6.450 Principles of Digital Communication
Wednesday, October 9, 2002
MIT, Fall 2002
Handout #23
Due: Wednesday, October 16, 2002
Problem Set 6
Problem 6.1
(a) Show that for any
T >
0 the set of functions
{
ψ
m,k
(
t
) =
e
2
πimt/T
sinc(
t

kT
T
)
}
is an
orthogonal set.
Hint: Show that the Fourier transforms of each of these functions are
orthogonal to each other; review the argument used for the
T

spaced truncated sinusoids.
(b) Derive the energy equation, Eq. (13) in Lecture 9.
Problem 6.2
The following exercise is designed to illustrate the sampling of an ap
proximately baseband waveform.
To avoid messy computation, we look at a waveform
baseband limited to 3/2 which is sampled at rate 1 (
i.e.
, sampled at only 1/3 the rate
that it should be sampled at). In particular, let
u
(
t
) = sinc(3
t
).
(a) Sketch ˆ
u
(
f
). Sketch the function ˆ
v
m
(
f
) = rect(
f

m
) for each integer
m
such that
v
m
(
f
)
6
= 0. Note that ˆ
u
(
f
) =
∑
m
ˆ
v
m
(
f
).
(b) Sketch the inverse transforms
v
m
(
t
) (real and imaginary part if complex).
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 Spring '08
 DanielLee

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