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

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics Analytics Discretization and reconstruction of signals Lecture 13 Systems and Control Theory 1

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 2

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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. Systems and Control Theory 3

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 : F( ω ) f(t) …  Sampling in frequency domain = convolution of signal spectrum and an impulse train. t ω Source: http://cnx.org/content/m46012/latest/?collection=col11510/latest Systems and Control Theory 4

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 5

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 6

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 7

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 8

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 9

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 10

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 often 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 Systems and Control Theory 11

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 15

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 16

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 17

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 18

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics Analytics First-order hold  Predictive- FOH can be used for signals that don’t change too quickly over 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 Systems and Control Theory 19

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 20

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 21

STADIUS - Center for Dynamical STADIUS - Center for Dynamical Systems, Systems, Signal Processing and Data Signal Processing and Data Analytics 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 Systems and Control Theory 22