Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 246

n o ¼ e jj :

2
1 þð 2pt Þ 2

Using duality of FT and part (A) above we get: F

k :

f

From Tables (scaling property) we have: s ð kt Þ$ jj S

Applying this to the above result we get:

()
¼ 2pe 2p jj

2
1 þð t Þ 2

F

Tutorial 13

Q: Determine the impulse response of the ideal bandstop filter whose frequency
response is shown below.

H ( f )
1

f , Hz

− f 2 − f 1 f 1 f 2

0

Solution: It is better that we arrange H(f) in terms of well-known functions. In
this question it can be written as follows:

H ð f Þ¼ 1 G ð f Þ;

where G(f) is shown below.
To find an explicit formula for G(f), let f 0 ¼ f 1 þ f 2
2

; B ¼ f 2 f 1 :

G ð f Þ¼ Y
B

ð f f 0 Þþ Y
B

) ð f þ f 0 Þ¼ X ð f f 0 Þþ X ð f þ f 0 Þ;

where X ð f Þ¼ Q B ð f Þ .
Now from Tables we have:

L Q L ð f Þ) L sinc ð Lt Þ F Q L ð f Þ [multiply both sides by L]
Hence, for this question we have: Bsinc ð Bt Þ
|{z}
x ð t Þ

1. sinc ð Lt Þ F

1

F Y B ð f Þ

:

|{z}
X ð f Þ

2. x ð t Þ cos ð 2pf 0 t Þ F 2 X ð f f 0 Þþ 2 X ð f þ f 0 Þ;

F X ð f f 0 Þþ X ð f þ f 0 Þ
|{z}
G ð f Þ

) 2x ð t Þ cos ð 2pf 0 t Þ

|{z}
g ð t Þ¼ 2B sinc ð Bt Þ cos ð 2pf 0 t Þ

) h ð t Þ¼ F 1

f

1 G ð f Þ

g¼ F 1 fg F 1 G ð f f ¼ d ð t Þ g ð t Þ
¼ d ð t Þ 2B sinc ð Bt Þ cos ð 2pf 0 t Þ: