Question 1 By considering it an ideal op-amp analysing the circuit as As vi = x(t) – v so the output voltage will be y(t) = A(x(t) – v1) And close loop again: A → ∞ And the same has been obtained from the analysis of the equivalent circuit thus: Yes, the described system is stable and regular. Question 3a)H(x, t) = 3x (t -1) - 2x(t + 1) i. System is not causal because the system depends on x(t + 1) ii. System is linear because both outputs are the same. iii. System is time invariant both are equal. iv. System is stable because it has a step response. v. System is regular b) H(x, t) = sin(2 pi x(t)) i. System is causal because H depends only on current or past time. ii. System is non-linear. iii.System is time invariant. iv.System is stable. v.System is regular. c) H(x, t) = t2x(t) i. System is causal ii. System is linear. iii. System is time invariant. iv. System is not stable. v. System is regular. d) H(x, t) = integral(cos(pi T) x (t + T) dT, -0.5,0.5) i.System is not causal. ii.System is linear. iii. System is time invariant. iv. System is stable. v. System is regular. Question 5a) Bounded sequence depends on the input Not periodic Absolutely summable Square summableb) Bounded Periodic Absolutely Summable Square SummableC) Bounded Periodic Not Absolutely Summable Not Square Summabled) Bounded Not Periodic Absolutely Summable Square Summable 0 0 4 Theory Assignment Answer in no more than 10 pages total Minimum 10pt font size October 26, 2014 1. (Multiplier) Consider the operational amplifier circuit in Figure 1. Draw an equivalent circuit using the model for an operational amplifier including input resistance Ri, output resistance Ro and open loop gain A (given in Figure 2.5 of the lecture notes). Analyse this circuit to obtain a relationship between the input voltage signal x and output voltage signal y. By taking limits as Ri →∞, A →∞ and Ro → 0 find an expression relating x and y assuming that the operational amplifier is ideal. Obtain the same expression directly using the rules for analysing ideal operational amplifiers. Is the system that describes this circuit stable? Is it regular? 2. (Properties of signals) Plot each of the following signals and show whether they are: bounded, periodic, absolutely integrable, square integrable. (a) x(t) = 1 (b) x(t) = u(t + 1)e−t where u(t) is the step function (c) x(t) = sin(2πt) cos(πt) (d) x(t) = sin2(πt) πt 3. (Properties of systems) State whether each of the following systems are: causal, linear, time invariant, stable, regular. Plot the impulse and step response of the systems whenever they exist. (a) H(x,t) = 3x(t− 1) − 2x(t + 1) (b) H(x,t) = sin ( 2πx(t) ) (c) H(x,t) = t2x(t) (d) H(x,t) = ∫ 1/2 −1/2 cos(πτ)x(t + τ)dτ 4. (Masses, springs, and dampers) Figure 2 depicts a mechanical system involving two masses, two springs, and a damper connected between two walls. Suppose that the spring K2 is ...

1. Question 1
By considering it an ideal op-amp analysing the circuit as
As vi = x(t) – v so the output voltage will be y(t) = A(x(t) –
v1)
And close loop again:
A → ∞
And the same has been obtained from the analysis of the
equivalent circuit thus:
Yes, the described system is stable and regular.
Question 3a)H(x, t) = 3x (t -1) - 2x(t + 1)
i. System is not causal because the system depends on x(t +
1)
ii. System is linear because both outputs are the same.
iii. System is time invariant both are equal.
iv. System is stable because it has a step response.
v. System is regular b) H(x, t) = sin(2 pi x(t))
i. System is causal because H depends only on current or past
time.
ii. System is non-linear.
iii.System is time invariant. iv.System is stable.
v.System is regular. c) H(x, t) = t2x(t)
i. System is causal
ii. System is linear.
iii. System is time invariant.
iv. System is not stable.

2. v. System is regular. d) H(x, t) = integral(cos(pi T) x (t
+ T) dT, -0.5,0.5)
i.System is not causal.
ii.System is linear.
iii. System is time invariant.
iv. System is stable.
v. System is regular.
Question 5a)
Bounded sequence depends on the input
Not periodic
Absolutely summable
Square summableb)
Bounded
Periodic
Absolutely Summable
Square SummableC)
Bounded
Periodic
Not Absolutely Summable
Not Square Summabled)
Bounded
Not Periodic
Absolutely Summable Square Summable
0
0
4

3. Theory Assignment
Answer in no more than 10 pages total
Minimum 10pt font size
October 26, 2014
1. (Multiplier) Consider the operational amplifier circuit in
Figure 1. Draw an equivalent circuit using
the model for an operational amplifier including input
resistance Ri, output resistance Ro and open loop
gain A (given in Figure 2.5 of the lecture notes). Analyse this
circuit to obtain a relationship between
the input voltage signal x and output voltage signal y. By taking
limits as Ri →∞, A →∞ and Ro → 0
find an expression relating x and y assuming that the
operational amplifier is ideal. Obtain the same
expression directly using the rules for analysing ideal
operational amplifiers. Is the system that describes
this circuit stable? Is it regular?
2. (Properties of signals) Plot each of the following signals and
show whether they are: bounded, periodic,
absolutely integrable, square integrable.
(a) x(t) = 1
(b) x(t) = u(t + 1)e−t where u(t) is the step function
(c) x(t) = sin(2πt) cos(πt)
(d) x(t) =
sin2(πt)
πt

4. 3. (Properties of systems) State whether each of the following
systems are: causal, linear, time invariant,
stable, regular. Plot the impulse and step response of the
systems whenever they exist.
(a) H(x,t) = 3x(t− 1) − 2x(t + 1)
(b) H(x,t) = sin
(
2πx(t)
)
(c) H(x,t) = t2x(t)
(d) H(x,t) =
∫ 1/2
−1/2 cos(πτ)x(t + τ)dτ
4. (Masses, springs, and dampers) Figure 2 depicts a mechanical
system involving two masses, two
springs, and a damper connected between two walls. Suppose
that the spring K2 is at rest when the
mass M2 is at position p(t) = 0. A force, represented by the
signal f, is applied to mass M1. Derive
a differential equation relating the force f and the position p of
mass M2. Suppose that H is a linear
time invariant system mapping f to p and suppose that M1 = K1
= K2 = B = 1 and M2 = 2. Find the
transfer function of H. Find the poles and zeros of H and draw a
pole zero plot. Determine whether H
is stable and/or regular. Find and plot the impulse response and
the step response of H if they exist.
5. (Properties of sequences) Plot each of the following
sequences and show whether they are: bounded,

5. periodic, absolutely summable, square summable.
(a) xn = 2
−nun where un is the step sequence
(b) xn = sin(πn/4) cos(πn/6)
(c) xn = sin(n)
(d) xn = un−1/n
1
−
+
x(t)
12kΩ
y(t)22kΩ
Figure 1: Operational amplifier circuit configured as a
multiplier
M1
BB
f(t)
M2
K1

6. p(t)
K2
Figure 2: Two masses, a spring, and a damper
2