1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

392 CHAPTE R 9 • z -TRANSFORMS AND APPLICATIONS

(a) Start with Yo = ~xo, Y1 = !x 1 + Hxo, and show by induction that

Yn = E~=O ~ ur Xn-1·

(b) Use the transfer function H(z) =! (•~i> and find the unit-sample

response h(n) = 3-^1 (H(z)].

(c) Verify that the general term in part (a) is given by the convolution

for mula y(n) = Yn = L;~h 1 Xn-i·

9. Show that the moving average filter y(n] = ~(x(n] + x [n -1) + x (n-2] + x[n -3] +

x[n - 4) + x [n -5)) is designed to zero out cos(mr), cos(2;n), sin(2;n), cos( in),

and sin(in).

1 0. Use the transfer function fl(z) = I^5 k i(1-·^6 )
6 Ek=O z = • ,_; and show that the moving
average filter in Exercise 9 has an alternative formula y(n] = ~(x[n) - x[n -6]) +
y[n- l].

11. 
    

    Use the t ransfer function H(z) = ~ Ek^7 =O z It = j{ll-• - z^8 ) and show t hat t he moving
    average filter in Example 9.24 has an alternative formula y[n) = Hx[n] - x(n -
    8)) + y (n - 1).

    
    


12. 
    

    (a) Construct a filter using t he zeros e i.i1!, • and e ±~2w. What signals are

    
    


13. 
    

14.

15.

"zeroed-out"?

( ) fi

~ ± 4 ,, ±32:r

b Construct a lt er using the zeros e • , e 2, and e. What signals

are "zeroed-out"?

±i1l' ±••

(a) Construct a filter using the zeros e-r and e-r. Wbat signals are

"zeroed-out"?

(b) Construct a filter using the zeros e ~

"zeroed-out"?

± ..

(a) Construct a filter using the zeros e-r

"zeroed-out"?

± i2'1f

and e :r. What signals are

± i.!:>'11'

and e-.-. What signals are

( ) C fi

%iw ±•1" ±tf>11'

b onstruct a lter using the zeros e <r , e • , and e ......-. What signals

are "zeroed-out"?

±ul' ±i'll"

(a) Construct a filter using the zeros e....--and e-,-. What signals are

"zeroe<l·out"?

.± ift' ±i&•

(b) Construct a filter using t he zeros e -r and e-.-. What signals are

"zeroed-out"?

16. Construct the combination filter using the zeros fc;e ±;• and -foe;" and poles ~e±if
    for attenuating cos(n-ir), cos( ~n), and sinGn) and "boosting up" some of the low
    frequencies, respectively.


17. (a) Construct a filter using t he zeros e±iSn/^4 and e_,,, = - 1 for "zeroing
    out" cos(3 4 " n), sin(3; n), and cos(7rn).

(b) Construct a filter using the poles ~e~ and ~e^0 ' = k for "boosting

up" signals near cos( tn) and sin( in) and low-frequency signals.

(c) Construct a filt er using t he zeros and poles in part.~ (a) and (b).