NONUNIFORM FAST COSINE TRANSFORM AND CHEBYSHEV …

Progress In Electromagnetics Research, PIER 28, 253–273, 2000

NONUNIFORM FAST COSINE TRANSFORM AND

CHEBYSHEV PSTD ALGORITHMS

B. Tian and Q. H. Liu

Department of Electrical and Computer EngineeringDuke UniversityDurham, NC 27708, USA

1. Introduction2. Formulation

2.1 Forward NUFCT Algorithm2.2 The Inverse Nonuniform Fast Cosine Transform (NUIFCT)

Algorithms2.3 Application of NUFCT in the Chebyshev PSTD Algorithm

3. Numerical Results3.1 Numerical Results of NUFCT Algorithms3.2 Numerical Results of One-Dimensional Chebyshev PSTD

Algorithms4. ConclusionsReferences

1. INTRODUCTION

In this work our main interests are to develop fast cosine transformalgorithms that are applicable to nonuniformly spaced data, and touse these algorithm to improve the numerical solution of time-domainMaxwell’s equations.

Fast cosine transform (FCT) has many applications in signal pro-cessing and computational electromagnetics. There are several well-developed efficient FCT algorithms [1–3]. The basic assumption of theregular FCT algorithms is that the input data has to distribute uni-formly. In reality, however, the nonuniform data is common in manyapplications. The nonuniformity may be a result of sampling error, of

254 Tian and Liu

convenience, or by intention. For example, to solve a wave propagationproblem in a medium with both electrically large and small regions, itis more efficient to sample nonuniformly. Under these circumstances,regular FFT or FCT algorithms do not apply. Unfortunately, the di-rect summation of the nonuniform discrete cosine transform (NUDCT)costs O(N2) arithmetic operations, where N is the number of samplepoints.

Following the idea of fast interpolation for the nonuniform fastFourier transform (NUFFT) algorithms [4–8], in this work we developnonuniform fast cosine transform algorithms. In particular, we usethe idea of least-squares interpolation of an exponential function ona nonuniform grid by some oversampled uniform points. During thisinterpolation, the regular Fourier matrices are derived [6, 8] which areindependent of the nonuniform points. As a result, the interpolationcan be performed efficiently. These algorithms have the complexity ofO(N log2 N).

For the nonuniform inverse cosine transforms, we find that the solu-tion can be obtained by the iterative conjugate-gradient FFT method.That is, the matrix equation is solved iteratively by the CG method;while in each CG iteration the matrix vector multiply is achieved ef-ficiently by the FFT algorithm for the convolution and correlation.The resulting inverse algorithm has a computational complexity ofO(KN log2 N) where K is the number of CG iterations.

The major application of the NUFCT algorithms is the numericalsolution of time-dependent Maxwell’s equations. Conventionally, thefinite-difference time-domain method [9–11] is used to solve these par-tial differential equations with a second-order accuracy. Higher orderfinite-difference methods have also been used [12–15]. The Fourier andChebyshev pseudospectral methods originally proposed by Kreiss andOliger [16] and by Orszag [17] have been widely used [18]. Because ofthe implicit periodicity in the FFT, the Fourier pseudospectral methodwas used for periodic problem. The Chebyshev pseudospectral methodis more widely used since it does not have such a limitation. For amedium with several regions, traditionally a multiregion approach isused so that each region is solved separated and boundary conditionsare used across adjacent regions.

Recently the perfectly matched layer (PML) [19] has been used forboth Fourier and Chebyshev pseudospectral methods [20–27]. It com-pletely removes the limitation of the wraparound effect associated with

Nonuniform fast cosine transform and Chebyshev PSTD 255

the Fourier pseudospectral method, and allows an effective modeling ofan unbounded medium in the Chebyshev pseudospectral method. Fol-lowing the convention of the FDTD method, we call these the Fourierand Chebyshev PSTD methods. For smooth fields, these methodsrequire only two points and π points per wavelength, respectively,rather than 10–20 points per wavelength in the FDTD method evenfor a moderate-size problem. The advantage of the Chebyshev PSTDmethod over the Fourier PSTD method is that it can be used to modelbounded media.

In this work, we proposed the application of our NUFCT algorithmsin the Chebyshev pseudospectral time domain (PSTD) method for thesolution of Maxwell’s equations. Instead of treating multiple regionsseparately, this method allows the complete treatment of multiple re-gions simultaneously. This advantage becomes particularly attractivewhen there are many regions with significant different scales. Further-more, for regions with different dielectric constants, this method allowsus to sample differently from region to region so that the memory re-quirement is efficient.

This paper is organized as follows. We first present the forwardand inverse fast cosine transform algorithms for nonuniformly sampledata. Then as an application, we apply these algorithms to repre-sent electromagnetic fields and their derivatives in order to solve theone-dimensional Maxwell’s equations. Finally we present the numeri-cal results for the NUFCT algorithms and the nonuniform ChebyshevPSTD method.

2. FORMULATION

2.1 Forward NUFCT Algorithm

For nonuniform points ck ∈ [0, N ] , two different discrete cosinetransforms (DCT’s) of real data αk can be defined as:

fj =N∑

k=0

αk cos(kπcj/N)

=�e

(N∑

k=0

αkei2πkc′j/N

)= �e(f̃j), j = 0, 1, 2, . . . , N (1)

256 Tian and Liu

gj =N∑

k=0

αk cos(jπck/N)

=�e

(N∑

k=0

αkei2πjc′k/N

)= �e(g̃j), j = 0, 1, 2, . . . , N (2)

where cj = 2c′j . Here we refer to the fast algorithm for (1), which hasnonuniform frequency points cj , as NUFCT-1. Similarly, NUFCT-2 refers to the algorithm for (2), which has nonuniform time samplepoints ck . Note that in order to use the regular Fourier matrices in [6],we have rewritten the cosine transforms as the real part of weightedsummation of exponential functions. In general, the regular fast cosinetransform algorithms do not apply. The direct summation of (1) and(2), unfortunately, costs O(N2) arithmetic operations.

Following the idea for the nonuniform fast Fourier transform(NUFFT) in [6], we interpolate an exponential function at nonuni-form points with some oversampled uniform points. Introducing accu-racy factor sk = cos(kπ/mN −π/2m) and interpolating each value ofske

i2πkmc′j/mN in terms of (q + 1) uniform points, we obtain

f̃j �N∑

k=0

βk

q/2∑

�=−q/2

x�(c′j)ei2kπ([mc′j ]+�)/mN , (3)

g̃jsj �N∑

k=0

αk

q/2∑

�=−q/2

x�(c′k)ei2πj([mc′k]+�)/mN , (4)

where βk = s−1k αk, x� are interpolation coefficients, m is the over-

sampling rate, and [mc′j ] denotes the integer nearest to mc′j .We first formulate the NUFCT-1 algorithm. We rearrange (3) as

f̃j =q/2∑

�=−q/2

x�(c′j)N∑

k=0

βkeikπ([mc′j ]+�)/mN =

q/2∑

�=−q/2

x�(c′j)µ([mc′j ] + �

)

(5)where

µ([mc′j ] + �

)=

N∑

k=0

βkei2kπ([mc′j ]+�)/mN (6)


is the regular FFT of array βk of length mN with (m − 1)N zerospadded.

Our first step is to find the interpolation coefficients x� which satisfythe following conditions:

skzkmc′j =

q/2∑

�=−q/2

x�(c′j)zk[mc′j ]+� (7)

where z = ei2π/mN .Through similar procedures as in [6], we can calculate the least-

square solution of the coefficients as x(c′j) = F−1a(c′j) where F =

A†A is the regular Fourier matrix, and a(c′j) = A†v(c′j) . The matrixA, F and vector v are listed bellow:

A =

1 1 · · · 1z[mc′j ]−q/2 z[mc′j ]−q/2+1 · · · z[mc′j ]+q/2

......

. . ....

zN([mc′j ]−q/2) zN([mc′j ]−q/2+1) · · · zN([mc′j ]+q/2)

F =

N + 11 − zN+1

1 − z· · · 1 − zq(N+1)

1 − zq

1 − z−(N+1)

1 − z−1N + 1 · · · 1 − z(q−1)(N+1)

1 − zq−1

......

. . ....

1 − z−q(N+1)

1 − z−q

1 − z−(q−1)(N+1)

1 − z−(q−1)· · · N + 1

andv(c′j) =

(s0 s1z

mcj s2z2mcj · · · sNzNmcj

)T

Here the regular Fourier matrix F is uniquely determined by (m, N, q)and is independent of cj . This is the key point of our algorithm asit makes the interpolation efficient. For given parameters (m, N, q) ,the regular Fourier matrix need to be calculated only once, making thesolution for the interpolating coefficients efficient.

Our second step is to find the solution to (6) by regular FFT. Equa-tion (6) can be rewritten as:

µj =N∑

k=0

βkei2πkj/mN, j = [mcj ] + �, (8)

258 Tian and Liu

Then through (5), we obtain the sequence f̃j (j = 0, 1, 2, . . . , N) , andhence the real part fj .

Similarly, the NUFCT-2 algorithm for (2) consists of following steps:1)Calculate the transform coefficients by

Tn =∑

�,k,[mc′k]+�=n

αkx�(c′k)

2) Use the regular FFT to calculate the following equation:

g̃jsj =mN∑

n=0

Tnei2πnj/mN

3) Scale the values by sj .The asymptotic number of arithmetic operations of these

NUFCT algorithms is O(2mN log2 N) , where m � N . Usually wechoose q = 8 and m = 2.

2.2 The Inverse Nonuniform Fast Cosine Transform

(NUIFCT) Algorithms

Corresponding to the two forward NUFCT algorithms, we also de-veloped two inverse algorithms, NUIFCT-1 and NUIFCT-2. These al-gorithms find αk from fj and gj , respectively, with O(N log2 N)arithmetic operations. In comparison, the direct solution requiresO(N3) arithmetic operations.

Equation (1) can also be written as f = Bα . From the elementarymatrix identities we know that B−1 = (B†B)−1B† , so the solution toinverse DCT is

α = (B†B)−1h, h = B†f (9)

where Bjk = cos kπcj

N . Note that (B†B)j� = 12(aj+� + bj−�) where

aj =N∑

k=0

cosjπck

N, j = 0, 1, 2, . . . , 2N

and

bj =N∑

k=0

cosjπck

N, j = −N,−N + 1, . . . , N − 1, N


which can be calculated by the NUFCT-2 algorithm with O(N log2 N)arithmetic operations. Similarly, h in (9) can also be obtained by usingNUFCT-2. Then the solution α in (9) can be obtained by using theconjugate-gradient FFT (CG-FFT) method [28, 29]. In the CG-FFTmethod, the solution of α is obtained iteratively. Each CG iterationinvolves operations such as discrete correlation and convolution

∑

�

(B†B)j�y� =12

{F−1[F(a)F∗(y) + F(b)F(y)]

}j

(10)

where F represents the fast Fourier transform.In summary, the procedures for the NUIFCT-1 algorithm are:1) Calculate the array h by using NUFCT-2;2) Find arrays aj and bj with NUFCT-2;3) Solve α in (9) via the CG-FFT method.Similarly, the representation of NUIFCT-2 can be written as:

f = B†α

Then the solution of NUIFCT-2 is:

α = (B†)−1f = B(B†B)−1f = Bd, d = (B†B)−1f

After using CG-FFT to calculate d , the transform coefficients can beeasily obtained by using NUFCT-1.

2.3 Application of NUFCT in the Chebyshev PSTD Algo-rithm

One of many applications of the NUFCT algorithms is the solutionof Maxwell’s equations by the Chebyshev PSTD method. Maxwell’sequations governing the electric and magnetic fields in the mediumwith permeability µ , electric permittivity ε and conductivity σ aregiven as:

∇× E = −µ∂H∂t

− M (11)

∇× H = ε∂E∂t

+ J + σE (12)

where J, M are the imposed electric and magnetic current densities,respectively. The conventional method to solve these equations is the

260 Tian and Liu

finite-difference time-domain method [9–11]. Here we use the NUFCTalgorithms to approximate the spatial derivatives and solve equations(11) and (12) by a leap-frog scheme. We demonstrate the nonuniformChebyshev PSTD method through a one-dimensional problem.

For a one-dimensional TEM wave propagating along the x direc-tion, the above equations become

∂Ey(x, t)∂x

= −µ∂Hz(x, t)

∂t− Mz(x, t) (13)

−∂Hz(x, t)∂x

= ε∂Ey(x, t)

∂t+ Jy(x, t) + σEy(x, t). (14)

To truncate the computation domain, we use the perfect matched layer(PML) as the absorbing boundary condition [19] under complex coor-dinates [20, 30, 31]

eη = aη + iωη

ω, η = x, y, z.

In frequency domain, we replace the operator ∇ in Maxwell’s equa-tions by

∇e =∑

η=x,y,z

η̂1eη

∂

∂η

Then in time domain equation (13) and (14) become:

ayε∂Ey(x, t)

∂t+ (ayσ + ωyε)Ey(x, t) + ωyσ

t∫

−∞

Ey(x, t)dt

= −∂Hz(x, t)∂x

− Jy(x, t) (15)

azµ∂Hz(x, t)

∂t+ ωzµHz(x, t) = −∂Ey(x, t)

∂x− Mz(x, t). (16)

In the above, aη and ωη are respectively the scaling and attenuationfactors of the PML. In a regular region of interest, aη = 1 and ωη = 0 ;while in the PML region surrounding the region of interest aη ≥ 1and ωη > 0 . The outer boundary of the PML region may assume thecondition of a perfect electric or magnetic conductor.

For the time integration of the (15) and (16), we use the centerdifferencing scheme with a temporal staggered grid. Here, the H field


is defined at (n + 1/2)∆t whereas the E field is defined at n∆t . Thetemporal derivatives are approximated by central difference which hasthe second-order accuracy. Following the same procedure as in [31],we achieve the time-stepping equations for E field and H field as:

Hz(x, n + 1/2)

=

(axµ

∆t− ωxµ

2

)Hz(x, n − 1/2) − ∂Ey(x, n)

∂x− Mz(x, n)

axµ/∆t + ωxµ/2(17)

Ey(x, n + 1)

=

(axε

∆t−ωxε

2−σ

2

)Ey(x, n)−ωyσEI

y−∂Hz(x, n+1/2)

∂x−Jy

(x, n+

12

)

axε/∆t + ωxε/2 + σ/2(18)

where EIy =

t∫−∞

Eydt is the time-integrated electric field.

In order to use the NUFCT algorithms for the spatial derivatives, weexpand the electric and magnetic field by the Chebyshev polynomials.For example, a function f(y)(y ∈ [1, 1]) can be approximated as:

f(y) =N∑

k=0

αkTk(y) (19)

where Tk = cos(k cos−1(y)) is the kth Chebyshev polynomial. To findthe coefficients α , we multiply Tm on both sides of equation (19) andintegrate over y from −1 to 1,

+1∫

−1

f(y)Tm(y)dy√1 − y2

=N∑

k=0

αk

+1∫

−1

Tk(y)Tm(y)dy√1 − y2

Applying the orthogonality of Chebyshev polynomials, we obtain therepresentation of αk as:

αk =2 − δk0

π

+1∫

−1

f(y)Tk(y)dy√1 − y2

.=2 − δk0

N(−1)k

N∑

j=0

f(yj) cos(kπcj/N)∆cj

(20)

262 Tian and Liu

In equation (20), we set y = − cos(

πcN

), c ∈ [0, N ] and δkj is the

Kronecker delta function. It is efficient to calculate α by using ourNUFCT-2 algorithm. From equation (20), it is also obvious that wecan use the regular FCT algorithm to compute αk if we let the valueof y to locate at the extrema points of Chebyshev polynomials y =− cos(πj/N), j = 1, 2, . . . , N . In the nonuniform case, cj are any realnumbers within [0, N ].

Similarly, the spatial derivative df/dx can also be expanded interms of Chebyshev polynomials:

df(y)dy

=d

dy

N∑

k=0

αkTk(y) =N∑

k=0

βkTk(y) (21)

From the recursion relation for the first derivative of Chebyshev poly-nomials

T ′n+1

n + 1− T ′

n−1

n − 1= 2Tn,

it is straightforward to get the relation between the coefficients αk andβk in (21). Specifically, equation (21) can be written as:

α0T′0+α1T

′1+· · ·+αNT ′

N = β0T′1+

β1

4T ′

2+· · ·+ βN

2

(T ′

N+1

N + 1−

T ′N−1

N − 1

)

Comparing the coefficients of the same Chebyshev polynomials, weobtain βk from the given coefficients αk from the following relations:

βN = 0βN−1 = 2NαN

βN−2 = 2(N − 1)αN−1

βi−1 = βi+1 + 2iαi, i = N − 2, N − 3, . . . , 2β0 = α1 + β2/2

Once βk are found, the spatial derivative of df(y)dy can be obtained

very efficiently with the NUFCT-1 algorithm.In the above derivation of the derivative using Chebyshev poly-

nomials, we have assumed that y ∈ [−1, 1] . However, we can ap-ply it to any computational domain by a simple linear transforma-tion. For given domain x ∈ [x1, x2] , we define x = ay + b , where


y ∈ [−1, 1], a = (x2 −x1)/2 and b = (x2 +x1)/2 . Thus the derivativeof f(x) is:

df(x)dx

=df(ay + b)

dy

dy

dx=

1a

df(ay + b)dy

.

Given the approximation of the derivatives, equations (17) and (18)form a leap-frog system on a centered grid where all field componentsare defined at the cell centers. Thus, either from the initial conditionsof the fields or from the continuous source excitation, the fields at alllater time steps can be obtained.

3. NUMERICAL RESULTS

3.1 Numerical Results of NUFCT Algorithms

For our forward NUFCT algorithms, we select N = 64, m = 2and q = 8 for the following examples. The input data {αk} and thesampling intervals are obtained by a pseudorandom number genera-tor with a uniform distribution within the range of [0, 1]. Fig. 1(a)compares the NUFCT-1 output data with the results evaluated by thedirect sum. Fig. 1(b) shows the absolute error between the NUFCTand direct results. The NUFCT algorithm has the L2 and L∞ errors[4] of E2 = 1.2760× 10−6 and E∞ = 8.07× 10−6 . In Fig. 2, we com-pare the numbers of arithmetic operations in our NUFCT-1 algorithmwith the theoretical behavior. It is clear that our algorithm has thecomplexity of O(N log2 N).

For the inverse algorithm NUIFCT-1, we use the same parametersN = 64, m = 2, q = 8 . The input data are also randomly distributedwithin [0, 1]. Fig. 3(a) shows the comparison of NUIFCT-1 result anddirect result. Fig. 3(b) is the absolute error. Here the L2 and L∞errors are E2 = 8.3375×10−5 and E∞ = 2.76×10−5 . From Fig. 4, weshow that our NUIFCT algorithm has the complexity of O(N log2 N) .Since the NUFCT-2 and NUIFCT-2 have a similar accuracy as theNUFCT-1 and NUIFCT-1, their results are omitted for brevity.

3.2 Numerical Results of One-Dimensional Chebyshev PSTDAlgorithm

First we show the NUFCT algorithm for the derivative of the func-tion. In this example, the function f(x) is the first derivative of theBalckman-Harris window function. Using N = 16 in the NUFCT

264 Tian and Liu

(a) (b)

Figure 1. (a) Comparison of the NUFCT-1 output data with the directevaluated results. (b) The absolute error of the NUFCT results.

0 200 400 600 800 1000 12000

0.5

1

1.5

2

2.5x 10

5

Ope

ratio

n N

umbe

r of

NU

FC

T1

Alg

orith

m

N

Actual Number Theoretical Number

Figure 2. Number of arithmetic operations as a function of N inthe NUFCT-1 algorithm. The theoretical curve is proportional toO(N log2 N).


0 20 40 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n

Out

put D

ata

0 20 40 600

1

2

3

4

5

6

7x 10

−5

n

Abs

olut

e E

rror

(a) (b)

Figure 3. (a) Comparison of the NUIFCT-1 output data with thedirect evaluated results. (b) The absolute error of the NUIFCT results.

0 200 400 600 800 1000 12000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

7

Ope

ratio

n N

umbe

r of

NU

IFC

T1

Alg

orith

m

N

Actual Number Theoretical Number

Figure 4. Number of arithmetic operations as a function of N . Thetheoretical curve is proportional to O(N log2 N).

266 Tian and Liu

Figure 5. The Numerical derivative df/dx on a nonuniform grid bythe NUFCT algorithm (N = 16) along with the exact solution.

algorithm with a nonuniform grid, the numerical derivative df/dx iscompared with the exact solution in Fig. 5.

We used the Chebyshev PSTD algorithm to simulate electromag-netic wave propagation in both homogeneous and inhomogeneous me-dia. An electric current source is used to excite the electromagneticwaves. Its time function is the first derivative of Blackman-Harris win-dow function with a center frequency fc = 200 MHz.

The first case is free space. The computation domain is [0, 12] withonly 64 uniformly distributed points. Therefore, the sampling densityis π cells per wavelength at f = 2.5fc . The source is located in thecenter of medium and receivers are distributed among the whole regionto measure the E field. Fig. 6 shows the comparison of cell size ∆xj

used here and the cell size used in the Chebyshev PS method. Fig. 7compares the E field with the analytical solution at all receivers whileFig. 8 shows the comparison of one receiver. The numerical resultagree well with the analytical solution.

In an inhomogeneous case, we have a two-layer medium: free spacefrom x = 0 m to x = 8 m and medium with εr = 4, µr = 1, σ = 0from 8 m to 12 m. The dipole is placed at the center of free space.


0 10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Cell Index

Cel

l Siz

e

Uniform PointsExtrema Points

Figure 6. The comparison of cell size ∆xj for the uniform distribu-tion and the conventional distribution at extrema points of Chebyshevpolynomials.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 108

0

2

4

6

8

10

12

14

16

18

t (s)

Nor

mal

ized

EF

ield

AnalyticalNumerical

Figure 7. The normalized E-field at every 4th grid point.

268 Tian and Liu

0 0.5 1 1.5 2 2.5 3 3.5 4

x 108

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

t (s)

Nor

mal

ized

EF

ield

AnalyticalNumerical

Figure 8. The comparison of normalized E-field at one receiver.

0 10 20 30 40 50 60 70 80 900.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Cell Index

∆ x

Figure 9. The cell size ∆xj as a function of the cell index.


Figure 10. The comparison of normalized E-field at all receivers inthe inhomogeneous case.

0 1 2 3 4 5 6 7 8 9

x 108

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

t (s)

Nor

mal

ized

Ey

Numerical Analytical

Figure 11. The comparison of normalized E-field at one receiver.

270 Tian and Liu

Here we use 88 points, also about π cells per wavelength. Fig. 9shows the cell size as a function of x . We have selected denser pointsaround the boundary to reduce the Gibbs’ phenomenon and uniformcells elsewhere. Fig. 10 is the comparison of E field at all receivers andFig. 11 compares results at one receiver. The agreement is also verygood.

4. CONCLUSIONS

Using the least-squares interpolation and the regular Fourier matrix,we develop nonuniform fast forward cosine transform (NUFCT) al-gorithms for nonuniformly spaced data. Based on forward algorithmsand CG-FFT, we also propose nonuniform inverse fast cosine transform(NUIFCT) algorithms. Numerical results show that these NUFCT al-gorithms are very accurate and require only O(N log2 N) arithmeticoperations.

NUFCT algorithms are used in the Chebyshev PSTD algorithm toallow an nonuniform grid so that the grid points need not locate at theextrema points of Chebyshev polynomials. This property is very usefulfor inhomogeneous media and enables us to take all layers of mediumwithin one region. As long as we select more points around each bound-ary and satisfy π cells per wavelength on the average, the numericalresults agree well with analytical results. However, without NUFCT,we would have to calculate the fields with a multiregion scheme andmatch these solutions at the interfaces between regions. Multidimen-sional results of the Chebyshev PSTD method will be reported in thenear future.

ACKNOWLEDGMENT

This work was supported by Environmental Protection Agency througha PECASE grant CR-825-225-010, and by the National Science Foun-dation through a CAREER grant ECS-9702195.


REFERENCES

1. Rao, K. R., and P. Yip, Discrete Cosine TransformAlgorithms,Advantages, Applications, Academic Press, London, 1990.

2. Chen, W. A., C. Harrison, and S. C. Fralick, “A fast computa-tional algorithm for the discrete cosine transform,” IEEE Trans.Commu., Vol. 25, No. 9, 1004–1011, 1977.

3. Lee, B. G., “A new algorithm to compute the discrete cosinetransform,” IEEE Trans. Acoust., Speech, Signal Processing, Vol.ASSP-32, No. 6, 1243–1245, 1984.

4. Dutt, A., and V. Rokhlin, “Fast Fourier transforms for nonequi-spaced data,” SIAM J. Sci. Comput., Vol. 14, No. 6, 1368–1393,November 1993.

5. Beylkin, G., “On the fast Fourier transform of functions withsingularities,” Appl. Computat. Harmonic Anal., Vol. 2, 363–382,1995.

6. Liu, Q. H., and N. Nguyen, “An accurate algorithm for nonuni-form fast Fourier transforms,” IEEE Microwave Guided WaveLett., Vol. 8, No. 1, 18–20, 1998.

7. Liu, Q. H., and X. Y. Tang, “Iterative algorithm for nonuniforminverse fast Fourier transform (NU-IFFT),” Electronics Letters,Vol. 34, No. 20, 1913–1914, 1998.

8. Nguyen, N., and Q. H. Liu, “The regular Fourier matrices andnonuniform fast Fourier transforms,” SIAM J. Sci. Compt., Vol.21, No. 1, 283–293, 1999.

9. Yee, K. S., ”Numerical solution of initial boundary value prob-lems involving Maxwell’s equations in isotropic media,” IEEETrans. Antennas Propagat., Vol. AP-14, 302–307, 1966.

10. Kunz, K. S., and R. J. Luebbers, Finite Difference Time DomainMethod for Electromagnetics, CRC Press Inc., Florida, 1993.

11. Taflove, A., Computational Electrodynamics: The Finite Differ-ence Time Domain Method, Artech House, Inc., Norwood, MA,1995.

12. Fang, J., “Time domain finite difference computation for Max-well’s equations,” Ph.D. dissertation, University of California,Berkeley, CA, 1989.

13. Hadi, M. F., and M. Piket-May, “A modified FDTD (2,4) schemefor modeling electrically large structures with high phase accu-racy,” 12th Annu. Rev. Progress Appl. Computat. Electromagnet.,Monterey, CA, March 1996.

14. Carpenter, M. H., D. Gottlieb, and S. Abarbanel, “The stabilityof numerical boundary treatments for compact higher-order finitedifference schemes,” Inst. Comput. Applicat. Sci. Eng., Rep. 91-71, 1991.

272 Tian and Liu

15. Young, J. L., D. Gaitonde, and J. J. S. Shang, “Toward the con-struction of a fourth-order difference scheme for transient EMwave simulation: staggered grid approach,” IEEE Trans. Anten-nas Propagat., Vol. 45, No. 11, 1573–1580, 1997.

16. Orszag, S. A., “Comparison of pseudospectral and spectral ap-proximation,” Stud. Appl. Math., Vol. 51, 253–259, 1972.

17. Gottlieb, D., and S. A. Orszag, Numerical Analysis of SpectralMethods, SIAM, Philadelphia, 1977.

18. Fornberg, B., A Practical Guide to Pseudospectral Methods, Cam-bridge University Press, New York, 1996.

19. Berenger, J.-P., “A perfectly matched layer for the absorptionof electromagnetic waves,” J. Computational Physics, Vol. 114,185–200, 1994.

20. Liu, Q. H., “The PSTD algorithm: a time-domain method requir-ing only two cells per wavelength,” Microwave Opt. Tech. Lett.,Vol. 15, No. 3, 158–165, 1997.

21. Liu, Q. H., “Large-scale simulations of electromagnetic andacoustic measurements using the pseudospectral time-domain(PSTD) algorithm,” IEEE Trans. Geosci. Remote Sensing, Vol.37, No. 2, 917–926, 1999.

22. Liu, Q. H., “PML and PSTD algorithm for arbitrary lossy aniso-tropic media,” IEEE Microwave Guided Wave Lett., Vol. 9, No.2, 48–50, 1999.

23. Liu, Q. H., and G.-X. Fan, “A frequency-dependent PSTD al-gorithm for general dispersive media,” IEEE Microwave GuidedWave Lett., Vol. 9, No. 2, 51–53, 1999.

24. Liu, Q. H., and G.-X. Fan, “Simulations of GPR in dispersivemedia using the PSTD algorithm,” IEEE Trans. Geosci. RemoteSensing, Vol. 37, No. 5, 2317–2324, 1999.

25. Liu, Q. H., “The PSTD algorithm for acoustic waves in inhomo-geneous, absorptive media,” IEEE Trans. Ultrason., Ferroelect.,Freq. Contr., Vol. 45, No. 4, 1044–1055, 1998.

26. Yang, B., D. Gottlieb, and J. S. Hesthaven, “On the use of PMLABC’s in spectral time-domain simulations of electromagneticscattering,” Proc. ACES 13th Annual Review of Progress in Ap-plied Computational Electromagnetics, Monterey, 926–933, 1997.

27. Yang, B., D. Gottlieb, and J. S. Hesthaven, “Spectral simulationsof electromagnetic waves scattering,” J. Comp. Phys., Vol. 134,216–230, 1997.

28. Sarkar, T. K., “Application of conjugate gradient method to elec-tromagnetics and signal analysis,” PIER 5, Progress in Electro-magnetics Research, Elsevier, New York, 1991.


29. Catedra, M. F., and R. P. Torres, J. Basterrechea, and E. Gago,The CG-FFT Method: Application of Signal Processing Tech-niques to Electromagnetics, Boston: Artech House, 1995.

30. Chew, W. C., and W. H. Weedon, “A 3D perfectly matchedmedium from modified Maxwell’s equations with stretched co-ordinates,” Microwave Opt. Tech. Lett., Vol. 7, 599–604, 1994.

31. Liu, Q. H., “An FDTD algorithm with perfectly matched layersfor conductive media,” Micro. Opt. Tech. Lett., Vol. 14, No. 2,134–137, 1997.

NONUNIFORM FAST COSINE TRANSFORM AND CHEBYSHEV …

Documents