[PPT] - reconstruction of signals Lecture 13 Systems and Control Theory 1 PowerPoint Presentation

SLIDE 1

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Discretization and reconstruction of signals

Lecture 13

1

SLIDE 2

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Discretizing signals

Sampling a continuous signal at discrete intervals
This can be achieved by multiplying the signal with a train of Dirac-

deltas.

Source: http://cnx.org/content/m46012/latest/?collection=col11510/latest 2

SLIDE 3

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Discretizing signals

Is it possible to sample a continuous signal without loss of

information?

Yes, as long as the signal has a limited bandwidth
Bandwidth = maximum frequency in a signal
How often should we sample the signal?
Nyquist theorem: If a signal has a bandwidth B then it can be fully

reconstructed after being sampled with a frequency 2B.

3

SLIDE 4

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Proof of Nyquist theorem

Sampling = multiplication by a train of Dirac-impulses
Fourier transform of an impulse train with period T is an impulse

train with period T-1:

Sampling in frequency domain = convolution of signal spectrum and

an impulse train.

f(t)

t ω Source: http://cnx.org/content/m46012/latest/?collection=col11510/latest 4

F(ω)

…

SLIDE 5

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Proof of Nyquist theorem

Convolution with an impulse

train is the same as shifting the signal by the offset of each impulse and adding the results.

The spectrum of the signal can

be fully reconstructed if there are no overlaps in the result.

Source: http://cnx.org/content/m46012/latest/?collection=col11510/latest 5

SLIDE 6

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Proof of Nyquist theorem

If the sampling frequency is

too low then information will be lost in the spectrum of the result.

If the sampling frequency is at

least twice the bandwidth then the signal can be reconstructed without a problem.

Source: http://cnx.org/content/m46012/latest/?collection=col11510/latest 6

SLIDE 7

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Aliasing

What happens if we sample too slowly?
Will the higher frequencies simply be removed from the signal?
The red sine wave below is being sampled at just over it’s

bandwidth, however the blue sine wave will be recreated as it fit’s all data points and is within the expected bandwidth.

Source: http://en.wikipedia.org/wiki/Aliasing#mediaviewer/File:AliasingSines.svg 7

SLIDE 8

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Aliasing

Aliasing is the effect where frequencies too high to be sampled are

folded onto lower frequencies

A too low sample rate doesn’t just lose information in the higher
frequencies. It also causes faulty values for in the lower frequencies.
The severity of aliasing depends on the application.

Source: http://www.cs.berkeley.edu/~sequin/CS184/IMGS/Sampl_Alias_F35.jpg 8

SLIDE 9

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Reconstruction

To retrieve the original spectrum of a sampled signal we have to

multiply the sampled signal with a block function:

This is the same as

convoluting the samples with the inverse Fourier transform of a block function which is also called the interpolation function:

Source: http://upload.wikimedia.org/wikipedia/commons/thumb/1/1f/ReconstructFilte png/400px-ReconstructFilter.png 9

SLIDE 10

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Reconstruction

Because convoluting with a shifted impulse simply shifts a signal

this can be rewritten as:

Source: http://sepwww.stanford.edu/public/docs/sep107/paper_html/node24.html 10

SLIDE 11

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Choosing a sampling time

There are a number of rules of thumb that can be used to choose a

sampling time:

Bandwidth
A minimal sampling frequency of twice the bandwidth is necessary

to achieve lossless sampling, however a higher sampling rate is

ften used so as to create a margin of error. A good rule of thumb

is 2,2 times the bandwidth.

The speed at which the system generating the signal can react to

inputs

Rise time
Settling time

11

SLIDE 12

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Interpolation of DT signals

In reality it is hard to achieve ideal reconstruction as the sinc

function requires an infinite time span.

Source: http://sepwww.stanford.edu/public/docs/sep107/paper_html/node24.html 15

SLIDE 13

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Zero-order hold

Keep the signal at a constant value for the duration of a sampling

period.

Transfer function of a ZOH-filter:

Source: http://en.wikipedia.org/wiki/Zero-order_hold#mediaviewer/File:Zeroorderhold.impulseresponse.svg & http://upload.wikimedia.org/wikipedia/commons/thumb/1/15/Zeroorderhold.signal.svg/585px-Zeroorderhold.signal.svg.png 16

SLIDE 14

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

First-order hold

Linear interpolation between samples.
Transfer function of a FOH-filter:

Source: http://en.wikipedia.org/wiki/First-order_hold#mediaviewer/File:Firstorderhold.impulseresponse.svg & http://en.wikipedia.org/wiki/First-order_hold#mediaviewer/File:Firstorderhold.signal.svg 17

SLIDE 15

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

First-order hold

Basic-FOH is a non-causal system.
Causal-FOH introduces a time delay.

Source: http://en.wikipedia.org/wiki/First-order_hold#mediaviewer/File:Delayedfirstorderhold.impulseresponse.svg & http://en.wikipedia.org/wiki/First-order_hold#mediaviewer/File:Delayedfirstorderhold.signal.svg 18

SLIDE 16

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

First-order hold

Predictive-FOH can be used for signals that don’t change too quickly
ver time.
The current and previous sample are used to predict the next

sample.

Source: http://en.wikipedia.org/wiki/First-order_hold#mediaviewer/File:Predictivefirstorderhold.impulseresponse.svg & http://en.wikipedia.org/wiki/First-order_hold#mediaviewer/File:Predictivefirstorderhold.signal.svg 19

SLIDE 17

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

What effect do ZOH and FOH have on the spectrum?

Some information will be permanently lost.

Source: http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec17.pdf 20

SLIDE 18

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Reconstruction: example

Original image:

Source: http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec17.pdf 21

SLIDE 19

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Reconstruction: example

Spatially sampled image:

Source: http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec17.pdf 22

SLIDE 20

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Reconstruction: example

Image reconstructed with ZOH:

Source: http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec17.pdf 23

SLIDE 21

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Reconstruction: example

Smoothing applied to ZOH:

Source: http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec17.pdf 24

SLIDE 22

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Reconstruction: example

Image reconstructed with FOH:

Source: http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec17.pdf 25

SLIDE 23

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Reconstruction: example

Larger gap between samples:

Source: http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec17.pdf 26

SLIDE 24

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Reconstruction: example

Reconstruction with ZOH:

Source: http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec17.pdf 27

SLIDE 25

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

STADIUS - Center for Dynamical

Systems, Signal Processing and Data Analytics

Reconstruction: example

Reconstruction with FOH:

Source: http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec17.pdf 28