This paper presents four fan-shaped antennas: U.S.-FAN, CROSS-FAN, CROSS-FAN-W, and CROSS-FAN-S. Each of these antennas stands upright above a ground plane, and has edges expressed by an exponential function and a circle function. The four antennas are investigated using frequencies from 1.5 GHz to 11 GHz. The CROSS-FAN is found to have a lower VSWR over a wide frequency band compared to the U.S.-FAN. The CROSS-FAN-W and CROSS-FAN-S are modified versions of the CROSS-FAN, each designed to have a stop band (a high VSWR frequency range) for interference cancellation. The stop band for the CROSS-FAN-W is controlled by a wire (total length 4Lwire) that connects the fan-shaped elements. The center frequency of the stop band fstop is close to the frequency corresponding to a wire segment length Lwire of half the wavelength. It is also found that the stop band in the CROSS-FAN-S can be controlled by four slots, one cut into each of the fan-shaped elements. The center frequency of the stop band fstop is close to the frequency corresponding to a slot length Lslot of one-quarter of the wavelength. Experimental work is performed to confirm the theoretical results, using the CROSS-FAN-S.
 Recent studies have revealed that an elliptical plate (or elliptical ring) antenna exhibits a wideband VSWR characteristic [Hattori et al., 2005; Nakano et al., 2005b; Baiju et al., 2006; Bahadori and Rahmat-Samii, 2006]. This antenna is composed of an elliptical radiating element (or elliptical radiating ring) and a ground plane, both made of a thin conducting plane and lying in the same plane (having a card-type structure). It is emphasized that the card-type structure facilitates the use of the antenna in information technology (IT) mobile devices, such as laptop computers and mobile phones. In addition to the card-type antennas, wideband antennas, where the radiating element stands upright above a ground plane, have been proposed [Taniguchi and Kobayashi, 2003; Suh et al., 2004; Lau et al., 2005; Nakano et al., 2006a]. These upright wideband antennas are suitable for use as a base station antenna installed on a wall or ceiling.
 A wideband antenna often encounters the requirement that it should not transmit signals over a specific frequency range (for example, from 5.15 GHz to 5.825 GHz for wireless LAN systems), not to cause any interference on a nearby wireless device. This is achieved, provided that the antenna has a high VSWR over the specific frequency range. However, there has been little work published on such anti-interference design; we are still at the stage where appropriate antennas that provide wideband characteristics and facilitate design for anti-interference behavior should be developed [Lee et al., 2005; Baiju et al., 2006; Bahadori and Rahmat-Samii, 2007].
 This paper presents wideband antennas having fan-shaped radiating elements. Analysis reveals the wideband characteristics and anti-interference properties of these antennas.
 First, the concept used for realizing a wideband characteristic is briefly summarized on the basis of a planar bi-triangle antenna having infinitely extended arms [Carrel, 1958]. Subsequently, a truncated triangle standing upright above a ground plane is presented and the VSWR is investigated. It is found that the VSWR can be decreased by changing the linear edges of the triangle to curved edges. This radiating element having the curved edges is designated as an upright single fan-shaped (U.S.-FAN) element.
 Second, the number of fan-shaped elements is increased from one to two, in order to further improve the VSWR of the U.S.-FAN. These two fan-shaped elements intersect at right angles and hence this antenna is called a CROSS-FAN. The VSWR of the CROSS-FAN is compared with that of the U.S.-FAN.
 Anti-interference properties in this paper are realized by designing the antenna such that the VSWR is as high as possible in the interference frequency range. Third, based on this idea, a conducting wire of total length 4Lwire is attached to the CROSS-FAN, which is named the CROSS-FAN-W. The effect of the wire on the VSWR is investigated with the aim of realizing a frequency band with high VSWR (stop band). The relationship between the center frequency of the stop band fstop and wire length 4Lwire is discussed. Note that complete elimination of the interference can be achieved with the help of external filter circuits.
 Finally, a CROSS-FAN with slots is analyzed. This antenna, designated as the CROSS-FAN-S, is distinguished from the aforementioned CROSS-FAN-W in that, rather than adding a conducting wire, slots are cut in the fan-shaped elements. How the slots generate a stop band in the frequency response of the VSWR is discussed. The current flowing on the fan-shaped element at fstop is compared with the currents at frequencies outside the stop band. Other radiation characteristics of the CROSS-FAN-S are also presented and discussed.
 Note that the antenna characteristics of the U.S.-FAN, CROSS-FAN, CROSS-FAN-W, and CROSS-FAN-S are calculated theoretically on the basis of the electric field E and magnetic field H within an analysis space including the antenna, where these E and H are obtained using the finite difference time domain (FDTD) method [Yee, 1966; Taflove, 1995]. Experimental work is performed to confirm the theoretical results, using the CROSS-FAN-S.
2. Wideband Antennas and U.S.-FAN Antenna
 The antennas presented in this paper are recognized as derivatives of the triangle antenna shown in Figure 1a. The two arms of this original triangle can be reduced to a single arm using a ground plane of infinite extent, as shown in Figure 1b, where the image arm exists under the ground plane, thereby reproducing the original triangle shown in Figure 1a. It is well known that the input impedance of the triangle shown in Figure 1b is frequency-independent if the arm is of infinite extent [Carrel, 1958]. However, in practice, the arm above the ground plane in Figure 1b cannot be infinite.
Figure 1c shows a modified version of the triangle shown in Figure 1b, where the top of the triangle is truncated using a circle function of radius rc. The ground plane has a finite diameter of DGP. At this stage, the triangle loses its inherent frequency-independent characteristic for the input impedance. However, it is expected that the truncated triangle will have a relatively wideband input impedance, because the antenna structure partially retains the original structure of Figure 1a.
 The structure of the truncated triangle (TT) in Figure 1c is symmetric with respect to the y-z plane as well as the x-z plane, where each plane is regarded as a magnetic wall. The electric fields E(r, t) and the magnetic fields H(r, t) at symmetric points with respect to the y-z plane are written as
Equations (1), (5), and (6) are called odd symmetry relationships, while equations (2), (3), and (4) are called even symmetry relationships. Note that the electric and magnetic fields at symmetric points with respect to the x-z plane have the following relationships: components Ey, Hx, and Hz have odd symmetry relationships; components Ex, Ez, and Hy have even symmetry relationships (note that these symmetry relationships with respect to both the x-z and y-z planes hold true for all the antennas that appear later in this paper).
 We use the FDTD method [Yee, 1966; Taflove, 1995] to obtain E and H for the input impedance calculation. Due to the abovementioned symmetry relationships, a practical FDTD calculation can be preformed using one-fourth of the entire FDTD analysis space of NxΔx × NyΔy × NzΔz, where Δx, Δy, and Δz are the side lengths of Yee's cell, and Nx, Ny, and Nz are integers representing the number of cells in the x, y, and z directions, respectively. This leads to a reduction in the computation time.
 The frequency domain input current Iin(ω) (used for calculating the input impedance Zin = Rin + jXin, from which the VSWR is calculated) is obtained by integrating the frequency domain H(r, ω), i.e., Fourier-transformation of the magnetic field H(r, t), around the conductor at the antenna input. Figure 2 shows the VSWR for the TT shown in Figure 1c, where the configuration parameters are chosen to be DGP = 320Δ for the diameter of the ground plane, (xP, yP, zP) = (100Δ, 0, 176Δ) for edge point P, and (xQ, yQ, zQ) = (0, 0, 1Δ) for bottom point Q, where Δ = Δx = Δy = Δz = λ4 / 400, with λ4 being the wavelength at a frequency of 4 GHz. The radius of the circle is rc = 116Δ. It is found that the VSWR (relative to a 50-ohm impedance) is relatively constant at frequencies above 4 GHz (note that the diameter of the ground plane DGP, edge point zP, and width 2xP at 4 GHz are 0.8 wavelength, 0.44 wavelength, and 0.5 wavelength, respectively). However, its value is too high to match to a 50-ohm transmission line. To solve this issue, we transform the linear edge QP of the TT into a curved edge defined by an exponential function, as shown in inset of Figure 3,
and x0 is an arbitrary constant. The value of x0 is optimized using a frequency of 4 GHz such that the VSWR is less than 2; we simply iterate calculating the VSWR, while gradually increasing x0 from 0.01 mm. It is found that the antenna exhibits small VSWRs (greater than approximately 1.1 and less than 1.2) between x0 = 0.1 mm and 1.2 mm. Based on this fact, we select x0 = 1 mm as the optimized value. The curve in Figure 3 shows the VSWR frequency response for the optimized x0. It is revealed that the antenna realizes a VSWR of less than 2 at frequencies above 3.1 GHz.
 From now on, we shall designate the antenna shown by inset of Figure 3 as an upright single fan-shaped (U.S.-FAN) antenna to distinguish it from the card-type fan-shaped antenna found in the work of Nakano et al. [2006b], where a fan-shaped element (sandwiched between thin dielectrics) and a ground plane, both made of a film conductor, are in the same plane, unlike the U.S.-FAN.
3. Cross Fan-Shaped Antenna (CROSS-FAN)
 We increase the number of fan-shaped elements NFAN from one to two (note that when NFAN is infinite, the structure constitutes a conducting body of revolution and is symmetric with respect to the z axis). These two fan-shaped elements intersect at right angles, as shown by inset in Figure 4. This structure is designated as a cross fan-shaped (CROSS-FAN) antenna. The curve in Figure 4 shows the VSWR of the CROSS-FAN; the VSWR is less than 2 at frequencies above 2.1 GHz. It can be said that the CROSS-FAN has a wideband VSWR characteristic.
 We calculate the radiation pattern using the equivalence principle [Harrington, 1961], adopting spherical coordinates (r, θ, ϕ) with (, , ) as unit vectors. For this, the electric current density Je(r′, t) = × H(r′, t) and magnetic current density Jm(r′, t) = E(r′, t) × on a surface enclosing the antenna are calculated, where E(r′, t) and H(r′, t) are, respectively, the time domain electric and magnetic fields obtained by the FDTD method and is the unit vector directed outward from the closed surface. Note that r′ is the position vector from the coordinate origin to a point on the closed surface. After Fourier-transforming these time domain Je(r′, t) and Jm(r′, t) functions, we obtain the frequency domain Je(r′, ω) and Jm(r′, ω) functions and then integrate them over the closed surface to obtain vectors Ne and Nm:
where k0 is the phase constant (k02 = ω2μ0ɛ0).
 Using Ni(ω) (i = e, m), the θ and ϕ components of the radiation field are calculated as
where Z0 is the free-space intrinsic impedance (Z0 ≈ 120π Ω).
 Representative radiation patterns for the CROSS-FAN are compared with those for the U.S.-FAN in Figure 5. These are normalized with respect to their maximum values. It is found that the radiation patterns of both the CROSS-FAN and U.S.-FAN are almost omni-directional around the z axis (see the radiation patterns for (θ, ϕ) = (30°, variable)). The cross polarization component at high frequencies for the CROSS-FAN is smaller than that for the U.S.-FAN, as desired (see Figure 5b). Note that the CROSS-FAN and U.S.-FAN show a monopole-like pattern in the x-z plane, where the beam is tilted upward above the ground plane. This is because the ground plane is finite. If the ground plane is of infinite extent, the beam is in the horizontal direction (θmax = 90°). The gain for the CROSS-FAN, which is related to the radiation pattern, is summarized as follows: the gain (relative to an isotropic antenna) observed in the fixed direction θmax = 60° is between −0.7 dBi and +3.2 dBi in the frequency range of 3.1 GHz to 10.6 GHz.
4. CROSS-FAN Surrounded by a Wire (CROSS-FAN-W)
 The insets (a), (b), and (c) in Figure 6 show CROSS-FAN antennas; each CROSS-FAN is surrounded by a conducting wire. This wire, designated as the control wire, is electrically connected (soldered) to an edge point of each of the four fan-shaped elements. The control wire has a total length of 4Lwire at a height z = zwire. This antenna is designated as the CROSS-FAN-W, which is distinguished from the CROSS-FAN-S to be discussed later in section 5.
 The curve in Figure 6 shows the VSWR for the structure shown by inset (a), where the total length 4Lwire is chosen to be 4 × 37.5 mm, and the height is zwire = zP = 33 mm. The control wire varies the wideband VSWR characteristic for the CROSS-FAN, generating a stop band. The center frequency of the stop band fstop is closely related to the segment length; Lwire is approximately one-half of the wavelength at fstop.
 The structures shown by insets (b) and (c) in Figure 6 are modified versions of the structure shown by inset (a). It is noted that the antenna mechanisms for (b) and (c) are the same as that for (a). The center frequency fstop for (b) is higher than that for (a) due to the fact that the control wire is shorter; fstop for (c) is even higher because the control wire is shorter than that for (b) (not shown in Figure 6).
 Each current along the control wire segment (of length Lwire) exhibits a sinusoidal distribution at fstop; the maximum value of the current appears at the end points of the wire segment and the minimum value (zero) appears at the middle point of the wire segment. It follows that the current along the control wire (of total length 4Lwire) has four maxima and four minima (zeros) at fstop. In other words, a high VSWR is generated when the length of the control wire (4Lwire) is approximately two wavelengths at fstop.
Figure 7 shows the radiation patterns for the CROSS-FAN-W shown by inset (a) in Figure 6; those are observed at the frequencies outside the stop band. It can be said that the control wire does not affect the radiation pattern; that is, the inherent radiation patterns obtained for the CROSS-FAN in the absence of the control wire are retained. This is attributed to the fact that, at frequencies outside the stop band (at nonresonant frequencies), the current for the CROSS-FAN-W does not flow along the control wire, but flows along the edges of the fan-shaped elements, as in the CROSS-FAN.
5. CROSS-FAN With Slots (CROSS-FAN-S)
 The inset of Figure 8 shows a CROSS-FAN, where four slots are cut in the fan-shaped elements. This antenna is designated as the CROSS-FAN-S, which is distinguished from the CROSS-FAN-W in section 4. The four slots of the CROSS-FAN-S have the same length Lslot. Note that the bottom end of each slot is open. The purpose of this section is to investigate the effects of the slot length on the antenna characteristics. For this, we again use the configuration parameters for the CROSS-FAN discussed in section 3.
 Each curve in Figure 8 presents the VSWR of the CROSS-FAN-S as a function of frequency, where three values are used for the slot length Lslot. It is revealed that, as in the case of the CROSS-FAN-W, a stop band is generated in the frequency response of the VSWR for each Lslot. As Lslot is increased, the center frequency of the stop band fstop decreases. The length Lslot is found to be approximately one-quarter of the wavelength at fstop. The validity of the theoretical results is checked by experimental work; see the white dots for Lslot ≈ 11.2 mm. Note that when each of the four bent slots shown in inset of Figure 8 is replaced with a straight slot, the width of the stop band increases (for example, from 4.3% to 7.8% for a VSWR = 20 criterion, where Lslot ≈ 11.2 mm). This indicates the fact that the shape of the slot controls the width of the stop band.
Figure 9 shows the current (absolute value of current density Js) on the fan-shaped element standing upright above the x axis (x-element), where Lslot ≈ 11.2 mm. The current density is calculated as Js = × Hs = Hs, z − Hs, x, where Hs is the magnetic field on the fan-shaped element surface, and , and are the unit vectors for the rectangular coordinates (x, y, z). Unfortunately, Hs is not obtained by the FDTD calculation, and hence Hs is extrapolated using two magnetic fields above the fan-shaped element surface.
 The extrapolation for the x-component of the magnetic field at y near the surface of the x-element is performed using the following equation;
where a and b are unknown constants to be determined. Hx at y = 0.5Δy and Hx at y = 1.5Δy are given as
Therefore, the magnetic field Hx at the surface (y = 0), Hs,x, is written as
Similarly, the z-component of the magnetic field at y = 0, Hs, z, is written as
The current in Figure 9a is observed at the center frequency of the stop band fstop (=6.88 GHz), and the currents in Figures 9b and 9c are observed outside the stop band: fL = 3.1 GHz in Figure 9b and fH = 10.6 GHz in Figure 9c. It can be clearly seen that the current is concentrated around the slot at fstop (in other words, the current resonates at fstop), while the currents at frequencies fL and fH (nonresonant frequencies) flow along the edges of the fan-shaped element. Note that the current on the fan-shaped element standing upright above the y axis (y-element) is the same as that shown in Figure 9, due to the structural symmetry of the antenna with respect to the antenna axis (z axis).
 The abovementioned current behavior leads to the fact that the radiation patterns for the CROSS-FAN-S at the frequencies outside the stop band are almost the same as those for the CROSS-FAN. The validity of the theoretical radiation patterns (each normalized to the maximum value) is checked by experimental work, as shown in Figure 10; the experimental results (black and white dots) are in good agreement with the theoretical results. It can be seen that the CROSS-FAN-S shows an almost omni-directional radiation pattern around the z axis at θ = 30°. Additional calculations reveal that the radiation patterns around the z axis at 30° ≤ θ ≤ 60° are also almost omni-directional (although not illustrated).
 Finally, we calculate the gain in the direction of (θ = 60°, ϕ = 0°) for the CROSS-FAN-S. It is found that a gain between −0.7 dBi and +3.2 dBi is obtained over a frequency range of 3.1 GHz to 10.6 GHz, excluding the stop band. It is worth noting that the gains for the CROSS-FAN, CROSS-FAN-S, and CROSS-FAN-W at nonresonant frequencies behave similarly, resulting from the fact that these three antennas have similar current distributions.
 This paper reveals the characteristics of four fan-shaped (FAN) antennas, including the input impedance (VSWR), radiation pattern, and gain. The number of fan-shaped elements classifies these antennas into two categories: U.S.-FAN and CROSS-FAN. The former (U.S.-FAN) is composed of a single FAN element standing upright on a ground plane, and the latter (CROSS-FAN) is a modified version of the U.S.-FAN and composed of two FAN elements intersecting each other at right angles above a ground plane. The CROSS-FAN exhibits lower VSWR values than the U.S.-FAN over a wide frequency range.
 The CROSS-FAN is further investigated for realizing high VSWR values in a specified frequency range (stop band). First, a thin conducting wire of length 4Lwire is used with the CROSS-FAN. The wire is parallel to the ground plane and electrically connected to the four edges of the FAN elements. This antenna is designated as the CROSS-FAN-W. It is found that the wire contributes to the generation of a stop band; the center frequency of the stop band is controlled by the wire length 4Lwire and located at a frequency corresponding to a wire length 4Lwire of approximately two wavelengths (i.e., Lwire is one-half of the wavelength). The radiation patterns for the CROSS-FAN-W at frequencies outside the stop band are very similar to those for the CROSS-FAN. In other words, the radiation characteristics, including the radiation pattern and gain, are not affected by the conducting wire when the operating frequency is outside the stop band.
 Subsequently, a CROSS-FAN-S antenna, where four slots are cut into the fan-shaped elements, is analyzed. It is found that the center frequency of the stop band is located at a frequency corresponding to a slot length Lslot of approximately one-fourth of the wavelength. It is also found that the radiation pattern and gain for the CROSS-FAN-S at the frequencies outside the stop band are close to those for the CROSS-FAN, resulting from the fact that both antennas have similar current distributions at frequencies outside the stop band.
 The authors thank V. Shkawrytko for his assistance in the preparation of this manuscript.