Data transmission system

To determine signal distortions along a DWG a low-complexity digital data transmission system as shown in Fig. 1 is assumed. In this communication system a data source generates an infinite pulse sequence

(1)$$T\!\sum\limits_{i = -\infty}^{\infty}\!d( i) \cdot \delta_0( t-iT) .$$

where d(i) represents the actual data to be transmitted. For example, d(i) can be a sequence of discrete voltage values of a sampled analog signal or output values of a digital data processing system. These data points are weighted by a Dirac impulse δ ₀. The time between each impulse is the sampling period T. Since the impulse sequence (1) would have an infinite bandwidth, an ideal low-pass filter

(2)$$ g( \omega ) = {\rm rect}\left({\omega\over \omega_{\!N}}\right) = \left \{ {\matrix {1&\text{for}&\vert \omega\vert < \omega_{\!N}\cr 0&\text{else}} $$

is used as pulse shaping filter to limit the signal bandwidth of the transmitted signal to 2ω_N. The cut-off frequency ω_N of the filter fulfills the Nyquist–Shannon sampling theorem f _N = ω_N/2π = 1/2T. After the pulse shaping filter, the band-limited signal is transmitted via DWG. Additionally, the signal is superimposed by white Gaussian distributed noise n(t). To fulfill the matched filter criterion, the distorted signal is filtered at the receiver by an ideal low-pass filter (receive filter) given by (2). The filtered receive signal r(t) is then sampled with the sampling period T. This results in a disturbed signal sequence at the receiver, where each received symbol r(iT) can be described by

(3)$$ \eqalign{\matrix r ( iT ) & = T\!\displaystyle\sum_{k = -\infty}^{\infty}\! d ( k ) \cdot h ( iT-k T ) + n ( iT ) \cr & = \underbrace{ T d ( i ) \cdot h ( iT ) }_{r_0 ( iT )} \cr & + \underbrace{T\!\displaystyle\sum_{\substack{k = -\infty \cr \vskip0.6pc \hskip-1.55pc k \neq 0}}^{\infty}\!d ( k ) \cdot h ( iT-k T ) }_{\Delta r ( iT ) } + \, n ( iT )}$$

where iT represents the i-th sample point for a sampling period of T = π/ω_N. The pulse shaping filter, the DWG as well as the receive filter are summarized as the channel impulse response h(t). It can be seen that the received symbol at time iT is not only the transmitted symbol d(i) disturbed by the channel impulse response (hereinafter called r ₀(iT)), it is also superimposed by the time-shifted channel responses of all symbols d(k) that were transmitted before and after the actual symbol d(i) (hereinafter called Δr(iT)). This effect is well known as ISI.

Fig. 1. Schematic representation of a low-complexity digital data transmission system using a DWG as a channel.

In order to evaluate the signal distortions by the channel, the SNR and the ISI of the received symbols r(iT) are determined. For uncorrelated random data d(i) the SNR can be expressed by

(4)$${\rm SNR} = {{\rm E} \{ \left\vert r_0( iT) \right\vert ^2 \} \over \sigma_N^2} = {\sigma_D^2\over \sigma_N^2} \left\vert h( 0) \right\vert ^2$$

where $\sigma _D^2$ and $\sigma _N^2$ are the variances of the transmitted data sequence d(i) and noise n(iT). The ISI can be quantified by

(5)$${\rm ISI} = {{\rm E} \{ \left\vert r_0( iT) \right\vert ^2 \} \over {\rm E} \{ \left\vert \Delta r( iT) \right\vert ^2 \} } = {\left\vert h( 0) \right\vert ^2\over \sum\limits_{\substack{k = -\infty \cr \vskip0.6pc \hskip-1.55pc k \neq 0}}^{\infty} \left\vert h( k T) \right\vert ^2 }$$

Equations (4) and (5) depend solely on the impulse response h(t) of the transmission channel including filters and DWG. Since the filter characteristics are already described by (2), a mathematical description of the DWG channel is needed to calculate the signal distortions caused by the DWG. Due to the large variety of possible diameters and materials for circular DWGs in the mm-wave frequency band, a general description of the transmission behavior of circular DWGs is presented in Section “DWG channel.” Based on this description, the channel impulse response h(t) is derived in Section “Signal distortions” and the resulting signal distortions are determined using SNR and ISI.

DWG channel

The frequency response of a DWG in the bandpass domain can be described by

(6)$$\underline{H}_{B P}( j\omega) = e^{-\gamma( \omega) l}$$

where γ(ω) is the frequency-dependent propagation constant

(7)$$\gamma( \omega) = \alpha( \omega) + j\beta( \omega) \qquad \alpha,\; \beta \in {\opf R},\; $$

l the length of the waveguide and j the imaginary unit. The attenuation per unit length of an electromagnetic wave propagating along the DWG is characterized by the attenuation constant α whereas the phase change per unit length is described by the phase constant β. In practice, both constants are frequency-dependent. These frequency dependencies cause signal distortions while transmitting data along such a waveguide. To describe the frequency dependency mathematically, in this paper, real and imaginary parts of the propagation constant are approximated by a Taylor series. The Taylor series is calculated at the center frequency ω₀ of the transmitted signal:

(8)$$ \eqalign{ \,\tilde{\gamma}( \omega) & = \tilde{\alpha}( \omega) + j\tilde{\beta}( \omega) \qquad \qquad {\rm with}\cr \tilde{\alpha}( \omega) & = \alpha_0 + \alpha_1 ( \omega-\omega_0) \cr \tilde{\beta}( \omega) & = \beta_0 + \beta_1 ( \omega-\omega_0) + \beta_2 ( \omega-\omega_0) ^2} $$

and

(9)$$\gamma_0 = \alpha_0 + j\beta_0,\; \quad \gamma_1 = \alpha_1 + j\beta_1,\; \quad \gamma_2 = j\beta_2$$

For the attenuation constant α, the Taylor coefficients are

(10)$$ \alpha_0 = \alpha( \omega_0) ,\; \quad \alpha_1 = \left.{d\alpha\over d\omega} \right\vert _{\omega = \omega_0} $$

The coefficient α₀ corresponds to the attenuation per unit length at the center frequency ω₀. The coefficient α₁ represents the first-order frequency dependency of the attenuation in the frequency range around the center frequency ω₀. For the phase constant β, the Taylor coefficients are

(11)$$ \beta_0 = \beta( \omega_0) ,\; \quad \beta_1 = \left.{d\beta\over d\omega} \right\vert _{\omega = \omega_0},\; \quad \beta_2 = {1\over 2}\left.{d^2\beta\over d\omega^2} \right\vert _{\omega = \omega_0} $$

where β₀ corresponds to the phase per unit length, β₁ characterizes the group delay τ_g(ω₀) per unit length, and β₂ is a measure for the waveguide dispersion at the center frequency. The approximation of the attenuation constant and the phase constant using a Taylor series of first and second order has been found to be highly accurate for relative bandwidths up to $5$–10%. For most DWGs, a larger relative bandwidth (up to $\approx \!30\percnt$) would result in a frequency dependency of α(ω) and β(ω) that is slightly higher than the frequency dependency calculated by the Taylor approximation in this paper.

The transmission behavior of a DWG and consequently its propagation constant γ(ω) depends on the geometry and materials used as well as the operating frequency. In case of circular DWGs, the diameter D of the core in relation to the free-space wavelength λ₀ significantly influences the transmission behavior. To illustrate this effect, the electric field E of the fundamental mode HE₁₁ in a circular DWG is shown for different wavelengths related to its diameter in Fig. 2. The figure shows the cross section of a circular DWG consisting of a core with a diameter D and a relative permittivity ɛ_r,1 surrounded by a cladding with the relative permittivity ɛ_r,2. At low frequencies, meaning large wavelengths in relation to the DWG diameter (D/λ₀ ≪ 1), the field of the fundamental mode HE₁₁ spreads widely into the cladding material. With increasing frequency, the field is further concentrated within the core of the DWG until the mode propagates almost completely in the core (D/λ₀ ≫ 1). In the mm-wave frequency range, DWGs are usually operated with a diameter smaller or approximately equal to the free space wavelength (D/λ₀ ≈ 1). Thus, the fundamental mode propagates in core and cladding. The ratio of the field in the core as well as in the cladding significantly influences the attenuation and phase behavior of a signal transmitted along a DWG. Since this ratio depends on the frequency, the attenuation and phase behavior can be strongly frequency-dependent, which in turn leads to signal distortions. In order to enable a general description of the propagation characteristics of various DWGs at different frequencies, Snitzer introduced the normalized frequency V and the material ratio Δ for circular DWGs [Reference Snitzer18]:

(12)$$V = {\omega\over c_0}{D\over 2}\sqrt{\varepsilon_{r, 1}-\varepsilon_{r, 2}}$$

(13)$$\Delta = {\varepsilon_{r, 1}-\varepsilon_{r, 2}\over \varepsilon_{r, 2}}$$

This allows a comparison of DWGs with different diameters and materials at different frequencies, despite the large number of possible design variations. Furthermore, the normalized frequency V allows a clear classification of single-mode and multi-mode waveguides. Circular waveguides with a combination of core diameter, core and cladding material that fulfill V < 2.405 for a certain frequency are single-mode DWGs. Normalized frequencies of V ≥ 2.405 indicate multi-mode DWGs. In the following, the normalized frequency V and the material ratio Δ are used to derive a general description of the Taylor coefficients α₀, α₁ and β₀, β₁, and β₂. This allows the reader to calculate these coefficients for a particular application without the need for FEM simulation tools or other complex numerical calculations.

Fig. 2. Illustration of the E-field distribution E of the fundamental mode HE₁₁ in a DWG at different wavelengths λ₀ related to its diameter D.

Attenuation constant

The attenuation constant α(ω) of DWGs depends not only on the materials used for core and cladding, but also on the field distribution in both regions. For circular DWGs, the attenuation constant can be calculated by

(14)$$\alpha( \omega) = {\pi\over \lambda_0} \left(\sqrt{\varepsilon_{r, 1}} R_1 \tan\delta_1 + \sqrt{\varepsilon_{r, 2}} R_2 \tan\delta_2 \right)$$

where tanδ₁, tanδ₂ are the material loss factors and R ₁, R ₂ are the geometric loss factors of core and cladding, respectively [Reference Yeh and Shimabukuro19] (pp. 339), with

(15)$$R_i = { \displaystyle\int_{A_i} {\bf E}_i \cdot {\bf E}_i^\ast \, dA \over \displaystyle\sqrt{\frac{\mu_0}{\varepsilon_0\varepsilon_{r, i}}} \left[\displaystyle\int_{A_1} {\bf E}_1 \! \times \! {\bf H}_1^\ast \, dA + \displaystyle\int_{A_2} {\bf E}_2 \! \times \! {\bf H}_2^\ast \, dA \right]} .$$

The geometric loss factors R ₁ and R ₂ represent the ratio between the power dissipation due to dielectric losses and the time-average power flow in core and cladding, respectively. E₁, H₁ and E₂, H₂ are the electric and magnetic vector fields of the fundamental mode HE₁₁ in the core area A ₁ and in the cladding area A ₂, respectively. Both geometric loss factors R ₁, R ₂ can by normalized with respect to V and Δ. Their respective values have been analytically calculated from the numerical solution of the dispersion relation for circular DWGs as described in [Reference Yeh and Shimabukuro19] (pp. 137). The obtained values for R ₁ and R ₂ are shown in Fig. 3(a) in dependency of the normalized frequency V for typical material ratios of Δ in the mm-frequency range. It can be seen that with increasing frequency the proportion of the wave in the core region increases (R ₁ increases, R ₂ decreases). Since the cladding material has a lower permittivity than the core material, and thus usually a lower loss factor, it seems desirable to design DWGs with the lowest possible values of R ₁ and the highest possible values of R ₂. However, for low values of R ₁ waves are weakly guided by the DWG which leads to unwanted radiation or mode coupling in case of discontinuities like bends. For this reason, higher values of R ₁ ($V \approx 2$–3) in combination with low-loss core materials are preferred in mm-wave applications.

Fig. 3. Geometric loss factors R ₁, R ₂ (a) and their respective derivatives dR ₁/dV, dR ₂/dV (b) as a function of the normalized frequency V for different material ratios Δ.

From (10) and (14), the Taylor coefficient α₀ at a certain frequency $V_0 = V( \omega _0) = ( {\omega _0}/{c_0}) ( {D}/{2}) \sqrt {\varepsilon _{r, 1} - \varepsilon _{r, 2}}$ and therefore the attenuation of a DWG ($20\log ( e^{\alpha _0 l})$ in dB) can be calculated by

(16)$$\alpha_0 = {\omega_0\over 2 c_0} \sqrt{\varepsilon_{r, 1}} \left[R_1( V_0) \tan\delta_1 + {R_2( V_0) \over \sqrt{1 + \Delta}} \tan\delta_2 \right]$$

Following the same procedure the Taylor coefficient α₁ and therefore the change of the attenuation over frequency can by calculated by

(17)$$ \alpha_1 = {\sqrt{\varepsilon_{r, 1}}\over 2c_0} \left[\tan\delta_1 \left(V_0 \left. {dR_1\over dV} \right\vert _{V = V_0} + \,R_1( V_0) \right)\right. + \left. {\tan\delta_2\over \sqrt{\Delta + 1}} \left(V_0 \left. {dR_2\over dV} \right\vert _{V = V_0} + \, R_2( V_0) \right)\right] $$

It can be seen that α₁ depends not only on the geometric loss factors R ₁ and R ₂, but also on their derivatives dR ₁/dV and dR ₂/dV. The derivatives for typical material ratios Δ are shown in Fig. 3(b) as a function of the normalized frequency V. Due to the negative values of the derivative of R ₂, it seems possible to compensate for the frequency-dependent attenuation by appropriate material selection. However, such a minimization of α₁ can only be achieved with a loss factor of the cladding material (tan δ₂) significantly higher than the loss factor of the core (tan δ₁) for typical material ratios Δ in the mm-wave frequency range. This would lead to a considerably higher insertion loss of the DWG. For this reason, α₁ is significantly influenced by the permittivity and loss factor of the DWG core in practice.

With the help of (16) and (17) as well as Figs 3(a) and 3(b), the Taylor coefficients α₀ and α₁ can be directly calculated for a wide range of DWGs in the mm-wave frequency range. In Table 1 these coefficients are given exemplarily for single-mode DWGs (V ₀ = 2.4) with air cladding (ɛ_r,2 = 1, tan δ₂ = 0) and cores made of Teflon (PTFE), polyethylene (PE), and polystyrene (PS). Comparing the PTFE and the PE waveguide, it can be seen that the higher core permittivity of PE not only increases the transmission losses (e.g. from $3.1$ to 3.5 dB/m at 140 GHz), it also increases their frequency dependency. In combination with a high loss tangent (see PS in Table 1), the frequency dependency is highly increased. Since it is still unclear how this frequency dependency affects a signal transmitted via DWG, the coefficients α₀ and α₁ are used in Section “Signal distortions” to calculate the signal distortions caused by the frequency-dependent attenuation of circular DWGs.

Table 1. Taylor coefficients α₀ and α₁ as well as β₀, β₁, and β₂ for some exemplarily chosen DWGs with air cladding (ɛ_r,2 = 1, tanδ₂ = 0)

Phase constant

Similar to the attenuation constant α(ω) the phase constant β(ω) depends on the materials and field distributions in core and cladding. The phase constant is given by

(18)$$\beta( \omega) = {2\pi\over \lambda_g} = {\omega\over c_0}\sqrt{\varepsilon_{r, ef\!f}( \omega) }$$

where λ_g is the guided wavelength and ɛ_r,eff is the effective relative permittivity with ɛ_r,2 ≤ ɛ_r,eff ≤ ɛ_r,1. The effective relative permittivity is normalized by

(19)$$B( V) = {\varepsilon_{r, ef\!f}( V) - \varepsilon_{r, 2}\over \varepsilon_{r, 1} - \varepsilon_{r, 2}}$$

which is referred as the normalized phase constant that ranges between values of 0 and 1 [Reference Yeh and Shimabukuro19] (p. 61). The normalized phase constant B can be used as a measure of the field concentration in core and cladding. It is shown as a function of the normalized frequency V for different values of Δ in Fig. 4(a). From (11), (18), and (19), the Taylor coefficient β₀ at a certain normalized frequency V ₀ and therefore the phase change per unit length at the center frequency ω₀ can be calculated by

(20)$$\beta_0 = {2V_0\over D}\sqrt{B( V_0) + {1\over \Delta}}$$

Following the same procedure the Taylor coefficient β₁, namely the group delay per unit length, can by calculated by

$$\beta_1 = {\sqrt{\varepsilon_{r, 1} - \varepsilon_{r, 2}} \over 2c_0} \cdot \left. D{d\beta\over dV} \right\vert _{V = V_0} \qquad {\rm with}$$

(21)$${d\over dV}\beta( V) = {2\left(\frac{1}{\Delta} + B( V) \right) + V\frac{d}{dV}B( V) \over D\sqrt{B( V) + \frac{1}{\Delta}}}$$

By multiplying the derivative of β with diameter D, a normalized value for the first-order frequency dependency of the phase constant is obtained which only depends on V and Δ. This value is shown as a function of V for different values of Δ in Fig. 4(b). From (21) and Fig. 4(b) it can be seen that β₁ and therefore the group delay τ_g varies strongly over frequency. In case of a broadband signal that is transmitted along such a waveguide, the different frequency components of this signal reach the receiver at different times. This effect is known as waveguide dispersion. To reduce the waveguide dispersion, the group delay should therefore be flat over frequency. According to Fig. 4(b) this is only possible for certain normalized frequencies in a range of 2 < V < 3 or V → ∞.

Fig. 4. Normalized phase constant B (a) as well as the first and second derivatives D(dβ/dV) (b) and D(d ²β/dV ²) (c) as a function of the normalized frequency V for different material ratios Δ.

The waveguide dispersion per unit length of a DWG is described by the Taylor coefficient

$$\beta_2 = {1\over 2}{D\left(\varepsilon_{r, 1} - \varepsilon_{r, 2} \right)\over 4c_0^2} \cdot \left. D{d^2\beta\over dV^2} \right\vert _{V = V_0} \qquad {\rm with}$$

(22)$${d^2\over dV^2}\beta( V) = {V\frac{d^2}{dV^2}B( V) + 2 \frac{d}{dV} B( V) \over D\sqrt{B( V) + \frac{1}{\Delta}}} - { V\left(\frac{d}{dV}B( V) \right)^2 \over 2D\left(B( V) + \frac{1}{\Delta}\right)^{{3}/{2}} }{}$$

that has been derived from (11), (18), and (19). Multiplying the second derivative of β with diameter D yields a normalized value for the second order frequency dependency which only depends on V and Δ (shown in Fig. 4(c)). By using (20)–(22) as well as Fig. 4(a)–4(c), the Taylor coefficients β₀, β₁, and β₂ can now be directly calculated for a wide range of DWGs in the mm-wave frequency range. From (22) and Fig. 4(c) it can be seen that the dispersion vanishes completely (β₂ = 0) for certain DWG designs. Thus, it is possible to realize dispersion-minimized circular DWGs in the normalized frequency range 2.5 < V < 2.8. Air-clad PTFE waveguides, for example, with core diameters of D = 3.13 mm, D = 1.79 mm, and D = 1.04 mm have negligible dispersion at 80 GHz, 140 GHz, and 240 GHz, respectively (V = 2.62, Δ = 1) (see Table I). Due to the rounded diameter values, the dispersion minimum is not precisely obtained, but a comparison between the single-mode DWGs and the dispersion-optimized DWG in Table I shows that the Taylor coefficient β₂ is considerably lower. It can be observed that dispersion-minimized designs are all multi-mode waveguides (V ≥ 2.405). In order to benefit from the low waveguide dispersion, it might be necessary to reduce the modal dispersion of these DWGs as described in [Reference Meyer, Turan and Schneider20]. However, one might also ask the question whether it is necessary to reduce the dispersion β₂ to zero to obtain negligible signal distortions. For this reason, the Taylor coefficient β₂ is used in Section “Signal distortions” to calculate the signal distortions caused by waveguide dispersion of circular DWGs.

Signal distortions

According to Heaviside [Reference Heaviside21] a channel is considered free of distortion if its propagation constant fulfills

(23)$$\tilde{\gamma}_{opt}( \omega) = \alpha_0 + j \beta_1\omega.$$

Since this condition cannot be fulfilled by a DWG, the distortions to be expected when transmitting a signal via DWG are analyzed in the following. To evaluate DWGs as communication channels, the baseband (ω₀ = 0) channel impulse response h(t) is used. The baseband impulse response can be determined by inverse Fourier transformation ${\large{\cal F}^{-1}}$ of the baseband frequency transfer function

(24)$$\underline{H}_{B B}( j\omega) = \left \{ \matrix {e^{-\left(\gamma_0 + \gamma_1\omega + \gamma_2\omega^2 \right)l} & \text{for} & \vert \omega\vert \leq \omega_{\!N}\ \cr 0\hfill&\text{else}} $$

where γ₀, γ₁, and γ₂ are the complex coefficients of the approximated propagation constant $\tilde {\gamma }( \omega )$ given in (9). As described in Section “Data transmission system,” a band-limited signal is assumed that is caused by an ideal low-pass filter as transmit and receive filter (see equation (2)). Thus, the baseband impulse response is

(25)$$h( t) = {\large{\cal F}^{-1}}\left\{\underline{H}_{B B}( j\omega) \right\} = {1\over 2\pi} \int_{-\omega_{\!N}}^{\omega_{\!N}} e^{-\tilde{\gamma}( \omega) l} \cdot e^{\,j\omega t} d\omega$$

For an analytical description of the impulse response h(t), the solution of (25) is separated into two parts. First, only distortions caused by a frequency-dependent attenuation are taken into account. In the second part, distortions due to a frequency-dependent phase constant, namely the dispersion, are considered.

Distortions caused by attenuation

The frequency-dependent attenuation of DWGs is modeled by the Taylor coefficients α₀ and α₁. Therefore, the baseband frequency response of the DWG given in (24) simplifies to

(26)$$ \underline{H}_{B B}( j\omega) = \left \{ \matrix {e^{-\left(\gamma_0 + \gamma_1\omega \right)l} & \text{for} & \vert \omega\vert \leq \omega_{\!N}\ \cr 0\hfill&\text{else}} $$

By applying the inverse Fourier transform according to (25), the baseband impulse response of the DWG channel is obtained:

(27)$$h( t) = {T\over \pi} {\sin\left[( j\gamma_1\,l + t) \omega_{\!N} \right]\over j\gamma_1\,l + t } e^{-\gamma_0 l}$$

This impulse response can now be used to determine the SNR and the ISI caused by the frequency-dependent attenuation of a DWG. For this considerations the Taylor coefficient β₁ is set to zero, since it only represents a time shift (namely by the group delay τ_g) of the impulse response h(t) and, therefore, does not affect SNR and ISI.

Following equation (4) the SNR depends on the magnitude squared of the impulse response of the wanted signal |h(0)|². For a DWG channel with frequency-dependent attenuation follows:

(28)$$\left\vert h( 0) \right\vert ^2 = {1\over \omega_{\!N}^2}\cdot {\sinh^2( \alpha_1 l\, \omega_{\!N}) \over ( \alpha_1 l) ^2} e^{-2\alpha_0 l}$$

Since α₁ l ω_N is much smaller than 1 for DWGs in the mm-wave frequency range (see Section “Attenuation constant,”) the small argument approximation of hyperbolic sine (sinh(x) ≈ x) can be applied. Thus, the SNR simplifies to

(29)$${\rm SNR} \approx {\sigma_D^2\over \sigma_N^2} \cdot e^{-2\alpha_0 l} \quad {\rm for} \quad \vert \alpha_1 l\, \omega_{\!N}\vert \ll 1$$

This clearly shows that the SNR at the receiver of a DWG-based transmission system is simply the SNR of a channel with purely additive white Gaussian-distributed noise (AWGN channel) reduced by the attenuation of the DWG. Thus, the effect of the frequency-dependent attenuation on the SNR is negligible.

In addition to the SNR, the ISI is of interest in order to determine the expected signal distortions along the transmission channel. Applying the definition of the ISI given by (5) leads to an analytical description of the ISI. The magnitude squared of the impulse response of the wanted signal |h(0)|² has already been given by (28). The sum of the magnitude squared of the impulse response of all signals that superimpose the wanted signal converges to

(30)\begin \displaystyle\sum_{\substack{k = -\infty \cr\vskip0.6pc \hskip-1.55pc k \neq 0}}^{\infty}\left\vert h( k T) \right\vert ^2 = & {1\over \omega_{\!N}^2}\cdot \sinh^2( \alpha_1 l\omega_{\!N}) e^{-2\alpha_0 l} \cr & \cdot \left[{\omega_{\!N}\over \alpha_1 l} \coth( \alpha_1 l\, \omega_{\!N}) - {1\over ( \alpha_1 l) ^2} \right]\end

(for derivation see Appendix A). Using (28) and (30) as well as the definition of the ISI given by (5) an analytical description of the ISI can be obtained:

(31)$${\rm ISI} = {1\over \alpha_1 l\, \omega_{\!N} \!\cdot\! \coth( \alpha_1 l\, \omega_{\!N}) -1 }$$

Figure 5(a) shows the ISI for the common range of Taylor coefficient α₁ in the mm-wave frequency range for different signal bandwidths 2f _N and a DWG length of l = 1 m. It can be seen that for typical values of the frequency-dependent attenuation in the mm-frequency range $\alpha _1 = 10^{-13}\rm { {..}}\,10^{-11}$ s/m a tolerable ISI can be achieved for all bandwidths over a DWG length of l = 1 m. For example, the single-mode DWG made of PTFE given in Table I of 1 m length would still achieve a very good ISI of 33 dB even at a bandwidth of 20 GHz. Using a single-mode DWG made of PS reduces the ISI to 17 dB due to higher frequency dependency of the attenuation. With increasing length, the signal distortions further increase. For example, if the length is changed from $1$ to 10 m, the ISI lowers by 20 dB. Therefore, for larger bandwidths and longer length l an optimization of the DWG toward low α₁ ≤ 10⁻¹² s/m is recommended. A detailed study is given in Section “DWG design for low signal distortions.”

Fig. 5. ISI as a function of Taylor coefficient α₁ (a) and β₂ (b) for different signal bandwidths 2f _N as well as α₁ and β₂ (c) at a signal bandwidth of 2f _N = 5 GHz.

Distortions caused by dispersion

The frequency-dependent phase constant of DWGs is modeled by the Taylor coefficients β₀, β₁, and β₂, which includes the modeling of group delay and dispersion. This yields the baseband frequency transfer function given by (24). Applying the inverse Fourier transform according to (25) leads to the impulse response of the dispersive DWG channel

(32)\begin h( t) = & {T\over 4\pi} \sqrt{{\pi\over \gamma_2 l}} e^{ {\left(\gamma_1 \, l - jt\right)^2\over 4\gamma_2 l} -\gamma_0\,l }\cr & \cdot \left[{\rm erf}\left({2\gamma_2 l\, \omega_{\!N} + \gamma_1\,l - jt \over 2\sqrt{\gamma_2 l}} \right)\right. \cr & - \left. {\rm erf}\left({-2\gamma_2 l\, \omega_{\!N} + \gamma_1\,l - jt \over 2\sqrt{\gamma_2 l}} \right)\right]\end

where erf() represents the error function

(33)$${\rm erf}( x) = {2\over \sqrt{\pi}} \int_{0}^{x} e^{-\tau^2}d\tau.$$

Similar to Section “Distortions caused by attenuation,” the Taylor coefficient β₁ is set to zero for further analysis, since it has no influence on the signal distortions. Accordingly, the magnitude squared of the wanted signal is

(34)\begin \left\vert h( 0) \right\vert ^2 = & {\pi\over 16\omega_{\!N}^2} {1\over \vert \beta_2\vert l} e^{- 2\alpha_0 l}\cr & \cdot \left\vert {\rm erf}\left({\,j2\beta_2 l\, \omega_{\!N} + \alpha_1\,l \over 2\sqrt{\,j\beta_2 l}} \right)\right. \cr & - \left. {\rm erf}\left({-j2\beta_2 l\, \omega_{\!N} + \alpha_1\,l \over 2\sqrt{\,j\beta_2 l}} \right)\right\vert ^2 \end

and the magnitude squared of the interfering signal is

\begin \displaystyle\sum_{\substack{k = -\infty \cr\vskip0.6pc \hskip-1.55pc k \neq 0}}^{\infty}\left\vert h( k T) \right\vert ^2 = & {\pi\over 16\omega_{\!N}^2} {1\over \vert \beta_2\vert l} e^{- 2\alpha_0 l}\cr & \cdot \displaystyle\sum_{\substack{k = -\infty \cr k \neq 0}}^{\infty} e^{-{\alpha \over \beta_2}{\pi\over \omega_{\!N}}k} \cdot \left\vert {\rm erf}\left(\zeta_1( k) \right)\right. - \left. {\rm erf}\left(\zeta_2( k) \right)\right\vert ^2\cr \end

with

(35)$$\eqalign {\begin &\zeta_1( k) = {\,j2\beta_2 l\, \omega_{\!N} + \alpha_1\,l - j\frac{\pi}{\omega_{\!N}}k \over 2\sqrt{\,j\beta_2 l}} \cr& \zeta_2( k) = {-j2\beta_2 l\, \omega_{\!N} + \alpha_1\,l - j\frac{\pi}{\omega_{\!N}}k \over 2\sqrt{\,j\beta_2 l}} \end}$$

The series (35) converges (for proof see Appendix B) and is therefore analyzed numerically to determine the ISI. First, the ISI is considered without the influence of frequency-dependent attenuation (α₁ = 0). The results are shown in Fig. 5(b). A considerably greater effect on the ISI compared to the frequency-dependent attenuation can be observed in Fig. 5(a). Especially with increasing signal bandwidth the ISI increases significantly. For typical values of |β₂| ≤ 10⁻²¹ s²/m², an acceptable ISI can only be achieved for smaller bandwidth and/or shorter length. If the single-mode DWG made of PTFE in Table I is considered, it can be seen that for a bandwidth of 20 GHz and a length of 1 m only an ISI in a range of $\approx \! 10 { {..}}\,20\, {\rm dB}$ is achieved. This is a significant reduction compared to the ISI caused by frequency-dependent attenuation. A transmission over l ≥ 10 m seems only possible for smaller bandwidths (e.g. 5 GHz). However, as already shown in Section “Phase constant,” the waveguide dispersion can be reduced to almost zero (|β₂| → 0) for certain circular DWGs. In this case, the ISI and therefore the transmission length would be infinite. Since in practical application, the frequency-dependent attenuation is not negligible (see Section “Distortions caused by attenuation”), the question remains whether a minimized waveguide dispersion actually results in a low overall ISI.

DWG design for low signal distortions

As shown in Sections “Distortions caused by attenuation” and “Distortions caused by dispersion,” frequency-dependent attenuation as well as waveguide dispersion have a significant effect on signals transmitted along a DWG. This is especially true for long distance communications (l ≥ 10 m). However, it has also been shown that for certain circular DWGs, the waveguide dispersion vanishes completely (β₂ = 0). An interesting question at this point is whether this dispersion minimum really leads to minimal signal distortions despite frequency-dependent attenuation (α₁ ≠ 0). To answer this question, equation (35) is evaluated numerically for values of α₁ ≠ 0. In order to keep the variation of the results low, only a bandwidth of 2f _N = 5 GHz is considered. The results for the ISI in dependency of α₁ and β₂ are shown in Fig. 5(c). It can be seen that for small values of α₁ (≤ 10⁻¹⁵ s/m) the ISI as a function of α₁ and β₂ in Fig. 5(c) is almost congruent with the ISI in Fig. 5(b) (B = 5 GHz), where only dispersion caused by frequency-dependent group delay (α₁ = 0) is considered. The influence of the frequency-dependent attenuation on the ISI can therefore be neglected for α₁ ≤ 10⁻¹⁵ s/m. With increasing values of α₁, however, a limitation in the maximum achievable ISI appears despite negligible waveguide dispersion (β₂ ≤ 10⁻²⁵ s²/m²). For example, a DWG with a Taylor coefficient of α₁ = 10⁻¹² s/m and 1 m length leads to an ISI of ISI ≈ 41 dB even if the waveguide dispersion is zero. This limitation in the maximum achievable ISI despite negligible waveguide dispersion corresponds to the values determined for the attenuation-dependent ISI in Fig. 5(a) (B = 5 GHz). Whether the optimization of a DWG to low waveguide dispersion also leads to a reduction of the expected ISI depends mainly on the frequency-dependent attenuation of the respective DWG. For example, for a DWG with a Taylor coefficient of α₁ = 10⁻¹² s/m, reducing the waveguide dispersion below |β₂| ≤ 3 × 10⁻²³ s²/m² would not further improve the ISI. This clearly shows that low signal distortions in DWGs can only be achieved if the frequency-dependent attenuation in addition to dispersion is reduced. This fact has not been considered in the majority of scientific studies so far. Furthermore, it is more challenging to reduce the frequency dependency of the attenuation than to reduce the waveguide dispersion of a DWG. Considering the mathematical description of the first-order frequency-dependent attenuation in (17), it is noticeable that the loss factors of core and cladding (tanδ₁, tanδ₂) occur with different prefactors. If both factors are considered separately, it can be seen that the prefactor of the cladding material's loss factor becomes negative for normalized frequencies of V ≥ 1.2 (see Fig. 6). Thus, a minimization of the Taylor coefficient α₁ to zero is theoretically possible. However, a reduction of the frequency dependency of the attenuation is only possible if the cladding material has significantly higher losses than the core material ($\tan \delta _2\approx 4\,{ {..}}\,5\times \tan \delta _1$). This would result in a significant increase of the total losses of the DWG and thus in a decrease of the SNR at the receiver.

Fig. 6. Core and cladding material prefactor in (17) as a function of the normalized frequency V for different material ratios Δ.

It can generally be said that for low material ratios Δ the frequency dependency of attenuation and group delay is reduced and the signal distortions decrease. However, lower values of Δ also result in a lower field concentration in the core of the DWG (see Fig. 3(a)). This requires a larger size of the DWG package (core + cladding). In addition, the sensitivity to discontinuities will be increased. Therefore, a compromise on practical issues must be found when reducing Δ. In most of nowadays DWG applications, the relative permittivity of the used materials is in a range of $\varepsilon _r \approx 2 { {-}}3$ and the loss factor in a range of $\tan \delta \approx 10^{-4} { {-}}10^{-3}$. Considering (17) and Fig. 6, the respective range of this Taylor coefficient is $\alpha _1 \approx 10^{-13} { {-}}10^{-11}$ s/m ($V \approx 2 { {-}}3$). For such DWGs, a minimization of the waveguide dispersion is recommended, since it also reduces the frequency-dependent attenuation to an acceptable value. For the waveguide-dispersion-minimized DWG in Table I with a Taylor coefficient of α₁ = 6.0 × 10⁻¹³ s/m, an ISI of ISI = 10 dB could still be achieved over lengths of 58 m (5 GHz bandwidth) or 14 m (20 GHz bandwidth). Reducing α₁ by a factor of 10 allows to increase the respective lengths by the same factor. For comparison, using the single-mode DWG made of PTFE in Table I with Taylor coefficients of α₁ = 6.4 × 10⁻¹³ s/m and β₂ = 2.0 × 10⁻²² s/m would only achieve a transmission length of 19 m (5 GHz bandwidth) or 1.3 m (20 GHz bandwidth). This clearly shows the great potential of dispersion optimization of DWGs in the mm-wave frequency range.