kitano.pdf

Demonstration of Negative GroupDelays in a Simple Electronic Circuit

Masao [email protected], Room #1213

Kyoto University

Workshop on Quantum Optics, UCSB p.1/34

Group Delays in Circuits

Group delays in lumped systems (L = 0 or L c/)Mitchell and Chiao: Am. J. Phys. 66, 14 (1998)

Bandpass Amplifier (LC + opamp)Arbitrary Waveform Generator

Nakanishi, Sugiyama, and MK: quant-ph/0201001(to appear in Am. J. Phys.)

Highpass Amplifier (RC + opamp)Baseband pulse (No carriers)Band limited signal from rectangular pulser +lowpass filter


Negative delay circuit

The output LED is lit earlier than the input LED. Negative delay

Time constants could be order of seconds. We can see it!


Experiment

button

LED

pulse

pulse generatorLED

input output

negative delay circuit

1. rectangular pulse generator + low-pass filter

2. negative delay circuit


Group delays ideal case

Group delay for base-band signals (delay time: td)

VOUT(t) = (hVIN)(t) = VIN(t td)

h(t) = (t td)Fourier Transformed: VOUT() = H()VIN()

H() = (Fh)() =

dt h(t)eit = exp(itd)

time

Vin(t) Vout(t)H()td


Positive and Negative delays

td Causality Physical realization> 0 causal distributed system= 0 (locally, mutually) causal lumped system< 0 non causal impossible

Positive delays are easy, if you have an appropriatespace.

Record and play is also possible.

No way to make ideal (unconditional) negative delays.

No lumped systems (L = 0) can produce ideal positiveor negative delays.


Approximate delay with lumped systems

Ideal response function H()

A() = |H()| = 1,() = argH() = td

Approximate realization #1 with lumped systems

H() =1+ iT1 iT (1 pole, 1 zero)

A() = 1 (flat response)() = 2tan1T 2T

Stability condition T 0 (only positive delays)


Approximate delay (2)

Approximate realization #2 with lumped systems

H() = 1+ iT (1 zero)

A() =

1+(T )2 1+ (T )2

2( distortion)

() = tan1T TNo sign restriction on T (positive and negative delay)


Asymmetry positive / negative

Transfer function positive delay T < 0 negative delay T > 0

H() = 1+ iT

0

zero

Re

Im

0

zero

Re

Im

H() =1+ iT1 iT

0

pole

zero

Re

Im

0

pole

zero

Re

Im

The arrows shows the direction of the phase increase of H().

Stability condition requires that poles can be only in the upper-

half plane, but zero can be anywhere.Workshop on Quantum Optics, UCSB p.9/34

Negative Group Delay Circuit

Vin Vout

V- R

1/iC

V =(iC)1

R+(iC)1Vout =

11+ iCR

Vout

Vin V for large gain of operational amplifierVout = (1+ iCR)Vin

A pole is converted into a zero by the feedback circuit.Workshop on Quantum Optics, UCSB p.10/34

Finite bandwidth

Transfer Function

H() = 1+ iT (T = CR > 0)A() = 1+O(2T 2),() = T +O(3T 3)

Spectral condition for input signals

| | < 1T


Bandwidth

Works only for band-limited signals otherwise outputs are distorted.

-80-60-40-20

020406080

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Am

plitu

deA(

)

Frequency T Frequency TPh

ase(

)/

degr

ee


Band-Limit Circuit (Low-pass filter)

VoutR1 R1

C1C1

(1)R2

R2

Vin

Bessel filter (2nd order; m = 2)

HLP() =

1+ iTLP(3)+(iTLP)2

TLP = R1C1, = (1+R3/R2) = 1.268

Cutoff frequency: C = 0.7861/TLP


Low-pass filters

A rectangular pulser and a series of lowpass filters (m stages)are used to generate band-limited pulses.

RectangularPulser

LPF 1 LPF 2 LPF m

....

Pushing the button (at t = 0) starts the event.The band-limited output has a smooth leading edge.

A delay comparable to the pulse width is unavoidable.


Low-pass filters(2)

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7

ampl

itude

/A.U

.

time c t

m=24 6 8 10


Circuit Diagram

TRGOUT

VCRESTHDIS

timer IC7555

10k R0

C0

20k

R3

R1 C1

9V 9V

R1R2

SW

C R

C

R

20k

10k C R

C

R

C1 C1C1

R3

R2

R1

R1

100

TL082 TL082

TL082 TL082

Pulse generator

100

LEDLED

Vin Vout

input output Negative delay circuit

Low-pass filters

Trec


Circuit parameters

Pulse generator Negative delay circuitR0 6.8MC0 0.22FTrec 1.5sR1 2.2MC1 0.22FR2 10kR3 2.2kc 1.6Hz

R 1MC 0.22FR 10kC 22nFT (= RC) 0.22s


Circuit Board


Experimental result

m = 4,n = 2

-0.5

0

0.5

1

1.5

2

2.5

-1 0 1 2 3 4 5 6 7

input

ampl

itude

/V

time / sWorkshop on Quantum Optics, UCSB p.19/34

Experimental result

m = 4,n = 2

-0.5

0

0.5

1

1.5

2

2.5

-1 0 1 2 3 4 5 6 7

inputoutput

ampl

itude

/V

time / sWorkshop on Quantum Optics, UCSB p.19/34

Interference in time domain

In time domain (cf. H() = 1+ iT )

vout(t) =(

1+Tddt

)vin(t) = vin(t)+T

dvindt

(t)

Two terms interfere constructively at the leading edge anddestructively at the trailing edge.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-4 -2 0 2 4

exp(-x**2)-2*x*exp(-x**2)

(1-0.2*x)*exp(-x**2)

constructive destructive

time

vindvin/dt

vout


All-passive circuit

i(t)

v(t) C R

i(t) =1R

v(t)+Cddt

v(t) =1R

(1+RC

ddt

)v(t)

This circuit gives negative group delays, if we consider v(t) as

the input and i(t) as the output.


Unbalanced interferometer

Reflectivity of beam splitters: R = 1/2 , Delay: = 2L/c

Edark(t)= (1R)Ein(t)REin(t) 2(

1+

4ddt

)Ein(t)

L

dark port

bright port

input

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-4 -3 -2 -1 0 1 2 3 4

ampl

itude

t/T

inputbrightdark

dark (blow up)

= 0.08, = 0.15T , T : gaussian pulse width Workshop on Quantum Optics, UCSB p.22/34

Cascading for larger advancement

Transfer function for n stages: Hn() = (1+ iT )n

An() 1+ n(T )2

2= 1+

(

nT )2

2, n() nT

-80

-60

-40

-20

0

20

40

60

80

-0.6 -0.4 -0.2 0 0.2 0.4 0.60.8

0.9

1

1.1

1.2

1.3

1.4

1.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

n = 1

n = 1

n = 2

n = 2

n = 3n = 3

Am

plitu

deA(

)

Frequency T Frequency T

Phas

ear

g[(

)]

Bandwidth is reduced as n increased.Workshop on Quantum Optics, UCSB p.23/34

Cascading (2)

Usable bandwidth decreases as 1/

n, if T is given.

For a given input pulse (Tw), T = CR must be reduced asTw/

n.

Total advancement scales 1/

n;

td n (Tw/

n) =

nTw

Order m of lowpass filters must be increasedaccordingly;

m > n


Cascading, n = 10

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7

input

norm

aliz

edam

plitu

de

time / s


Cascading, n = 10

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7

inputoutput

norm

aliz

edam

plitu

de

time / s


Problems in cascading

It is possible to increase the advancement as large as thepulse width or more, but

The advancement increases slowly; 1/

n.

The order m of lowpass filters must be increased.

The system becomes very sensitive to noises. (A hugegain outside of the band)


Pushing the limit

0 5 1015 20 25

300

2

4

6

80.5

0

0.5

1

Total=nT

time t/sec

n

0 510 15

20 253

0

10

20

30

40

500.5

0

0.5

Ttotal=sqrt(n)T

time t/sec

n

Simple-minded cascading 1/

n cascading


Pushing the limit two pulse case

0 5 10 15 20 25 30 352

1.5

1

0.5

0

0.5

1

1.5

2

2.5

time t/sec

input

output(n=49)

output(linear,n=7)


Out-of-band gain

Outside gain Significant obstacle toward large n.

Realistic transfer function (finite gain at = ):

H() =1+ iT

1+ iT/, > 1

() = (11)T

A() =

1+(T )2

1+(T/)2

{1 ( = 0) ( = )

For = 0.5, n = 50; An() = n = 250 = 1015!


Role of lowpass filters

Predictability Your move will be predicted if somerestrictions are imposed.

JERKLIMIT

0.034ACCEL.LIMIT

10 0.11SPEEDLIMIT

35

...

dmxdtm

Resemblance of functions A negative delay circuitreveals the difference.

dist( f ,g) = max0 jm

d j fdt j djg

dt j


Role of filters (2)

On the condition m > n

0 1 2 3 4 5 6 7 8

0

0.5

n=8n=9n=10n=11n=12n=13

time t/sec 0 0.02 0.04 0.06 0.08 0.1 0.12

4

2

0

2

4

6

x 103

n=8n=9n=10n=11n=12n=13

time t/sec


Causality

Pulser LPF (m) ND (n)

A B C

O

O

A BC

t

Causal relation in a casual sense

O A B ?CCausal relation in a strict sense

A B C ( A)Workshop on Quantum Optics, UCSB p.32/34

Conclusion

A stand-alone (battery-operated) demonstration boxBattery operatedNo extra equipment such as an oscilloscope or afunction generator needed

Large negative delays 0.5s are achievedexperimentally.

25% of pulse width

Larger delay with cascading is possible (in priciple).Advancement can be larger than the pulse width.

Filter stage plays an essential role.The start of event (cause) is located before thefilters.


Group Delays in CircuitsNegative delay circuitExperimentGroup delays -- ideal casePositive and Negative delaysApproximate delay with lumped systemsApproximate delay (2)Asymmetry --- positive / negativeNegative Group Delay CircuitFinite bandwidthBandwidthBand-Limit Circuit (Low-pass filter)Low-pass filtersLow-pass filters(2)Circuit DiagramCircuit parametersCircuit BoardExperimental resultInterference in time domainAll-passive circuitUnbalanced interferometerCascading --- for larger advancementCascading (2)Cascading, $n=10$Problems in cascadingPushing the limitPushing the limit --- two pulse caseOut-of-band gainRole of lowpass filtersRole of filters (2)CausalityConclusion

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