AFSjap06.pdf

Extremely wideband signal shaping using one- and two-dimensional

nonuniform nonlinear transmission lines

E. Afshari

Electrical Engineering, California Institute of Technology, 136-93 Pasadena, California 91125

H. S. Bhat

Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027

A. Hajimiri

Electrical Engineering, California Institute of Technology, 136-93 Pasadena, California 91125

J. E. Marsden

Control and Dynamical Systems, California Institute of Technology, 107-81 Pasadena, California 91125

Received 16 May 2005; accepted 20 January 2006; published online 1 March 2006;

corrected 3 March 2006

We propose a class of electrical circuits for extremely wideband

EWB

signal shaping. A

one-dimensional, nonlinear, nonuniform transmission line is proposed for narrow pulse generation.

A two-dimensional transmission lattice is proposed for EWB signal combining. Model equations for

the circuits are derived. Theoretical and numerical solutions of the model equations are presented,

showing that the circuits can be used for the desired application. The procedure by which the circuits

are designed exemplifies a modern, mathematical design methodology for EWB circuits. ©

2006

American Institute of Physics

DOI:

10.1063/1.2174126

I. INTRODUCTION

As the name implies, signal shaping means changing

certain features of incoming signals, such as the frequency

content, pulse width, and amplitude. By extremely wideband

EWB

, we mean frequencies from dc to more than 100

GHz. EWB signal shaping is a hard problem for several rea-

sons. If we attempt to solve the problem with transistors, we

are limited by the highest possible transistor cutoff frequency

, the maximum efficiency of the transistor, and also its

breakdown voltage. For example, these bottlenecks arise in

high-frequency fully-integrated power amplifier design.

1,2

These same considerations hold for the wider class of

active devices. Even if we restrict consideration to silicon-

based technologies, active devices are technology dependent,

making it difficult to port the design from one complemen-

tary metal-oxide semiconductor

CMOS

technology to an-

other. Therefore, active device solutions to the signal shaping

problem will be limited in both functionality and portability.

Existing high-frequency circuits typically use either

tuned circuits

e.g.,

tank

or microwave techniques

e.g.,

transmission lines as impedance transformers

. These ap-

proaches are inherently narrowband and cannot be used in

applications such as ultrawideband impulse radio and ultra-

wideband radar

e.g., ground penetrating radar

, pulse sharp-

ening, jitter reduction, or a wideband power amplifier.

We propose a solution to the EWB signal shaping prob-

lem, using passive components only, that overcomes these

limitations. This solution is an extension of ideas presented

in our previous work.

The circuits we propose consist of

artificial transmission lines as well as extensions to two-

dimensional lattices. An artificial transmission line consists

of a number of

blocks connected as in Fig. 1. By choos-

ing the elements properly, we can ensure that signals incident

on the left boundary of the line are shaped in a particular way

as they propagate to the right. In what follows, we will ex-

plain that by tapering the values of the inductance

and

capacitance

in the line, along with introducing voltage

dependence in the capacitors

, we can make circuits that

perform a variety of tasks. Extending the line to a two-

dimensional lattice, we can use similar ideas to design cir-

cuits that combine the power in an array of incoming signals.

In this paper, we consider millimeter-scale on-chip trans-

mission lines on a semiconductor substrate, e.g., silicon. The

relative resistance of each element on the chip is small

enough to be neglected, so we do not consider the effect of

loss. The effect of energy loss in transmission lines has been

discussed in our earlier work.

Philosophically, we are motivated by developments in

the theory of nonlinear waves, especially solitons. Solitons

are localized pulses that arise in many physical contexts

through a balance of nonlinearity and dispersion. Since the

1970s, various investigators have discovered the existence of

solitons in nonlinear transmission lines

NLTLs

, through

Electronic mail: ehsan@caltech.edu

Electronic mail: hsb2106@columbia.edu

Electronic mail: hajimiri@caltech.edu

Electronic mail: marsden@cds.caltech.edu

FIG. 1. 1D artificial transmission line.

JOURNAL OF APPLIED PHYSICS

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both mathematical models and physical experiments. Re-

cently, NLTLs have proven to be of great practical use in

EWB focusing and shaping of signals.

3,4

A. Survey

Here we offer a brief, selective survey of the transmis-

sion line literature relevant to our application. Before pro-

ceeding, let us make a few definitions that will help label the

transmission lines under consideration.

Definitions

Linear

Capacitors and inductors are constant with

respect to changes in voltage.

Nonlinear

Capacitors are voltage dependent and/or

inductors are current dependent.

Uniform

Identical capacitors and inductors are used

throughout the line.

Nonuniform

Different capacitors and inductors are used

in different parts of the line.

In the present work, we do not consider current-dependent

inductors because of implementation issues.

Scott’s classical treatise

was among the first to treat the

physics of transmission lines. Scott showed that the

Korteweg–de Vries

KdV

equation describes weakly nonlin-

ear waves in the uniform NLTL described above. If the non-

linearity is moved from the capacitor parallel to the shunt

branch of the line to a capacitor parallel to the series branch,

the nonlinear Schrödinger

NLS

equation is obtained

instead.

At the other end of the spectrum, nonuniform linear

transmission lines have been extensively used by the micro-

wave community for impedance matching and filtering. In

fact, the idea of a nonuniform linear transmission line goes

back to the work of Heaviside in the 19th century

see Kauf-

man’s bibliography

for details

Model equations for lines that combine nonuniformity,

nonlinearity, and resistive loss have been derived in the

literature,

but these models were not analyzed and the pos-

sible applications of a nonuniform NLTL were not explored.

In other work, numerics and experiments

indicated that a

nonuniform NLTL could be used for “temporal contraction”

of pulses.

Extensions to two dimensions have been briefly consid-

ered. For the description of long waves in a two-dimensional

lattice consisting of one-dimensional

lines coupled

together by capacitors, one obtains a modified Zakharov-

Kuznetsov

equation.

It should be mentioned that in

Sec. 2.9 of Scott’s treatise,

precisely this sort of lattice is

considered, and a coupled mode theory is introduced. These

lattices consist of weakly coupled 1D transmission lines, in

which wave propagation in one direction is strongly and in-

herently favored.

When a small transverse perturbation is added to the

KdV equation, one obtains a Kadomtsev-Petviashvili

model equation. Dinkel

et al.

carry out this procedure for a

uniform nonlinear 2D lattice, and mention that the circuit

may be useful for “mixing” purposes; however, no physical

applications are described beyond this brief mention in the

paper’s concluding remarks.

B. Present work

We review one-dimensional transmission line theory

with the aim of clarifying the effects of discreteness, nonuni-

formity, and nonlinearity. Continuum equations that accu-

rately model these effects are derived. We show analytically

that a linear nonuniform transmission line, with constant de-

lay but exponentially tapered impedance, can be used for

combination of signals. The speed and amplitude of outgoing

signals are analyzed directly from the continuum model. We

show numerically that introducing weak nonlinearity causes

outgoing pulses to assume a solitonlike shape. Practical ap-

plications of this are described.

We generalize the notion of a transmission line to a two-

dimensional transmission lattice. For a linear nonuniform lat-

tice, we write the continuum model and derive a family of

exact solutions. A continuum model is also derived for the

nonlinear nonuniform lattice. In this case, we apply the re-

ductive perturbative method and show that a modified KP

equation describes the weakly nonlinear wave propagation in

the lattice.

For the two-dimensional lattices, we present a variety of

numerical results. We choose the inductance and capacitance

of lattice elements in a particular way, which we call an

electric lens or funnel configuration. We solve the semidis-

crete model of the lattice numerically, and show that the

resulting solutions have physically useful properties. For ex-

ample, our numerical study predicts that a linear nonuniform

lattice can focus up to 70% of the power of input signals

with frequency content in the range of 0–100 GHz. We

present numerical studies of nonlinear lattices as well. In this

case, power focusing is present alongside frequency upcon-

version, or the ability of the lattice to

increase

the frequency

content of input signals. The numerical studies show that

nonlinear nonuniform lattices can be used for EWB signal

shaping applications.

II. UNIFORM NONLINEAR 1D

In this section we review a few facts about uniform

NLTLs and their use for

pulse narrowing

see Fig. 1

.At

node

in the transmission line, Kirchoff’s laws yield the

coupled system of ordinary differential equations

ODEs

−

+1/2

−1/2

−

+1/2

Here

+1/2

is the magnetic flux through the inductor

that is between nodes

and

+1, and

is the

charge on the varactor at node

. Using this,

can be

rewritten and combined into

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−2

−1

Starting from this semidiscrete model, we develop a con-

tinuum model in the standard way.

Write

as the position of node

along the line; assume

that the nodes are equispaced and that

−

is small.

Then, define

such that

. This means that

. We Taylor expand to fourth order in

and find that

is equivalent to

Let

and

be, respectively, the induc-

tance and capacitance per unit length. Then

becomes

We regard this as a continuum model of the transmission line

that retains the effect of discreteness in the fourth-order term.

A. Discreteness generates dispersion

Considering small sinusoidal perturbations about a con-

stant voltage

, we compute the dispersion relation

for

1−

where

=1/

. We see that for

depends

nonlinearly on

. Wave trains at different frequencies move

at different speeds.

In the applied mathematics/physics literature, one finds

authors introducing dispersion into transmission lines

through the use of shunt-arm capacitors. This is unnecessary.

Experiments on transmission lines we have described, with-

out shunt-arm capacitors, reveal that dispersive spreading of

wave trains due to the discrete nature of the line is a com-

monly observed phenomenon. Accurate continuum models

of the transmission line we have considered should include

this discreteness-induced dispersion. Therefore, we use infor-

mation about the

=0 case only if it leads to mathematical

insights about the

0 case, which is what truly concerns

us.

B. Traveling wave solutions

Retaining

as a small but nonzero parameter, we search

for traveling wave solutions of

of the form

where

−

. Using this ansatz and the varactor model

1−

, we obtain the ODE,

−

where

−2

and primes denote differentiation with re-

spect to

. Now integrating twice with respect to

,weob-

tain

−

We search for a localized solution, for which

→

0as

→

. This forces the constants to be zero:

=0. Now

multiplying

by 2

, integrating with respect to

, and

again setting the constant to zero,

−

where

−

and

The first-order ODE

can be integrated exactly. Taking the

integration constant to be zero, we obtain the single-pulse

solution,

−

sech

−

The sech

form of this pulse is the same as for the soliton

solution of the KdV equation. Indeed, applying the reductive

perturbation method to

, we obtain KdV in the unidirec-

tional, small-amplitude limit.

C. Reduction to KdV

Starting with

and again modeling the varactors by

1−

, introduce a small parameter

1 and

change variables via

1/2

−

3/2

with

−2

. Writing

−1/2

−3/2

we find that

1/2

and

3/2

−

1/2

Using the formula for

, we rewrite the left-hand side of

1−

−2

−

Using this and

, we rewrite

in terms of the long space

and time variables

and

−2

−

Now introducing the formal expansion

̄

the order

terms in

cancel. Keeping terms to lowest

order,

,wefind

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=0,

In what follows, we abuse notation by using

to denote

Integrating

with respect to

yields the KdV equation,

=0.

The KdV equation has been investigated throughly and many

of its properties are well known, including solution by in-

verse scattering, complete integrability, and geometric struc-

ture. Hence we will not pursue these topics here.

D. Remark 1: Zero-dispersion case

If we had a purely continuous transmission line, we

would take the

→

0 limit of

and obtain

This equation, which in general yields discontinuous shock

solutions, has been studied before

and we will not repeat

the general analysis. However, note that if we carry out the

reductive perturbation method on

, we end up with the

→

0 limit of

, which is the inviscid Burgers equation,

=0.

It is well known

that for any choice of initial data

no matter how smooth, the solution

develops

discontinuities

shock waves

in finite time. Meanwhile, for

large classes of initial data, the KdV equation

possesses

solutions that stay smooth globally in space and time.

What is intriguing is this: suppose we keep

as an arbi-

trary, nonzero parameter and solve

analytically, using

the inverse scattering method, we obtain a function

In the work of Lax and Levermore,

it was shown that in the

zero-dispersion

→

0 limit, the sequence

does

not

converge to a solution of Burgers’ equation

. Therefore,

we conclude that the

0 continuum model allows funda-

mentally different phenomena than the

=0 model. In the

nonlinear regime, we must keep track of discreteness.

E. Remark 2: Linear case

Note that if

is constant, we arrive at the linear,

dispersive wave equation,

−

This equation can be solved exactly using Fourier trans-

forms. In fact, we will carry out this procedure for a similar

equation in the following section.

F. Frequency response

So far we have discussed special solutions of

and the

KdV equation. Our primary concern is the transmission lines

for the mixing of EWB signals. The physical setup requires

that an incoming signal enter the transmission line at, say, its

left boundary. The signal is transformed in a particular way

and exits the line at, say, its right boundary.

Various authors have examined the initial-value problem

for the KdV equation. It is found that, as

→

, the solution

of the KdV equation consists of a system of interacting soli-

tons. Therefore, we expect that the incoming sinusoidal sig-

nals will be reshaped into a series of solitonlike pulses. Sup-

pose we wish to determine the precise frequency response in

the nonlinear regime, given an input sinusoid of frequency

we expect to see solitons of frequency

at the output end

of the line. We will address the problem of quantitatively

determining

in future work.

For now, we mention that a comprehensive mathematical

analysis of the

quarter-plane

problem,

xxx

=0,

19a

=0,

19b

19c

for the KdV equation is not possible at this time. This in-

cludes the frequency response problem for which

sin

. Inverse scattering methods applied to

yield

information only in the simplest of cases, i.e., when

is a

constant.

The problem is that in order to close the evolution

equations for the scattering data associated with

, one

needs to postulate some functional form for

and

. It does not appear possible to say

a priori

what

these functions should be.

One approach

is to postulate that these functions van-

ish identically for all

. They obtain approximate closed-form

solutions in the case where

is a single square-wave

pulse, with

0 for

. In future work, we will inves-

tigate whether this is possible if

is a sinusoidal pulse.

In this paper, we attempt an analytical solution of the

frequency response problem only in the

linear

regime. For

the

nonlinear

regime, we discuss special solutions and the

solution of the initial-value problem for the underlying

model equations to gain a qualitative understanding of the

models. For quantitative information about the general non-

linear, nonuniform frequency response problem, we use di-

rect numerical simulations of the semidiscrete model equa-

tions.

III. NONUNIFORM 1D

In this section, models for nonuniform transmission lines

will be derived and their dynamics will be discussed. We

study the one-dimensional case because they can be solved

exactly; these solutions will be used in our analysis of the

two-dimensional case. By nonuniform, we mean that the in-

ductance

and capacitance

varys as a function of

position,

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A. Linear case

For now, assume that the line is linear,

=0.

Then, modifying

, we obtain the exact, semidiscrete

model,

−

+1/2

20a

−1/2

−

+1/2

20b

which can be combined into the single second-order equa-

tion,

+1/2

−1

−

−1/2

−

−1/2

+1/2

Let

and

be, respectively, the inductance and ca-

pacitance per unit length at the position

along the transmis-

sion line. This yields the relations

and

, and allows us to expand,

+1/2

−1/2

−

Expanding

as before, we retain terms up to fifth order in

on both sides,

−

xxxx

−

xxx

−

xxx

−

where we have used subscripts to denote differentiation. We

now assume that

varies slowly as a function of space, so

that

. Hence our continuum model is

−

LCV

−

xxxx

−

xxx

To be clear, we specify that

→

R

and

→

R

are smooth and positive. The parameter

is a measure of

discreteness, which as discussed above contributes disper-

sion to the line.

1. Physical scenario

We are interested in solving the following

signaling

problem

: the transmission line is dead

no voltage, no cur-

rent

=0, at which point a sinusoidal voltage source is

switched on at the left boundary. We assume that the trans-

mission line is long, and that it is terminated at its

physical

right boundary in such a way that the reflection coefficient

there is very small. This assumption means that we may

model the transmission line as being semi-infinite.

We formalize this as an initial-boundary-value problem

IBVP

. Given a transmission line on the half-open interval

, we seek a function

→

R

that

solves

LCV

xxxx

−

xxx

24a

=0,

24b

=0,

24c

sin

24d

=0,

24e

where

and

are arbitrary constants, while

must be posi-

tive.

2. Nondimensionalization

Examining the form of problem

, we expect that

when

the uniform case

, it may be possible to find

exact traveling wave solutions. Hence we exploit the linear-

ity of

24a

and seek solutions when

is a slowly varying

function of

In order to carry this out, we must first nondimensional-

ize the continuum model

. Suppose that the transmission

line consists of

sections, each of length

. This gives a

total length

. Next, suppose that we are interested in

the dynamics of

on the time scale

. Using the constants

and

, we introduce the rescaled, dimensionless length, and

time variables,

and

We then nondimensionalize

by writing it in terms of the

variables

LCd

−

For the purposes of notational convenience, we omit primes

from now on.

3. Exponential tapering

The general nonuniform problem, with arbitrary

and

, may not be classically solvable in closed form. Here we

consider the exponentially tapering given by

27a

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−

27b

where

, and

are positive constants. In our discussion

of 2D transmission lattices, we will see that a generalization

of this tapering solves certain EWB signal-shaping problems.

Using

, the nondimensionalized 1D equation

simpli-

fies considerably to

−2

xxxx

−

xxx

We will analyze this equation subject to the previously dis-

cussed initial/boundary conditions

24b

–

24e

4. Perturbative solution

We will find solutions of

accurate to first order in

Let us begin by solving the

=0 case. Note that the case

=0 arose in our discussion of the uniform problem

see

From the setup of the problem, it is clear that the solu-

tion will consist of a wave train moving to the right at some

finite speed. Hence we try the ansatz,

−

Substituting this into

gives

−2

where

−

. Integrating twice with respect to

and set-

ting integration constants to zero gives a second-order ODE,

which has the general solution,

−

sin

−

−2

−

Now imposing the boundary condition

24d

, we solve for

the amplitude and phase:

−

. We also obtain the

dispersion relation

1±

1−

−2

Because this is a dispersion relation for a non-

dimensionalized equation,

and

are unitless,

as is the

parameter

. For a physical solution, the phase velocity must

be positive

, so we raise the above equation to the

1/2 power and discard the negative root. Putting everything

together, we have the two fundamental modes,

sin

−

30a

1±

1−

−2

1/2

30b

By linearity of

, the general solution of the

=0 equation

is the superposition,

=−

−

where

. Applying the second boundary condition

24e

we have

−

=−

−

5. Discussion

Using the dispersion relation

30b

, we can calculate the

cutoff frequency of the line. This is the frequency

for

which

becomes imaginary, which in the case of

30b

gives the relation

Here we used the definition

−1

, where

and

Taking

=0 in the above formula produces the classical

solution to the linear wave equation, with the single right-

moving mode,

Taking

0in

, we find three effects of discreteness.

The first is dispersion: though the phase velocity equals the

group velocity of the outgoing signal, viz.,

we see from

30b

that both of these velocities are nonlinear

functions of

, the frequency of the incoming signal. Second,

there are now two right-moving modes, one fast and one

slow, corresponding to

and

−

. Finally, discreteness

causes a decrease in the maximum speed of the wave train;

this follows immediately from

6. General case

We examine

with

0. On physical grounds we

expect that the voltage grows as a function of distance from

its starting point

=0. Accordingly, we introduce the ansatz,

= exp

where

−

. Inserting

, we obtain

−2

−

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Choose

/2 to eliminate the

terms exactly. Two of

the

terms cancel. We further ignore all terms of order

2, obtaining

−2

which is precisely the equation we solved in the

=0 case.

Hence an approximate solution of

, correct to

,is

given by

= exp

with

defined in

–

7. Discussion

The qualitative effect of a small but positive value of

is now clear. The frequency and speed of propagation for the

outgoing signal is unchanged from the

=0 case. The only

effect we expect to observe is growth of the initial sinusoid

as it propagates down the line.

Since it appears from

that we have produced a volt-

age that is unbounded in space, we remind the reader that in

reality, the transmission line is terminated at its right bound-

ary, say at

. So long as the resistive termination is cho-

sen so that the reflection coefficient is nearly zero, we may

use

to predict the wave form at any point

8. Remark

Exact solutions of

can be obtained computationally.

Let us outline the procedure in this case. First, we write the

full expression of

in the form

=0,

where

−

/64

/12

−

−2

−

/12

Here we use the convention that

. One

way to determine a unique solution is to specify the four

initial conditions

, where

=0, 1, 2, 3. We leave it as an

exercise to show that the four conditions

24b

–

24e

also

determine the solution uniquely. Then

can be solved via

the matrix exponential. Let

R

have coordinates

−1

for

=1, 2, 3, 4. Now write

as the first-order

system,

where

0100

0010

0001

−

The solution to

is then

In practice, given particular values of all required constants,

the solution can be found easily using

MATLAB

. As a final

remark, note that we do not need to compute the entire vec-

tor

, but merely the first component

B. Nonlinear case

Of course, we can build transmission lines that are both

nonuniform and nonlinear. To model such lines, we recog-

nize that

20b

is no longer time independent.

Hence combining

20a

and

20b

in the nonlinear case,

where

0, we find

+1/2

−1

−

−1/2

−

−1/2

+1/2

From here, the derivation of the continuum model proceeds

precisely as in the linear case. The end result is

−

xxxx

−

xxx

Suppose we take

1−

and

−2

Then introducing the change of variables

, we may again

use

to rewrite our equation. We note that in order to

balance terms, we must treat the inductance in a particular

way. We first write

−1/2

−3/2

so that

−3/2

In this case, the order

equation is

−

=0.

By introducing the time variable

and taking

, we remove

from the equation. We integrate with

respect to

, obtaining

sss

−

=0,

where as before we use

to denote

, the leading-order

contribution in the expansion

. Equation

has been

studied before as a “variable-depth” KdV equation. The now

classical result

is that for

small but positive, the usual

soliton wave form of the KdV equation is modified by a shelf

of elevation that trails the solitary wave. That is, the solution

is no longer a symmetric sech

pulse, but instead the wave

decays at its left boundary with a larger height than at its

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right boundary. The shelf elevation is

while its length is

−1

. The detailed

→

dynamics have been analyzed

by way of the transformation,

exp

which is used to convert

to a variable-coefficient KdV

equation,

sss

=0.

It is found that on a sufficiently large time scale, the trailing

shelf degenerates into a train of small-amplitude solitary

waves.

C. Numerics

We have performed direct numerical simulations of real-

istic transmission lines, using a finite-difference scheme. Af-

ter describing the numerical scheme, we discuss different test

cases. Our first goal is to validate our continuum models by

comparing their predictions against numerical solutions of

the underlying semidiscrete equations. Our second goal is to

demonstrate useful applications through carefully selected

numerical experiments.

1. Scheme

Equations

and

are, respectively, linear and non-

linear continuum models of the semidiscrete system

Continuum models are very useful for analytical studies; for

numerical studies, we work directly with the semidiscrete

system

, which we rewrite here,

+1/2

−

+1/2

0,1,2,

¼

44a

−1/2

−

+1/2

1,2,

¼

−1

44b

As in the continuum model, we take the line to be initially

dead,

= 0 and

=0,

and we also incorporate sinusoidal forcing at the left bound-

ary,

sin

However, for obvious reasons, the computational transmis-

sion line cannot be semi-infinite. We terminate the line at

node

, necessitating the right boundary condition,

−1/2

where

is the termination resistance. We choose

such that

the reflection coefficient at the right boundary is practically

zero. Taking

–

together, we have a closed system for

the interior voltages and inductances. We solve this system

directly using the standard ode45 Runge-Kutta method in

MATLAB

2. Remark

The procedure described above is entirely equivalent to

solving the partial differential equations

PDEs

and

by the method of lines combined with a finite-difference

spatial discretization. The method is accurate to second order

in space and fourth order in time.

3. Results

First we simulate a linear exponentially tapered line. As

predicted by the perturbative theory, we see two modes

propagating inside an exponentially shaped envelope. As

shown in Fig. 2, the amplitude of the wave increases slowly

as a function of element number.

Next we simulate both uniform and nonuniform NLTLs.

In the nonuniform case, we use the exponential tapering de-

scribed above. We observe in Fig. 3

that sinusoids are now

converted to solitonlike wave forms. If we switch on nonuni-

formity, multiple pulse generation is suppressed, as shown in

Fig. 3

. That is, fewer solitonic pulses are generated from

the same incoming sinusoidal signal.

The nonuniformity also allows us to narrow the width of

pulses considerably, as demonstrated in Fig. 4. Note that Fig.

4 also shows that the resulting pulses are not symmetric, as

predicted by theory. The asymmetry appears on the left

trail-

ing

side of the pulse.

To summarize,

the nonuniform linear transmission

line can be used for pulse compression/voltage amplification.

However, the frequency and speed of outgoing waves cannot

be significantly altered using a linear circuit.

The nonuni-

form nonlinear transmission line can increase both the volt-

age amplitude and the frequency content of incoming waves.

We now generalize 1D transmission lines to 2D transmission

lattices. The extra dimension allows us to create a circuit that

can simultaneously upconvert and combine incoming sig-

nals.

FIG. 2.

Color online

Voltage

vs element number

=10nsfora1D

nonuniform linear transmission line with parameters:

=100,

=0.1 nH,

and

=1 pF. The input, at the left end of the line

, is a sinusoid with

frequency

=5 GHz.

054901-8 Afshari

etal.

J. Appl. Phys.

, 054901

2006

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