} -e^{\frac{t-100}{100}}) \,u (t)$$. The system is ____________
a) Causal and stable
b) Causal but not stable
c) Stable but not causal
d) Neither causal nor stable

Explanation: For t<0, h (t) = 0.
Therefore from the definition of causality, we can infer that the system is Causal.
Now, $$\int_{-∞}^∞ |h(t)| \,dt = ∞$$
From the definition of stability, we can infer that the system is unstable.
Hence, the given system is causal but not stable.

6. The impulse response of a continuous time LTI system is H (t) = e-|t|. The system is ___________
a) Causal and stable
b) Causal but not stable
c) Stable but not causal
d) Neither causal nor stable

Explanation: For t<0,
H (t) ≠ 0
Therefore the system is not causal
Again, $$\int_{-∞}^∞ |h(t)| \,dt$$ = $$\frac{1}{3}$$ < ∞
The system is stable.

7. The impulse response of a continuous time LTI system is H (t) = e-t u (3-t). The system is __________
a) Causal and stable
b) Causal but not stable
c) Stable but not causal
d) Neither causal nor stable

Explanation: For t<0, h (t) ≠ 0
Therefore the system is not causal.
Again, $$\int_{-∞}^∞ |h(t)| \,dt = \int_{-∞}^∞ e-t \,u(3-t) \,dt = ∞$$
System is unstable.

8. The impulse response of a continuous time LTI system is H (t) = e-t u (t-2). The system is __________
a) Causal and stable
b) Causal but not stable
c) Stable but not causal
d) Neither causal nor stable

Explanation: Since, h (t) = 0 for t<0, so the system is causal.
Again, $$\int_{-∞}^∞ |h(t)| \,dt = \int_{-∞}^∞ e-t \,u(t-2) \,dt$$ < ∞
The system is stable.

9. The continuous time convolution integral y(t) = cos πt [u (t+1) – u (t-1) * u(t)] is __________
a) $$\frac{sin⁡πt}{π}$$ [u (t+1) – u(t-1)]
b) $$\frac{sin⁡πt}{π}$$ u(t-1)
c) $$\frac{sin⁡πt}{π}$$ u(t+1)
d) $$\frac{sin⁡πt}{π}$$ u(t)

Explanation: For t<-1, y (t) = 0
For t<1, y (t) = $$\int_{-1}^t cos⁡πt \,dt = \frac{sin⁡πt}{π}$$
For t>1, y (t) = $$\int_{-1}^t$$ cos⁡ πt dt = 0
y (t) = $$\frac{sin⁡πt}{π}$$ [u (t+1) – u(t-1)].

10. The continuous time convolution integral y(t) = e-3tu(t) * u(t+3) is ___________
a) $$\frac{1}{3}$$[1 – e-3(t+3)] u(t+3)
b) $$\frac{1}{3}$$[1 – e-3(t+3)] u(t)
c) $$\frac{1}{3}$$[1 – e-3t] u(t)
d) $$\frac{1}{3}$$[1 – e-3t] u(t+3)

For t≥-3, y (t) = $$\int_{-∞}^3 e^{-3t} \,dt$$
= $$\frac{1}{3}$$[1 – e-3(t+3)]
y(t) = $$\frac{1}{3}$$[1 – e-3(t+3)] u(t+3).