Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
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2. Similar to Example 4.2,compute the Fourier series coefficients for the signal
xt cosV
0
t:
3. Find and sketch the Fourier transform of the following signals:
(a) The rectangular signal defined as
xt A,
jtj t
0,
t > t.
&
(b) The periodic impulse train defined as
dt
I
kÀI
dt À kT
0
:
(c) xt dt.
(d) xt 1.
4. Find the z-transform and ROC of the following sequences:
(a) xn 1, n ! 0.
(b) xn e
Àan
, n ! 0:
(c) xn Àa
n
, n À1, À 2, F F F , À I
0,
n ! 0.
&
(d) xn sin!n, n ! 0:
5. The z-transform of an N-periodic sequence can be expressed in terms of the z-transform of its
first period. That is,if
x
1
n xn, 0 n N À 1
0,elsewhere
&
denotes the sequence of the first period of the periodic sequence x(n). Show that
Xz
X
1
z
1 À z
ÀN
, jz
N
j > 1,
where X
1
z ZTx
1
n.
6. Given the periodic signal
xn f1,1,1, À 1, À 1, À 1,1,1,1, À 1, F F Fg
of period equal to 6. Find Xz.
EXERCISES
177
Summary :
0: (c) xn Àa n , n À1, À 2, F F F , À I 0, n ! Given the periodic signal xn f1,1,1, À 1, À 1, À 1,1,1,1, À 1, F F Fg of period equal to 6.
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amp,find,periodic,signal,period,sequence,ztransform,first,following,defined,fourier,42compute,gien