Digital Signal Processing Reference
In-Depth Information
(c)
Write a mathematical function for the waveform of Figure P2.19(b), using the re-
sults of Part (a).
2.20.
(a)
Prove the time-scaling relation in Table 2.3:
q
q
1
ƒ a ƒ
L
d(at)dt =
d(t)dt.
L
-
q
-
q
(
Hint
: Use a change of variable.)
(b)
Prove the following relation from Table 2.3:
t
u(t - t
0
) =
L
d(t - t
0
)dt.
-
q
(c)
Evaluate the following integrals:
q
(i)
cos(2t)d(t)dt
L
-
q
q
sin(2t)d(t -
p
/
4
)dt
(ii)
L
-
q
q
cos[2(t -
p
/
4
)]d(t -
p
/
4
)dt
(iii)
L
-
q
q
(iv)
sin[(t - 1)]d(t - 2)dt
L
-
q
q
(v)
sin[(t - 1)]d(2t - 4)dt
L
-
q
2.21.
Express the following functions in the general form of the unit step function
u(;t - t
0
):
(a)
(b)
(c)
(d)
In each case, sketch the function derived.
u(2t + 6)
u(-2t + 6)
u(
4
+ 2)
u(
4
- 2)
2.22.
Express each given signal in terms of
u(t - t
0
).
Sketch each expression to verify the
results.
(a)
(b)
(c)
(d)
u(-t)
u(3 - t)
tu(-t)
(t - 3)u(3 - t)
2.23.
(a)
Express the output
y
(
t
) as a function of the input and the system transformations,
in the form of (2.56), for the system of Figure P2.23(a).
(b)
Repeat Part (a) for the system of Figure P2.23(b).
