4

FUNDAMENTALS OF RADAR

This chapter reviews basic theories of radar signal processing. Section 4.1 describes the principles of radar operation. Section 4.2 covers the basic configuration of a radar system. Section 4.3 and 4.4 discuss radar range equations, radar cross section, and clutter. Section 4.5 explains the Doppler frequency shifts from the wave propagation viewpoint. Radar resolution and ambiguity function are covered in Section 4.6.

4.1 PRINCIPLES OF RADAR OPERATION

By transmitting and receiving electromagnetic waves at regular time intervals, the radar is capable of identifying the distance from the target to the radar and/or the moving speed of the target. The detection and ranging functions can be accomplished either by timing the delay between transmission of a radio pulse and its subsequent return, or by calculating the difference in frequency between the transmitted signal and the echo signal. The pulse-based timing delay normally employs the amplitude modulation (AM) technique. Conversely, the frequency-difference-based ranging and detection method usually employs the linear frequency modulation scheme.

Consider the simple case of a radar system where a sequence of an AM pulse is transmitted. In this case, the information signal is a single pulse repeated at regular time intervals. The commonly used radar carrier-modulated pulse train is shown in Fig. 4.1.

In Fig. 4.1, the pulse repetition interval (PRI) is represented as T, which is the time interval between the start of one pulse and the start of the next pulse. The pulse repetition frequency f _PRF is the inverse of PRI; that is, f _PRF = 1/PRI. The pulse repetition interval T is equal to the sum of the pulsewidth time T_p and the receiver listening time. If the target delay time is τ, then its range can be computed by the following equation with c as the speed of light or EM wave:

FIGURE 4.1 Transmitter and receiver pulse trains.

The minimum and the maximum unambiguous detection range can be computed as follows:

Range ambiguity occurs when continuous pulse streams are sent and continuous echoes from the targets are received by the radar. Since all the pulses are identical, the radar cannot identify which echo originated from which pulse, and this gives rise to an ambiguity in the range measurement.

Figure 4.2 shows the problem of range ambiguity due to short pulse repetition period T, where a series of transmitted pulses and received echoes are displayed. In this case, target 2 exceeds the maximum unambiguous range. Echoes from target 1 show up after every transmitted pulse, but echoes from target 2 appear only after the second pulse was transmitted. For this type of pulsed radar waveform, the radar does not have sufficient data to decide which range corresponds to the correct target.

The most common method for resolving range ambiguity involves using multiple f _PRF values. Since the amount of range ambiguity depends on the f _PRF, it will change when the f _PRF is changed, as stated in Eq. (4.1b). Switching f _PRF gives the additional advantage of uncovering targets that are at the blind range for one or some of the f _PRF values. Such targets would be invisible at one or some f _PRF levels but clearly visible at the others.

FIGURE 4.2 Pulse repetition period and range ambiguity.

Range resolution is the ability of a radar system to distinguish between two or more targets at different ranges. Two pulses with duration T_p and separated by a time interval Δt are resolvable if Δt ≥ T_p. As shown in Fig. 4.3, the range resolution is then equal to one-half of the pulsewidth time, or

where the factor comes from the round-trip traveling time of the radar pulse.

FIGURE 4.3 Range resolution.

The duty cycle of a radar pulse stream is defined as the ratio of the time ON to the time OFF of the transmitter. The equation for computing a duty cycle is

4.2 BASIC CONFIGURATION OF RADAR

A typical radar system can be functionally divided into six essential blocks as shown in Fig. 4.4. These blocks and their interrelationships are depicted in Fig. 4.4.

FIGURE 4.4 Block diagram of a radar system.

4.2.1 Waveform Generator

The waveform generator produces and controls the waveform to be modulated and transmitted by the transmitter. Depending on the application, amplitude modulation, linear frequency modulation, and stepped frequency modulation are three examples of signal modulation techniques provided by waveform generators.

4.2.2 Transmitter

The transmitter generates powerful pulses and/or waveforms of electromagnetic energy at precise time intervals and sends them to the antenna block. Radar transmission normally utilizes either resonating microwave tubes or solid-state devices for high-powered transmission.

4.2.3 Antenna System

The antenna system routes the energy from the transmitter and radiates it in a directional beam. It also picks up the returning echo energy and passes it to the receiver with minimal loss of energy. The antenna system includes a transmitting function and a receiving function. The transmitting function has an antenna, as well as a transmission line or waveguide from the transmitter to the antenna. The receiving function consists of the transmission line or waveguide from the antenna to the receiver. When the transmitter and receiver operate on separate antennas for signal transmission and receiving, the system is called a bistatic radar system. When the same antenna is used for both transmitter and receiver, it is called a monostatic radar system. In this situation, a duplexer, which is essentially a microwave circuit or electronic switch, is required. The duplexer connects the antenna to the transmitter and disconnects the antenna from the receiver for the duration of the transmitted pulse. After the transmitting pulse has ended, the duplexer immediately disconnects the transmitter and connects the receiver to the antenna.

4.2.4 Receiver

The receiver receives the target reflected energy from the antenna block. It then performs amplification, filtering, and demodulation on the received signal. The preprocessed signal is then sent to a computer or signal processor for further processing.

4.2.5 Computer/Signal Processor

The computer/signal processor performs complex mathematical computations on the demodulated signal to extract target velocity and/or range information.

4.2.6 Timing and Control

The timing-and-control block provides timing information to synchronize various signals and to control the operation of other radar components.

After the echo signal is processed, the targets' range and/or speed information are/is sent out for display on cathode ray tube–liquid crystal display (CRT/LCD)-based devices.

4.3 THE RADAR RANGE EQUATION

The power density at range R from the transmitter is defined as the transmitted power divided by the surface area 4πR² of an imaginary sphere of radius R. Let P_T be the power transmitted by the antenna through an isotropic radiator (one that radiates uniformly in all directions); then the power density can be expressed as

Power density from an isotropic antenna in watts per square meter (W / m²).

Let G_T be the radar transmitter antenna gain; the power density S_I received by the target from a transmitter with antenna gain G_T then becomes

Let σ be the target's scattering coefficient or radar cross section (which is in units of m² and will be discussed later); then the power density back at the input of the receiver antenna will be

In Chapter 3 we defined the effective area of receiving antenna A_e as the ratio of the received power at the antenna terminals to the power density of the incident wave. We also mentioned that A_e affects the received power. Therefore, let S be the power received by the receiving antenna; then

Let G_R be the gain of receiving antenna, from Eq. (3.34), the gain G_R is then related to its effective area by the expression

The total received power S scattered by a target is then

For a monostatic radar where the antenna performs both transmitting and receiving functions, then G_T = G_R = G. Therefore

There is always some energy loss due to circuit or equipment power dispassion. This energy loss will reduce the total amount of power received. They are lumped together and are denoted by L, where L < 1. The total received power then becomes

This equation can also be expressed in dB values as

Figure 4.5 illustrates the power density relationships between the transmitter, the target, and the receiver.

FIGURE 4.5 Key elements of radar range equation.

Equation (4.2b) or (4.2c) is called the Radar range equation, which can be used to compute the range of the target. The maximum detection range R_max is the range at which the received power just equals the minimum detectable signal S_min. By equating R_max for S = S_min, the maximum range can be computed as follows, from Eq. (4.2b):

Example 4.1 Calculation of Maximum Radar Range Given the following parameters of a range detection radar,

• Antenna gain: 34 dB
• Transmit peak power: 2 kW
• Radar carrier frequency f_c: 10 GHz
• Overall system losses: 5 dB
• Target RCS: 5 m²
• Minimum detectable signal power: 1.6 × 10⁻¹³ W

The maximum detection range of the radar can be calculated as follows:

For a 10-GHz carrier frequency, the wavelength λ = c/f = 3 × 10^–2 m. By applying the radar range equation (4.2c), with S_dB = − 128, one obtains

Therefore

R_max 15,848 m.

4.4 CROSS SECTION AND CLUTTER

4.4.1 Target Cross Section

The “cross section” of a target relates the amount of transmitting power that strikes the target to the amount of power that is reflected back to the receiver. Let the power density of a plane wave incident on the target be S_i W/m², and the amount of power scattered isotropically be S_c; then the cross section σ is defined in terms of S_i and S_c as follows:

S_c = σ S_i.

Notice that the incident power density S_i is referred back to the transmitting antenna. For a large value of range R, the power density received at the target can be roughly considered as the amount of power received at the target. The reflected power S_c is therefore a fraction of the incident power S_i.

The power density S_r of the scattered wave at the receiving antenna is

Here S_r is referred back to the target.

The cross section can be rearranged in terms of the power densities S_r and S_i as

To ensure that the receiving antenna is in the far field and that the waves are planar, the cross section is expressed as

In terms of the various fields that make up EM waves, this equation can be expanded as follows:

Here E_r and H_r are the respective electric and magnetic fields at the radar receiver, while E_i and H_i are the respective electric and magnetic fields at the target.

4.4.2 Cross Section and the Equivalent Sphere

If a target scatters power uniformly over all angles, its cross section would be equal to the area from which power was extracted from the incident wave. Since the sphere has the ability to scatter isotropically, it is convenient to interpret its cross section in terms of the projected area of an equivalent sphere.

A sphere with radius a λ will intercept power contained in πa² of the incident wave. The cross section of a sphere is therefore equal to the area πa² even though the actual area that returns power to the receiver is a very small area where the surface lies parallel to the incident wavefront. The cross section of a sphere is therefore one-quarter of the surface area, and it can be shown that the average cross section of any large object that consists of continuous curved surfaces will be one-quarter of its total surface area.

4.4.3 Cross Section of Real Targets

Target cross section is a complicated function of the viewing aspect, incident wave frequency, and polarization. It is often related to target physical size, and under certain circumstances, it may be significantly larger. The effective surface roughness of a target (as a function of λ) also plays an important role in determining its cross section. There are two popular reflection mechanisms: diffuse reflection and specular reflection. The diffuse reflector, such as a cornfield, scatters the EM waves in all directions and returns only a fraction of the energy back to the radar antenna. The specular reflector, such as a smooth paved surface, acts as a mirror and reflects the energy from the EM wave in one well-defined direction. The energy returned to the radar antenna could be either nonexistent or very strong, depending on whether the reflecting surface is at a right (90°) angle to the radar beam. These two reflection mechanisms determine individually, or in combination, the target reflection characteristics.

4.4.4 Radar Cross Section (RCS)

The radar cross section (RCS) of an object is a measure of its size as seen at a particular radar wavelength and polarization. RCS has units of m² and is often expressed in decibels relative to a square meter (dBsm),

σ(dBsm) = 10 log₁₀[σ(m²)].

The characteristics of RCS, or scattering coefficient, depend on the field polarity of the transmitter and the receiver. Since a radar receiver detects electric fields, the RCS is a function of polarity on E^r and E^t. Here E^r and E^t refer to the receiver and transmitter E field, respectively. The relationship between RCS and polarity of the transmitted and received electric fields can be expressed as

where the subscripts V and H refer to the respective vertical polarized and horizontal polarized E fields. The first subscript of σ _HH refers to the scattering coefficient σ due to a transmitted horizontal polarized E field, or E^t_H. The second subscript of σ _HH relates the scattering coefficient σ to the received horizontal polarized E field, or E^r_H. Similar relations apply to σ _VH, σ _HV, and σ _VV. For most applications, the radar transmitter and receiver are arranged to possess the same polarity. The target RCS is therefore simplified as a single value that corresponds to either σ _HH or σ _VV depending on whether the transmitted polarization is horizontal (H) or vertical (V).

The following list displays the radar cross section of some well-known objects for comparison purpose:

4.4.5 Clutter

The EM waves reflected from the objects around the target are referred to as “clutter.” These returns may be from the surface surrounding the target (ground), or from the volume of space around it (rainfall) as shown in Fig. 4.6.

FIGURE 4.6 Surface clutter and volume clutter.

Clutter is thus defined as the undesired return from a physical object or a group of objects. Clutter may be divided into sources distributed over a surface (land or sea), within a volume (weather or chaff), or concentrated at discrete points (structures, birds, or vehicles). The magnitude of the signal reflected from the surface is a function of the material, roughness, and angle. Two main scattering types of clutter are diffuse and specular, as described before. Rain and dust are the two main contributors of volume backscatters.

In addition to detecting the target range, radar is also used extensively to detect the speed of a target. The detection of target speed is primarily based on the detection of Doppler frequency, which is discussed next.

4.5 DOPPLER EFFECT AND FREQUENCY SHIFT

For stationary and separate wave receivers and wave sources, such as sound or light, the frequency or wavelength generated by the wave source will be the same at the receiver site; that is, f_r = f_s or λ_r = λ_s, where the subscripts r and s refer to wave receiver and wave source, respectively. Only electromagnetic waves are considered in this book. Figure 4.7 shows a wave source S and a wave receiver R at some distance away. The concentric circles are the wavefronts emitted from the wave source S located at the center of the circles. The symbol r_i, for i = 1, 2, 3, denotes the radius corresponding to the wavefront generated at time T_i = (i − 1) T with T = 1/f_s as the wave period. The symbol S_i is the origin of the circle r_i. For a stationary wave source, S_i = S for every T_i, and the distance between any two circles has the same wavelength λ_s = λ_r.

FIGURE 4.7 Wave propagation for stationary source and stationary receiver.

FIGURE 4.8 Wave propagation for moving source and stationary receiver.

4.5.1 Doppler Frequency

When the wave source moves, either toward or away from the receiver, the center of the concentric circles will move as well. As a consequence, a frequency difference occurs between the wave source and the wave receiver. This frequency difference or shift is called the Doppler frequency.

Figure 4.8a shows a situation where the wave source S is moving at the speed V_s toward the receiver R. For a wave period of T, the origin of the wave source S has moved to two locations, from S₁ to S₂ at time T₂ = T, and to S₃ at time T₃ = 2T. The distances between S₁, S₂, and S₃ are S₂ − S₁ = d and S₃ –S₂ = d.

Figure 4.8b shows the effects of the waveforms generated at different locations by the moving source. Consider the circle of r₁, which has origin S₁ at time t = 0. At time t = T, the source S moves to a new position S₂ with a radius of r₂ and having d = V_sT away from S₁. A dotted circle corresponding to the stationary case generated at time T with the same radius r₂ is shown for comparison. The difference between radii r₁ and r₂ along the direction between S and R is the wavelength as seen by the receiver. In other words, the received wavelength λ_r is related to the source wavelength λ_s by

where c is the speed of the wave, in this case the speed of light.

Since λ = c/f, the received frequency can be computed from the source frequency by

The frequency difference between receiver and source, or Doppler frequency f_D, is

The positive value of f_D means that the received frequency is higher than the frequency emitted by the approaching source. If the receiver is stationary, then, after the source passes the receiver, the “speed of approach” V_s becomes negative, and the frequency recorded by the receiver becomes lower than the frequency emitted by the now-receding source. The Doppler frequency, caused by relative movement between source and receiver, is also called the Doppler frequency shift.

The same principle applies when the source is stationary but the receiver is approaching it at a speed V_r. For one receiver wave period T, which equals to 1/f_r, the receiver will move a distance of d = V_rT′. This means that the receiver will receive the source wave by less distance d than when the receiver is stationary. Figure 4.9 displays an example of a receiver moving toward the wave source at a speed V_r.

FIGURE 4.9 Wave propagation for stationary source and moving receiver.

The new received wavelength is related to the source wavelength by

Since the moving receiver now determines the period of the wave, the received frequency is related to the source frequency by

The frequency difference between the receiver and source or Doppler frequency is

Consider the case when both the wave source and the wave receiver are moving and V_s and V_r are the speeds with which they are approaching each other. Figure 4.10 displays a scenario in which the source moves toward the receiver at speed V_s for a time interval t = T = 1/f_s, with a distance d₁ = V_sT. It also shows that the receiver moves toward the source at speed V_r for the same time interval T with a distance d₂ = V_rT.

FIGURE 4.10 Wave propagation for moving source and moving receiver.

The received wavelength λ_r is related to the source wavelength λ_s by

The received frequency is related to the source frequency by

where V_rs = V_r + V_s. For V_rs c,

The frequency difference between receiver and source, or Doppler frequency, is

Notice that the value of V_rs, and therefore f_D, is positive if the wave source and wave receiver are moving toward each other; otherwise, it is negative.

For most Doppler frequency detectors, both the transmitter and the receiver are stationary. Figure 4.11 illustrates one possible application where the transmitter (source) and the receiver are stationary, while the target is moving at speed V.

FIGURE 4.11 Doppler radar with separate source and receiver.

As shown in the figure, the radial velocity of the target relative to the transmitter (source) is − V cos θ_s, and the radial velocity of the target relative to the receiver is −V cos θ_r. The Doppler frequency arising under these circumstances can be calculated by assuming that

• The target is moving away from the transmitter with a velocity −V cos θ_s.
• The receiver is moving away from the target with a velocity −V cos θ_r.

These two assumptions are equivalent to the receiver moving away from the transmitter (source) with velocity − V cos θ_s − V cos θ_r, even though both are stationary.

The Doppler frequency of the separated transducers described above can be computed as follows, from Eq. (4.7):

If the transmitter (source) and the receiver are collocated, then θ_r = θ_s = θ and the formula for the Doppler frequency becomes

Equation (4.9) is a very important formula for computing the Doppler frequency, which serves as the basis for detecting the speed of a target.

An alternative way to derive the Doppler frequency and the corresponding relative velocity between the radar and target is as follows. Assume that the distance between radar and target is R, and that the wavenumber over the transmitted and reflected paths is then 2R/λ_s, where λ_s is source wavelength. Since one wavelength λ_s corresponds to an angular phase of 2π, the total phase is 4πR/λ_s. The rate of change in with respect to time t is the angular Doppler frequency ω_D, which is then

Therefore

Here f_D is the Doppler frequency and V_r is the relative velocity of the target with respect to the radar. The value of V_r, and therefore f_D, is positive if the radar and target are moving toward each other; otherwise, it is negative.

Example 4.2 Computation of Doppler Frequency Figure 4.12 shows a radar-carrying vehicle flying along the y axis at (velocity) V = 800 kilometers per hour (km/h), with a ground target located at (r₀, 0). Let r₀ = 20 km and the carrier frequency of radar be 3 GHz. The distance from the three radar positions A, B, and C to the target are r₁, r₀, and r₃, respectively. Position B is at (0, 0), and the angles between the x axis and positions A and C are α and β, respectively. Letting α = β = 30°, the Doppler frequency at positions A, B, and C can be computed as follows. The wavelength of a radar signal is computed as

FIGURE 4.12 Example of Doppler frequency.

The radial velocity V_r at position A is

Given V_r and λ, from Eq. (4.10), the Doppler frequency can be obtained as

The radial velocity at position B equals zero; therefore the Doppler frequency f_D = 0. The radial velocity at position C is

therefore,

4.6 RADAR RESOLUTION AND AMBIGUITY FUNCTION

The primary function of a modern radar system is to detect targets of interest while estimating each target's position and velocity. Range and velocity determinations depend on the accuracy of measuring the time delay and the Doppler frequency, which in turn are related to the fundamental properties of radar waveform. The continuous-wave (CW) waveform and the pulse waveform are two of the most frequently used techniques in the radar applications. The advantage of CW radar is unambiguous Doppler measurement; that is, each target velocity produces a single unique Doppler frequency of the CW carrier. However, the target range measurements using CW radar are entirely ambiguous. This is because that all the returned waveforms are continuous; the radar cannot identify which echo came from which waveform, and thus an ambiguity in the range measurement occurs.

Most modern radars employ a pulse waveform technology, where a single antenna serves for both transmitting and receiving functions. A pulse-based radar waveform will provide both the range and the velocity information for a target. The waveform of a pulse-based radar can be single-carrier-frequency-based, frequency-modulated, or phase-modulated, as described in more detail in next chapter.

The target range of a pulse-based radar, with echo delay time τ, is computed as R = cτ/ 2.

Letting T be the pulse repetition period, the maximum unambiguous target range R occurs at τ = T, namely, R_max = cT/ 2. The range ambiguity arises if the target is located at a range exceeding R_max.

The Doppler frequency f_D, which causes carrier frequency shift, is induced by a target having velocity V relative to the radar, specifically, f_D = 2V/λ.

For moving targets, the differential phase change between successive pulses, denoted as Δ, is given by Δ = 2π (2ΔR/λ), where ΔR is the change in target range between successive radar pulses. When the phase change between radar pulses exceeds 2π, the Doppler measurement becomes ambiguous.

The most common method for resolving range and Doppler ambiguities involves using multiple f_PRF values. This has the effect of changing the apparent target range estimated by each pulse or pulse burst, and allows for some ambiguity resolution in either Doppler frequency or range, depending on the particular application.

The ambiguity function (AF) is widely used in radar waveform analysis which shows the distortion of a matched filter output due to the Doppler shift of the echo signal from a moving target. It reveals the range–Doppler position of an ambiguous response and defines the range and Doppler resolution. The ambiguity function χ(τ, f_D) is defined as the cross-correlation between a waveform s(t) and s*(t − τ) exp (j2πf_Dt), which is a time-delayed and complex-conjugated, frequency-shifted replica of s (t):

Along the delay τ axis, χ(τ, f_D) represents the autocorrelation function of s(t); along the f_D axis, χ(τ, f_D) is proportional to the spectrum of s²(t):

It is common to refer to the absolute value of χ(τ, f_D) as the ambiguity surface of the waveform. A normalized expression of the ambiguity surface requires that

A short monotone pulse will now be used as an example to show the ambiguity surface |χ(τ, f_D)|. Let the envelope of the rectangular pulse be

where

The rectangular pulse and its frequency spectrum are shown in Fig. 4.13. The frequency spectrum is a sinc function with its maximum occurring at the origin, while the first zero is located at 1/T_p and − 1/T_p away from the origin. It is clear that the narrower the rectangular pulse is, the wider the bandwidth of the spectrum becomes.

FIGURE 4.13 Rectangular pulse and its frequency spectrum.

The ambiguity surface |χ(τ, f_D)| of the rectangular pulse can be expressed as

where x = f_DT_p(1 −|τ |/T_p).

FIGURE 4.14 Ambiguity function of a rectangular pulse in 3D view.

The ambiguity surface |χ(τ, f_D)| is a function of both the time delay τ and the Doppler frequency f_D. For |τ | < T_p and at a given value of τ, the sinc function has its maximum occurring at the origin and zeros located at x = ± n, where n = 1, 2, 3, ....

Figure 4.14 is a 3D display of |χ(τ, f_D)|. The magnitude of the sinc function decreases along the delay-time τ axis and reaches zero when |τ | equals T_p. The bandwidth of the sinc function along the f_D axis is stretched as the value of |τ | increases from zero to T_p.

Figure 4.15 displays the cross-sectional view of Fig. 4.14. The waveform in Fig. 4.15a shows the magnitude of |χ(τ = 0, f_D)| against the Doppler frequency with τ = 0. The waveform in Fig. 4.15b shows the magnitude of |χ(τ = 0.5T_p, f_D)| against the Doppler frequency with τ = 0.5T_p. The 3-dB bandwidth doubles at τ = 0.5T_p when compared with the one at τ = 0. The Doppler frequency is used to compute the speed of target, which has higher magnitude and narrow bandwidth at τ = 0 than do those at τ = 0.5T_p. This implies that the accuracy of the target's speed measurement is adversely affected when the delay time increases.

Figure 4.16 displays another cross-sectional view of Fig. 4.14. The waveform in Fig. 4.16a shows the magnitude of |χ(τ, f_D = 0)| against the time delay with f_D = 0. The waveform in Fig. 4.16b shows the magnitude of |χ(τ, f_D = 2.5/T_p)| against the time delay with f_D = 2.5/T_p. As can be seen, the magnitude of |χ(τ, f_D)| equals zero at time delay τ = T_p and τ = –T_p in both cases. The function |χ(τ, f_D = 0)| corresponds to the output of pulse compression, which is used to measure the target's range. As shown in Fig. 4.16b, the outputs of pulse compression values are reduced at increased f_D, and therefore cause errors or ambiguity in target range measurement.

FIGURE 4.15 Cross-sectional view of Fig. 4.14 with τ = 0 (a) and τ = 0.5T_p(b).

FIGURE 4.16 Cross-sectional view of Fig. 4.14 with f_D = 0 (a) and f_D = 2.5/T_p(b).

FIGURE 4.17 A 3-dB contour of ambiguity function of a rectangular pulse in 3D view.

The 3-dB energy contour of the magnitude of |χ(τ, f_D)| is shown as a 3D view in Fig. 4.17. This is the region where the target's speed and range should both be measured.