Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 248

(B) In this part, the frequency response reminds us of the modulation property. It
can be written as:

H ð f Þ¼ G ð f f c Þþ G ð f þ f c Þ:

Hence, using Tables or Tutorial 13 we get:

h ð t Þ¼ 2g ð t Þ cos ð 2pf c t Þ¼ 2f c sinc 2 ð f c t Þ cos ð 2pf c t Þ:

Tutorial 15

Q: Show that:

(A) x ð t Þ y ð t Þ F X ð f Þ Y ð f Þ
(B) x ð t Þ y ð t Þ F X ð f Þ Y ð f Þ

Solution:

(A)

Ff x ð t Þ y ð t Þg¼ Z

1

x ð t Þ y ð t Þ e j2pft dt

1

¼ Z

1

F 1 f X ð f Þg y ð t Þ e j2pft dt

1

2 3

¼ Z

1

Z

1

4 5 y ð t Þ e j2pft dt

X ð k Þ e þ j2pkt dk

1 1

2 3

¼ Z

Z

1 1

4

y ð t Þ e j2p ð f k Þ t dt

5

X ð k Þ dk

1 1

|{z}
Y ð f k Þ

¼ Z

1

X ð k Þ Y ð f k Þ dk ¼ X ð f Þ Y ð f Þ:

1