![Page 1: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/1.jpg)
Signals and Systems Fall 2003
Lecture #1321 October 2003
1. The Concept and Representation of Periodic Sampling of a CT Signal2. Analysis of Sampling in the Frequency Domain3. The Sampling Theorem —the Nyquist Rate4. In the Time Domain: Interpolation
5. Undersampling and Aliasing
![Page 2: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/2.jpg)
SAMPLING
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, 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
—…
How do we perform sampling?
![Page 3: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/3.jpg)
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) betw
een sampling points are lost. Key Question for Sampling:
Under what conditions can we reconstructthe original CT signal x(t)
from its samples?
![Page 4: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/4.jpg)
Impulse Sampling—Multiplying x(t) by the sampling function
![Page 5: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/5.jpg)
Analysis of Sampling in the Frequency Domain
Multiplication Property =>
=Sampling Frequency Important to
note:
![Page 6: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/6.jpg)
Illustration of sampling in the frequency-domain for a band-limited (X(jω)=0 for |ω| > ωM) signal
![Page 7: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/7.jpg)
Reconstruction of x(t) from sampled signals
If there is no overlap
between shifted
spectra, a LPF can
reproduce x(t) from xp(t)
![Page 8: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/8.jpg)
Suppose x(t) is bandlimited, so that
X(jω)=0 for |ω| > ωM
Then x(t) is uniquely determined by its
samples {x(nT)} if
where ωs = 2π/T
ωs > 2ωM = The Nyquist rate
![Page 9: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/9.jpg)
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
![Page 10: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/10.jpg)
Observations (Continued)
(2) Sampling is fundamentally a time varyingoperation, 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 (ωs > 2ωM).
(3) What if ωs <= 2ωM? Something different: more later.
![Page 11: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/11.jpg)
Time-Domain Interpretation of Reconstruction
of Sampled Signals—Band-Limited Interpolation
The lowpass filter interpolates the samples assuming x(t)
containsno energy at frequencies >= ωc
![Page 12: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/12.jpg)
Graphic Illustration of Time-Domain Interpolation
Original
CT signal
After Sampling
After passing the LPF
![Page 13: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/13.jpg)
Interpolation Methods
Bandlimited Interpolation Zero-Order Hold First-Order Hold —Linear interpolation
![Page 14: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/14.jpg)
Undersampling and Aliasing
When ωs ≦ 2ωM => Undersampling
![Page 15: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/15.jpg)
Undersampling and Aliasing (continued)
Xr (jω)≠X(jω) Distortion because of aliasing
— Higher frequencies of x(t) are “folded back” and take on the “aliases” of lower frequencies
— Note that at the sample times, xr(nT) = x(nT)
![Page 16: Signals and Systems Fall 2003 Lecture #13 21 October 2003 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling](https://reader036.vdocument.in/reader036/viewer/2022062511/5515033a5503465e608b4588/html5/thumbnails/16.jpg)
A Simple Example
X(t) = cos(wot + Φ)
Picturewould be
Modified…
Demo: Sampling and reconstruction of coswot