1,The Concept and Representation of Periodic
Sampling of a CT Signal
2,Analysis of Sampling in the Frequency Domain
3,The Sampling Theorem — the Nyquist Rate
4,In the Time Domain,Interpolation
5,Undersampling and Aliasing
Signals and Systems
Fall 2003
Lecture #13
21 October 2003
We live in a continuous-time world,most of the signals we
encounter are CT signals,e.g,x(t),How do we convert them into DT
signals x[n]?
SAMPLING
How do we perform sampling?
— Sampling,taking snap shots of x(t) every T seconds.
T – sampling period
x[n] ≡ x(nT),n =,..,-1,0,1,2,..,— regularly spaced samples
Applications and Examples
— Digital Processing of Signals
— Strobe
— Images in Newspapers
— Sampling Oscilloscope
— …
Why/When Would a Set of Samples Be Adequate?
Observation,Lots of signals have the same samples
By sampling we throw out lots of information
– all values of x(t) between sampling points are lost,
Key Question for Sampling:
Under what conditions can we reconstruct the original CT signal
x(t) from its samples?
Impulse Sampling — Multiplying x(t) by the sampling function
Analysis of Sampling in the Frequency Domain
Important to
note,ω
s
∝1/T
Illustration of sampling in the frequency-domain for a
band-limited (X(jω)=0 for |ω|> ω
M
) signal
No overlap between shifted spectra
Reconstruction of x(t) from sampled signals
If there is no overlap between
shifted spectra,a LPF can
reproduce x(t) from x
p
(t)
The Sampling Theorem
Suppose x(t) is bandlimited,so that
Then x(t) is uniquely determined by its samples {x(nT)} if
Observations on Sampling
(1) In practice,we obviously
don’t sample with impulses
or implement ideal lowpass
filters.
— One practical example:
The Zero-Order Hold
Observations (Continued)
(2) Sampling is fundamentally a time-varying operation,since we
multiply x(t) with a time-varying function p(t),However,
is the identity system (which is TI) for bandlimited x(t) satisfying
the sampling theorem (ω
s
> 2ω
M
).
(3) What if ω
s
≤ 2ω
M
Something different,more later.
Time-Domain Interpretation of Reconstruction of
Sampled Signals — Band-Limited Interpolation
The lowpass filter interpolates the samples assuming x(t) contains
no energy at frequencies ≥ ω
c
T
h(t)
Graphic Illustration of Time-Domain Interpolation
Original
CT signal
After sampling
After passing the LPF
Interpolation Methods
Bandlimited Interpolation
Zero-Order Hold
First-Order Hold — Linear interpolation
Undersampling and Aliasing
When ω
s
≤ 2 ω
M
Undersampling
Undersampling and Aliasing (continued)
— Higher frequencies of x(t) are,folded back” and take on the
“aliases” of lower frequencies
— Note that at the sample times,x
r
(nT) = x(nT)
X
r
(jω)≠ X(jω)
Distortion because
of aliasing
Demo,Sampling and reconstruction of cosω
o
t
A Simple Example
Picture would be
Modified…
Sampling of a CT Signal
2,Analysis of Sampling in the Frequency Domain
3,The Sampling Theorem — the Nyquist Rate
4,In the Time Domain,Interpolation
5,Undersampling and Aliasing
Signals and Systems
Fall 2003
Lecture #13
21 October 2003
We live in a continuous-time world,most of the signals we
encounter are CT signals,e.g,x(t),How do we convert them into DT
signals x[n]?
SAMPLING
How do we perform sampling?
— Sampling,taking snap shots of x(t) every T seconds.
T – sampling period
x[n] ≡ x(nT),n =,..,-1,0,1,2,..,— regularly spaced samples
Applications and Examples
— Digital Processing of Signals
— Strobe
— Images in Newspapers
— Sampling Oscilloscope
— …
Why/When Would a Set of Samples Be Adequate?
Observation,Lots of signals have the same samples
By sampling we throw out lots of information
– all values of x(t) between sampling points are lost,
Key Question for Sampling:
Under what conditions can we reconstruct the original CT signal
x(t) from its samples?
Impulse Sampling — Multiplying x(t) by the sampling function
Analysis of Sampling in the Frequency Domain
Important to
note,ω
s
∝1/T
Illustration of sampling in the frequency-domain for a
band-limited (X(jω)=0 for |ω|> ω
M
) signal
No overlap between shifted spectra
Reconstruction of x(t) from sampled signals
If there is no overlap between
shifted spectra,a LPF can
reproduce x(t) from x
p
(t)
The Sampling Theorem
Suppose x(t) is bandlimited,so that
Then x(t) is uniquely determined by its samples {x(nT)} if
Observations on Sampling
(1) In practice,we obviously
don’t sample with impulses
or implement ideal lowpass
filters.
— One practical example:
The Zero-Order Hold
Observations (Continued)
(2) Sampling is fundamentally a time-varying operation,since we
multiply x(t) with a time-varying function p(t),However,
is the identity system (which is TI) for bandlimited x(t) satisfying
the sampling theorem (ω
s
> 2ω
M
).
(3) What if ω
s
≤ 2ω
M
Something different,more later.
Time-Domain Interpretation of Reconstruction of
Sampled Signals — Band-Limited Interpolation
The lowpass filter interpolates the samples assuming x(t) contains
no energy at frequencies ≥ ω
c
T
h(t)
Graphic Illustration of Time-Domain Interpolation
Original
CT signal
After sampling
After passing the LPF
Interpolation Methods
Bandlimited Interpolation
Zero-Order Hold
First-Order Hold — Linear interpolation
Undersampling and Aliasing
When ω
s
≤ 2 ω
M
Undersampling
Undersampling and Aliasing (continued)
— Higher frequencies of x(t) are,folded back” and take on the
“aliases” of lower frequencies
— Note that at the sample times,x
r
(nT) = x(nT)
X
r
(jω)≠ X(jω)
Distortion because
of aliasing
Demo,Sampling and reconstruction of cosω
o
t
A Simple Example
Picture would be
Modified…
