6.42.      Consider a causal discrete-time FIR filter described by the impulse response

h[ n ] = {2,2, - 2, - 2}

( a)   Sketch  the  impulse  response   h[ n ]  of  the   filter.

( b)   Find  the  frequency  response  H(f!) of  the filter.

(c)     Sketch  the magnitude  response  IH(O)I and  the phase  response  8(0) of  the filter.

(a)         The impulse response h[ n ] is sketched in Fig.  6-27(a).  Note  that  h[ n ]  satisfies the condition ( 6.164 ) with N = 4.

(b)        By definition (6.27)

 

L Â

00

H( fl ) =          h [ n ] e -iOn = 2 + 2e -;u - 2e -i20 - 2e -i30

n = -oo

= 2(1 -e -i30 ) + 2( e -m - e -iW )

= 2 e -i30!2( eim12 _ e -im12 )  + 2 e -i30!2( eifl / 2 _ e -in12 )

n        3!1)

= j e -im;i (sin 2 + sin 2  = Hr(  fl ) ei[)- 3012)1                                ( 6.171)

3

where                                                H,( fl ) = sin{    ) + sin{       )

 

 

(c)         From Eq. ( 6.171)

 

 

3

I H( fl)I = I H,( fl ) I = Isin{     ) + sin{       ) I

 

 

           

which are sketched in Fig. 6-27( b ). We see that the system is a bandpass FIR filter with linear phase.

 

 

 

IH(fl)I

 

 

 

-11'

 

11'

2

 

0                 11'

2

 

1T      fl

 

 

 

 

 

 

 

11'

 

 

0(0)

 

 

 

 

 

 

 

 

(a)

-11'

 

 

 

 

Fig. 6-27

 

(b)