The Large N Harmonic Oscillator as a String Theory

a r X i v :h e p -t h /0408180v 2 2 J a n 2005hep-th/0408180

PUTP/2130

The Large N Harmonic Oscillator as a String Theory Nissan Itzhaki and John McGreevy Department of Physics,Princeton University,Princeton,NJ 08544We propose a duality between the large-N gauged harmonic oscillator and a novel string theory in two dimensions.

August,2004

1.Introduction

Since’t Hooft’s work thirty years ago[1]it is generally believed that large-N gauge theories admit a dual string theory description.Given the utility of such dualities for both sides,and the di?culty of?nding them,it is of interest to?nd more examples which are under precise control.The aim of this paper is to explore a new and remarkably simple duality of this kind.We address an old question that was resurrected recently[2]:what is the string theory dual of the large-N gauged harmonic oscillator?

A clue comes from the fact that the large N inverted harmonic oscillator is dual to a certain string theory in two dimensions(for reviews see[3,4,5]).Therefore,we should?nd the meaning in string theory of rectifying the inverted matrix potential.This is done in section4,after discussing some of the properties and symmetries of the large-N harmonic oscillator in sections2and3.The physics of the harmonic oscillator and the inverted oscillator are very di?erent.The latter has a continuous spectrum,while in the former the spectrum is discrete.This implies that,unlike in the standard2d-string/matrix model duality,there is no need to take a double scaling limit to?nd a continuum dual string description.Finite N corresponds to nonzero string coupling constant.In this sense, the duality we propose works more like the AdS/CFT duality[6]than the standard2d-string/matrix model duality.Since the matrix model is well-de?ned at?nite N,we will be able to de?ne and study interesting?nite-coupling e?ects in this bosonic string theory.

The observables of interest in the large-N harmonic oscillator are the overlap ampli-tudes between resonances.In section5we explain how the resonances come about on the string theory side.In section6we calculate these overlap amplitudes on both sides of the duality and show that the large-N limit of the harmonic oscillator exactly agrees with the relevant sphere amplitudes on the string theory side.In this section,we also make contact with a normal matrix model.In section7we show that the duality passes non-trivial tests involving the1/N corrections to the leading large-N behavior.Section8is devoted to a heuristic picture of the duality.

In addition to providing a new example of open/closed string duality,the relation we propose is interesting for a completely di?erent reason.The matrix harmonic oscillator is closely related to the quantum hall e?ect(QHE)and therefore the dual stringy description might be useful for understanding various open questions in the QHE,like what is the e?ective description of the quantum phase transition from one plateau to the next.In section9we brie?y speculate on this and other possible applications and generalizations.

2.The Matrix Model

The model we wish to study is the gauged quantum mechanics of an N ×N matrix harmonic oscillator,

S =1

2

.(2.2)Since there are N fermions the vacuum energy is

E 0=1

2+...+(2N ?1)

2.(2.3)

The Hilbert space is spanned by states labelled by N integers k n such that 0≤k 1

H |k 1,k 2,...,k N =

N 2

∞ k 1=0∞ k 2=k 1+1...∞ k N =k N ?1+1q N n =1k n .(2.5)

Performing the sums sequentially we get

[8]

Z =q N 2/2

N n =11

2+

N n =1α?n αn ,(2.7)

and we are using stringy conventions for the commutators

[αm,αn]=nδn+m.(2.8)

Later on we shall argue that this chiral boson is(up to normalization)equivalent to the target-space?eld of the string description.Note that for?nite N,the momentum modes of the chiral boson are truncated in a clean way.This is another example of the stringy exclusion principle[9,10],about which we will say more in section5.

To understand this boson description in more detail,we introduce matrix raising and lowering operators

a i j=1

2 X i j+iP i j ,a?i j=12 X i j?iP i j ,(2.9)

where P is the momentum conjugate to the matrix X.These operators satisfy

[a i j,a?k l]=δi lδk j,[H,a]=?a,[H,a?]=a?.(2.10) The vacuum is de?ned by a i j|0 =0.The states

|{m i} ≡c{m i}

r

i=1tr(a?m i)|0 ,(2.11)

with m1≥m2≥...≥m r provide a useful basis for the Hilbert space.From(2.10)we see that indeed the spectrum is evenly-spaced.Note that the stringy exclusion is the statement that

m i≤N(2.12) in order that these states be linearly independent.

An orthonormal basis of states can be constructed from(2.11)as

|R({m i}) =c{m i}χR({m i})(a?)|0 (2.13) (see e.g.[11]or[12]1)where R({m i})is the representation of U(N)corresponding to the Young tableau with columns of lengths(m1,m2,...,m r)andχR(U)is the character of U

in this representation.The bound m i≤N guarantees that this is indeed a tableau for a U(N)representation.

In terms of the free-fermion description,the wavefunction for the state(2.13)is a Slater determinant constructed as follows.Look at the tableau{m i}sideways,de?ne k n to be the row-lengths;note that there are at most N rows by(2.12).The many-fermion wavefunction is then

z1|? z2|··· z N|R({m i}) =det

n,l=1,..,N ψk

n+N?n+1

(z l)(2.14)

whereψn(z)= n|z =H n(z)e?|z|2/2are the single-particle harmonic-oscillator wave-functions.For example,for the empty tableau,this?lls the lowest N energy levels with

fermions.The state tr(a?)k|0 corresponds to exciting k fermions by one level each–it makes a hole in the fermi sea at level N?k.For m=N,the hole is at the lowest state.For

m>N,there is no such single-particle description,in accord with the stringy exclusion

principle.

This is the same Hilbert space as that of two-dimensional Yang-Mills theory on a

cylinder(see e.g.[13]).The Hamiltonian for the matrix oscillator is,however,the number

of boxes in the tableau,

(H?E0)|{m i} =

r

i=1m i|{m i} (2.15)

rather than the second Casimir C2(R).

3.Symmetries

As is well-known(see e.g.§12of[4])there are in?nitely many conserved charges associated with the harmonic oscillator.These charges generate the w∞algebra that controls much of the physics of the system,and was studied quite intensively in the case of the inverted harmonic oscillator[14]in relation with the c=1theory.Although most of our discussion can be obtained from these papers by plugging i’s in the right places,we ?nd it useful to be explicit because the harmonic oscillator physics is quite di?erent from the inverted oscillator physics.

At large N,a semi-classical description exhibits most of the relevant physics in a simple way.Semi-classically,the eigenvalues form a Fermi sea in phase space.It is most convenient to parametrize the phase space by

U=(X+iP)

2

,V=U?=

(X?iP)

2

,(3.1)

which satisfy the Poisson bracket

{U,V}P B=i.(3.2) The Hamiltonian is H=UV,and thus

U(t)=e?it U(0),V(t)=e it V(0),(3.3) which implies that following charges are conserved

Q n,m=e i(n?m)t U n V m.(3.4) These charges form the w∞algebra

{Q n,m,Q n′,m′}P B=i(nm′?mn′)Q n+n′?1,m+m′?1.(3.5) The ground state is obtained by?lling states in the phase space up to the Fermi surface

UV=N,(3.6)

so that the area in units of the Poisson brackets(3.2)is equal to the number of fermions.

Excitations above the ground state can be described using area-preserving transfor-mations of the phase space.The area is preserved since it corresponds to the number of fermions.Area-preserving transformations can be described using a single scalar function h(U,V),a basis for which are

h nm=U n V m,(3.7)

n and m must be integers in order to preserve the connectivity of the fermi surface.h nm determines a vector?eld that is associated with an in?nitesimal area-preserving transfor-mation of the U-V plane,

B nm =

?h nm

?V

?U.(3.8)

The Lie brackets of the B nm’s form a w∞algebra.

The vector?eld B nm clearly depends both on n and m.However,when acting on the ground state(3.6), B nm depends only on|s|=|n?m|.Let us see how this comes about. Eq.(3.8)implies that h nm generates the following in?nitesimal deformation

δV=??U h=?nU n?1V m,δU=???V h=??mU n V m?1.(3.9)

Therefore,to leading order in?,we?nd that

(V+δV)(U+δU)=UV+?(n?m)U n V m.(3.10)

If UV is to begin with a constant(like in the ground state)then(3.10)implies that the deformation of the Fermi surface is

δ(UV)～? U

with a as in(2.9).Momentarily we shall claim that these are dual to excitations of the closed string”tachyon”.As should be clear from the semi-classical discussion above,there are other single trace operators in the theory that involve both a?and a.These are associated with the h nm with m=0.For example,consider the operator

tr(a(a?)2).(3.15)

Acting with this operator on the ground state has the same e?ect as acting with tr(a?). Namely,both take the ground state to the?rst excited state.However clearly these operators are not the same.In fact we will see that it is natural to relate these operators to discrete states in the dual stringy description.

At this point we encounter a small puzzle.At the quantum level there seem to be many more excitations than at the semi-classical level,which clearly makes no sense.To see this we note that there are many di?erent single trace operators that at the semi-classical level are associated with the same h nm.The simplest example is tr(aa?aa?)and tr(a2a?2). On the one hand,since in a non-Abelian theory the order inside the trace matters,these are di?erent operators.On the other hand semi-classically they are both associated with h2,2.The fact that the theory is gauged resolves this issue:in one dimension there is no electric or magnetic?eld and so the current that couples to A must vanish on-shell,2

j=[X,D0X]=i[a,a?]=0.(3.16)

This implies that operators which di?er by such commutators actually give the same result when acting on physical states.As a result,the inequivalent single-trace operators are in1-to-1correspondence with w∞generators.In section5we discuss further these operators at the quantum level.

Eq.(3.16)is reminiscent of a normal matrix model(for a recent discussion in relation with the c=1theory see[15]).We will elaborate on this connection in§6.

4.The dual string theory

Generally speaking it is not an easy task to?nd the stringy dual description of some large-N gauge theory.In our case a natural candidate comes from the well-studied duality between2d string theory with a Liouville direction and matrix quantum mechanics.The matrix quantum mechanics dual to2d string theory is closely related to the one that we are considering.The kinetic term is the same and the sign of the potential term is?ipped. Namely,the free fermions experience the celebrated inverted quadratic potential,rather then the harmonic oscillator potential of our case.Starting from that well-tested duality, what we have to do is?gure out what it means,on the string theory side,to?ip the potential,and see if what we get makes sense.In this section we discuss the meaning of?ipping the potential on the string theory side.In the rest of the paper we test our conjecture for the dual stringy description.

As was emphasized in[3]the curvature of the potential in the matrix model is related to the tension of the dual string theory by

U(x)=

1

α′(:?X?X:?:????:)+

Q

α′

?2?,Q=b+1/b=2.(4.3)

Adding the Liouville term,we end up with the following worldsheet action

S=1

g ??αX?αX+?α??α?+√α′ .(4.4)

According to the reasoning above,to?nd the candidate dual to the large-N harmonic oscillator we could either apply(4.2)or double Wick rotate

X→iX,?→i?.(4.5)

Either way we get a CFT whose stress tensor is

T(z)=1

√

4πα′ d2σ√α′R(2)Q?+μ0e i2b?/√

in the free theory,μ0=0.However,as we shall see in the next section,in the interacting theory1/μ0is the parameter that controls the genus expansion.

From(4.9)we also see that now,again unlike the usual case,we can compactify?. The allowed radii of compacti?cation seem to be

R=m/2,(4.10)

where m is an integer.However,since non-perturbatively there are D-branes and open strings e?ects in the theory,the open-string coupling constant should be well-de?ned.Since g2o=g s we?nd that m is an even number.So the smallest possible radius is1

?～?+2π.(4.11)

The dual string theory we propose to the large-N harmonic oscillator is(4.8)with this periodicity condition(4.11).

A useful way to think about the radius of the?direction is that?is dual toθof

(3.12),which has the same periodicity.The relation we are proposing between the large-N harmonic oscillator and that string theory is simple:X should be identi?ed with the quantum mechanics time and?should be identi?ed with the angular variable,θ,in phase-space.This seems to make sense since excitations in both descriptions travel at the same speed(1in our units)regardless of their energy.Note,however,that on the string theory side,at least naively,there are both left-movers and right-movers(in the target space), while on the quantum mechanics side there are only left-movers.In the next section we will see how to resolve this apparent contradiction.

5.Correlation functions

In this section we describe the map between the quantum mechanics and the string theory degrees of freedom and compare their correlation functions.This section is devoted to a more qualitative illustration of the dictionary between the two sides of the duality. More details can be found in the next section.

Let us start with the quantum mechanics side.The observables of interest are the overlaps between di?erent states

O(bra;ket)= bra|ket .(5.1)

Since the time evolution in the theory is trivial,in principle,all other gauge invariant observables are determined by these overlap amplitudes.The simplest case corresponds to what we loosely speaking call a two-point function,namely(5.1)with

bra|= 0|tr(a k1),|ket =tr((a?)k2)|0 .(5.2) By counting index lines,we see that the leading contribution in the large-N limit goes like

O(k1;k2)～δk1,k2N k1.(5.3)

As usual there are1/N corrections.What is special about this large-N theory is that here the1/N corrections are truncated after a?nite number of https://www.doczj.com/doc/094924540.html,ly,

O(k1;k2)=δk1,k2[k1/2]

l=0C l(k1)N k1?2l,(5.4)

where the C l’s depend on k1but not on https://www.doczj.com/doc/094924540.html,ter on we shall see how this comes about on the string theory side.

The analog of a three-point function is(5.1)with

bra|= 0|tr(a k),|ket =tr((a?)k1)tr((a?)k2)|0 .(5.5)

In that case the leading contribution in the large-N limit is

O(k;k1,k2)～δk,k1+k2N k?1.(5.6)

We now turn to the string theory side.The ghost-number-two cohomology contains the tachyon vertex operator

T±k=cˉc e?ik(X±?)e i2b?,(5.7)

where the factor e i2b?is the string coupling that multiplies the tachyon wave function, and k is an integer due to the periodicity(4.11).There are four kinds of modes,with the following interpretations

T+

k>0

:incoming leftmover,T?k>0:incoming rightmover,

T+

k<0

:outgoing leftmover,T?k<0:outgoing rightmover.

Note that to make the relation with the matrix model simple,the energy in the X direction (rather than in the ?direction which is actually the time direction in this background)determines in our terminology if a wave is incoming or outgoing.

S-matrix amplitudes are constructed in the usual fashion,and take the form

A (k 1,k 2,...,k n +;k n ++1,...,k n ++n ?)= D X D ?e ?S n ++n ?

i =1 d 2σ√

n !

( d 2σ√ge ik i ?±ik i X e i 2b?,

where

S 0=1g ?αX?αX ??α??α?+iR (2)Q? .(5.10)

Eq.(5.9)now takes the form of a sum over amplitudes in the free theory with the same tachyon (or any other closed string mode)insertions plus additional insertions of the screen-ing operator, d 2σ√2(k +tot ?k ?tot )=0,(5.12)

where g is the genus and n is the number of insertions of the screening operator.This relation is the equivalent of momentum conservation in the ?direction.The last two terms need no explanation.The rest of the terms are usually called the background charge,as

they are induced by the coupling of?0to the integral of the world sheet curvature which is

χ=2?2g?h,

where h is the number of holes;in our case h corresponds to the total number of closed string https://www.doczj.com/doc/094924540.html,bining(5.12)and(5.11)we get

k?tot=?k+tot=2?2g?(n+n++n?).(5.13)

This relation implies that1/μ0is indeed the genus expansion parameter:Given a certain amplitude determined by k i,n+and n?,we see that as we increase g,we decrease n,the power ofμ0.The phases arising from the complex dilaton conspire to makeμ?10the string coupling constant.In fact,comparing(5.13)to the’t Hooft counting of powers of N,we see that we can identify

μ0～N.(5.14) Now we are ready to compare some stringy amplitudes with the quantum results mentioned at the beginning of this section.Let us start with1→1amplitudes.There are two possible ways to satisfy the energy conservation(5.11)and momentum conservation(5.12) conditions on the sphere(g=0).The?rst is to take n=0and k?tot=k+tot=0.This can be achieved by having two particles with the same chirality and opposite energy(say n+=2and n?=0).Such amplitudes vanish since they involve only two closed string insertions,which do not su?ce to saturate the ghost zero modes on the sphere.Indeed there is no dual quantum mechanical amplitude for these scattering amplitudes.

The second way to saturate(5.11),(5.12)is more interesting.We take one particle with positive chirality and energy k and another particle with the negative chirality and energy?k.So n+=1and n?=1.From(5.13)we see that n=k and so the amplitude scales like

A(k;?k)～μk0～N k.(5.15) This amplitude does not vanish since(due to the insertions of the screening operators) it involves more than two closed string insertions on the sphere.It is natural to make the following identi?cation between the closed string modes and the quantum mechanics operators

T+

k>0?tr((a?)k),T?k<0?tr(a k).(5.16)

Indeed we see that(5.15)is in agreement with the harmonic oscillator result(5.3).Notice

and T?k>0vanish.3This resolves the puzzle arising from that amplitudes that involve T+

k<0

the naive expectation that we should have twice as many stringy modes as excitations of the Fermi surface,raised at the end of the previous https://www.doczj.com/doc/094924540.html,ly,just like in the harmonic oscillator,we only see tachyon wavefunctions with p?=?i??negative.In this ’imaginary’version of Liouville theory,there is a sense in which the Seiberg bound becomes the fact that the target-space?eld is chiral.

Stringy exclusion

As usual in string theory there are corrections from higher-genus worldsheets.Since n≥0we?nd from(5.13)non-vanishing amplitudes only for

g≤ k

3In the next section we shall see how this comes about in higher point functions as well.

5.1.Discrete states

The closed-string ghost-number-two cohomology contains other states,known as dis-crete states,that are the remnants of the massive modes of the string.In the usual c=1 theory they appear as non-normalizable modes with imaginary energy(or real Euclidean energy)and imaginary Liouville momentum.In our case they become normalizable prop-agating modes that are as important as the tachyon modes discussed above.

Let us review how the discrete states come about.The simplest way to think about them is in terms of the chiral SU(2)current algebra

J±(z)=e±2iX(z),J3(z)=i?X(z).(5.19)

The highest weight?elds with respect to this algebra areΨj,j(z)=e2ijX where j= 0,1/2,1....With the help of J?0= dz

(2j++2j?)=0,(5.22)

2

and so the sphere amplitude scales like

.(5.23)

A(m?,j?;m+,j+)～δm++m?μj++j?

On the quantum mechanics side we can follow the same steps.The SU(2)generators

are

J+=1

2

tr(a2),J3=

1

The simplest way to calculate1→m overlap amplitudes

O(k;k1,k2,...k m)= 0|tr(a k)tr((a?)k1)tr((a?)k2)...tr((a?)k m)|0 ,(6.3)

in the large-N limit is to use the w∞algebra m times to move tr(a k)to the right.From (3.5)5we get

O(k;k1,k2,...k m)=δk,k1+k2+...k m k(k?1)(k?2)...(k?m+2)k1k2...k m N k?m+1.(6.4)

To match this on the string theory side,we have to compute the relevant sphere amplitudes in full detail.As explained in the previous section the relevant calculation is equivalent to scattering amplitudes in the free theory(μ0=0)with extra insertions of the screening operator.In particular,on the sphere we get for the1→m tachyon scattering amplitude

A(k;?k1,?k2,...,?k m)μ

0=

μn0

Γ(k i)

πr?2

(k?m+1)!(k?1)! μ0Γ(1)Γ(k i).(6.7) This expression contains some zeros and some in?nities that must be dealt with before comparing with the harmonic oscillator results.There are two kinds of zeros.The?rst comes from the(1/Γ(0))k?m+1and is due to the screening operators.Recall that to obtain

?nite results in the Liouville theory one needs to renormalize the cosmological constant when b→1(see e.g.[31]).To be precise we need to takeμ0→∞and b→1while keeping

μ=πμ0

Γ(b2)

Γ(0) in that equation byμ/π.A precise way to think about this procedure is the following. Take b=1+?/2and twist the boundary condition on the tachyon?eld T(?)=e2πi?T(?+ 2π).That twisting has the e?ect of lifting the tachyon zero mode to k=?so that the μ0Γ(1)/Γ(0)in(6.7)becomesμ0Γ(1+?)/Γ(?)and in the limit?→0we simply getμ/π.

The formula(6.7)also vanishes when one of the outgoing momenta,?k i,is positive. This is exactly as it should be since there are no operators on the harmonic oscillator side that correspond to T?k>0(see discussion after eq.(5.18)).

In?nities arise from theΓfunctions when k i>0.The reason for these in?nities is that the amplitudes are sitting on a resonance.Physically there is no region of interaction,the particles are always nearby and they keep interacting forever.In the harmonic oscillator side we avoided this by not propagating in time each of the operators.To realize the same e?ect on the string theory side we need to introduce leg factors.Heuristically,these can be viewed as putting back the external legs that are missing in the amputated S-matrix elements calculated by string theory amplitudes.The way to?nd the relevant leg-factors is to match the two-point function on both sides of the duality.Then one can test the duality by comparing the higher point functions.The relevant leg-factors are7

f ket(k)=?1

Γ(?k)

,(k<0)and f bra(k)=πΓ(k)Γ(k+1),(k>0).(6.9)

Putting all of this together one?nds that the relation between the string theory scattering amplitudes and the matrix model overlap amplitudes for the1→m processes is

O(k;k1,k2,...k m)=f bra(k)

m

i=1f ket(k i)A(k;?k1,?k2,...?k m)μ,(6.10)

?Γ(n+1)

+O(?0))but that would have made the relation with the matrix model a bit more clumsy.

withμ=N.This relation holds in the1→1cases by construction.The fact that it holds for the other cases(m>1),that have non-trivial dependence on the k’s(see eq.(6.4))is an encouraging consistency check of the proposed duality.

So far we did not consider the winding modes.Since?is compacti?ed,modular invariance implies that there must be winding modes in the theory.But there are no candidate dual operators for the winding modes in the quantum mechanics.8This is exactly the same puzzle we have encountered with the momentum modes with opposite chirality.In that case the resolution was that scattering amplitudes with at least one opposite-momentum mode vanish.The same is expected to happen for the winding modes. As in the case of the momentum modes with opposite chirality,we do not know how to prove this in general,but we demonstrate it here for some cases.Consider,for example, scattering amplitudes of m winding modes.The analog of(5.12)in that case is

2?2g?(n+m)=0,(6.11)

where again n is the number of insertions of the screening operator.We see that no amplitude with m>2can satisfy this relation.For m=1and m=2the relation can be satis?ed on the sphere with n=1and n=0respectively.But both cases vanish since they involve only two insertions of closed strings on the sphere.

6.1.Normal matrix models and matrix integral representation

We end this section with a discussion on a relation with normal matrix models.A normal matrix model(NMM)is an integral over complex matrices Z such that[Z,Z?]=0. The fact that Z commutes with its conjugate implies that,much like in the case of a single matrix,in many interesting cases the model can be described in terms of the eigenvalues of Z and Z?.The NMM that appears to be related to the quantum mechanics described here takes the form

Z NMM(J,J?)= [Z,Z?]=0d N2Zd N2Z?e?tr(Z?Z+ZJ?+Z?J).(6.12) The normal-matrix constraint in the path integral can be imposed with a Lagrange mul-tiplier matrix A,

δN2 i[Z,Z?] = dA e tr[Z,Z?]A,(6.13)

随机过程作业

第三章 随机过程 A 简答题： 3-1 写出一维随机变量函数的均值、二维随机变量函数的联合概率密度（雅克比行列式）的定义式。 3-2 写出广义平稳（即宽平稳）随机过程的判断条件，写出各态历经随机过程的判断条件。 3-3 平稳随机过程的自相关函数有哪些性质功率谱密度有哪些性质自相关函数与功率谱密度之间有什么关系 3-4 高斯过程主要有哪些性质 3-5 随机过程通过线性系统时，输出与输入功率谱密度之间的关系如何 3-6 写出窄带随机过程的两种表达式。 3-7 窄带高斯过程的同相分量和正交分量的统计特性如何 3-8 窄带高斯过程的包络、正弦波加窄带高斯噪声的合成包络分别服从什么分布 3-9 写出高斯白噪声的功率谱密度和自相关函数的表达式，并分别解释“高斯”及“白”的含义。 3-10 写出带限高斯白噪声功率的计算式。 B 计算题： 一、补充习题 3-1 设()()cos(2)c y t x t f t πθ=?+，其中()x t 与θ统计独立，()x t 为0均值的平稳随机过程，自相关函数与功率谱密度分别为：(),()x x R P τω。 ①若θ在（0，2π）均匀分布，求y()t 的均值，自相关函数和功率谱密度。 ②若θ为常数，求y()t 的均值，自相关函数和功率谱密度。 3-2 已知()n t 是均值为0的白噪声，其双边功率谱密度为：0 ()2 N P ω= 双，通过下图()a 所示的相干解调器。图中窄带滤波器（中心频率为c ω）和低通滤波器的传递函数1()H ω及2()H ω示于图()b ，图()c 。

试求：①图中()i n t （窄带噪声）、()p n t 及0()n t 的噪声功率谱。 ②给出0()n t 的噪声自相关函数及其噪声功率值。 3-3 设()i n t 为窄带高斯平稳随机过程，其均值为0，方差为2 n σ，信号[cos ()]c i A t n t ω+经过下图所示电路后输出为()y t ，()()()y t u t v t =+，其中()u t 是与cos c A t ω对应的函数，()v t 是与()i n t 对应的输出。假设()c n t 及()s n t 的带宽等于低通滤波器的通频带。 求()u t 和()v t 的平均功率之比。

机场集团战略规划

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ADS_Harmonic_Balance

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