Digital Signal Processing Reference
In-Depth Information
Verify each preceding result, using the symbolic mathematics of MATLAB. Simplify
each expression to agree with Table 4.3.
4.10.
Use (4.23) to calculate the Fourier coefficients for the signals in Figure P4.10. Evaluate
for each waveform, and verify these values directly from the waveform; L'Hôpital's
rule is useful in some cases.
C
0
4.11.
Using Table 4.3, find the Fourier coefficients for the exponential form for the signals of
Figure P4.11. Evaluate all coefficients.
4.12.
Consider the signals of Figure P4.11(a) and (d).
(a)
Change the period of to Use Table 4.3 to find the Fourier coefficients
of the exponential form for this signal.
(b)
Use Table 4.3 to find the Fourier coefficients of the exponential form for
(c)
Consider the signal
x
a
(t)
T
0
= 4.
x
d
(t).
x(t) = a
1
x
a
(t) + b
1
x
d
(t - t),
where
x
a
(t)
is defined in part (a). By inspection of Figure P4.11(a) and (d), find
a
1
,
b
1
and such that
x
(
t
) is constant for all time; that is,
t
x(t) = A,
where
A
is a con-
stant. In addition, evaluate
A
.
(d)
Use the results of parts (a) and (b) to show that all the Fourier coefficients of
x
(
t
)
in part (c) are zero except for
C
0
= A.
4.13.
Let
x
a
(t)
be the half-wave rectified signal in Table 4.3. Let
x
b
(t)
be the same signal de-
layed by
T
0
/2.
(a)
Find the coefficients in the exponential form for
x
b
(t).
(
Hint:
Consider time delay.)
(b)
Show that the Fourier coefficients of the sum
[x
a
(t) + x
b
(t)]
are those of the full-
wave rectified signal in Table 4.3.
4.14.
(a)
Use Table 4.3 to find the exponential form of the Fourier series of the impulse
train in Figure P4.14. The magnitude of the weight of each impulse function is
unity, with the signs of the weights alternating.
(b)
Verify the results of part (a) by calculating the Fourier coefficients, using (4.23).
4.15.
Consider the waveforms and in Figure P4.10. Let the difference of these sig-
nals be Then is the same waveform of Example 4.2, except for the average
value. Show that the Fourier coefficients of
x
c
(t)
x
d
(t)
x
s
(t).
x
s
(t)
are equal to those of
x
(
t
) in Example 4.2,
x
s
(t)
except for the average value.
4.16.
A signal has half-wave symmetry if For example, has half-
wave symmetry, as does the triangular wave of Figure P4.11(a). Show that a signal with
half-wave symmetry has no even harmonics; that is,
x(t - T
0
/2) =-x(t).
sin v
0
t
C
k
= 0, k = 0, 2, 4, 6, Á .
4.17.
Consider the signals in Figure P4.11. For
k
sufficiently large, the Fourier coefficient of
the
k
th
harmonic decreases in magnitude at the rate of Use the properties in
Section 4.4 to find
m
for the signals shown in the following figures:
1/k
m
.
(a)
Figure P4.11(a)
(b)
Figure P4.11(b)
(c)
Figure P4.11(c)
(d)
Figure P4.11(d)
