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Eq. (2.35)and property(1.46)of 6 [ n - & ] w e have
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( b ) Similarly, we have
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- n , , ]= x [ n - n o ]
(c) By Eq. (2.35) and definition (1.44) of u [ n - k ] we have
( d l In a similar manner, we have x[n]*u[n-no]=
rn -m
n -ntl
x[k]u[n-k-no]=
x[k]
CHAP. 2 1
LINEAR TIME-INVARIANT SYSTEMS
2.28. The input x [ n ] and the impulse response h [ n ] of a discrete-time LTI system are given
(a) Compute the output y[n] by Eq. (2.35). ( b ) Compute the output y [ n ] by Eq. (2.39).
( a ) By Eq. (2.35) we have
Sequences x [ k ] and h[n - k ] are shown in Fig. 2-20(a) for n < 0 and n > 0. From Fig. 2-20(a) we see that for n < 0, x [ k ] and h [ n - k ] do not overlap, while for n 2 0, they overlap from k = 0 to k = n . Hence, for n < 0, y [ n ] = 0. For n r 0, we have
Changing the variable of summation k to m
=n -k
and using Eq. (1.901,we have
Fig. 2-20
LINEAR TIME-INVARIANT SYSTEMS
[CHAP. 2
Thus, we can write the output y [ n ] as
which is sketched in Fig. 2-20(b). ( b ) By Eq. (2.39)
Sequences h [ k l and x [ n - k ] are shown in Fig. 2-21 for n < 0 and n > 0.Again from Fig. 2-21 we see that for n < 0, h [ k ] and x[n - k l d o not overlap, while for n 2 0, they overlap from k = 0 to k = n. Hence, for n < 0, y [ n ] = 0. For n 2 0, we have
Thus, we obtain the same result as shown in Eq. (2.134).
Fig. 2-21
2.29. Compute y [ n ] = x [ n ]* h [ n ] ,where
( a ) x [ n ] = cunu[n],h [ n ]= p n u [ n ]
( b ) x [ n ] = cunu[n],h [ n ] = a - " u [ - n ] , 0 < a < 1
CHAP. 21
LINEAR TIME-INVARIANT SYSTEMS
(a) From Eq. (2.35) we have
Since we have
u [ k ] u [ n- k ] =
Osksn otherwise
Using Eq. (1.90), we obtain
y[n]=
x [ k ] h [ n- k ] =
f aku[k]a-'"-*'u[-(n - k ) ]
For n 1 0 , we have
u [ k ] u [ k- n J = Osk otherwise
Thus, using Eq. (1.911, we have
For n
r 0, we have
u [ k ] u [ k- n ] = n s k otherwise
Thus, using Eq. (1.921, we have
LINEAR TIME-INVARIANT SYSTEMS
[CHAP. 2
) Combining Eqs. ( 2 . 1 3 6 ~and (2.I36b), we obtain (2.137)
which is sketched in Fig. 2-22.
-2-10 1 2 3
Fig. 2-22
2.30. Evaluate y [ n ] = x [ n ]* h [ n ] , where x [ n ] and h [ n ] are shown in Fig. 2-23, ( a ) by an analytical technique, and ( b ) by a graphical method.
- 1 0 1 2 3
Fig. 2-23
(a) Note that x [ n ]and h [ n ]can be expressed as x [ n ] = 6 [ n ] + 6 [ n l ] + 6 [ n- 2 ] + 6 [ n -31 h [ n ]= 6 [ n ]+ S [ n - 1 ]
+ S [ n - 21
Now, using Eqs. (2.38), (2.130), and (2.I3l), we have
x [ n ]* h [ n ]= x [ n ]* { S [ n ]+ 6 [ n - 1 ]
+ 6 [ n- 21) n S [ n ]+ x [ n ]* S [ n - I ] + x [ n ]* S [ n - 2 1 ) *]
= x [ n ]+ x [ n - 1 ] + x [ n - 21
Thus,
y [ n ] = S [ n ]+ S [ n - I ]
+ S [ n - 21 + 6 [ n - 31
+6[n- 1]+8[n-2]+6[n-3]+6[n-41
+S[n-2]+6[n-3]+6[n-4]+6[n-5]
or or
y [ n ] = S [ n ]+ 2 S [ n - 1 ] + 36[n - 21 + 3 6 [ n - 31 + 2 6 [ n - 41 + 6 [ n - 51
Y [ ~ I ={ 1 , 2 , 3 , 3 , 2 , l }
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