Signals and Systems ��. ���� ��� � Telecommunications Engineering, KMITL

Homework 2 SOLUTIONS

1. ��ก��������� � (signal waveform) ������� ����� u(t) ��� step function ��� r(t) ��� ramp

function

(a) x(t) = u(t+1) – 2u(t) + u(t-1)

SOLUTION:

(b) y(t) = r(t+2) – r(t+1) –r(t-1) + r(t-2)

SOLUTION:

2. ���� H ��ก���� !"����"��" H1, H2, H3, H4 ��#� $#% ��ก ��! ����&��'���%!� #�(�&)�-

�� +�&)�,�#������� - y(t) = H{x(t)} ���

H1: y1(t) = x1(t)x1(t-1)

H2: y2(t) = |x2(t)|

H3: y3(t) = 1 + 2x3(t)

H4: y4(t) = cos(x4(t))

H1

H3

H2

H4

Σ

Σ

x(t)

y(t)

+

+

+

-

Signals and Systems ��. ���� ��� � Telecommunications Engineering, KMITL

SOLUTION: �ก�����������

�������ก���� (5) ������ก����� �! �"#�����$%�!��-�%�#!��'%$��((��)

3. $ ก���� LTI (linear time-invariant system) � ��� # �����.���(�&)� x1(t) �, �.�����$���

�� ��&)� y1(t) ��#�

$#% �� ��&)� y2(t) ��� y3(t) - �(�&)� �/� x2(t) ��� x3(t)

� $� )� ���'��������

�%�#!����� �*%

�%��ก���� (1) +�� (3) �+��,� (4)

H (LTI)

-1 t

0 1 2

1

x3(t) x2(t)

t 0 1 2 3 4

1

-1

x1(t)

t 0 1 2

1

y1(t)

t 0 1 2

1

2

Signals and Systems ��. ���� ��� � Telecommunications Engineering, KMITL

SOLUTION:

�'��� x2(t) = x1(t) – x1(t-2) ��*�%$�ก��((����+(( LTI ��������� y2(t) = y1(t) – y1(t-2)

,��-�%$�����ก �

�'��� x3(t) = x1(t+1) + x1(t) ��������� y3(t) = y1(t+1) – y1(t)

4. - y(t) = [cos(3t)] x(t) $#% !� ����������)������(�%�� ���%������

(a) Memoryless

SOLUTION: YES. �!�� y(t) '.)�ก (%�!�������� t ���� )�

(b) Time-invariant

SOLUTION: NO.

�ก y1(t) = [cos(3t)]x1(t)

��*�%�%�!��: ,�� x2(t) = x1(t-t0)

����� y2(t) = [cos(3t)]x2(t) = [cos(3t)] x1(t-t0)

��*�%��%�#!��: y1(t-t0) = [cos(3 t-3t0)]x1(t-t0)

����*�%�%�!���� t0 +���%�#!�������� ���,/�ก���*�%��%�#!������� t0

(c) Linear

SOLUTION: YES.

,�� x(t) = ax1(t) + bx2(t)

����� y(t) = cos(3t)[ ax1(t) + bx2(t)]

= ax1(t)cos(3t) + bx2(t)cos(3t)

= ay1(t) + by2(t)

y2(t)

t 0 1 2 3 4

1

2

1

2

y1(t)

t -1 0 1 2

Signals and Systems ��. ���� ��� � Telecommunications Engineering, KMITL

(d) Causal

SOLUTION: YES. ��((����+((�������- 0�� )�ก���%$������((+(( causal ����

(e) Stable

SOLUTION: YES. 1�,�� |x(t)| ≤ M, ∀t

|y(t)| = |cos(3t) x(t)| ≤ |cos(3t)|| x(t)| ≤ 1 ⋅ M = M, ∀t

5. H �/����� linear +���� �! ����&��'� �(�&)�-�� +�&)���#���#.��� ��� #

(a) H �/�������� causal %������

SOLUTION: NO. 1���((����+(( causal +��� 1�%�!�������ก��� t0 �%�#!������%$�������ก�%�

���� )� +���ก x2(t) +�� y2(t) �������� x2(t) ������� t=1 +�� y2(t) ����ก�%���� t=0

(b) H �/�������� time-invariant %������

SOLUTION: NO. �!�� x3(t) = x1(t-1) +�� y3(t) ≠ y1(t-1)

H

x2(t)

t 0 1 2 3 4

1

H

H

x1(t)

t 0 1 2

1

x3(t)

t 0 1 2

1

y1(t)

t 0 1 2 3 4

1

y3(t)

t 0 1 2 3 4

1

y2(t)

t 0 1 2 3 4

1

(c) H �/�������� memoryless

SOLUTION: NO. 1�������((�������- ����%$������((+((

��((+(( causal 0�� )�ก���������((�������-������

(d) - �(�&)� x(t) ,�#����

SOLUTION: �'��� x(t) = x

��*�%$�ก��((����+((�/$���� �������

y(t) = y1(t) + 2y3(t)

6. �� ��&)� y[n] ,�#��������������

(a) $#% �� ��&)� y[n] $ ก�(�&)�

SOLUTION:

y

1

2

Signals and Systems

Telecommunications Engineering, KMITL

memoryless %������

1�������((�������- ����%$������((+(( causal ���� +�������+�������((���,/�

0�� )�ก���������((�������-������

,�#���� H �/���#� $#% �� ��&)� y(t)

x(t) = x1(t) + 2x3(t)

��*�%$�ก��((����+((�/$���� �������

(t)

,�#�������������� LTI +��������(�&)� x[n] = δδδδ[n] �/���#�

$ ก�(�&)� x[n] = δδδδ[n-1]

-1

x(t)

0 1 2

1 2

(t)

t 0 1 2 3 4

Systems ��. ���� ��� � Telecommunications Engineering, KMITL

���� +�������+�������((���,/�

x(t)

t 0 1 2

(b) $#% �� ��&)� y[n] $ ก�(�&)�

SOLUTION:

7. $#% convolution sum

h[n] �/���#�

SOLUTION:

8. MATLAB: (�0�"�"ก)

Signals and Systems

Telecommunications Engineering, KMITL

$ ก�(�&)� x[n] = 2δδδδ[n]- δδδδ[n-2]

convolution sum ���� � ∑ ����� � �� �� - �(�&)� x[n] ���

1

2

3

-1

-2

-3

y[n] ����� $���

Systems ��. ���� ��� � Telecommunications Engineering, KMITL

��� impulse response