3 Signal Waveform Variations in Ultrawideband Wireless Systems: Causes and Aftereffects

The majority of conventional wireless systems have a bandwidth much smaller than their carrier frequency. These systems use a sinusoidal carrier signal and transmit information by changing their amplitude and frequency (or phase). The theory and practice of modern wireless systems are based on the well-developed body of theory of conventional wireless systems. Ultrawideband (UWB) radio and radar systems with spatially short nonsinusoidal waveforms present a new set of problems that conventional theory does not cover.

This chapter discusses what happens when a UWB signal has a duration of τ seconds and a physical length (spatial longitude) of cτ (where c indicates the velocity of light in the media) shorter than the radiating and receiving antennas and radar target sizes. In this case, τ is the duration of a simple signal or the width of the autocorrelation function if a signal has an intraimpulse modulation. In the latter case, cτ is the physical length of a signal when taking into account its matched processing result. Under these conditions, the signal waveform will substantially change during radiation, target reflection, and reception. This presents a much different case than that found in conventional narrowband (NB) signals. We will see how the signal waveform can change, in very surprising ways, depending on the angle from the antenna, the position of target “bright spots,” and other reasons. As a result of this waveform modification, we must understand what happens to the UWB signal to optimize system designs and exploit the benefits of an information-rich signal. This chapter presents an analysis of the signal waveform changes found in UWB wireless systems with very short-duration (nanosecond, picosecond) signals or signals with short autocorrelation functions.

3.1.2 Signal Waveforms, Bandwidth, and Propagation Effects

Conventional radio-based systems use continuous signals with narrow frequency bandwidths set by the information content and modulation method. The work of Shannon and Weaver [1] demonstrated how expanding the signal bandwidth would increase the information capacity of the system with more communications data content or smaller radar spatial resolution. Expanding the capabilities of radar systems means that we must find ways to produce instantaneously wide bandwidth signals without interfering with conventional NB systems. This need to expand the radar capabilities drove the development of UWB technology. We find that there are major differences in the signal processing of conventional NB and UWB systems due to the nonsinusoidal nature of the UWB waveforms.

Harmonic sinusoidal waveforms have many convenient properties because the shape does not vary during the most common types of signal transformation, for example, addition, subtraction, differentiation, and integration. Because nonsinusoidal UWB signals lack this convenient property, we must take a different analytical approach. The following sections explain those differences and provide useful relationships for analysis and design of UWB systems.

Increasing the radar system’s resolution and information content requires generating a short-duration signal, which does not follow the rules for sinusoidal signals. Shortening the signal to a duration or autocorrelation period τ implies a large bandwidth because τ = 1/Δf, where Δf indicates the signal bandwidth.

Our signal pulse has a length of cτ in space. Decreasing the pulse duration or autocorrelation period can make the signal’s physical length or spatial resolution smaller than the physical sizes of the antenna aperture and/or the longitudinal size L of the illuminated target. The resulting processes in this case become obviously apparent when cτ ≪ L. This occurs when a UWB system has a large antenna aperture and a narrow beam width. For the cτ ≪ L case, the UWB signal waveform will change during the radiation, target reflection, and reception processes and, as a result, becomes unknown.

Our analysis also shows how the signal waveform varies with the angle with respect to the antenna. This means that the antenna’s radiation pattern changes its location in space during the travel of the current pulse along the aperture. The receiving antenna’s pattern will demonstrate the same behavior during the travel of an incident UWB pulse of the field along its length. These different field shapes at radiation and reception do not allow using the principle of reciprocity for evaluating the parameters of receiving antennas based on their parameters in the regime of radiation.

The unknown shapes of the UWB radar signal field incident on the target and the field reflected to the receiving antenna do not allow us to determine the target radar cross-section (RCS) of a target by classical methods. This unknown received signal waveform does not permit using the classical correlator with a reference signal or a matched filter for optimum processing.

As a result, we cannot use conventional NB analysis methods, which assume a sinusoidal harmonic carrier oscillation.

In this chapter, we analyze the propagation of UWB signals resulting from the cτ ≪ L or the short-pulse condition. Chapter 1 of Ultra-Wideband Radar Technology [2] also discusses this problem.

3.2 Change of UWB Signal Waveforms during Radiation

3.2.1 Analysis of UWB Radiation: Introduction and Background

Many Russian and foreign publications have examined the processes involved in antenna excitation and signal radiation for the cτ ≪ L condition, for both series excitation of the entire aperture (in traveling-wave antennas) and parallel excitation (in aperture antennas). Other names for this excitation condition include nonstationary (time-dependent) excitation. See Immoreev and Sinyavin in Antenny (Russian) for a review of these studies [3].

These previous studies found rigorous solutions to the exterior electromagnetic problems. These solutions can describe the electromagnetic field radiated by the antenna in the far zone for a specified UWB waveform of the exciting current in the cτ ≪ L or “short-pulse” condition. Unfortunately, these cumbersome and complicated solutions prevent a thorough understanding of the physical processes occurring in the antenna.

3.2.2 How Antennas Radiate UWB Signals

To understand what happens when an antenna is excited by a short pulse, we can divide the antenna into elementary radiators called Hertzian dipoles whose sizes (dimensions) are small enough to suppose a uniform current distribution along their length. We can represent the radiation field as the sum of fields radiated by these dipole antennas.

First, we must determine the dependence of the Hertzian dipole’s field E(θ, t) on the excitation current i(t), where θ is the angle from the antenna to the observation point. The majority of publications define this dependence in the case of sinusoidal current. To find all the field parameters for cτ ≪ L, we must determine the field dependence on the excitation current of an arbitrary time law for the Hertzian dipole in the far zone. In the far field, the magnetic component of the field differs from its electric component by the multiplier 1/120π. Goldstein and Zernov, in Electromagnetic Fields and Waves [4], derived Equation 3.1 for the electric component of the Hertzian dipole as the far-field dependence of this field on the excitation current i(t) for the arbitrary time law as

E (θ, t) = Z 0 sin θ 4 π c R d d t [i (t - R c)] Δ L, (3.1)

$E (θ, t) = \frac{Z_{0} \sin θ}{4 π c R} \frac{d}{d t} [i (t - \frac{R}{c})] Δ L, (3.1)$

where Z₀ is the wave impedance of free space, θ is the angle between the dipole axis and the direction toward the observation point, ΔL is the Hertzian dipole length, and R is the distance to the observation point.

Using Equation 3.1, let us consider the field of a simplest linear radiator with length L, excited from one end by the pulse current source as shown in Figure 3.1 [5]. Let us divide this antenna into elementary radiators—Hertzian dipole—with length ΔL. As shown in Figure 3.1, we define ΔL_j as the jth radiating element radiator with coordinate L_j. The current pulse i(t) appears at point O, propagates along the antenna, and sequentially excites the radiating elements.

After the current pulse arrives at point O, the first elementary radiator will radiate. The electromagnetic field in the far-zone will have of the form [3,5,6]

E 1 (t, θ) = Z 0 sin θ 4 π c R d d t [i (t - L 1 c 0 - R - L 1 cos θ c)] Δ L 1 . (3.2)

$E_{1} (t, θ) = \frac{Z_{0} sin θ}{4 π c R} \frac{d}{d t} [i (t - \frac{L_{1}}{c_{0}} - \frac{R - L_{1} \cos θ}{c})] Δ L_{1} . (3.2)$

The second term in parentheses defines the signal delay in the radiator (where c₀ indicates the velocity of current pulse propagation along the radiator) and the third term relates to the signal delay in space taking into account the deviation of the observation line from the antenna’s aperture by angle θ.

Images

FIGURE 3.1
The radiating antenna model for the analysis and calculation of the observed field from each ΔL_j element excited by the current pulse i(t). Summation of the fields from each element gives the total radiation field.

In the time interval ΔL/c, the current pulse excites the second elementary radiator and produces the second electromagnetic field E₂(t, θ), which has an expression similar to Equation 3.2 in which L₁ is replaced with L₂.

The current pulse moves along the linear radiator and successively excites each subsequent radiation element. To have a relatively simple physical picture of the processes occurring in the antenna, we can introduce some simplifications that will not change the final result and that coincide with the results of the rigorous theory. For analysis purposes, we can disregard the loss in the antenna wire and assume no delay in the antenna wire so that c₀ = c. Further, we assume a matched radiator condition so that the current pulses radiate into ambient space without reflecting back to the transmitter.

3.2.3 UWB Antenna Far-Field Radiation Effects

In this case, the total far field of all elementary radiators has the following form:

E Σ (t, θ) = Z 0 sin θ 4 π c R Σ j - 1 N d d t [i (t - L j c - R - L j cos θ c)] Δ L . (3.3)

$E_{Σ} (t, θ) = \frac{Z_{0} \sin θ}{4 π c R} Σ_{j - 1}^{N} \frac{d}{d t} [i (t - \frac{L_{j}}{c} - \frac{R - L_{j} \cos θ}{c})] Δ L . (3.3)$

We can move from the discrete antenna representation to a continuous representation by reducing the length of an elementary radiator toward zero (ΔL → 0), which sends the number N of radiators toward infinity (N → ∞). Then, we can transform the summation in Equation 3.3 into the integral:

E Σ (t, θ) = Z 0 sin θ 4 π c R \int 0 L d d t ⎡ ⎣ ⎢ i ⎛ ⎝ ⎜ t - L c - R - L cos θ c ⎞ ⎠ ⎟ ⎤ ⎦ ⎥ d L . (3.4)

$E_{Σ} (t, θ) = \frac{Z_{0} \sin θ}{4 π c R} \int_{0}^{L} \frac{d}{d t} [i (t - \frac{L}{c} - \frac{R - L \cos θ}{c})] d L . (3.4)$

Equation 3.4 describes the electric component of the far electromagnetic field of an extended antenna excited from one end by the arbitrary current waveform i(t). The field described by Equation 3.4 considers the delays of the current pulse in the antenna wire and the field pulse in space. The expression in parentheses shows the current time with consideration for this delay. After calculating the derivative of this time with respect to dL, we can perform the following change of variables in Equation 3.4 with

d t = cos θ - 1 c d L .

$d t = \frac{cos θ - 1}{c} d L .$

As a result, we obtain the integral of a function derivative with respect to the integration variable. Such an integral is equal to the function itself. Then, we get

E Σ (t, θ) = Z 0 sin θ 4 π R 1 cos θ - 1 [i (t - L c - R - L cos θ c)] L 0 = Z 0 sin θ 4 π R 1 cos θ - 1 [i (t - L c - R - L cos θ c) - i (t - R c)] . (3.5)

$\begin{matrix} E_{Σ} (t, θ) & = \frac{Z_{0} \sin θ}{4 π R} \frac{1}{\cos θ - 1} [i (t - \frac{L}{c} - \frac{R - L \cos θ}{c})]_{0}^{L} \\ = \frac{Z_{0} \sin θ}{4 π R} \frac{1}{\cos θ - 1} [i (t - \frac{L}{c} - \frac{R - L \cos θ}{c}) - i (t - \frac{R}{c})] . \end{matrix} (3.5)$

For the general case, Equation 3.5 shows that the field generated by the antenna consists of two parts, negative and positive, which have different delays. We can show that the shape of each part of this field repeats the shape of the exciting current pulse. However, we cannot divide these parts; it is impossible to do so because they are a single whole.

To determine the total antenna field for a specific case, we shall assume that the exciting pulse i(t) has a Gaussian curve waveform with unit amplitude as shown in Figure 3.2 and the waveform shown in Equation 3.6 as

i (t) = exp [- 4 (t τ) 2], (3.6)

$i (t) = \exp [- 4 (\frac{t}{τ})^{2}], (3.6)$

where τ is the pulse duration at a level of 0.5. The derivative of this pulse is a symmetric bipolar pulse as shown in Figure 3.2, curve di(t)/dt. Substituting Equation 3.6 into 3.5, we get

E Σ (t, θ) = Z 0 sin θ 4 π R ⎡ ⎣ ⎢ ⎢ 1 cos θ - 1 ⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ exp ⎡ ⎣ ⎢ ⎢ - 4 ⎛ ⎝ ⎜ ⎜ t - L c - R - L cos θ c τ ⎞ ⎠ ⎟ ⎟ 2 ⎤ ⎦ ⎥ ⎥ - exp ⎡ ⎣ ⎢ ⎢ - 4 ⎡ ⎣ ⎢ ⎢ t - R c τ ⎞ ⎠ ⎟ ⎟ 2 ⎤ ⎦ ⎥ ⎥ ⎫ ⎭ ⎬ ⎪ ⎪ ⎪ ⎪ ⎤ ⎦ ⎥ ⎥ . (3.7)

$E_{Σ} (t, θ) = \frac{Z_{0} \sin θ}{4 π R} [\frac{1}{\cos θ - 1} {\exp [- 4 (\frac{t - \frac{L}{c} - \frac{R - L \cos θ}{c}}{τ})^{2}] - \exp [- 4 [\frac{t - \frac{R}{c}}{τ})^{2}]}] . (3.7)$

Equation 3.7 shows how the shape of the total radiation field depends on the relationship between the antenna length L and the pulse length in space cτ. This shape also depends on the observation angle θ.

We can show the formation of the total field E_Σ(t, θ) from Equation 3.7 for the short pulse cτ ≪ L by using the discrete antenna model shown in Figure 3.1. The current pulse moves along the antenna and arrives at each elementary radiator with a different delay. The delay between each element will time-shift the fields formed at point M. Figure 3.3 uses a solid line to show these fields for the observation angle θ = 90°. The fields have positive and negative half-waves with equal areas. Therefore, in the process of summation, these half-waves partially cancel each other. The degree of this cancellation depends on the ratio of antenna length L and the pulse duration in space cτ. Complete cancellation of fields starts at the time t_c = τ . Only some portion of the fields of elementary radiators located near the excitation point and at the radiator end remains, as shown in Figure 3.3 by the dashed line. Therefore, at cτ ≪ L, the total radiator field E_Σ(t, τ, θ) has two individual fields, one radiated at the instant when the pulse arrives at the antenna’s excitation point and the other at the instant when the pulse reaches the antenna end. Some treat this process as radiation from the excitation point and the antenna end.

Images

FIGURE 3.2
The Gaussian current pulse (solid line) and its derivative (dashed line) closely resembled the typical excitation pulses used in many short-range radar systems and serves as a typical model for analysis.

Images

FIGURE 3.3
UWB antenna far-field determination when L/cτ ≫ 1. Adding the individual element fields at a point M located at an angle 90° to the antenna axis produces the total field.

Images

FIGURE 3.4
As the pulse length cτ approaches the antenna length L and cτ ≤ 1, then the number of radiating elements decreases and produces the field E_Σ as shown at θ = 90°. The shape of the total field approaches the shape of the derivative of the current pulse exciting the antenna.

As the ratio L/cτ decreases and the pulse length approaches the antenna length L, the time interval in which the fields cancel each other decreases relative to the duration of the field pulse. Finally, at L/cτ ≫ 1, the compensation practically terminates and separated fields merge as shown in Figure 3.4. In this case, the radiation occurs simultaneously from the entire aperture, and the shape of the total field approaches the shape of the derivative of the current pulse exciting the antenna.

Figure 3.5 shows the total antenna fields E_z(t, τ, θ) for cτ ≪ L and different observation angles θ. Notice how the shape of the total field pulse depends on the observation angle because the length of the projection of the antenna depends on the observation angle. We can determine the boundary angle at which the shape of the field pulse can be treated as two separate pulses with different polarities from the expression θb=π−arc cos[cτL−1] $θ_{b} = π - arc cos [\frac{c τ}{L} - 1]$ .

Images

FIGURE 3.5
UWB antenna total antenna fields E_z(t, τ, θ) for a pulse length shorter than the antenna size cτ ≪ L and different observation angles θ.

The boundary angle θ_b, depends on the ratio L/cτ. If cτ > 2L, then the angle θ_b, takes negative values. This means that we cannot separate the far field into two pulses at all observation angles θ.

If the antenna does not match the impedance of free space, then the current pulse reflected from the antenna’s end returns to the point of excitation and is absorbed at this point. Under this unmatched condition, the expression for the field will take the following form:

E Σ (t, θ) = A 1 [i (t - L c - R - L cos θ c) - i (t - R c)] + A 2 [i (t - L c - R - L cos θ c) - i (t - 2 L c - R c)], (3.8)

$E_{Σ} (t, θ) = A_{1} [i (t - \frac{L}{c} - \frac{R - L \cos θ}{c}) - i (t - \frac{R}{c})] + A_{2} [i (t - \frac{L}{c} - \frac{R - L \cos θ}{c}) - i (t - \frac{2 L}{c} - \frac{R}{c})], (3.8)$

where

A 1 = Z 0 sin θ 4 π R 1 cos θ - 1 and A 2 = Z 0 sin θ 4 π R 1 cos θ + 1 .

$A_{1} = \frac{Z_{0} \sin θ}{4 π R} \frac{1}{\cos θ - 1} and A_{2} = \frac{Z_{0} \sin θ}{4 π R} \frac{1}{\cos θ + 1} .$

For the symmetric antenna not matched with space as shown in Figure 3.6, the expression for the field becomes more complicated:

E Σ (t, θ) = A 1 [i (t - L c - r - L cos θ c) - i (t - r c)] - A 2 [i (t - L c - r - L cos θ c) - i (t - 2 L c - r c)] + A 2 [i (t - r c) - i (t - L c - r + L cos θ c)] - A 1 [i (t - 2 L c - r c) - i (t - L c - r + L cos θ c)] . (3.9)

$\begin{array}{l} E_{Σ} (t, θ) = A_{1} [i (t - \frac{L}{c} - \frac{r - L \cos θ}{c}) - i (t - \frac{r}{c})] - A_{2} [i (t - \frac{L}{c} - \frac{r - L \cos θ}{c}) - i (t - \frac{2 L}{c} - \frac{r}{c})] \\ + A_{2} [i (t - \frac{r}{c}) - i (t - \frac{L}{c} - \frac{r + L \cos θ}{c})] - A_{1} [i (t - \frac{2 L}{c} - \frac{r}{c}) - i (t - \frac{L}{c} - \frac{r + L \cos θ}{c})] . \end{array} (3.9)$

Images

FIGURE 3.6
The UWB symmetric antenna presents a much more complicated far-field radiation pattern as shown in Equation 3.9 for E_Σ(t, θ). The far field at any angle θ and time t results from the summation of the signals from each antenna element.

In all cases, the total field consists of independent fields radiated from the points of excitation and antenna ends. The shape of the total radiated field depends on the ratio between the antenna length L and the pulse length cτ.

The literature review carried out by Sinyavin and Immoreev (myself) [3] in Antenny shows that not only the linear radiators but also the aperture radiators radiate from their excitation point and ends.

In any case, for the short-pulse condition cτ ≪ L, the antenna far zone consists of the summation of an electromagnetic pulse sequence separated by intervals, which depend on the antenna geometry. This exciting current pulse shape forms the radiated signal sequence, which will have a different shape from the original exciting pulse because of the antenna size delays and additions of each increment in the far field. The nonsinusoidal waveforms and summation effects differ from the conventional NB expectations.

The method shown here can find universal applications in defining the far field for antenna of any size and exciting signal waveforms.

3.3 Radiation Patterns of UWB Antenna

3.3.1 How the Radiated Waveform Varies with the Observation Angle

Figure 3.5 shows how the pulse waveform will vary with the observation angle θ and time t. We can show this dependence by using the antenna shown in Figure 3.1, which is excited by the Gaussian pulse shown in Figure 3.2. Equation 3.7 describes the field produced by the Gaussian pulse. The resulting field E_Σ(t, θ) depends on the time from excitation, the angular direction, the shape of the exciting signal, and the antenna length. For our case, we can fix the shape of the exciting signal and the antenna length. To find the dependence of the field on the angular direction, we shall select several instants t₀, t₁, t₂, t₃, . . . in the interval during which the current pulse moves along the antenna. Using Equation 3.7, we can determine the field E_Σ(t, θ) as a function of θ at each of these instants to find the instantaneous field patterns. Figure 3.7 shows a family of instantaneous patterns obtained at different time intervals for a pulse length shorter than the antenna length, that is, cτ ≪ L.

3.3.2 Time-Domain Radiated UWB Field

Figure 3.7 shows how the pattern maximum changes direction during the time interval in which the field exists. At the initial instant, the field maximum follows the antenna axis. As the pulse of current moves along the antenna, the pattern moves in space from the antenna axis toward the normal to the antenna axis and decreases. The behavior of the instantaneous patterns shown in Figure 3.7 correlates with the results obtained when Zakharov and Sugak [7] did these calculations with a rigorous linear antenna model.

Images

FIGURE 3.7
Instantaneous patterns of an antenna with series excitation.

In the case of a more complex (multiple pulse) shape excitation, the radiated field motion of the instantaneous pattern becomes more complicated but retains the time dependence. This time dependence of the field pattern makes it unsuitable for calculation of the parameters of a wireless system because this dependence prohibits determining antenna parameters such as the directivity factor and the beam width. In early publications, for example, Harmuth [8] described other variants of the radiation pattern models, namely, models based on the peak amplitude, the peak power, and the steepness. However, for practical applications, the energy pattern is the most suitable model.

We can find the spatial energy pattern W_T(θ, ϕ) by averaging the power radiated in each angular direction over the time required for the pulse to travel along the antenna. In this case, θ describes a one-dimensional pattern or a cross-section two-dimensional pattern on this angle. Two angles θ and ϕ describe a two-dimensional spatial pattern. This pattern describes the spatial distribution of the density of the radiated energy flux as a function of angles θ and ϕ, which describe the coordinate system two-dimensional spatial axes as [9]

W T (θ, ϕ) = 1 Z 0 \int - \infty + \infty E 2 Σ (θ, ϕ, t) d t . (3.10)

$W_{T} (θ, ϕ) = \frac{1}{Z_{0}} \int_{- \infty}^{+ \infty} E_{Σ}^{2} (θ, ϕ, t) d t . (3.10)$

The infinite limits of integration in time allow the application of this expression to current pulses of any shape and to any antenna length.

For comparison of the antenna characteristics, we can use the normalized energy pattern

W TN = W T ( θ , ϕ ) W T max, (3.11)

$W_{TN} = \frac{W_{T} (θ, ϕ)}{W_{T \max}}, (3.11)$

where the maximum direction pattern value has the form

W T max = 1 Z 0 ⎡ ⎣ ⎢ \int - \infty + \infty E 2 Σ (θ, ϕ, t) d t ⎤ ⎦ ⎥ max . (3.12)

$W_{T \max} = \frac{1}{Z_{0}} [\int_{- \infty}^{+ \infty} E_{Σ}^{2} (θ, ϕ, t) d t]_{\max} . (3.12)$

Figure 3.8 shows the normalized energy patterns for the example considered in this section at different values of ratio L/cτ. If cτ ≫ 1, then the energy pattern coincides with the radiation pattern of a half-wave vibrator. As the ratio L/cτ increases, the pattern maximum deviates from the normal, and at cτ ≪ L, a linear antenna radiates along its axis. During these changes, the value of the pattern maximum increases and the pattern width decreases.

We define the directivity factor of an antenna radiating a UWB signal as the ratio of the energy-flux density of the antenna in the direction of the maximum radiation (W_Tmax) and the energy-flux density of an equivalent isotropic antenna (W_T0) receiving the same input energy as the first antenna. This gives the directivity factor D_T as

D T = W T max W T 0 . (3.13)

$D_{T} = \frac{W_{T \max}}{W_{T 0}} . (3.13)$

The total energy of the field radiated by the antenna through a sphere of radius R is

W = R 2 Z 0 \int 0 2 π \int 0 π \int - \infty + \infty E 2 Σ (θ, φ, t) sin θ d θ d φ d t . (3.14)

$W = \frac{R^{2}}{Z_{0}} \int_{0}^{2 π} \int_{0}^{π} \int_{- \infty}^{+ \infty} E_{Σ}^{2} (θ, φ, t) \sin θ d θ d φ d t . (3.14)$

Images

FIGURE 3.8
One-dimensional (ϕ = constant) energy patterns of an antenna with series excitation. When L/cτ ≪ 1, the radiation pattern resembles a half-wave vibrator and radiates perpendicularly to the antenna axis.

By dividing this energy by the surface area of a sphere surrounding the antenna, we get the energy-flux density of the equivalent isotropic antenna as

W T 0 = W 4 π R 2 = 1 4 π Z 0 \int 0 2 π \int 0 π \int - \infty + \infty E 2 Σ (θ, ϕ, t) sin θ d θ d ϕ d t . (3.15)

$W_{T 0} = \frac{W}{4 π R^{2}} = \frac{1}{4 π Z_{0}} \int_{0}^{2 π} \int_{0}^{π} \int_{- \infty}^{+ \infty} E_{Σ}^{2} (θ, ϕ, t) \sin θ d θ d ϕ d t . (3.15)$

The obtai ned expressions allow us to determine the energy directivity factor of the antenna as

D T = 4 π ⎡ ⎣ ⎢ \int - \infty + \infty E 2 Σ ( θ , ϕ , t ) d t ⎤ ⎦ ⎥ max \int 0 2 π \int 0 π \int - \infty + \infty E 2 Σ ( θ , ϕ , t ) sin θ d θ d φ d t . (3.16)

$D_{T} = 4 π \frac{[\int_{- \infty}^{+ \infty} E_{Σ}^{2} (θ, ϕ, t) d t]_{max}}{\int_{0}^{2 π} \int_{0}^{π} \int_{- \infty}^{+ \infty} E_{Σ}^{2} (θ, ϕ, t) sin θ d θ d φ d t} . (3.16)$

The antenna shown in Figures 3.1 and 3.6 as discussed earlier used series excitation. For an antenna simultaneously excited by a short pulse (as in the case of an aperture antenna), the far fields generated at different angles have different shapes. We can find the instantaneous field patterns of such an antenna from Equation 3.5 or 3.8 or 3.9 upon removal of term L/c, which determines the delay of the current pulse for the travel time of this pulse along the aperture, from the expression in parentheses. Figure 3.9 shows these patterns for different instants t₀,t₁,t₂,t₃, . . . Unlike the preceding case, the axis of the instantaneous pattern does not change its spatial orientation. However, during the time when the current pulse exists in the antenna, the pattern has two separate diverging beams. Figure 3.10 shows a family of normalized energy patterns of an aperture antenna for different values of ratio L/cτ.

Images

FIGURE 3.9
The variation of UWB antenna pattern with time. This shows the normalized instantaneous energy patterns for an aperture antenna at different times t₀, t₁, t₂, t₃, . . .

Images

FIGURE 3.10
The variation of UWB antenna pattern with the ratio of the antenna length L to the pulse length cτ. This shows how the energy pattern of an aperture antenna shifts depending on the ratio of L/cτ for 5, 10, 20, 30, and 40.

3.4 Determination of Variation in the UWB Signal Waveform during Reception and the Resulting Antenna Reception Pattern

3.4.1 How UWB Pulse Waveforms Vary with the Observation Angle

The previous discussion showed how the radiated signal waveform varies with the observation angle from the antenna axis, as shown in Figure 3.5. In the same way, the received signal waveform will depend on the angle between the receiving and the transmitting antennas. The current pulse waveform induced by the incident UWB field and measured at the receiving antenna output depends on the mutual position and angular orientation of the antennas. Therefore, the receiving antenna pattern depends on these positions and angles. Therefore, for the short-pulse case where cτ ≪ L, a UWB antenna will have different patterns in the radiation and reception modes. UWB and conventional NB antenna theories differ in this respect. Therefore, we cannot apply conventional NB antenna theory to UWB systems without considering these effects.

Figure 3.11 shows the positions of a UWB transmitting L_T and receiving L_R antenna and sets the geometry for our analysis of how the received waveform varies with the antenna’s positions and orientations. We start by placing the receiving antenna at a distance R from the transmitting antenna, far zone at the point as shown in Figure 3.11. For simplicity, we assume the perfect case of matched load conditions and the load and the antenna ends so that we have no reflected energy. The transmitting antenna radiates the field electric component toward the receiving antenna at angle θ_T between the line of the antenna aperture and the direction to the receiving antenna. The signal wave front hits the receiving antenna at angle θ_R between the line of the antenna aperture and the direction to the transmitting antenna.

Let us consider the transmitting antenna, which has to match with free-space impedance. As against Equation 3.8, equation of this antenna looks like

E Σ (t, θ T, L T) = A 1 [i (t - L T c - R - L T cos θ T c) - i (t - R c)] + A 2 [i (t - R c) - i (t - L T c - R + L T cos θ T c)] . (3.17)

$E_{Σ} (t, θ_{T}, L_{T}) = A_{1} [i (t - \frac{L_{T}}{c} - \frac{R - L_{T} \cos θ_{T}}{c}) - i (t - \frac{R}{c})] + A_{2} [i (t - \frac{R}{c}) - i (t - \frac{L_{T}}{c} - \frac{R + L_{T} \cos θ_{T}}{c})] . (3.17)$

Images

FIGURE 3.11
The transmitting and receiving antenna geometry for determining the effects of antenna orientation on a UWB signal sent between the antennas under the short-pulse condition cτ/L ≪ 1. This analysis assumes perfect matching of the transmitting L_T and receiving L_R antennas with outputs and space, which produces no reflections.

For our analysis, we divide the receiving antenna into elementary segments dL_R—Hertzian dipoles. The electromotive force induced in an elementary segment is directly proportional to the projection of electric vector component E_Σ(t, θ_R, L_T) of the field incident onto this segment, so dE = E_Σ(t, θ_R, L_T) sin θ_RΔL_R. In each elementary segment, this incident UWB field generates the elementary current dI = dE/Z, where Z indicates the total impedance of the antenna circuit and includes the radiation resistance Z_R and load impedance Z_L. We shall assume that Z_R = Z_L and impedance Z is frequency independent to simplify the calculations.

The elementary currents induced in each segment flow toward the load and toward the antenna’s matched dipole ends, which produce no reflections. Therefore, the part of these currents that flows toward the antenna ends creates the secondary radiation of the antenna. The currents flowing toward the load create a voltage drop across this load.

Note that, as in the case of radiation, the signal delay of the receiving antenna wire and in the space (if the field is incident at some angle to the dipole aperture) depends on the angle θ_R so that for the right side of a dipole

i r ⎛ ⎝ ⎜ t, θ R, L R ⎞ ⎠ ⎟ = sin θ R Z \int 0 L R E Σ ⎛ ⎝ ⎜ t - L R c + L R cos θ R c ⎞ ⎠ ⎟ d L (3.17a)

$i_{r} (t, θ_{R}, L_{R}) = \frac{\sin θ_{R}}{Z} \int_{0}^{L_{R}} E_{Σ} (t - \frac{L_{R}}{c} + \frac{L_{R} \cos θ_{R}}{c}) d L (3.17a)$

and for the left side of a dipole

i 1 ⎛ ⎝ ⎜ t, θ R, L R ⎞ ⎠ ⎟ = sin θ R Z \int - L R 0 E Σ ⎛ ⎝ ⎜ t - L R c - L R cos θ R c ⎞ ⎠ ⎟ d L . (3.17b)

$i_{1} (t, θ_{R}, L_{R}) = \frac{\sin θ_{R}}{Z} \int_{- L_{R}}^{0} E_{Σ} (t - \frac{L_{R}}{c} - \frac{L_{R} \cos θ_{R}}{c}) d L . (3.17b)$

The total current flowing through the load of the receiving antenna is

i ⎛ ⎝ ⎜ t, θ R, L R ⎞ ⎠ ⎟ = sin θ R Z ⎡ ⎣ ⎢ \int 0 L R E Σ ⎛ ⎝ ⎜ t - L R c + L R cos θ R c ⎞ ⎠ ⎟ d L + \int - L R 0 E Σ ⎛ ⎝ ⎜ t - L R c - L R cos θ R c ⎞ ⎠ ⎟ d L ⎤ ⎦ ⎥ . (3.18)

$i (t, θ_{R}, L_{R}) = \frac{\sin θ_{R}}{Z} [\int_{0}^{L_{R}} E_{Σ} (t - \frac{L_{R}}{c} + \frac{L_{R} \cos θ_{R}}{c}) d L + \int_{- L_{R}}^{0} E_{Σ} (t - \frac{L_{R}}{c} - \frac{L_{R} \cos θ_{R}}{c}) d L] . (3.18)$

This gives a receiving antenna voltage across the load Z_L:

u Σ ⎛ ⎝ ⎜ t, θ R, L R ⎞ ⎠ ⎟ = sin θ R 1 2 ⎡ ⎣ ⎢ \int 0 L R E Σ ⎛ ⎝ ⎜ t - L R c + L R cos θ R c ⎞ ⎠ ⎟ d L + \int - L R 0 E Σ ⎛ ⎝ ⎜ t - L R c - L R cos θ R c ⎞ ⎠ ⎟ d L ⎤ ⎦ ⎥ . (3.19)

$u_{Σ} (t, θ_{R}, L_{R}) = \sin θ_{R} \frac{1}{2} [\int_{0}^{L_{R}} E_{Σ} (t - \frac{L_{R}}{c} + \frac{L_{R} \cos θ_{R}}{c}) d L + \int_{- L_{R}}^{0} E_{Σ} (t - \frac{L_{R}}{c} - \frac{L_{R} \cos θ_{R}}{c}) d L] . (3.19)$

Let us substitute in this formula a total field E_Σ(t, θ_T) from Equation 3.17:

U Σ (t, θ T, θ R, L T, L R) = sin (θ R) 1 2 \times ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ \int 0 L R A 1 [i (t - L T c - r - L T cos ( θ T ) c - L R c + L R cos ( θ R ) c) - i (t - r c - L R c + L R cos ( θ R ) c)] d L + \int 0 L R A 2 [i (t - r c - L R c + L R cos ( θ R ) c) - i (t - L T c - r + L T cos ( θ T ) c - L R c + L R cos ( θ R ) c)] d L + \int - L R 0 A 1 [i (t - L T c - r - L T cos ( θ T ) c - L R c + L R cos ( θ R ) c) - i (t - r c - L R c + L R cos ( θ R ) c)] d L + \int - L R 0 A 2 [i (t - r c - L R c - L R cos ( θ R ) c) - i (t - L T c - r + L T cos ( θ T ) c - L R c + L R cos ( θ R ) c)] d L ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ . (3.20)

$\begin{array}{l} \begin{matrix} U_{Σ} (t, θ_{T}, θ_{R}, L_{T}, L_{R}) \end{matrix} \\ = sin (θ_{R}) \frac{1}{2} \times (\begin{matrix} \int_{0}^{L_{R}} A_{1} [i (t - \frac{L_{T}}{c} - \frac{r - L_{T} cos (θ_{T})}{c} - \frac{L_{R}}{c} + \frac{L_{R} cos (θ_{R})}{c}) - i (t - \frac{r}{c} - \frac{L_{R}}{c} + \frac{L_{R} cos (θ_{R})}{c})] d L \\ + \int_{0}^{L_{R}} A_{2} [i (t - \frac{r}{c} - \frac{L_{R}}{c} + \frac{L_{R} cos (θ_{R})}{c}) - i (t - \frac{L_{T}}{c} - \frac{r + L_{T} cos (θ_{T})}{c} - \frac{L_{R}}{c} + \frac{L_{R} cos (θ_{R})}{c})] d L \\ + \int_{{- L}_{R}}^{0} A_{1} [i (t - \frac{L_{T}}{c} - \frac{r - L_{T} cos (θ_{T})}{c} - \frac{L_{R}}{c} + \frac{L_{R} cos (θ_{R})}{c}) - i (t - \frac{r}{c} - \frac{L_{R}}{c} + \frac{L_{R} cos (θ_{R})}{c})] d L \\ + \int_{{- L}_{R}}^{0} A_{2} [i (t - \frac{r}{c} - \frac{L_{R}}{c} - \frac{L_{R} cos (θ_{R})}{c}) - i (t - \frac{L_{T}}{c} - \frac{r + L_{T} cos (θ_{T})}{c} - \frac{L_{R}}{c} + \frac{L_{R} cos (θ_{R})}{c})] d L \end{matrix}) . \end{array} (3.20)$

Figure 3.12 shows the dependence of voltage U_Σ(t, θ, L) on angle θ_R. Our calculations assume that the Gaussian current pulse of Equation 3.6 has duration τ such that cτ ≪ L_T excites the antenna and the antennas have equal length L_R = L_T. We select the angle θ_T = 30° because at this angle, the incident field pulse waveform on the receiving antenna is closest to the transmitting-antenna exciting pulse waveform as shown in Figure 3.5 and fulfills the receiving antenna condition cτ ≪ L_R.

Figure 3.12 shows how the receiving antenna voltage across the load is the sum of two pulses; each of these resembles the incident field pulse waveform. These pulses have mirror images because the currents flowing into the load from different antenna arms have opposite directions. The interval between these pulses becomes especially pronounced at small angles θ_R, in which case the field falls on different parts of the receiving antenna with a significant delay. The variation of the voltage pulse waveform across the antenna load as a function of the angle of the current pulse incident onto the antenna has a similar variation in the shape of the field pulse of the transmitting antenna as a function of the observation angle. The receiving antenna’s field pattern (determined as the dependence of the voltage at the load on angular coordinates) demonstrates a similar behavior. As the incident field pulse travels along the receiving antenna, this pattern moves in space, that is, it becomes time-dependent. As a result, in our analysis, we must use the energy pattern of the receiving antenna. We determine the receiving antenna pattern by averaging of the power received from each angular direction during the time required for the pulse to travel along the antenna aperture. The received energy pattern W_R describes the distribution of the energy-density flux received by the antenna from space such that

Images

FIGURE 3.12
For a receiving antenna and the shorter pulse condition cτ ≪ L, the output voltage of the receiving antenna shown in Figure 3.11 depends on the angle of incidence of the wave front with the antenna axis θ_R. This change of received waveform can change the signal spectrum and receiver design.

W R (θ, ϕ) = 1 Z L \int - \infty + \infty u 2 Σ ⎛ ⎝ ⎜ θ, ϕ, t ⎞ ⎠ ⎟ d t . (3.21)

$W_{R} (θ, ϕ) = \frac{1}{Z_{L}} \int_{- \infty}^{+ \infty} u_{Σ}^{2} (θ, ϕ, t) d t . (3.21)$

We can normalize the energy pattern as

W R N (θ, ψ) = W R ( θ , φ ) W R max, (3.22)

$W_{R N} (θ, ψ) = \frac{W_{R} (θ, φ)}{W_{R \max}}, (3.22)$

where

W R max = 1 Z L ⎡ ⎣ ⎢ \int - \infty + \infty U 2 Σ (θ, φ, t) d t ⎤ ⎦ ⎥ max .

$W_{R max} = \frac{1}{Z_{L}} [\int_{- \infty}^{+ \infty} U_{Σ}^{2} (θ, φ, t) d t]_{max} .$

Figure 3.13 presents a family of one-dimensional (ϕ = constant) energy patterns for the above example when L_R/cτ = 1, 3, and 10. If L_R/cτ is close to unity, the energy pattern coincides with the pattern of a half-wave oscillator. As the ratio L_R/cτ increases, the pattern shape changes and it splits into two beams, each having a smaller width.

Images

FIGURE 3.13
UWB receiver antenna patterns depend on the angle of incidence θ_T and the ratio of the antenna length to the pulse length L_R/cτ. (a) When L_R/cτ = 1, the received radiation pattern resembles a half-wave oscillator antenna; (b) and (c) indicate the effects of shortening the pulse length for the cases L_R/cτ = 3 and 10.

3.4.2 UWB Antenna Directivity

The antenna directivity factor D_R of the receiving antenna is defined as the ratio of density W_Rmax of the energy flux received by the antenna in the direction of the pattern maximum and the density W_R0 of the energy flux received by an equivalent isotropic antenna at the same energy-flux density received from space, which means that

D R = W R max W R 0 . (3.23)

$D_{R} = \frac{W_{R \max}}{W_{R 0}} . (3.23)$

We can find the total energy of the field incident onto the antenna through a sphere with radius R as

W = R 2 Z L \int 0 2 π \int 0 π \int - \infty + \infty U 2 Σ (θ, φ, t) sin θ d θ d ϕ d t . (3.24)

$W = \frac{R^{2}}{Z_{L}} \int_{0}^{2 π} \int_{0}^{π} \int_{- \infty}^{+ \infty} U_{Σ}^{2} (θ, φ, t) \sin θ d θ d ϕ d t . (3.24)$

Dividing this energy by the surface area of the sphere surrounding the antenna, we obtain the density of the energy flux received by the equivalent isotropic antenna as

W R 0 = W 4 π R 2 = 1 4 π Z L \int 0 2 π \int 0 π \int - \infty + \infty u 2 Σ (θ, φ, t) sin θ d θ d ϕ d t . (3.25)

$W_{R 0} = \frac{W}{4 π R^{2}} = \frac{1}{4 π Z_{L}} \int_{0}^{2 π} \int_{0}^{π} \int_{- \infty}^{+ \infty} u_{Σ}^{2} (θ, φ, t) \sin θ d θ d ϕ d t . (3.25)$

We can use Equations 3.24 and 3.25 to find expressions that determine the energy directivity factor of a receiving antenna as

D R = 4 π ⎡ ⎣ ⎢ \int - \infty + \infty u 2 Σ ⎛ ⎝ ⎜ θ , ϕ , t ⎞ ⎠ ⎟ d t ⎤ ⎦ ⎥ max \int 0 2 π \int 0 π \int - \infty + \infty u 2 Σ ( θ , φ , t ) sin θ d θ d ϕ d t . (3.26)

$D_{R} = 4 π \frac{[\int_{- \infty}^{+ \infty} u_{Σ}^{2} (θ, ϕ, t) d t]_{\max}}{\int_{0}^{2 π} \int_{0}^{π} \int_{- \infty}^{+ \infty} u_{Σ}^{2} (θ, φ, t) \sin θ d θ d ϕ d t} . (3.26)$

3.5 UWB Antenna Reception and Transmission Patterns and the Reciprocity Principle

Comparison of the formulas for patterns of the considered antennas at radiation and reception shows how the field energy E_Σ(θ) and voltage U_Σ(θ) depend on different values of the delays L_T and L_R in each antenna and the mutual observation angles θ_T and θ_R. This means that the current waveforms induced in these antennas will differ for the general case. Therefore, the antenna radiation and reception patterns will differ.

For example, Figure 3.14 shows the energy patterns of two antennas for the short-pulse case L_T = L_R = 10 cτ. Figure 3.14a shows Case 1, where the antenna, transmitting W_T(θ) and the receiving W_R(θ) demonstrates patterns when antenna No. 1 radiates a short signal at observation angle θ_T = 60° and antenna No. 2 receives this signal at observation angle θ_R = 120°. Figure 3.14b shows Case 2 when antenna No. 2 radiates a short signal at observation angle θ_T = 120° and antenna No. 1 receives this signal at observation angle θ_R = 60°.

Images

FIGURE 3.14
The reciprocity theorem for NB antennas applied to UWB antennas. However, in this case, we receive different patterns for the reception and transmission cases. This illustrates the radiation patterns of two antennas of equal length L_T = L_R = 10 cτ in the shorter pulse condition radiating and receiving at the angles defined in Figure 3.11. (a) Case 1: Antenna No. 1 radiating at θ_T = 60° to No. 2 receiving at θ_R = 120°. (b) Case 2: Antenna No. 2 radiating at θ_T = 120° and No. 1 receiving at θ_R = 60°.

In accordance with the principle of reciprocity, different antennas working in the same regime (radiation or reception) form identical patterns regardless of the antenna’s number and arrangement. In the case of short-pulse UWB signals where cτ ≪ L, we find that the radiation and reception patterns differ greatly. This feature means that we cannot determine the UWB receiving antenna pattern by its pattern in the transmitting regime as we could for NB signal reception and radiation. Despite this, we see that the reciprocity principle holds good for UWB antennas too.

3.6 Special Problems in UWB Signal Detection

High-resolution UWB signals can form the image of a large object with enough detail to permit target recognition. This assumes a pulse length much smaller than the overall size of the target, as opposed to the larger pulse length as in the case of NB radar systems. Before we can image and recognize a target, we must detect the signal. The variations in UWB waveforms depending on the relation of the pulse to antenna length and observation angle discussed earlier may interfere with detection.

Earlier, we examined the variations in the UWB signal waveforms during transmission and reception for the case of a pulse length shorter than the antenna length, which means cτ ≪ L.

Reflection from the target will cause additional signal waveform variations. We can relate the incident signal spectrum U_inc(ω) on the target and reflected signal spectrum U_ri(ω) from the ith local target scattering center as

u r i (ω) = u i n c (ω) K i (ω), (3.27)

$u_{r i} (ω) = u_{i n c} (ω) K_{i} (ω), (3.27)$

where K_i(ω) is the complex frequency characteristic of the ith K_i(ω) local scattering center.

In the case of a relatively narrow frequency band signal, we can assume that the local scattering center has a uniform frequency response and a linear phase response. Therefore, the signal spectrum and the shape of a reflected NB signal from a local scattering center will not change, and we can assume that

u i n c (ω) \approx u r i (ω) . (3.28)

$u_{i n c} (ω) \approx u_{r i} (ω) . (3.28)$

Differences arise only in the changed amplitude and initial phase of the reflected signal. This fact allows processing of a NB signal with the use of a correlator with a reference signal or a matched filter.

In the case of a wide frequency band, we must account for the nonuniform frequency response and nonlinear phase response of a local scattering center. These factors cause changes in the spectrum of a UWB signal during its reflection from the target, which means U_inc(ω) ≠ U_ri(ω). Changes in the signal spectrum cause an additional change in the signal shape. Therefore, under the UWB short-pulse conditions, cτ ≪ L, we must detect a signal with an unknown waveform.

In 2008, Chernyak and I contributed an article for Radiotekhnika [10], which presented a theoretical validation of an optimal detection algorithm for a signal in the form of a packet of identical pulses with an unknown shape and a known repetition rate in the presence of white Gaussian noise. We synthesized the detection algorithm using a priori information about the radiating pulse repetition period T_p.

We assumed that the pulses of duration τ illuminated the target and also assumed a signal pulse repetition rate, which gave us a packet of M identical pulses during an interval small enough to consider the target immobile. With these assumptions, we modeled the signal u_s(t) as the packet of M identical pulses shown in Figure 3.15. This took the form

u s (t) = Σ k = 0 M - 1 u r (t - k T p), (3.29)

$u_{s} (t) = Σ_{k = 0}^{M - 1} u_{r} (t - k T_{p}), (3.29)$

Images

FIGURE 3.15
Detection of reflected UWB signals requires finding ways to identify unknown waveforms produced by the nonuniform frequency and nonlinear phase response of a scattering center for the short pulse cτ ≪ L condition. This shows an unknown signal with a constant repetition period T_p and consistent waveforms used to derive a detection algorithm.

where u_s(t) represents the signal (a pulse with an unknown shape, duration T, and energy E_s) reflected from the target.

If the individual pulses of a packet do not overlap, then

\int - \infty \infty u r (t - k T p) u r (t - m T p) d t = ⎧ ⎩ ⎨ ⎪ ⎪ E s, k = m 0, k \neq m . (3.30)

$\int_{- \infty}^{\infty} u_{r} (t - k T_{p}) u_{r} (t - m T_{p}) d t = {_{0, k \neq m}^{E_{s}, k = m} . (3.30)$

We assumed Gaussian noise with a zero mean value, which means that the input signal is the mean value of the sum of the signal and noise.

For the known signal, the logarithm of the ratio of likelihood functional has the form (see Kazarinov’s Radio-Engineering Systems [11])

ln Λ = ln W s / n [ u ( t ) ] W n [ u ( t ) ] = Σ k = 0 M - 1 \int - \infty \infty u (t) u r (t - k T p) d t - 1 2 Σ k = 0 M - 1 Σ l = 0 M - 1 \int - \infty \infty u r (t - k T p) u r (t - l T p) d t, (3.31)

$\ln Λ = \ln \frac{W_{s / n} [u (t)]}{W_{n} [u (t)]} = Σ_{k = 0}^{M - 1} \int_{- \infty}^{\infty} u (t) u_{r} (t - k T_{p}) d t - \frac{1}{2} Σ_{k = 0}^{M - 1} Σ_{l = 0}^{M - 1} \int_{- \infty}^{\infty} u_{r} (t - k T_{p}) u_{r} (t - l T_{p}) d t, (3.31)$

where u(t) is the received realization of the sum of the signal and noise (or only noise) with duration T₀ > (M − 1)T_p + T.

Taking into account Equation 3.30, we can simplify Equation 3.31 to this form

ln Λ = Σ k = 0 M - 1 \int - \infty \infty u (t) u r (t - k T p) d t - 1 2 Σ k = 0 M - 1 \int - \infty \infty u 2 r (t - k T P) d t . (3.32)

$\ln Λ = Σ_{k = 0}^{M - 1} \int_{- \infty}^{\infty} u (t) u_{r} (t - k T_{p}) d t - \frac{1}{2} Σ_{k = 0}^{M - 1} \int_{- \infty}^{\infty} u_{r}^{2} (t - k T_{P}) d t . (3.32)$

We can apply the adaptive approach because the unknown signal u_r(t) is not random and has some predictability. The essence of our approach lies in application of an algorithm that is optimal for known signal parameters. In this algorithm, we can replace the unknown parameters with their maximum-likelihood estimates [12]. In this case, we should estimate the whole signal as a function of time rather than individual signal parameters [13].

In order to obtain the estimate, we consider the logarithm of the likelihood functional [14]:

$ℜ_{1} = - \frac{1}{2} \int_{- \infty}^{\infty} [u (t) - Σ_{k = 0}^{M - 1} u_{r} (t - k T_{P})]^{2} d t . (3.33)$

The maximum of the functional ℜ₁, when considered as a function of $Σ_{k = 0}^{M - 1} u_{r} (t - k T_{P})$ , corresponds to the minimum of the expression in square brackets. Since the duration of received realization u(t) of the sum of the signal and noise comprises all arriving signals, the minimum of the expression in square brackets occurs under the condition that, within each time interval containing the signal, this signal is equal to the received realization:

$\begin{matrix} \begin{matrix} {\hat{u}}_{0} (t - k T_{P}) = u (t) & for \end{matrix} & t \in (k T_{P}, k T_{P} + T) . \end{matrix} (3.34)$

Let us perform the change of variables in Equation 3.33 using t₁ = t - KT_p. This gives us

û₀ (t₁) = u(t₁ + kT_p) for t₁ ∈ (0, T).

Estimates of the same function u_r(t₁) obtained within the time intervals spaced by repetition period T_p are statistically independent because Gaussian noise within these intervals is independent. Hence, for t₁ ∊ (0, T), the maximum-likelihood estimate û₀(t₁) refined with the use of M measurements has the form

$\begin{matrix} \begin{matrix} \bar{{\hat{u}}_{r} (t_{1})} = \frac{1}{M} Σ_{k - 0}^{M - 1} u (t_{1} + k T_{Π}) & for \end{matrix} & t_{1} \in (0, T) . \end{matrix} (3.35)$

After the change of variables t₁ = t - KT_p in Equation 3.32, we obtain

$\ln Λ = Σ_{k = 0}^{M - 1} \int_{0}^{T} u (t_{1} + k T_{P}) u_{r} (t_{1}) d t_{1} - \frac{1}{2} Σ_{k = 0}^{M - 1} \int_{0}^{T} u_{r}^{2} (t_{1}) d t_{1} . (3.36)$

Replacing $\begin{matrix} \bar{{\hat{u}}_{r} (t_{1})} \end{matrix}$ with the obtained estimate of u_r(t₁) from Equation 3.35, by omitting the subscript of t₁ and the inessential factor 1/M and performing some simple transformations, we obtain from Equation 3.36 an adaptive detection algorithm that is optimal according to the criterion of the generalized likelihood ratio:

$ℜ = U_{out} = \int_{0}^{T} [Σ_{k = 1}^{M - 1} u (t + k T_{P})]^{2} d t_{<}^{>} U_{t h r e s h o l d}, (3.37)$

where U_threshold defines the detection threshold.

Thus, the optimum algorithm consists of the summation of the segments (with duration T each) of the received realization within those time intervals where we expect signals. The calculation of the energy of this sum and comparison of the obtained energy with a threshold determined by the specified probability of a false alarm indicates the presence of an unknown waveform signal. This describes the well-known energy detector that uses the energy of each pulse from the packet and the energy of correlation coupling between these pulses. Figure 3.16 shows the block diagram of a detector for unknown waveform signals.

In the case of a sufficiently rapid motion of the target relative to the radar, the reflected signal packet pulses u_s(t) become different. When this happens, we may assume at least two adjacent pulses that have identical waveforms. Therefore, let us consider the important practical case of processing two signals, where M = 2. Figure 3.17 shows the block diagram of a detector for this specialized case. At M = 2, we can represent the optimum algorithm from Equation 3.37 in the form

$ℜ = U_{o u t} = \int_{0}^{T} u^{2} (t) d t + \int_{0}^{T} u^{2} (t + T_{P}) d t + 2 \int_{0}^{T} u (t) u (t + T_{P}) d t_{<}^{>} U_{t h r e s h o l d} . (3.38)$

The first and second integrals in Equation 3.38 describe an algorithm for calculation of the energy of signals received within two adjacent periods, and the third integral determines an algorithm of mutual correlation processing of these signals.

It is seen from Equation 3.38 how the analyzed detector can take the form of two suboptimal detectors as shown in Figure 3.18. The first detector uses the sum of the energies of realization segments corresponding to expected signals in the first and second repetition periods. The second detector works as an inter-period correlation detector (IPCD). My colleague Fedotov and I proposed the IPCD in Radiotekhnika in 1998 [15]. The block diagrams in Figures 3.17 and 3.18 demonstrate these features of an optimal detector:

Images

FIGURE 3.16
The signal detector block diagram showing the summation of each signal period, squaring the summation and integrating to get a detectable output.

Images

FIGURE 3.17
Block diagram of the signal detector for M = 2 pulses.

Since the noises arriving from adjacent repetition periods are independent, the integrator input takes either the squared sum of the independent segments of normal processes, as shown in Figure 3.17, or the sum of two squared segments and the product of these segments, as shown in Figure 3.18. In both cases, the integrator input has a probability distribution substantially different from a normal distribution.
As the integration time increases, the probability distribution of the process at the integrator output approaches a normal distribution. This time depends on the duration T = nτ of the signal reflected from the target rather than the duration τ of the radiated signal. Hence, the target’s radial length, L = ncτ/2 determines the integration time; the degree of normalization of the distribution of the process at the integrator output; and, as a result, the level of the detector threshold.

Let us compare the efficiencies of the optimal detector of Equation 3.38 and suboptimal detectors. Figure 3.19 presents detection characteristics obtained via calculations with MATLAB®, for the optimum algorithm (curve 2), the IPCD algorithm (curve 3), and the sum of energy detectors (curve 4). For comparison, (curve 1) shows the detection characteristic for a classical detector of an exactly known (deterministic) signal. The probability of a false alarm is P_fa = 10^-3. We evaluated the suboptimum detector characteristics for a target whose radial length comprises 16 resolution elements. With such accumulation in the process of integration, the output characteristics of both optimal and suboptimal detectors can be approximately considered Gaussian quantities. For M = 2, curve 3 merges with curve 4. As seen from the figures, the absence of information on the signal shape causes substantial loss. This loss is due to the absence of a noise-free reference of the detected signal in the receiver.

Images

FIGURE 3.18
The combination of three suboptimum detectors in the case of M = 2 UWB-pulsed signal. The integrator of this detector takes either the squared sum of the individual processes or the product of those two segments as shown in Figure 3.18. The integrator input has a probability distribution substantially different from a normal distribution.

Images

FIGURE 3.19
Suboptimum detector’s probability of detection versus SNR (dB) for probability of false alarm P_fa = 10^-3. Curve 1: classical detector characteristics for a deterministic signal; Curve 2: optimum algorithm; Curve 3: ICPD detector; Curve 4: sum of energy detector coincides with the ICPD detector curve 3.

It follows from curves in Figure 3.19 that for the considered (typical) case, the suboptimal detectors have an efficiency that is only slightly lower than that of the optimum detector. Therefore, in some situations, a sufficiently simple IPCD algorithm can find a practical application.

Notice that the block diagrams in Figures 3.17 and 3.18 have optimal characteristics for detecting an immobile target and ideal conditions; for example, all received pulses have identical shapes, a known radial length, and a known signal duration T. This ideal case helps to estimate the loss of efficiency for a moving target with an unknown length. Similarly, the classical theory allows estimating the loss of detection efficiency for unknown signal parameters by comparing the case with the detection characteristics of a deterministic signal with completely known parameters. In real situations, with unknown signal, target velocity, and length, this case will require a multichannel system. Detection could possibly use a time sequential search along the signal delay to determine the target range; however, this also requires a parallel search along the duration of the integration interval produced by the target’s radial length.

3.7 Target RCS Measurement with UWB Signals

Conventional radar signals generally have a pulse length of ct ≪ L, which is much greater than the target size L. Illuminating a target produces multiple overlapping reflections in space from the different target scattering points and appears as a summation signal at the reception point. For NB signals, the RCS of a far-field target has the well-known expression:

$σ = 4 π R^{2} \frac{E_{2}^{2}}{E_{1}^{2}}, (3.39)$

where R indicates the target range, E₁ indicates the transmitted signal field intensity amplitude at the target point scattering surface, and E₂ indicates received scattered field intensity amplitude of the of target scattered signal at a reception point.

In the case of UWB signal where cτ ≪ L with distances Δl > cτ/2 between the target’s reflective points, the signals reflected from these target points appear separated in space. The received signal will appear as a sequence of pulses and time-shifted relative to each other by different time intervals. Each reflected pulse will have different amplitudes and polarities. As a result, the signal amplitude E₂ at the reception point becomes analytically indeterminate.

In the case of the NB signal, we can use the concept of a generalized RCS, σ, where [16]

$σ = 4 π R^{2} \frac{W_{2}}{W_{1}}, (3.40)$

where W₁ = ∫_τ₁ Π₁ (t)dt represents the signal energy of the radar signal at the target point, W₂ = ∫_τ₁ Π₁ (t)dt shows signal energy scattered by the target at the reception point, Π₁(t) indicates the illuminating signal Poynting vector (the energy-flux density) acting during time τ₁ at the target point, and Π₂(t) shows the scattered signal Poynting vector (the energy-flux density) acting during time τ₂ at the reception point.

To evaluate the applicability of Equation 3.40, we can compare the RCSs obtained in the cases of a target illuminated with an NB signal and a UWB signal. Our analysis uses a simple target of clustered scattering points represented by two elementary radiators with distance Δ_l between them as shown in Figure 3.20a. We shall assume that the reflectors do not influence each other.

Figure 3.20b shows the NB signal Δ_l ≪ cτ and the well-known result. The field pulse illuminating the target has carrier e₁(t) = E₁cosωt and a secondary radiator scattered field at the receiver:

$e_{1} (t) = E_{2 (1)} \cos ω (t - t_{1}) + E_{2 (2)} \cos ω (t - t_{2}) = E_{2} \cos (ω t - ϕ), (3.41)$

Images

FIGURE 3.20
Determining the target RCS for an UWB pulse with cτ < Δl requires a different analysis. (a) The multiple scattering center target model. (b) The reflected signal from an NB signal cτ > Δl. (c) The UWB pulse reflection model showing the separate reflections from each scattering center.

where the subscript in parentheses denotes the reflector number and $φ = \frac{4 π Δ l}{λ} .$ .

This makes the total amplitude of field E₂ at the receiver

$E_{2}^{2} = E_{2 (1)}^{2} + E_{2 (2)}^{2} + 2 E_{2 (1)} E_{2 (2)} \cos φ . (3.42)$

In accordance with Equation 3.36, the target RCS is

$σ_{Σ 2} = 4 π R^{2} \frac{E_{2}^{2}}{E_{1}^{2}} = σ_{1} + σ_{2} + 2 \sqrt{σ_{1} σ_{2}} \cos φ . (3.43)$

If φ = 0 and σ₁ = σ₂ = σ, the RCS has a maximum value of σ_Σ = 4σ. Accordingly, in the case of N reflectors with equal RCSs σ and equal-phase reflected fields, the RCS of a clustered radiator is σ_ΣN = N²σ. In the general case of a random phase ϕ with arbitrary values ranging from 0 to 2π, the NB RCS by a NB signal becomes the mean value

$σ_{N B} = {\bar{σ}}_{Σ N} = N σ . (3.44)$

In the case of a UWB signal where Δl > cτ, the considered secondary radiator sees an illuminating field pulse with a duration t and an energy-flux density Π₁(t). Because of the “small” pulse length Δl > cτ, we have two nonoverlapping field pulses reflected from two reflectors to the receiver. Generally, these pulses will have different energy-flux densities Π₂₍₁₎(t) and Π₂₍₂₎(t) with different durations τ₁ and τ₂.

In this case, Equation 3.40 describes the RCS of each secondary radiator as

$\begin{matrix} \begin{matrix} σ_{1} = 4 π R^{2} \frac{W_{2 (1)}}{W_{1}} & and \end{matrix} & σ_{2} = 4 π R^{2} \frac{W_{2 (2)}}{W_{1}}, \end{matrix} (3.45)$

where W₁ = ∫_τ Π₁ (t)dt shows the energy of the radar sounding signal at the target point and the terms W₂₍₁₎ = ∫_τ₂ Π₂₍₁₎(t)dt and W₂₍₂₎ = ∫_τ₂ Π₂₍₂₎(t)dt describe the signal energies, which arrive at the receiver from the first reflector at time t₁ and from the second reflector at time t₂ as shown in Figure 3.20b.

When signals from each reflector arrive at the receiver separately, then the reflector causing the highest amplitude determines the apparent RCS of the entire radiator. In this case, we do not use the energy arriving at the receiver from the smaller reflector. We can treat this circumstance as a loss. If the number of radiators increases, this loss likewise increases.

In order to maximize the RCS of a target illuminated by a UWB signal, we need to use the energy of the signal at the output of the optimal processing system as described in Section 3.6. We can estimate the RCS of the given secondary radiator consisting of two identical reflectors (M = 2) in the case of optimal processing of two pulses traveling one after another with an interval equal to the repetition period. After substituting two reflected pulses in each period (where the subscript denotes the reflector number) into Equation 3.40, we obtain an expression proportional to the energy of the received signal at the output of the processing system:

$\begin{array}{l} U_{o u t} = \int_{0}^{T} u_{1}^{2} (t) d t + \int_{0}^{T} u_{2}^{2} (t) d t + \int_{0}^{T} u_{1}^{2} (t + T_{r}) d t + \int_{0}^{T} u_{2}^{2} (t + T_{r}) d t + 2 \int_{0}^{T} u_{1} (t) u_{2} (t) d t \\ + 2 \int_{0}^{T} u_{1} (t + T_{r}) u_{2} (t + T_{r}) d t + 2 \int_{00}^{T} u_{1} (t) u_{2} (t + T_{r}) d t + 2 \int_{0}^{T} u_{2} (t) u_{1} (t + T_{r}) d t + 2 \int_{0}^{T} u_{1} (t) u_{1} (t + T_{r}) d t . \end{array} (3.46)$

The first four integrals determine the energy of the first and second signal pulses in the first and second repetition periods. The next six integrals determine the mutual energy of various pulse pairs. The first four of these six integrals describe the mutual energy of the pulses nonoverlapping in time and, therefore, take values of zero. The remaining two integrals determine the mutual energy of two overlapping (after a delay T_r) pulse pairs as shown in Figure 3.20c. This gives the total energy of the received signal, which determines the output voltage of the processing system as

$\begin{array}{l} W_{2} = \int_{0}^{T} u_{1}^{2} (t) d t + \int_{0}^{T} u_{2}^{2} (t) d t + \int_{0}^{T} u_{1}^{2} (t + T_{r}) d t + \int_{0}^{T} u_{2}^{2} (t + T_{r}) d t + 2 \int_{0}^{T} u_{1} (t) u_{1} (t + T_{r}) d t \\ + 2 \int_{0}^{T} u_{2} (t) u_{2} (t + T_{r}) d t = W_{2 (1, 1)} + W_{2 (1, 2)} + W_{2 (2, 1)} + W_{2 (2, 2)} + 2 W_{2 (1, 1 - 2, 1)} + 2 W_{2 (1, 2 - 2, 2)} \end{array} (3.47)$

where the first and second subscripts in parentheses indicate the period number and the reflector number, respectively.

If the energy of the radar’s sounding signal at the target point is W₁, then after substituting the obtained energies into Equation 3.40, we have

$σ_{Σ 2} = 4 π^{2} \frac{W_{2 (1, 1)} + W_{2 (1, 2)} + W_{2 (2, 1)} + W_{2 (2, 2)} + 2 W_{2 (1, 1 - 2, 1)} + 2 W_{2 (1, 2 - 2, 2)}}{W_{1}} = σ_{1} + σ_{1} + σ_{2} + σ_{2} + 2 σ_{1 - 2} + 2 σ_{2 - 1} . (3.48)$

If σ₁ = σ₂ = σ_1-2 = σ_2-1 = σ, then we get σ_Σ2 = 8σ. We get this RCS value by considering the energy reflected from the target within two sounding periods. For one sounding period, the ratio of the energy at the reception point to the energy at the target point gives the following RCS: σ_Σ2 = 4σ. Accordingly, for N secondary radiators with equal RCSs σ, the total RCS of a clustered radiator is

$σ_{Σ N} = N^{2} σ . (3.49)$

Comparing RCSs from Equations 3.44 and 3.45 that correspond to illumination of the same target consisting of N reflectors by NB and UWB signals, respectively, we see that the inequality $σ_{Σ N} \geq {\bar{σ}}_{Σ N}$ is satisfied. Thus, for a given target σ_UWB ≥ σ_NB, which implies a larger RCS for the UWB signal than for an NB signal. Returning to our earlier discussion and Figure 3.19, this enhanced RCS characteristics might compensate for the reduced probability of detection at a given signal-to-noise ratio (SNR) and provide some advantages in detection at longer ranges.

In the general case of arbitrary RCSs of individual reflectors of the target, target RCS σ_Σ depends on the amplitude of the output signal of the optimal processing system, which is proportional to the signal energy. Since the energy of the signal reflected from the target determines the RCS below, we refer to σ_Σ as the energy RCS.

3.8 UWB Radar Range Equation: Limitations and Features of UWB Radar Applications

The traditional range equation of narrowband radio systems usually involves parameters related to the signal harmonic (or quasi-harmonic) shape. This signal amplitude determines the peak and mean signal power and the target RCS. The radiation and reception patterns, which determine the directivity factors of the receiving and transmitting antennas, result from the spatial interference of harmonic oscillations radiated by the elementary antenna elements.

In UWB wireless systems, when cτ ≪ L the shape of the signal varies during the process of transmission and reception. Therefore, we should determine the range R of these systems with the help of the parameters expressed through the signal energy.

For UWB radars, the range equation has the following form:

$R = \sqrt[4]{\frac{{WD}_{T} σ_{Σ} S}{(4 π)^{2} ρ q N_{0}}}, (3.50)$

where W = ∫_τ p(t)dt gives the energy of the radiated signal when p(t) indicates the radiated power as a function of time, D_T the energy directivity factor of the transmitting antenna, σ_Σ the energy RCS of the target, S the effective (energy) surface area of the receiving antenna that determines the receiving antenna energy directivity factor, ρ the total losses in all systems of the radar, q the threshold value of the energy SNR, and N₀ the spectral density of the noise power.

For communication systems, we can write this equation as

$R = \sqrt[2]{\frac{{WD}_{T} S}{(4 π)^{2} ρ q N_{0}}} . (3.51)$

Energy range Equations 3.50 and 3.51 indicate a generalized relationship because this equation applies to the calculation of the wireless systems parameters at any relationships between signal duration cτ and antenna (and/or target) dimension L. However, if cτ ≫ L, then only Equations 3.50 and 3.51 apply for range calculations.

For example, let us consider the factors limiting the range of UWB radars. The UWB radars with nanosecond and picosecond pulses can attain high information efficiencies. In the case of simple short pulse, the radiation energy required for target detection means increasing the peak power P_peak assuming that the other radar and target parameters remain the same.

Simple calculations show how UWB radar target detection at large distances (100 km and more) requires a transmitter with a peak power of several gigawatts or even several tens of gigawatts. Unfortunately, gigawatt pulse generators pose many production problems and require construction with high-strength electrical components and dielectric materials. They can also generate unwanted environmental side effects from the emission of X-rays and direct high field strength electromagnetic radiation.

In principle, lowering the peak power in long-range UWB radars via passage to complex coded signals with large values of the bandwidth-duration product and subsequent matched filtering can provide a solution to higher signal energy. However, the variations in the shape of the UWB signal during radar observation prohibit the use of this method.

For high-power UWB radars, we can get the required performance from active pulse antenna arrays without the problems of gigawatt generators. However, the antenna array requires ensuring high system time stability for precise control of the multiple array generators, which will have individual characteristics. Note that the application of active pulse antenna arrays in the UWB radars makes these radars complex and expensive until the development of better manufacturing methods. (Editor’s Note: see Chapters 14, through 16 for examples of UWB antenna array design, signal acquisition, and a time calibration system, which measures the delay in each transmitter and receiver channel. The imaging software then applies measured time delay information to range measurements and imaging processes.)

Nevertheless, the main obstacle to the development of UWB radars with relatively long ranges lies in government restrictions on the emitted spectrum power. The high-information density of UWB systems requires wide segments of the frequency spectrum. The wide spectra of UWB radars could generate interference for many other radio-based systems operating in the same segments of the spectrum, for example, satellite navigation systems (Global positioning system [GPS], GLObalnaya NAvigatsionnaya Sputnikovaya Sistema or GLObal NAvigation Satellite System [GLONASS], GALILEO, etc.), satellite search and emergency systems, and air-traffic-control systems, and so on. Therefore, the electromagnetic compatibility of the UWB radars with other radio systems becomes one of the main problems limiting the development of these systems. Unlicensed short-range UWB radars now have limitations set by the First Report and Order of the U.S. Federal Communications Commission (FCC), published April, 2002 and supplemented in 2003 and 2004 [17,18]. This document sets the maximum UWB system radiation level limitations in different frequency bands in the form of the so-called masks. Figure 3.21 shows an example of a mask. Today, this document has become a reference in many countries. The regulations cover many different classes of unlicensed equipment, for example, surveillance systems, through-wall radar, ground-penetrating radar, medical imaging, and so on. Higher power equipments operating in specialized roles in remote locations might receive special permissions or licenses for needs such as defense-related systems. Chapter 4 presents the legal restrictions on different classes of unlicensed UWB systems set by American and European Union regulating organizations.

For long-range UWB radars, for example, space-based UWB radars, the electromagnetic compatibility with other radio systems operating in the same segments of the spectrum raises many questions. At this time, UWB radar technology development has emphasized low-power and short-range systems operating from centimeter ranges to tens of meters in air and other media such as the ground. Such radars can find wide specialized use and have become promising commercial products.

Images

FIGURE 3.21
American and European regulating bodies specify the radiation levels for unlicensed UWB equipment by masks showing spectrum areas and the maximum permitted power in a specified bandwidth (usually 50 MHz) in that region. Notice the low section between 0.96 and 1.61 GHz to prevent interference with GPS and communications systems. Chapter 4 has a complete listing of regulations for all covered classes of UWB equipments.

3.9 Design of Short-Range UWB Radar for Practical Applications

R&D Center of UWB Technologies of Moscow Aviation Institute has developed short-range radars for remote and contactless measurements of human physiological vital signs including respiratory and heart rates. These short-range radars have many practical applications including medical investigations; monitoring of patients; searching for people behind walls or buried under ruined building or avalanches; measurement of human vital signs in extreme or hazardous conditions and in hospital wards, and in medical diagnosis.

The operating feature of such radars is the combination of a high repetition rate of the sounding pulses and low velocity of the observed object. This combination allows coherent accumulation of large pulse packets (containing several hundred thousand or several million sounding pulses) within sufficiently large time intervals (of about 0.1 s) in which the observed objects are considered stationary. This circumstance allows substantial lowering of the peak and mean power of the radar transmitter, a possibility that is especially important for UWB radars operating under stringent requirements on the electromagnetic compatibility with radio equipment operating in the same frequency band.

Another feature of such radars is related to the back-and-forth motion of the observed objects (the human thorax, the human heart), which is unusual for radars. This feature creates special conditions for the radar observation of such objects. The point is that the shape (and the spectrum) of the output signals of quadrature channels substantially depends on the ratio of amplitude ΔR of the back-and-forth motion and wavelength λ of the radiated signal. If ΔR < λ, the output signal of quadrature channels has shapes that are close to the real trajectory of the object. However, when the amplitude of this motion becomes comparable with the wavelength of the radiated signal, the quadrature output signals (and their spectra) acquire complex shapes that are substantially different from the actual trajectory of the moving object. This case requires the application of special processing that restores the signal shape corresponding to the true object motion via calculation of the arctangent of the ratio of the output signals in two quadrature channels [19].

Separation of the signals corresponding to the respiratory rate and heart rate that are received by the UWB radar during observation of biological subjects likewise requires a nonstandard approach. The period of the thorax’s motion and the period of the heart’s motion are unstable over a long time interval. This violation of the periodicity of motion (or variability of the rate) is an important factor in medicine. Therefore, frequency separation of the respiratory and heart signals cannot be performed with conventional (analog or digital) filters, which perform averaging of the signals in the course of filtering and, hence, remove the information on the signal variability. We can use the so-called temporal filtering to separate signals with nonperiodic characteristics. In this filtering, one (usually, low-frequency) signal is approximated in each period by the segments of polynomials of different orders. The obtained approximation is the thorax’s motion separated from the sum signal. We can separate the heart’s motion from the sum signal by subtracting the approximated thorax motion. This filtering method retains all violations of the periodicity of motion, and the obtained data can be used for diagnostics.

As an example, we present some short-range UWB radars developed and manufactured at the R&D Center of UWB Technologies of Moscow Aviation Institute (http://www.uwbgroup.ru) and describe the characteristics and the operating conditions of these radars.

3.9.1 UWB Radar for 24-h Measurement of Patients’ Heart and Respiration Rates

The radar can remotely measure human respiration and heart rates at distances from 30 cm to 3.5 m, as shown in Figure 3.22a. The 2-ns radiated pulse duration with a repetition frequency of 2 MHz and the pulsed power of 0.4 W has a 240-μW mean power output. The radar delivers numerical data about cardiac and respiration rates at given time intervals (e.g., 1 min), which are sent to the medical staff via a wired or wireless communication channel. A computer monitor can also display the radar information as curves showing movements of a human chest and heart. The radar can give an alarm if the heart or respiration rates change outside the given thresholds. We designed this radar for use in resuscitation, burn, and postnatal wards.

Images

FIGURE 3.22
Example of UWB radars for biological monitoring. (a) The patient monitoring radar for heart and respiration rate measurements in hospital wards. (b) Prototype operator-monitoring radar for use in hazardous environments, at implementation of tests on computer or physical training simulator. (c) A through-wall radar for detecting the presence of living humans behind walls.

3.9.2 UWB Radar for Remote Human Physiological Measurements

We designed this radar to monitor the condition of human operators performing dangerous work, such as locomotive drivers and men on duty at atomic power stations or missile installation control panels. This radar can continuously and noninvasively monitor human conditions at ranges up to 10 m. Automatic diagnostic algorithms can provide the quantitative estimate of a human operators reactions and health upon changes in environmental conditions, which is expressed in conventional units of measurement—with numbers of the 10-point system approved for investigations of different human groups.

The radar registers human physiology parameters at implementation of tests on a computer, as shown in Figure 3.22b, or on a physical training simulator. It can display information on a computer monitor in the form of curves showing movements of the operators’ chest and heart.

3.9.3 UWB Radar for Through-Wall Detection of People

This radar can detect moving and stationary persons by measuring the chest movements of persons behind brick or concrete walls with thickness of up to 50 cm or in smoke, fog, and similar obscurants as shown in Figure 3.22c. The radar operator can set up the radar at a distance of up to 10 m in front of an obstacle and detect persons at the distance of 10 m behind it. The radar resolution with respect to the distance is 60 cm. Information about detected men is displayed on the illuminated display with a scale indicating the range beyond an obstacle. The radar can also be connected to a computer monitor for display at a remote location. This low-cost radar features simple operation.

The modified version of the radar can take multiposition measurements for exactly locating persons trapped under damaged buildings. Chapter 14 describes several through-wall radar systems.

Chapter 9 discusses some cases of practical medical applications of UWB radars, including my work on monitoring of patients’ vital signs. It includes a complete description of the patient monitoring radar developed at the R&D Center of UWB Technologies of Moscow Aviation Institute together with Dr. Teh-Ho Tao from Industrial Technology Research Institute (Taiwan) during 2008. The chapter also describes other applications of UWB radars in medical measuring and imaging. Several UWB radar-based devices have also gone into commercial development.

3.10 Conclusions

One of the originators of the UWB area of research, Dr. Harmuth, wrote as follows in 1981 [8]:

The relative bandwidth η can have any value in the range 1 ≥ η ≥ 0. Our current technology is based on a theory for the limit η → 0. Both theory and technology that apply to the whole range 1 ≥ η ≥ 0 will have to be more general and more sophisticated than a theory and technology that apply to the limit η → 0 only. Taking the Radar Handbook (Skolnik, 1970) as a representative summary for the limit η → 0 gives some idea of what one should reasonably expect for the general case 1 ≥ η ≥ 0. It took 40 years of radar development, including several technology-advancing wars, before the Radar Handbook could be compiled, and this indicates that a great deal of patience will be required before a comparable summary can be assembled for the general case.

These words are valid even today. Now, almost 30 years later, the insufficiency of the theoretical basis for the development of UWB technique and technology as a system and as a tool for the design of individual devices, especially antenna systems, remains an obstacle to further progress. Therefore, I hope to attract the interest of the specialists working in this field to the viewpoint presented here, especially because the new capabilities provided by UWB wireless systems for improving the quality and increasing the amount of the transmitted information attract yearly increasing funds and intellectual resources in many countries for the development of such systems.

Acknowledgments

I am grateful to Lev Astanin and Viktor Chernyak (Russia), as well as James D. Taylor and Stephen Johnson (USA), for their continued support of this study and their abundance of friendly advice. This work was executed with financial support of the Ministry of Education and Science of the Russian Federation.

References

1. Shannon, C. and Weaver, W., The Mathematical Theory of Communication, University of Illinois Press, Urbana, IL, 1949.

2. Immoreev, I., Main features of UWB radars and differences from common narrowband radars. In J.D. Taylor (ed.) Ultra-Wideband Radar Technology, CRC Press, Boca Raton, FL, 2000, Chapter 1, pp. 1-33.

3. Immoreev, I. and Sinyavin, A., Radiation of ultra-wideband signals. Antennas, Radiotekhnika, 2001, 1, p. 8 [in Russian].

4. Goldstein, L. and Zernov, N., Electromagnetic Fields and Waves, Soviet Radio, 1971 [in Russian].

5. Zaiping, N., Radiation Characteristics of traveling-wave antennas excited by nonsinusoidal currents. IEEE Transactions on Electromagnetic Compatibility, 25, 24, 1983.

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12. Repin, V.G. and Tartakovskii, G.P., Statistical Synthesis under Prior Uncertainty and Adaptation of Information Systems. Sovetskoe Radio, Moscow, 1977 [in Russian].

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14. Van Trees, H.L., Detection, Estimation and Modulation Theory, vol. 1. Wiley, New York, 1968.

15. Immoreev, I. and Fedotov, D., Optimum processing of radar signals with unknown parameters, Radiotekhnika, 1998, p. 84 [in Russian].

16. Brikker, A.M., Zernov, N.V. and Martynova, T.E., Scattering properties of antennas at action of nonharmonic signals, Radiotekhnika i Elektronika, 45, 5, 2000. [in Russian].

17. Federal Communications Commission (FCC), First Report and Order in the Matter of Revision of Part 15 of the Commission’s Rules Regarding Ultra Wideband Transmission Systems, FCC 02-48, ET Docket 98-153, April 2002.

18. Federal Communications Commission (FCC), Second Report and Order and Second Memorandum Opinion and Order, FCC 04-285, ET Docket 98-153, December 2004.

19. Immoreev, I., Radar observation of objects, which fulfill back-and-forth motion, Ultra-Wideband, Short-Pulse Electromagnetics 9, Springer, New York, 2010. Part 7, p. 435.

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Table of Contents for 3 Signal Waveform Variations in Ultrawideband Wireless Systems: Causes and Aftereffects

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3 Signal Waveform Variations in Ultrawideband Wireless Systems: Causes and Aftereffects