Asimplepredictorbasedondelay‐inducednegativegroupdelay
Revised8/19/201613:16:00
HenningU.Voss
WeillCornellMedicalCollege,CitigroupBiomedicalImagingCenter
516East72ndStreet,NewYork,NY10021,USA
e‐mailaddress:[email protected]
Averysimplelinearsignalpredictorthatusespastpredictedvaluesratherthanpastsignalvalues
forpredictionispresented.Man‐madeornaturalsystemsutilizingthispredictorwouldnotrequire
a memory of input signal values but only of already predicted, internalized states. This delay‐
inducednegativegroupdelay DINGD predictoraffordsreal‐timepredictionofsignalswithoutthe
needforaspecificsignalmodel.Itspropertiesarederivedanalyticallyandarenumericallytested
onvarioustypesofbroadbandinputdata.
Keywords:Prediction,forecasting,negativegroupdelay
Introduction
Negativegroupdelay NGD ofaninput/outputsystemcausestheoutputsignaltoanticipateorpredict
characteristics of the input signal. NGD and the related concept of negative group velocity have been
theorized and experimentally found in systems with anomalous dispersion 1‐4 , metamaterials 5‐7 ,
transmissionlines 8,9 ,andelectroniccircuits 10‐13 .Recently,ithasbeenshownthatnegativegroup
delay can also occur in continuous‐time systems with time‐delayed feedback, or mathematically, non‐
autonomousdelay‐differentialequations 14 .Timedelaysareatypicalcomponentofbiologicalneuronal
networks,anditisreasonabletohypothesizeapossiblerelevanceofthis delay‐inducednegativegroup
delay DINGD mechanisminneuronalcomputations 15 involved,forexample,inhumanmotorcontrol
16 .
The pioneering paper of Mitchell and Chiao 17 experimentally demonstrated NGD for Gaussian
waveformsinanelectroniccircuitandalsoshowedthatcausalityisretained.Theyusedtheconceptof
transferfunctions,fromwhichthefrequencydependentgroupdelaycanbederivedforanyinputsignal
waveformindependentofitsshape.Therefore,somesystemswithNGDcanbeviewedasreal‐timesignal
predictors 18 ,whichcanbeunderstoodbyanalyzingtheirtransferorfrequencyresponsefunctions.The
absolutevalueofthegroupdelaythendefinesthepredictionhorizon,i.e.,thetimetheoutputy t ,here
calledapredictor,predictstheinputx t aheadoftime.
In this tutorial‐style manuscript very simple, probably the simplest possible, DINGD predictors are
introduced.Theyaregivenbydiscrete‐timesystems,whichsimplifiesnumericalsimulationsandwould
allow for digital signal processing implementation. They are still delay‐induced NGD predictors in the
followingsense:Itwillturnoutthatthesepredictorsdonotusepastinputsignalvaluesx t‐1 ,x t‐2 ,…
forprediction,asmostconventionalpredictorsdo,butonlypreviouslypredictedoutputvaluesy t‐1 ,y t‐
2 ,…,alongwiththepresentinputvaluex t .Thepreviouslypredictedoutputvaluesaredelayedfeedback
inputstothepredictor.Thisschemecouldhaveadvantagesinnaturalorman‐madeapplications.
In the following, discrete‐time DINGD predictors will be described, theoretically analyzed, and their
performance will be illustrated with various numerical simulations of real‐time, broadband signal
prediction.
TheDINGDpredictor
ThesimplestDINGDpredictorisdefinedasthediscrete‐timenon‐autonomouslinearsystem
y t
bx t
cy t τ ,
1
wherex t isascalarinputsignalwhoseforthcomingvaluesaretobepredictedbyy t ,banon‐zeroinput
scaling parameter, c a non‐zero delayed feedback gain, and τ a positive integer, a time delay. Time is
restrictedtomultiplesofasamplingtimeintervalΔt,i.e.,t …,‐Δt,0,Δt,2Δt,….Forsimplicity,wesetΔt
1inthefollowing,suchthatt …,‐1,0,1,2,…andτ 1,2,….
In order to understand how the discrete‐time DINGD predictor predicts, it is necessary to derive its
frequency‐dependentgroupdelay.AlthoughEq. 1 looksquitesimple,itwasnotpossibleformetoderive
itspredictionpropertiesbyanyother,moreintuitive,means. Anattempthasbeenmadeforthetime‐
continousanalogueofEq. 1 inRef. 14 ,whereaheuristicexplanationwasprovidedtorelateitsbehavior
to anticipatory synchronization 19 , called „anticipatory relaxation dynamics“. In Ref. 20 it had been
conjecturedthatanticipatorysynchronizationisrelatedtothefindingsofMitchellandChiaobutitwasnot
specifiedhowexactly. The frequency response function defines the input/output relationship between x t and y t under
steady‐stateconditionsinFourierspaceas
Y ω
H ω X ω ,
whereω 2πfisthefrequencyinrad/time,fisfrequencyinoscillations/time,x t
X ω e dω,and
y t
Y ω e dω.ThelattertwoexpressionsareinverseFouriertransforms;thesignconventionhere
isoppositetoRef. 17 ,followingthemajorityoftheliterature.
Thefrequencyresponsefunctioncanbewrittenintermsofphaseandgainas
H ω
|H ω |e
.
Thefrequencyresponsefunctionofthediscrete‐timeDINGDpredictor 1 canbefoundbyinsertingthe
inverseFouriercomponentsofxandyintothepredictor.Itis
b
b
2
H ω
1 c cos ωτ
i c sin ωτ
1 ce
β ω
with
β ω
1
Itsgainis
|H ω |
anditsgroupdelay
δ ω
dΦ ω
dω
2ccos ωτ
|b|
β ω
c .
,
cτ c
cos ωτ
β ω
.
3
Thelatterexpressionisproperlydefinedoutsideofthepolesofthefrequencyresponsefunctiononly.
Inordertomakepredictionsonesamplestepormoreahead,weareseekingtoobtainanintegergroup
delayδ ‐1forτ 1 causalsystem and0 c 1 toavoidinstability .Itismostinstructivetofirst
considerthezerofrequencycase ω 0 andfromtheretoderivethegroupdelayforgeneralfrequencies.
Thegroupdelayforzerofrequenciesis
cτ
4
δ 0
.
1 c
Specificcasesforthetimedelayτareconsidered:
•τ 1:Thereisnointegersolutionforδ 0 forany0 c 1. ThisisaspecialcaseoftheelementaryNGD
IIRfilterintroducedbyRavelo 21 b 0inEq. 1 there,Ts 1 . •τ 2:Forc 1thegroupdelayatzerofrequencyisδ 0 ‐1.Furthermore,thegroupdelayis‐1forall
frequencieswhereitisdefined,i.e.,outsideofthepolesofthefrequencyresponsefunction.Thefrequency
response function has poles defined by the zeros of its denominator. The poles are located at ω 2n 1 π/2 n 0,1,… .ThehighestfrequencyforasignalsampledwithΔt 1isωN π,sothereisone
pole,atf1 ω1/2π ¼.Thispoleseparatestwofrequencybandswithqualitativelydifferentproperties:
Duetoaphasejumpatf1thesecondbandcausesasignreversaloftheoutputandthuscannotbeused
togetherwiththefirstbandforprediction.
Figure1showsintheupperleftpanelthefrequencyresponsefunctionexpressedthroughitsgainand
phase,aswellasthegroupdelayoverfrequencyforthiscase.FromthisfigureitisclearthattheDINGD
predictordepends,asallNGDbasedprediction,onthefrequencycontentofthesignaltobepredicted.
•τ 4:Forc 1thegroupdelayatzerofrequencyisδ 0 ‐2.Furthermore,thegroupdelayis‐2forall
frequencies for which it is defined. The poles of the frequency response function are located at ω 2n 1 π/4 n 0,1,… .Therearetwopoles,atf1 ω1/2π 1/8andf2 ω2/2π 3/8.Thesetwopoles
separatethreefrequencybandswithqualitativelydifferentproperties:Thefirstbandandthethirdband
canbeusedcombined.Thesecondbandcausespredictionwithreversedsignandcannotbecombined
withtheothertwobands.However,thisbandcouldalsobeuseful,too.Touseit,onecansetb 0inEq.
1 tocompensateforthesignreversal.
Figure1showsinthelowerleftpanelthefrequencyresponsefunctionandgroupdelay.
•Larger,evenτ:Itisstraightforwardtogeneralizetolargerdelaysaslongasc 1.Ingeneral,asthedelay
increases,therewillbemorepoles.Thismeansthefrequencybandswillbesplitupintomoresections.
Also,ingeneralthepredictionhorizonwillalwaysbehalfofthepredictorfeedbackdelayτasperEq. 4 .
Inordernottocauseinstabilityofsystem 1 andalsoinphysicalimplementationsavalueofc 1might
notbefeasibleanditismoreusefultoproceedwithc 1‐ε,withε≪1.Thishastheadditionaladvantage
thatthepolesofthefrequencyresponsefunctionareresolvingintomereresonances i.e.,thegainisnot
diverging .Figure1showsintherighttwopanelsthefrequencyresponsefunctionaswellasthegroup
delayforc 0.99andτ 2,4.
= 2, c = 1
4
2
2
0
0
-2
0
0.1
0.2
0.3
0.4
= 4, c = 1
4
= 2, c = 0.99
4
Gain |H| -2
Phase
0
0.1
Group Delay =-d /d
2
0
0
-2
0.3
0.4
= 4, c = 0.99
4
2
0.2
-2
0
0.1
0.2
0.3
0.4
normalized frequency f
0
0.1
0.2
0.3
0.4
normalized frequency f
Figure1:TheoryI.
Gainandphaseoffrequencyresponsefunction 2 aswellascorrespondinggroupdelay 3 forfeedback
delaysτ 2 toprow andτ 4 bottomrow ,forfeedbackgainc 1 leftcolumn andc 0.99 right
column .Themeaningofthegraphsisshowninthelegend.NotethatalthoughtheNGDisconstantovera
widefrequencyrange,thegainincreasesforhighfrequencies,thusrestrictingapplicabilityoftheDINGD
predictortosignalscontainingonlylowerfrequencies.ThisisdemonstratedintheapplicationsinFigs.2
to5.
Applicationexamples
Theperformanceofthediscrete‐timeDINGDpredictorwillbeillustratedwiththehelpoffourexamples,
Figs.2to5.Theexamplesaredescribedinthefigurecaptions.Inallexamples,c 1‐ε ε≪1 isused.All
computationswereperformedwithMATLABR2015a TheMathWorks,Inc.,Natick,MA .
Figure2:Band‐limitednoiseI.
Firstrow:Aband‐limitednoisesignalx t black,thickline anditspredictionsignaly t red ,theoutput
ofthepredictor 1 .Outof1000simulatedtimepoints,100areshown.
Second row, left image: The cross‐
correlation function CCF τ between
x t and y t peaks at a lag of ‐1. Its
peak value is CCF ‐1 0.98. This
0.2
shows that y t‐1 x t , or,
0
equivalently, y t x t 1 .
Therefore, y t is a predictor of x t .
x(t)
-0.2
Center and right image: Scatterplots
y(t)
between x t and y t and between
210 220 230 240 250 260 270 280 290 300
x t and y t‐1 . These plots confirm
time t
that y t is more correlated with
1
0.4
0.4
x t 1 than with x t , although the
0.2
0.2
DINGDpredictor,Eq. 1 ,dependson
0.5
0
0
x t onlyandnotonx t 1 .
-0.2
-0.2
0
Third row: Power spectral density
-5
0
5
-0.4
0
0.4
-0.4
0
0.4
function estimates for x t black,
x(t)
x(t)
thickline andy t red .Onecansee
that frequency components of Y ω 0.4
that are closer to the resonance,
|P x |
0.3
wherethegainincreases seefourth
|P y |
row , are amplified more, which
0.2
causesthemorejitteryappearanceof
0.1
the predictor time series y t as
comparedwiththesignaltimeseries
Est. Gain |H|
0
0.1
0.2
0.3
0.4 Phase
0.5
Est.
x t inthefirstrow.
Gain
|H|
4
Phase
Fourth row: Gain and phase of the
Group Delay
frequency response function as well
2
ascorrespondinggroupdelay.Shown
areanalyticvaluesfromEqs. 2 and
0
3 ,aswellasgainandphasevalues
estimatedfromthedata,asshownin
-2
thelegend.
0
0.1
0.2
0.3
0.4
0.5
normalized frequency f
Signal and prediction parameters:
The signal x t consists of 1000
samplesofwhitenoise,low‐passfilteredwithaButterworthfilterofseventhorderwithacutofffrequency
of0.15.TheparametersofEq. 1 areb 1.50,c 0.95,andτ 2.
Figure3:ChirpsignalI.
1
Firstrow:Theleftplotshowsthefirst
1
x(t)
200 data points of a chirp signal a
y(t)
0.5
frequency‐swept sine function x t 0
0
and its prediction signal y t . The
right plot shows the last 20 data
-0.5
points. Whereas an enlargement of
-1
the left plot would show prediction
50
100
150
200
985
990
995
1000
with reduced amplitude , too,
time t
time t
predictionismoreevidentintheright
1
1
1
plot with the higher frequency
oscillations. The DINGD predictor
0.5
0
0
predicts the signal with the same
0
group delay of ‐1, for both very low
-1
-1
and high frequencies. This is an
-5
0
5
-0.5 0 0.5 1
-0.5 0 0.5 1
example for a non‐stationary signal
x(t)
x(t)
thatstillcanbepredictedinrealtime.
4
Second row, left: The cross‐
|P x |
correlationfunctionhasapeakvalue
3
|P y |
of CCF ‐1 0.99. This shows that
2
y t‐1 x t ,or,equivalently,y t 1
x t 1 .Therefore,y t isapredictor
of x t . Center and right: The
Est. Gain |H|
0
0.1
0.2
0.3
0.4
0.5
scatterplotsconfirmthaty t ismore
Est. Phase
correlatedwithx t 1 thanwithx t .
Gain |H|
4
Phase
Group Delay
Third row: Power spectral density
2
function estimates for x t black,
thickline andy t red .Itisevident
0
that frequency components of the
signalthatareclosertotheresonance
-2
of the frequency response function
0
0.1
0.2
0.3
0.4
0.5
fourth row , where its gain
normalized frequency f
increases,areamplifiedmorerelative
tolowerfrequencycomponents.This
explains the relatively smaller amplitude of the predictor for low frequencies. In other words, the
parameterbhasbeensettoprovidecorrectamplitudesforhighfrequenciesonly.Thelargeestimation
errors of gain and phase for frequencies 0.15 are due to the lack of input signal power for those
frequencies.
Fourth row: Gain and phase of the frequency response function as well as corresponding group delay.
ShownareanalyticvaluesfromEqs. 2 and 3 ,aswellasgainandphasevaluesestimatedfromthedata,
asshowninthelegend.
Signalandpredictionparameters:Thesignalx t consistsof1000samplesofalinearchirpsignalwith
cutofffrequencyof0.15.TheparametersofEq. 1 areb 1.25,c 0.95,andτ 2.
Figure 4: Band‐limited noise II –
PredictionwiththesecondNGDband
andagroupdelayof‐2.
0.2
Firstrow:Aband‐limitednoisesignal
x t black, thick line and its
0
prediction signal y t red , the
-0.2
output of the predictor 1 . Out of
x(t)
1000 simulated time points, 50 are
705 710 715 720
y(t) 725 730 735 740 745 750
shown.Itisevidentthatthesignalis
time t
predicted two time steps ahead.It is
1
also evident that, although it is not
0.2
0.2
0.5
smooth, the signal is predicted with
0
0
0
high accuracy, including patterns of
-0.2
-0.2
datapointsthatarenotformedbythe
-0.5
envelopeofanoscillatorysignal.
-0.4
-0.4
-4
0
4
-0.4
0
0.4
-0.4
0
0.4
x(t)
x(t)
Second row, left image: The cross‐
correlation function CCF τ between
x t and y t peaks at a lag of ‐2. Its
|P x |
0.2
peak value is CCF ‐2 0.98. This
|P y |
0.15
shows that y t‐2 x t , or,
0.1
equivalently, y t x t 2 .
0.05
Therefore, y t is a predictor of x t .
Center and right image: Scatterplots
0
0.1
0.2
0.3
0.4
0.5
between x t and y t and between
x t and y t‐2 . These plots confirm
6
that y t is more correlated with
4
x t 2 thanwithx t .
Est. |H|
2
Est.
Third row: Power spectral density
|H|
0
function estimates for x t black,
-2
thickline andy t red .
0
0.1
0.2
0.3
0.4
0.5
Fourth row: Gain and phase of the
normalized frequency f
frequency response function as well
ascorrespondinggroupdelay.Shown
areanalyticvaluesfromEqs. 2 and 3 ,aswellasgainand phasevaluesestimatedfromthedata,as
showninthelegend.Again,outsideofthesignalfrequencybandtheestimatesnaturallyhavealargeerror.
Signal and prediction parameters: The signal x t consists of 1000 samples of white noise, band‐pass
filteredwithaButterworthfilterofseventhorderwithcutofffrequenciesof0.18and0.32.Theparameters
ofEq. 1 areb ‐1.50,c 0.90,andτ 4.NotethatthephasebehaviorofthesecondNGDbandrelative
tothefirstbandrequiresanegativevaluefortheparameterb.
Figure 5: Neuronal signal, predicted
withagroupdelayof‐8.
2
First row: The neuronal signal x t 1
black, thick line , a local field
potential from the left hippocampus
0
ofarat,obtainedfromCRCNS.org 22 -1
andfilteredasdescribedinRef. 15 .
Here 1200 out of 6250 used data
200
400 x(t)
600
800
1000
1200
y(t)
points are shown. The prediction
time t
1
signal y t red predicts the input
2
2
x t eight time steps ahead, with an
0.5
occasionalsmalloscillatoryerror.
0
0
0
Second row, left image: The cross‐
-2
-2
-0.5
correlation function CCF τ between
-50
0
50
-2 -1 0 1
-2 -1 0 1
x t and y t peaks at a lag of ‐8. Its
x(t)
x(t)
peakvalueisCCF ‐8 0.99.Agroup
delayofδ ‐8correspondstoatime
4
of6.4msintheoriginaltimescaleof
the data. In Ref. 15 this signal had
2
beenpredictedwithagroupdelayof‐
7.2msbyusingamorespecificmodel
0
based on delayed‐leak integrators,
which also can have NGD, caused by
the mechanism of anticipatory
-2
Est. Gain |H|
relaxationdynamics 14 .Centerand
Est. Phase
right image: Scatterplots between
Gain |H|
-4
x t and y t and between x t and
Phase
y t‐8 .
Group Delay
-6
Third row: Gain and phase of the
frequency response function as well
-8
0
0.1
0.2
0.3
0.4
0.5
ascorrespondinggroupdelay.There
normalized frequency f
are now τ/2 1 9 distinct
frequencybandswithNGD.Duetothe
choice of c 0.92 the value of δ ‐
8.00isnotcompletelyattained;however,inpracticethesignalispredictedwithapredictionhorizonof8
duetotheintegersamplingtime.
Predictionparameters:TheparametersofEq. 1 areb 1.75,c 0.92,andτ 16.
Cascading
Asithasbeenshownabove,onewaytoincreasetheNGDisbyincreasingthefeedbackdelayτoftheDINGD
predictor. But there is another way: Feeding the output of the predictor into another predictor, or
“cascading” 12, 13, 23 predictors. This way, it is possible to increase the NGD without increasing the
feedbackdelay.Thishastheadvantagethatonecanworkwithadelayofτ 1;althoughtheNGDwithout
cascadingwouldbe0.5,aswehaveseenbefore,withcascadingonecanagainobtainintegervalues.
Cascadingm 1DINGDpredictorsinthetimedomainmeansthattheoutputy t ym t relatestothe
inputx t via
5
y t
bx t
cy t τ ,
y t
by t
cy t τ ,
...
y t
by
t
cy t τ .
Thefrequencyresponsefunctionis
H ω
H ω
Itsgainis
|H
anditsgroupdelay
δ
ω
ω |
|H ω | e
|b|
β ω
dΦ ω
dω
/
.
6
,
mδ ω ,
7
definedwhereverδ ω isdefined.
Weconsiderthespecialcaseofm 2,τ 1first.Again,thisisaspecialcaseoftheIIRfilterconsideredin
Ref. 21 ,thistimewithcascading.Asbefore,inordertomakepredictionsonesamplestepormoreahead,
weneedtoobtainanintegergroupdelayδ ‐1.Sincethegroupdelayforzerofrequenciesisjustmtimes
thegroupdelayforthecaseofnocascading,forc 1thegroupdelayatzerofrequencyisδ2 0 ‐1.
Furthermore,forc 1thegroupdelayisconstant‐1forallfrequencieswhereitisdefined.Thepolesof
thefrequencyresponsefunctionarelocatedatω 2n 1 π n 0,1,… .Therefore,thereisonlyonepole
atf1 ω1/2π ½,attheedgeofthefrequencyrange.
Insummary,forτ 1andm 2thegroupdelayisnegativethroughouttheentirefrequencyrange.This
meansthatcascadingallowsforusingadelayofτ 1withoutasplitoffrequencybands,andtheNGDcan,
intheidealcase,extendovertheentirefrequencyband.Still,thevariablegainofthefrequencyresponse
functionpreventsthepredictionofsignalswithcertainfrequencies.
Itisstraightforwardtoderivecaseswithτ 1andm 2etc.However,thecaseofτ 1standsoutasthe
NGDbandsarenotsplitupintoseparatebands.
Figure6showsexamplesform 2andm 4,againforc 1andc 1‐ε,withε≪1.Figure7provides
anexample.
m = 2, c = 1
4
m = 2, c = 0.99
4
2
2
0
0
-2
Gain |H| -2
Phase
0
0.1
Group Delay =-d /d
0
0.1
0.2
0.3
0.4
m = 4, c = 1
4
2
0
0
-2
-2
0
0.1
0.2
0.3
0.4
normalized frequency f
0.3
0.4
m = 4, c = 0.99
4
2
0.2
0
0.1
0.2
0.3
0.4
normalized frequency f
Figure6:TheoryII–Cascading.
Gain and phase of frequency response function 6 as well as corresponding group delay 7 for the
cascadedsystemwithτ 1andcascadinglevelm 2 toprow andm 4 bottomrow ,forfeedback
gainc 1 leftcolumn andc 0.99 rightcolumn .NotethatalthoughtheNGDisconstantoverawide
frequencyrange,thegainincreasesforhighfrequencies,restrictingapplicabilityoftheDINGDpredictorto
signalscontainingonlylowerfrequencies.ThisisdemonstratedintheexampleFigure7.
Figure 7: Chirp signal II – Cascading
fourpredictors.
1
Firstrow:Theleftplotshows50data
0.5
points of a chirp signal x t and its
prediction signal y t . The right plot
0
0
shows the last 20 data points. The
-0.5
DINGD predictor predicts the signal
-1
with the same group delay of ‐2, for
x(t)
560 570 580 590 600
985
990
995
1000
bothlowandhighfrequencies.Thisis
y(t)
time t
time t
an example for a non‐stationary
1
1
1
signal that still can be predicted in
realtime.
0.5
0
0
Second row, left: The cross‐
0
-1
-1
correlationfunctionhasapeakvalue
of CCF ‐2 0.89. This shows that
-2
0
2
-1 -0.5 0 0.5
-1 -0.5 0 0.5
y t‐2 x t ,or,equivalently,y t x(t)
x(t)
x t 2 .Therefore,y t isapredictor
of x t . Center and right: The
2.5
|P x |
scatterplotsconfirmthaty t ismore
2
|P y |
correlatedwithx t 2 thanwithx t .
1.5
1
Third row: Power spectral density
Est. Gain |H|
0.5
functionestimatesforx t andy t .It
Est. Phase
isevidentthatfrequencycomponents
Gain
0
0.1
0.2
0.3
0.4 |H|
0.5
Phase
of the signal that are closer to the
Group Delay
4
resonanceofthefrequencyresponse
function, which equals the Nyquist
2
frequency 0.5, are amplified more
relative to lower frequency
0
components. This explains the
relatively smaller amplitude of the
-2
predictorforlowfrequencies.
0
0.1
0.2
0.3
0.4
0.5
normalized frequency f
Fourth row: Gain and phase of the
frequency response function as well
ascorrespondinggroupdelay.Shown
areanalyticvaluesfromEqs. 6 and 7 aswellasgainandphasevaluesestimatedfromthedata,asshown
inthelegend.Thelargeestimationerrorsofgainandphaseforfrequencies 0.25areduetothelackof
inputsignalpowerforthosefrequencies.
Signalandpredictionparameters:Thesignalx t consistsof1000samplesofalinearchirpsignalwith
cutoff frequency of 0.25. Note that the chirp frequency sweeps a larger range than in Figure 3, made
possiblebythemoveoftheresonancetohigherfrequencies.TheparametersofEq. 5 arem 4,b 1.40,
c 0.85,andτ 1.
Shapingthefrequencyresponsebymultipledelays
CascadingtwoDINGDsystemscorrespondstoatotaldelaythatislargerthanthedelaysofthesubsystems.
Similarly,onecouldtrytoshapethefrequencyresponsefunctionbyusingmorethanonedelayinasingle,
non‐cascaded,system.Therearemanypossibilities,andhereonlythreespecialcaseswithtwodelaysand
usefulpredictionpropertiesarepresented:Ahighpass,alowpass,andabandpasssystemwithNGD.Rather
than providing a detailed analysis, which would not add much to the already obtained insights, these
systemsarejuststated,theirfrequencyresponsefunctionsareshown,andtheyaretestedonnumerical
examples.TheDINGDsystemwithtwodelaysisgivenby
y t
bx t
c y t τ
c y t τ .
8
Thecoefficientsareassumedtobenon‐zeroandthedelaysareintegers≥1,asbeforeforthebasicDING
predictor.Thetwo‐delayfrequencyresponsefunctionis
b
H ω
.
1 c e
c e
With
β ω
1 c
c
2c c cos ω τ
τ
2c cos ωτ
2c cos ωτ itfollows
|b|
|H ω |
β ω
and
c τ
c τ cos ωτ
τ cos ω τ
τ
dΦ ω
c τ
c τ cos ωτ
c c τ
δ ω
.
dω
β ω
•Lowpasssystemwithc1 2,c2 1,τ1=1,τ2=2.Thephase,gain,andgroupdelayareshowninFigure8,
first row. This system has a group delay of ‐1. The numerical simulation data in Figure 9 has a low‐
frequencysinusoidaladdedtoalowpassfilterednoisesignalinordertodemonstratethatlowfrequency
componentsdonotdetrimentallyaffectpredictionofthehigh‐passcomponents.TheCCF,validatingthe
groupdelayof‐1,looksquiteremarkable.
•Highpasssystemwithc1 ‐2andotherwisesameparametersasbefore.Thissystemhasagroupdelay
of‐1.Thephase,gain,andgroupdelayareshowninFigure8,secondrow.Anumericalexampleisprovided
inFigure10.
•Bandpasssystemwithc1 ‐2,c2 1,τ1=2,τ2=4.Thephase,gain,andgroupdelayareshowninFigure
8,thirdrow.Thissystemhasagroupdelayof‐2.Tousethissysteminsteadofthelow‐orhighpasssystems
canhaveadditionaladvantagesinadditontothehigherNGD;thebehaviorofthegainallowsforsomewhat
higher/lower frequency components of the signal as in the lowpass/highpass systems. Also, compared
withthesingle‐delaysysteminFigure4,thistwo‐delaysystemallowsforalargersignalbandwidth.This
isshowninthenumericalexampleofFigure11.Ofallexamples,thesignalhereappearstobethemost
complexone.Thatthissignalcanbepredictedinrealtimetwotimepointsaheadbythesimplemechanism
ofdelay‐inducednegativegroupdelay,Eq. 8 ,isnottrivial.
|H|,
,
|H|,
,
|H|,
,
Figure8:TheoryIII‐Multipledelays.
Firstrow:LowpassDINGDsystemwithgroupdelayof‐1.Shownaregainandphaseoffrequencyresponse
functionaswellascorrespondinggroupdelayforthelowpasssystemwithparametersstatedinthetext
andalsoprovidedinthefiguretitles b 1 .Again,theleftcolumnhastheoreticalparametersthatcause
aconstantNGDforallfrequencies,andtherightcolumnhasmorerealisticparameters. Thegraphsinthe
leftcolumnareidenticaltothegraphsofthecascadedsysteminFigure6toprow,leftcolumn .
Secondrow:HighpassDINGDsystemwithgroupdelayof‐1.
Thirdrow:BandpassDINGDsystemwithgroupdelayof‐2.Thetwodelaysarenowtwiceaslargeasbefore,
i.e.,τ1=2andτ2=4.Forlegends,pleaserefertoFigure6.
|H|,
,
Power
y(t - 1)
y(t)
CCF
x, y
Figure9:LowpassDINGDsystem.
Firstrow:Thefirst100datapointsof
a lowpass filtered noise signal x t withanaddedsinusoidalsignalwith
frequency 0.01 and its prediction
signal y t . Due to the very low
frequency component of the
sinusoidal, relatively long transients
ofthepredictionsignalarevisiblefor
smallt.
Second row, left: The cross‐
correlationfunctionhasapeakvalue
of CCF ‐1 0.94. This shows that
y t‐1 x t ,or,equivalently,y t x t 1 .Therefore,y t isapredictor
of x t . Center and right: The
scatterplotsconfirmthaty t ismore
correlatedwithx t 1 thanwithx t .
Third row: Power spectral density
functionestimatesforx t andy t .
Fourth row: Gain and phase of the
frequency response function as well
ascorrespondinggroupdelay.Shown
areanalyticvaluesasprovidedinthe
textaswellasgainandphasevalues
estimatedfromthedata,asshownin
the legend. The large estimation
errors of gain and phase for
frequencies 0.35areduetothelack
of input signal power for those
frequencies.
Signalandpredictionparameters:Thesignalx t consistsof1000samplesofwhitenoise,low‐passfiltered
with a Butterworth filter of seventh order with a cutoff frequency of 0.30 with an added sine signal of
amplitude0.5andfrequency0.01.TheparametersofEq. 8 areb 3.00,c1 1.65,c2 0.80,τ1=1,τ2=
2.
|H|,
,
Power
y(t - 1)
y(t)
CCF
x, y
Figure10:HighpassDINGDsystem.
First row: A highpass filtered noise
signal x t and its prediction signal
y t . Out of 1000 simulated time
points, 50 are shown. Similar to
Figure4,itisevidentthatpatternsof
datapointsthatarenotformedbythe
envelope of an oscillatory signal are
predictedwell.
Second row, left: The cross‐
correlationfunctionhasapeakvalue
of CCF ‐1 0.89. This shows that
y t‐1 x t ,or,equivalently,y t x t 1 .Therefore,y t isapredictor
of x t . Center and right: The
scatterplotsconfirmthaty t ismore
correlatedwithx t 1 thanwithx t .
Third row: Power spectral density
functionestimatesforx t andy t .
Fourth row: Gain and phase of the
frequency response function as well
ascorrespondinggroupdelay.Shown
areanalyticvaluesasprovidedinthe
textaswellasgainandphasevalues
estimatedfromthedata,asshownin
the legend. The large estimation
errors of gain and phase for
frequencies 0.15areduetothelack
of input signal power for those
frequencies.
Signal and prediction parameters:
The signal x t consists of 1000
samplesofwhitenoise,low‐passfilteredwithaButterworthfilterofseventhorderwithacutofffrequency
of0.20.TheparametersofEq. 8 areb ‐2.00,c1 ‐1.92,c2 0.96,τ1=1,τ2=2.
|H|,
,
Power
y(t - 2)
y(t)
CCF
x, y
Figure 11: Band‐limited noise III –
Bandpass DINGD system with group
delayof‐2.
First row: A bandpass filtered noise
signal x t and its prediction signal
y t . Out of 1000 simulated time
points,50areshown.
Second row, left: The cross‐
correlationfunctionhasapeakvalue
of CCF ‐2 0.90. This shows that
y t‐2 x t ,or,equivalently,y t x t 2 .Therefore,y t isapredictor
of x t . Center and right: The
scatterplotsconfirmthaty t ismore
correlatedwithx t 2 thanwithx t .
Third row: Power spectral density
functionestimatesforx t andy t .
Fourth row: Gain and phase of the
frequency response function as well
ascorrespondinggroupdelay.Shown
areanalyticvaluesasprovidedinthe
textaswellasgainandphasevalues
estimatedfromthedata,asshownin
the legend. The large estimation
errors of gain and phase for
frequencies between 0.1 and 0.4 are
duetothelackofinputsignalpower
for those frequencies. Remarkably,
the groupdelayis constant ‐2for all
frequencies.
Signal and prediction parameters:
The signal x t consists of 1000
samples of white noise, bandpass filtered with a Butterworth filter of seventh order with a cutoff
frequenciesof0.12and0.38.TheparametersofEq. 8 areb ‐2.50,c1 ‐1.90,c2 0.94,τ1=2,τ2=4.
TheMATLABcodeforgeneratingthisfigureisavailablefromtheauthoronrequest.
Conclusionsanddiscussion
Averysimplediscrete‐timepredictorbasedondelayedfeedback‐inducedNGDhasbeendescribed.This
delay‐induced NGD or DINGD predictor predictsfuture signal values by using present input and past
output,oralreadypredicted,signalvalues.Itthusdiffersfrommostotherpredictorsorforecastingmodels,
whichdonottakeintoaccountpastpredictedbutonlypast inputsignalvalues 24,25 . Anexception
would be systems that use anticipatory synchronization for prediction 26 . In other words, most
conventionaltimeseriespredictorscanbewrittenintheform
Conventionalpredictor:predictedx t 1
f x t , x t 1 , x t 2 , … ,
inwhichf . isaspecificmodelofthetimeseriestobepredicted.Theseconventionalpredictorsdepend
onpriorobservationsbutnotpriorpredictions,andusuallycontaincoefficientsresultingfromafittoa
fixed learning data set or which are continuously being updated 27 . In contrast, the DINGD predictor
cannotbewritteninthisexplicitform,asitdependsalsoonalreadypredictedvalues.Forexample,all
DINGDpredictorswithpredictionhorizonof1consideredabovecanbewrittenas
DINGDpredictor:predictedx t 1
y t
g x t , y t 1 , y t 2 , … ,
where g . is a function depending on the spectral content of the time series to be predicted. This
expressionisfundamentallydifferenttotheanalogousexpressionforconventionalpredictionabove.For
example,Eq. 1 withτ 2andc 1,whenthegroupdelayis‐1andthepredictionhorizonequals1,could
bewrittenas
predictedx t 1
y t
bx t
y t 2 .
Itisapparentlynotpossibleinthiscasetoexpressthepredictedx t 1 inclosedformasafunctionofa
finite number of past values of x t only. Rather, the DINGD predictor resembles a dynamic form of
prediction, also called anticipatory relaxation dynamics 14 , than conventional prediction. It is this
dynamicoriginofpredictionthatmakestheDINGDpredictor,andpossibleotherNGD‐basedprediction,
socounterintuitive.However,avoidingpastinputvaluesmightbeadvantageousforpredictionbynatural
and artificial neuronal networks as only already predicted, internalized states need to be laid down in
memory.
Inaddition,theDINGDpredictordoesnotrequirealearningdatasetbutisatruereal‐timepredictoronce
thecoefficientshavebeenfixedandthedynamicshavesettledintoasteadystate.Asthenon‐stationary
chirpsignalexampleshaveindicated,itisworthlookingintothereal‐timeadvantagesofthisprediction
schemecomparedtoadaptivepredictionalgorithmsaswell.Furthermore,ithasbeendemonstratedhere
thattheDINGDpredictorisabletopredictcomplexnon‐smoothsignals,too,addinginsightsbeyondthe
contemporaryunderstandingofNGD 7,10,28 .
ItwouldbeinterestingifDINGDpredictorscouldbeappliedorphysicallyimplementedinsomeway.Of
particularinterestcouldbethefactthatonlypastpredictedsignalsareusedforprediction.Itmeansfor
exampleforapredictiveagentsuchasarobot 29 orthebrain 30,31 thatonlyinternalstatevariables
areneededforprediction,withouthavingtostoresensoryinputsinmemory.Further,themainingredient
fortheDINGDpredictoraretimedelays,whichareabundantinthecentralnervoussystemincludingthe
cerebellum 32,33 .
Acknowledgment
IwouldliketothankBlaiseRavelofordiscussions.
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