false

Textbook Notes
(368,781)

United States
(206,101)

University of Maryland
(1,931)

ENEE 322
(3)

Steven Marcus
(3)

Chapter

Unlock Document

Electrical & Computer Engineering

ENEE 322

Steven Marcus

Spring

Description

Homework #2
1.17 y(t) = x(sin(t))
(a) Since sin(t) 2 [▯1;1], y(t) only depends on x(▯);▯ 2 [▯1;1].
For example,
y(▯▯) = x(sin(▯▯)) = x(0)
Thus, y(t) is not causal.
(b) Consider two arbitrary inputs x (t1 and x (t)2
The response to x (t) is y (t) = x (sin(t)):
1 1 1
The response to x (2) is y 2t) = x 2sin(t)).
Let x 3t) be a linear combinition of x 1t) and x 2t), that is
x3(t) = ax1(t) + bx2(t)
where a and b are two arbitrary complex constants.
If x3(t) is the input to the given system, then the corresponding output y3(t)
is
y3(t) = x3(sin(t))
= ax1(sin(t)) + bx2(sin(t))
= ay1(t) + by2(t)
Therefore, the system is linear.
1.19 (b, d)
2
(b) y[n] = x [n ▯ 2]
(i) Consider two arbitrary input x [1] and x [2].
x1[n] ! y1[n] = x1[n ▯ 2]
x2[n] ! y2[n] = x2[n ▯ 2]
Let x 3n] be a linear combinition of x 1n] and x 2n], that is
x 3n] = ax 1n] + bx 2n]
where a and b are arbitrary complex constants.
If x3[n] is the input, the corresponding output y 3n] is
2
y3[n] = x3[n ▯ 2]
2
= (ax1[n ▯ 2] + bx2[n ▯ 2])
2 2 2 2
= a x 1n ▯ 2] + b x 2n ▯ 2] + 2abx 1n ▯ 2]x 2n ▯ 2]
6= ay1[n] + by2[n]
1 Therefore, the system is not linear.
(ii) Consider an arbitrary input x [1]. Let
2
y1[n] = x1[n ▯ 2]
be the corresponding output. Consider a second input x [n] o2tained by
time shifting:
x2[n] = x 1n ▯ N]
The output corresponding to the input x [n2 is
2 2
y2[n] = x2[n ▯ 2] = x 1n ▯ 2 ▯ N]
Also note that
2
y1[n ▯ N] = x 1n ▯ 2 ▯ N]
Thus,
y1[n ▯ N] = y 2n]
The system is time-invariant.
(d) y(t) = Odfx(t)g
(i) Linearity
Consider two arbitrary inputs x (1) and x (2).
x 1t) ▯ x1(▯t)
x1(t) ▯! y 1t) = Odfx (1)g =
2
x (t) ▯! y (t) = Odfx (t)g = x 2t) ▯ x2(▯t)
2 2 2 2
Let x 3t) be a linear combinition of x 1t) and x 2t). That is,
x3(t) = ax1(t) + bx2(t)
where a and b are arbitrary scalars.
If x (t) is the input to the given system,then the corresponding output y (t)
3 3
is
y3(t) = Odfx 3t)g
= Odfax (1) + bx (2)g
1
= 2 ((ax1(t) + bx2(t)) ▯ (ax1(▯t) + bx2(▯t)))
= aOdfx (1)g + bOdfx (t2g
= ay1(t) + by2(t)
2 Therefore, the system is linear.
(ii) Time-Invariance
Consider an arbitrary input x (1). Let
x (t) ▯ x (▯t)
y1(t) = Odfx 1t)g = 1 1
2
be the corresponding output.
Consider a second input x (2) obtained by shifting x (1) in time:
x2(t) = x1(t ▯ ▯)
The output corresponding to the input x (t2 is
y2(t) = Odfx (2)g
x (t) ▯ x (▯t)
= 2 2
2
= x 1t ▯ ▯) ▯ x 1▯t ▯ ▯)
2
Also note that
x1(t ▯ ▯) ▯ x1(▯t + ▯)
y 1t ▯ ▯) = 6= 2 (t)
2
Therefore, the system is not time-invariant.
1.27 (a,d,e,f)
(a) y(t) = x(t ▯ 2) + x(2 ▯ t)
(1) Since the value y(t) depends on the value x(t ▯ 2) and x(2 ▯ t), the
system is not memoryless
(2) Consider the input:
x (t) = x (t ▯ ▯)
2 1
The corresponding output to x (t) is
2
y2(t) = x 2t ▯ 2) + x2(2 ▯ t)
= x 1t ▯ 2 ▯ ▯) + x 12 ▯ t ▯ ▯)
Note that
y1(t ▯ ▯) = x1(t ▯ 2 ▯ ▯) + x1(2 ▯ t + ▯) 6= 2 (t)
Thus, the system is not time-invariant.
(3) Consider two arbitrary input x (t1 and x (t2,
3 x 1t) ▯! y1(t) = x1(t ▯ 2) ▯ 1 (2 ▯ t)
x 2t) ▯! y2(t) = x2(t ▯ 2) ▯ 2 (2 ▯ t)
Let x (t) be the linear combinition of x (t) and x (t)
3 1 2
x3(t) = ax1(t) + bx2(t)
The response to the input x (t) is
3
y3(t) = ax1(t ▯ 2) + bx2(t ▯ 2) + ax1(t ▯ 2) + bx2(2 ▯ t)
= a(x1(t ▯ 2) + x1(2 ▯ t)) + b(2 (t ▯ 2) + 2 (2 ▯ t))
= ay1(t) + by2(t)
Thus, the system is linear.
(4) Let t = 0, y(0) = x(▯2) + x(2)
Thus, the system is not causal.
▯
0; t < 0
(d) y(t) =
x(t) + x(t ▯ 2) t ▯ 0
(1) Since the value of y(t) depends on x(t▯2), the system is not memoryless.
(2) Consider an arbitrary input x 1t). Let
▯
0; t < 0
y1(t) =
x 1t) + x 1t ▯ 2); t ▯ 0
be the corresponding output.
Consider a second input x 2t) obtained by shifting x 1t) in

More
Less
Related notes for ENEE 322

Join OneClass

Access over 10 million pages of study

documents for 1.3 million courses.

Sign up

Join to view

Continue

Continue
OR

By registering, I agree to the
Terms
and
Privacy Policies

Already have an account?
Log in

Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.