Sampling Theorem
Challenge 02
Lesson 03: Review of the Sampling Theorem
dditional Sample Theorem related topics
Additional Sample Theorem related topics
Challenge 03
What’s it all about!
Shannon’s (Nyquist) Sampling Theorem.
Signal reconstruction (interpolation).
Practical interpolation.
Sampling modalities (critical, over, under).
pg
(
,
,
)
Comment:
What is this thing called compressed sampling?
Lesson 03
hallenge 02
Challenge 02
1
st
Order Impulse Invariant System
A simple firstorder
RC
circuit is show.
The relationship between the input forcing function
v
(
t
) and voltage
developed across the capacitor, denoted
v
o
(
t
), is defined by the 1
st
order
rdinary differential equation
ordinary differential equation
1
=
1
)
(
0
t
t
t
dv
What is the equivalent discretetime model of the circuit’s impulse
)
(
)
(
0
RC
RC
dt
response based on a sample rate of f
s
=1 k Sa/s, and RC=10
2
?
Lesson 03
hallenge 02
Challenge 02
It immediately follows that the system’s impulse response is given by:
f th
li
i d i
th
lti
di
t
i
l
)
(
1
=
)
(
/

t
u
e
RC
t
h
RC
t
If the sampling period is
T
s
, the resulting discretetime impulse
response is:
or, for
k
0,
)
(
]
[
s
s
d
kT
h
T
k
h
RC
kT
T
T

C
k
s
C
s
d
RC
e
RC
k
h
s
/
=
]
[
RC
T
e
/
;
α
Lesson 03
hallenge 02
Challenge 02
For
RC
=10
2
, and
f
s
=1000 Hz, the following results.
0
0
;
904837
.
0
2
3
1
.
0
10
10
*
10
1
3
2
C
e
e
e
1
.
0
10
/
10
/
RC
T
s
k
s
s
s
d
e
RC
T
kT
h
T
k
h
)
(
/
.
)
(
]
[
1
0
k
)
.
(
.
904837
0
1
0
Lesson 03
hallenge 02
Challenge 02
For
RC
=10
2
, and
f
s
=1000 Hz, the following difference equation results.
y[k]  0.904837y[k1] = 0.1x[k]
or
[k] = 0 904837y[k ] + 0 1x[k]
y[k] = 0.904837y[k 1] + 0.1x[k]
[k]
If x[k]=
[k], then y[k]:
x[k]
T
y[k]
y[0] = 0.1
y[1] = (0.1)(0.9)
[2]= (0 1)(0 9)(0 9)=
0.1
0.904837
y[2]= (0.1)(0.9)(0.9)=
=(0.1)(0.9)
2
…
Lesson 03
>> den=[1  .9094837]; num=[0.1];
den [1 0.9094837]; num [0.1];
>> x=[1 0 0 0 0 0 0 0 0 0 0 0 0 ] ;
% impulse
>> h=filter(num,den,x);
% impulse response
(,,
)
; p
p
>> plot(h)
[k]
h[k]
Lesson 03
SP
gift from Claude Shannon
DSP – a gift from Claude Shannon
hat were his accomplishments?
(MS (MIT) Sampling Theorem,
What were his accomplishments?
Who was his patron?
Ph.D. (MIT) Information Theory
The telephone company (Bell Labs) – does
this explain the interest in sampling and
information theory?
Lesson 03
DSP System Architecture
The Sampling Theorem is core to understanding DSP. The theorem both
enables and constrains the performance of a DSP system consisting of an
DC DAC di it l
DSP
l
l
i
l
diti
i
filt
ADC, DAC, digital or DSP processor, plus analog signal conditioning filters
(
i.e
., antialiasing and reconstruction filter).
Typical signal processing stream.
Lesson 03
Shannon
Factoid:
Some attribute the sampling theorem to
Claude
Shannon
, and others to
Harry Nyquist
.
Nyquist suggested the sampling theorem in 1928, which was
th
ti
ll
b Sh
i 1949 (MS Th
i )
mathematically proven by Shannon in 1949 (MS Thesis).
ome use the term "Nyquist Sampling Theorem" and others use
Some use the term Nyquist Sampling Theorem , and others use
"Shannon Sampling Theorem" to refer to the underlying samplilng
theory.