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2.161 Signal Processing: Continuous and Discrete
Fall 2008
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f
(
t
)
1
Massachusetts
Institute
of
Technology
Department
of
Mechanical
Engineering
2.161
Signal
Processing
 Continuous
and
Discrete
Fall
Term
2008
Lecture
10
1
Reading:
•
Class
Handout:
Sampling
and
the
Discrete
Fourier
Transform
•
Proakis
&
Manolakis
(4th
Ed.)
Secs.
6.1
–
6.3,
Sec.
7.1
•
Oppenheim,
Schafer
&
Buck
(2nd
Ed.)
Secs.
4.1
–
4.3,
Secs.
8.1
–
8.5
The
Sampling
Theorem
Given
a
set
of
samples
{
f
n
}
and
its
generating
function
f
(
t
),
an
important
question
to
ask
is
whether
the
sample
set
uniquely
deﬁnes
the
function
that
generated
it?
In
other
words,
given
{
f
n
}
can
we
unambiguously
reconstruct
f
(
t
)?
The
answer
is
clearly
no,
as
shown
below,
where
there
are
obviously
many
functions
that
will
generate
the
given
set
of
samples.
In
fact
there
are
an
inﬁnity
of
candidate
functions
that
will
generate
the
same
sample
set.
The
Nyquist
sampling
theorem
places
restrictions
on
the
candidate
functions
and,
if
satisﬁed,
will
uniquely
deﬁne
the
function
that
generated
a
given
set
of
samples.
The
theorem
may
be
stated
in
many
equivalent
ways,
we
present
three
of
them
here
to
illustrate
diﬀerent
aspects
of
the
theorem:
•
A
function
f
(
t
),
sampled
at
equal
intervals
Δ
T
,
can
not
be
unambiguously
reconstructed
from
its
sample
set
{
f
n
}
unless
it
is
known
apriori
that
f
(
t
)
contains
no
spectral
energy
at
or
above
a
frequency
of
π/
Δ
T
radians/s.
•
In
order
to
uniquely
represent
a
function
f
(
t
)
by
a
set
of
samples,
the
sampling
interval
Δ
T
must
be
suﬃciently
small
to
capture
more
than
two
samples
per
cycle
of
the
highest
frequency
component
present
in
f
(
t
).
•
There
is
only
one
function
f
(
t
)
that
is
bandlimited
to
below
Δ
T
radians/s
that
is
satisﬁed
by
a
given
set
of
samples
{
f
n
}
.
1
copyright
±
c
D.Rowell
2008
10–1
±
±
Note
that
the
sampling
rate,
F
s
=1
/
Δ
T
,
must
be
greater
than
twice
the
highest
cyclic
frequency
F
max
in
f
(
t
).
Thus
if
the
frequency
content
of
f
(
t
)
is
limited
to
Ω
max
radians/s
(or
F
max
cycles/s)
the
sampling
interval
Δ
T
must
be
chosen
so
that
π
Δ
T<
Ω
max
or
equivalently
1
Δ
2
F
max
The
minimum
sampling
rate
to
satisfy
the
sampling
theorem
F
N
=Ω
max
/π
samples/s
is
known
as
the
Nyquist
rate
.
1.1
Aliasing
Consider
a
sinusoid
f
(
t
)=
A
sin(
at
+
φ
)
sampled
at
intervals
Δ
T
,
so
that
the
sample
set
is
{
f
n
}
=
{
A
sin(
an
Δ
T
+
φ
)
}
,
and
noting
that
sin(
t
sin
(
t
+2
kπ
)
for
any
integer
k
,
2
πm
f
n
=
A
sin(
an
Δ
T
+
φ
A
sin
a
+
n
Δ
T
+
φ
Δ
T
where
m
is
an
integer,
giving
the
following
important
result:
Given
a
sampling
interval
of
Δ
T
,
sinusoidal
components
with
an
angular
frequency
a
and
a
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.161 taught by Professor Derekrowell during the Fall '08 term at MIT.
 Fall '08
 DerekRowell

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