5

RADAR MODULATION AND TARGET DETECTION TECHNIQUES

In Chapter 4, we mentioned that various waveforms can be used by radars to detect the range and/or speed of a target. In this chapter, we discuss several types of these waveforms. Section 5.1 reviews amplitude modulation (AM)-based radars, including continuous–wave radar and pulse–Doppler radar. Section 5.2 describes target detection techniques on AM-based radar. Frequency modulation (FM)-based radars are covered in Section 5.3, including linear frequency modulation (LFM), pulsed LFM, continuous-wave LFM (CWLFM), and stepped frequency modulation (SFM). The target detection techniques for FM-based radar, which utilize the in-phase–quadrature-phase demodulator and matched filtering, are discussed in Section 5.4.

5.1 AMPLITUDE MODULATION (AM) RADAR

Amplitude modulation (AM) is a modulation technique in which the amplitude of the carrier is varied in accordance with some characteristics of the baseband modulating signal. It is the most common form of modulation because of the ease with which the baseband signal can be recovered from the transmitted signal.

An AM signal p(t) can be described in terms of carrier frequency f_c and baseband signal p_b(t) by the following equation:

Letting the modulating signal p_b(t) be a sinusoidal signal as p_b(t) = A_m cos (2πf_at), Eq. (5.1) then becomes

In general A_m < 1; otherwise a phase reversal occurs and demodulation becomes more difficult. The extent to which the carrier's amplitude has been modulated is expressed in terms of a percentage modulation that is calculated by multiplying A_m by 100.

5.1.1 Continuous-Wave (CW) Radar

A radar system that continuously transmits a signal is referred to as continuous-wave (CW) radar. CW radar is normally used to detect the speed of a moving target. When a CW radar waveform continuously strikes a target that is moving either toward or away from the transmitting radar, the frequency of the reflected waveform is changed; this is known as the Doppler frequency. To detect the moving target speed, the CW radar receiver mixes (or homodynes) the received signal with a replica of the transmitted signal. After lowpass filtering, the only remaining component is the Doppler frequency, which can be used to calculate the speed of the target. The continuous-wave radar is the best means of detecting fast-moving objects that do not require range resolution. The disadvantage of the CW Doppler radar system is that it does not determine the range of the object, nor is it able to differentiate between objects when they lie in the same direction and are traveling at the same speed.

5.1.2 Pulse Modulation Radar

The pulse–doppler radar is identical to the CW radar except that the transmitted signal is a sequence of pulses. This technique allows the radar to measure both range as well as velocity of the target. A key requirement for any Doppler radar is coherence; that is, the transmitted signal frequency and the locally generated reference signal frequency must be kept the same with fixed phase difference. This reference signal is then used to detect both the range and the Doppler frequency of the received signal.

The pulse-modulated radar uses a sequence of narrow pulses to serve as a modulating signal and modulates the carrier frequency to form the transmitting waveform. Let p_N(t) be the pulse train, consisting of N pulses with period T, and assume that each pulse has time duration T_p and amplitude A. Then p_N(t) can be expressed as

and

FIGURE 5.1 Transmitter block diagram of a pulse-modulated radar system.

The pulse-modulated signal then becomes

The pulse modulation radar transmits radiofrequency energy in very short bursts, usually on the order of 0.1 to approximately 50 microseconds (μs). It is sometimes called the pulse–doppler radar. The carrier frequency f_c normally ranges from several hundred megahertz to tens of gigahertz. Figure 5.1 shows the transmitter block diagram of a typical pulse-modulated radar system.

The spectrum of a narrow pulse, as discussed in Chapter 1, is a sinc function. The first minimum (or zero) of the sinc function is located at both sides of the peak, which are displaced from the peak by 1/T_p with T_p as the pulsewidth. For a pulsed radar signal, the relationship between frequency bandwidth B and T_p can therefore be represented as

The radar range resolution is determined as follows, from Eq. (4.1c):

Figure 5.2 shows the pulse-modulated signal and its corresponding spectrum. As can be seen, the frequency spectrum of the pulse train is discrete and centered on the carrier frequency f_c. The envelope of the spectrum is a sinc function. For a periodic signal of period T, the frequency resolution is equal to 1/T. Figure 5.2 is plotted with T = 2T_p, and the first zero of the sinc function appears at f = 1/T_p away from the peak center f_c.

Another type of high-resolution radar is the impulse or short-pulse radar. The time duration of these radars is usually 0.25–1 nanoseconds (ns). The pulses are transmitted without carriers, so they are often called carrierless impulses or baseband video pulses. The resulting pulses are often called monocycle pulses. These types of radar are used for stationary target ranging only. No radial speed detection is required. Two popular pulses and their corresponding power spectra are shown in Fig. 5.3.

FIGURE 5.2 Time- and frequency-domain waveforms of pulse-modulated radar signal.

Since radar range resolution depends on the bandwidth of the received signal, and because the bandwidth of a time-gated sinusoid is inversely proportional to the pulse duration, the short pulse provides better range resolution. However, since short-pulse duration requires a high-powered transmitter in order to ensure good reception, this type of radar has the disadvantage of causing hardware problems and safety issues.

5.2 TARGET DETECTION TECHNIQUES OF AM-BASED RADAR

This section presents some techniques for detecting Doppler frequency in a received signal based on amplitude modulation.

5.2.1 Doppler Frequency Extraction

The transmitted signal of an amplitude modulated radar pulse can be represented as

p(t) = A_t cos(2πf_ct),

where f_c is the carrier frequency. The corresponding received signal from a single moving target will be

s(t) = A_r cos[2π (f_c + f_D)t + ],

FIGURE 5.3 Time- and frequency-domain waveforms of two video pulses.

where f_D is the Doppler frequency, is a phase term dependent on the distance to the target, and A_r < A_t.

To extract the Doppler frequency f_D, the p(t) and s(t) are multiplied together:

By passing the signal described above through a lowpass filter, one can remove the high-frequency component at 2f_c and leave only the Doppler frequency signal

A block diagram illustrating this Doppler frequency extraction procedure is shown in Fig. 5.4.

FIGURE 5.4 Block diagram of Doppler frequency extraction.

5.2.2 Motion Direction Detection

The Doppler process discussed above provides an absolute value of frequency difference; it contains no information regarding the direction of target motion. A number of techniques can be applied to preserve this directional information. Among them is a simple offset carrier demodulation method based on the AM technique, described next.

As shown in Eq. (4.8), the Doppler frequency f_D is positive if the target moves toward the receiver, while f_D becomes negative if the target moves away from it. This also implies that with amplitude modulation, the returned echo signal, which consists of frequency component f_c + f_D, will be greater than f_c if f_D is positive, and less than f_c if f_D is negative.

Let the received signal s(t) be represented as

s(t) = A_r cos[2π(f_c + f_D)t + ],

where A_r and are the amplitude and phase of the received signal. Let p₀(t) be a reference signal and represented as

p₀(t) = A₀ cos[2π(f_c + f₀)t],

where A₀ and f₀ are the amplitude and frequency of the reference signal, and f₀ is chosen to be greater than the maximum possible value of f_D.

The offset carrier demodulation method mixes the received signal s(t) with the reference signal p₀(t), and the resulting signal becomes

This signal is then passed through a lowpass filter to remove the high-frequency part of 2 f_c + f₀ + f_D. If the target is approaching the receiver, then f_D is positive and the filter output will have frequency f₀ − f_D < f₀. If the target is moving away from the receiver, then f_D is negative and the filter output will have frequency f₀ − f_D > f₀. Figure 5.5 is a block diagram illustrating the process discussed above.

FIGURE 5.5 Block diagram of an offset carrier demodulation.

As an example, Fig. 5.6 shows the block diagram of a pulse–doppler radar, which can be used to detect the range and speed of multiple targets. A 5.0-GHz sinusoid wave is gated by a narrow pulse generator with period T = 0.1 ms to produce a pulsed transmitter waveform. The pulse generator is controlled by the “Timing & sync” circuit. The 5.0-GHz sinusoid wave is also used to mix with a 30-MHz coherent oscillator (COHO) to generate a 4.97-GHz signal, which is then mixed with the returned echo signal. The returned echo signal consists of the 5.0-GHz Tx signal plus a Doppler frequency. Multiple targets will cause multiple echoes with multiple Doppler frequencies. The output of the receiver mixer, which consists of the 30-MHz intermediate-frequency (IF) signal with Doppler frequencies, is fed to another mixer. The output of this mixer, which is the baseband target echo signal with Doppler frequency, is then fed to an A/D converter and the range gate. The dashed-line box that includes the mixer, A/D converter and range gate, 2D data buffer, FFT, and decision circuit can be implemented by a firmware-based DSP processor.

FIGURE 5.6 Block diagram of a pulse–Doppler radar system.

The A/D converter outputs 30 samples per radar pulse period. The magnitude of each sample (or range bin) corresponds to the strength of the returned echo signal at that range. The resolution of each range bin depends on the sampling frequency of the A/D converter. The 30 range samples per period will be stored as one row in the 2D databank. After 25 radar pulse periods, a 25 × 30 data array will be processed by fast Fourier transform. Each column of the 2D array corresponds to one target response. The output of column-based FFT is used to determine the value of the Doppler frequency for the corresponding target. Once the Doppler frequency is found, the moving target's speed can be computed accordingly. The number of samples in each column of the 2D databank represents the number of pulse periods needed for computation. To increase the frequency resolution without increasing the number of radar pulse periods, the time-domain zero-padding method, as described in Chapter 2, can be used. This is a combination of time- and frequency-domain co-processing on a pulse–Doppler radar and is commonly used in many modern radar systems.

A method that has the advantage of both the high energy of a long pulsewidth and the high resolution of a short pulsewidth is discussed next.

5.3 FREQUENCY MODULATION (FM)-RADAR

Frequency modulation (FM) is a technique in which the frequency of the carrier is varied in accordance with some characteristic of the baseband modulating signal p_b(t):

The instantaneous frequency of p(t) can be obtained by differentiating the instantaneous phase of p(t):

The baseband-modulating signal p_b(t) normally is a sinusoidal function. Therefore Eq. (5.6) can be expressed as

p(t) = A₀ exp [j (2π f_ct + βsinω_at)],

where f_c is the carrier frequency and β is referred to as the modulation index and is the maximum value of phase deviation.

The linear frequency modulation (LFM) technique involves a transmitter frequency that is continually increasing or decreasing from a fixed reference frequency. That is, the transmitted FM signal is modified so that the frequency is modulated in a linear manner with time. In other words, from Eq. (5.7), one obtains

f = f_c + αt

and

βp_b(t) = 2π αt,

where α is the frequency changing rate or chirp rate, and is defined as

Substituting these equations into Eq. (5.6), one obtains

Two types of LFM signals are commonly used in radar applications: the pulsed LFM (also called “chirp”) and the continuous-wave LFM (CWLFM). The pulsed LFM signal is described first.

5.3.1 Pulsed Linear Frequency Modulation (LFM) Radar

Let a₁(t) and a₂(t) be defined as

where Rect(t) is a rectangular gate function, defined as

A pulsed symmetric LFM signal a(t) with duration T_p can be written as follows, from Eq. (5.8):

The pulsed nonsymmetric LFM signal p(t) with duration T_p can be written as

A baseband nonsymmetric pulsed LFM waveform with pulse duration T_p and period T is shown in Fig. 5.7a. The corresponding frequency–time relationship is shown in Fig. 5.7b. As can be seen, the LFM signal frequency increases from 0 to f_max during the pulse duration time T_p, where f_max = αT_p and the frequency bandwidth is B = f_max = αT_p.

FIGURE 5.7 Time-domain waveform (a) and time–frequency relation (b) of a pulsed LFM signal.

Analytically, the frequency spectrum of p(t), in the nonsymmetric LFM case, can be considered as the convolution of the two spectra A₁(f) and A₂(f), where A₁(f) and A₂(f) are the Fourier transform of a₁(t) and a₂(t), respectively; that is

where

Let −u² = jαx², or ; then

Therefore

and

where the symbol represents convolution.

In practical applications, the frequency spectrum of LFM-based signals is generated by applying DFT on the signals. As an example, the following parameters are used to generate the pulsed LFM waveforms together with their frequency spectra. Both symmetric and nonsymmetric LFM signals are covered:

Frequency chirp rate α = 3 × 10¹²

Pulse duration time T_p = 5 × 10⁻⁶ s

Period of pulse train T = 1.024 × 10⁻⁵ s

Sampling frequency f_s = 50 × 10⁶ Hz

Bandwidth: T_p × α = 15 MHz

Figure 5.8 displays the symmetric pulsed LFM signal and its frequency spectrum. The real and imaginary parts of the time-domain pulsed LFM waveform are shown in Figs. 5.8a and 5.8b. Both drawings are symmetric around the center of pulse duration. Figures 5.8c and 5.8d show the real and imaginary parts of the frequency-domain spectrum. Figures 5.8e and 5.8f illustrate the absolute value (or magnitude) and phase of the frequency spectrum. As can be seen, the spectrum is almost flat from frequency bins 175 to 325 for a total of approximately 150 bins. Notice that the frequency spectrum was shown with the origin located at the center of display, or realigned between −f_s/2 and f_s/2. Given the sampling frequency f_s = 50 MHz, which corresponds to frequency bin 512, the bandwidth of the pulsed LFM is approximately (325 – 175) × 30/512 15 MHz. The “sinc”-like shape at the two band edges of the LFM spectrum is caused by the convolution of the sinc function of A₁(ω) with A₂(ω), which is a constant in magnitude and is a band-limited signal.

Figure 5.9 displays the nonsymmetric pulsed LFM signal and its frequency spectrum. Figures 5.9a and 5.9b show the real and imaginary parts of the time-domain pulsed LFM waveform. Both drawings are no longer symmetric around the center pulse duration. Figures 5.9c and 5.9d show the real and imaginary parts of the frequency-domain spectrum; Figs. 5.9e and 5.9f illustrate the magnitude and phase of the frequency spectrum. Again, the spectrum is almost flat between frequency bins 260 and 410, for a total of ~150 bins. The frequency display is centered around the origin, or from −f_s/2 to f_s/2. The bandwidth of the pulsed nonsymmetric LFM is ~15 MHz. Notice that no negative frequency component exists in this case, which differs from the symmetric LFM case.

FIGURE 5.8 Time- and frequency-domain waveforms of a pulsed symmetric LFM signal.

FIGURE 5.9 Time- and frequency-domain waveforms of a pulsed nonsymmetric LFM signal.

FIGURE 5.10 Block diagram of a PLFM radar system.

A block diagram of pulsed LFM radar system is shown in Fig. 5.10. Here the transmitter consists of a pulse generator, chirp signal generator, mixer, power amplifier, and local oscillator. The receiver consists of a low-noise amplifier (LNA), mixer, in-phase–quadrature-phase detector (or demodulator), local oscillator, analog–digital (A/D) converter, and signal processor. The antenna is connected to either the transmitter or the receiver through a duplexer. The in-phase–quadrature-phase (I–Q) detector and other target-detection-related processing are discussed in later sections.

5.3.2 Continuous-Wave Linear Frequency Modulation Radar

Continuous-wave linear frequency modulation (CWLFM) radar transmits and receives signals continuously. The transmitting frequency is first increased then decreased, and the same pattern is repeated at time interval T. Figure 5.11 is the block diagram of a CWLFM radar system.

FIGURE 5.11 Block diagram of a CWLFM radar system.

FIGURE 5.12 Time–Frequency relationship of a CWLFM radar signal.

A CWLFM signal has a triangular shape of frequency change with respect to time. The upsweep and downsweep frequencies of the triangle are defined as

where f _max = f_c + αT_p and α, β are the two chirp rates of the upsweep and down-sweep frequencies. If T₀ = 2T_p, then the triangle is symmetric and α = β. In this case, the signal frequency increases from f_c to f _max during the first half of the triangle, and decreases from f_max to f_c for the second half. The sweep frequency bandwidth B = f_max − f_c = αT_p. Figure 5.12 displays the time–frequency relationship of a CWLFM signal when α = β.

A third type of frequency-modulation-based technique for radar application is stepped frequency modulation, discussed next.

5.3.3 Stepped Frequency Modulation Radar

The stepped frequency modulation (SFM) radar has been used extensively in short-range measurements to study the scattering properties of geophysical surfaces and other nontransparent media. It computes the target range by measuring the magnitude and phase response over a number of stepped frequencies (N) within a given frequency band.

FIGURE 5.13 Waveforms of (a) a CWSFM radar signal and (b) a pulsed SFM radar signal.

There are two types of stepped-frequency-based radar: the continuous-wave stepped frequency (CWSFM) radar and the pulsed SFM (PSFM) radar. Both the CWSFM radar and PSFM radar transmit and receive a sequence of N-frequency waveforms changed by the frequency step Δf. However, each section of the N-frequency waveforms is consecutive for the CWSFM radar, but in a pulsed form for the PSFM radar. Figure 5.13 shows the SFM waveforms in the time domain; Fig. 5.13a displays the CWSFM signal, while Fig. 5.13b shows the PSFM signal.

The starting frequency of the CWSFM or PSFM signal called the base frequency f₀ is the lowest frequency transmitted. The frequency change or step is a constant increment from the preceding frequency, and is denoted by Δf. The bandwidth B of a stepped frequency radar is f_U − f₀ = (N − 1) Δf, where f_U is the highest frequency and N is the number of frequency steps. Each single stepped frequency signal lasts for a period of time T, which is also considered as the pulse repetition interval (PRI), which serves as the time window where both transmitting and receiving signals are used to detect the targets. The N-frequency section group is considered as one burst, and each burst lasts for a time period NT.

Figure 5.14 displays the time–frequency relationship for the N-frequency CWSFM waveform (Fig. 5.14a) and PSFM waveform (Fig. 5.14b), respectively. The target detection and range computation schemes for both CWSFM and PSFM signals are quite similar; only the hardware implementation is different.

FIGURE 5.14 Time–frequency relationship of (a) CWSFM radar signal and (b) a PSFM radar signal.

The block diagram of a stepped-frequency-based radar system is shown in Fig. 5.15. The receiver box normally consists of an intermediate-frequency (IF) demodulator as shown in Fig. 5.6. Compare this with the CWLFM case shown in Fig. 5.11, where an extra system box, namely, the quadrature mixer, or the I–Q demodulator, is used. The I–Q demodulator generates the in-phase and quadrature-phase signals from the received signal, discussed next.

FIGURE 5.15 Block diagram of a stepped frequency modulation radar.

5.4 TARGET DETECTION TECHNIQUES OF FM-BASED RADAR

5.4.1 In-Phase Quadrature-Phase Demodulator

Consider the LFM signal represented by Eq. (5.8):

p (t) = A₀ exp [j (2π f_ct + π αt²)].

The observed and real-valued signal of this equation is represented as

In general, the carrier frequency f_c has a high value and αt has a relative low value. The phase term (t) = παt² is presented for illustration purposes. In a digital communication system, a complex-valued signal is needed to perform functions such as phase detection of a QAM signal, adaptive equalization, and many other functions. The complex-valued signal is required to transmit the time-varying phase information, and an in-phase–quadrature-phase (I–Q) demodulator is used to receive the correct phase information.

Figure 5.16 displays a typical I–Q demodulator where the input signal v(t) is a general LFM signal as described above, and the output are the real and imaginary parts of the phase signal (t). The two carrier signals, cos(2πf_ct) and sin(2πf_ct), are locally generated. The lowpass filter is designed to retain the low-frequency component (t), but will filter out the high-frequency component 4πf_ct + (t). The A/D converter is used to digitize the real and imaginary parts of the low-frequency phase signal (t).

FIGURE 5.16 In-phase–quadrature-phase (I–Q) demodulator.

The intermediate signals v_1c(t), v_1s(t), v_2c(t), and v_2s(t) can be derived as follows:

The phase signal (t) can then be computed as follows:

5.4.2 Matched Filter and Pulse Compression

The matched filter technique is widely used in digital communication fields to recover a signal that is corrupted by additive white Gaussian noise. In radar applications, a matched filter is applied on received signals to identify a target by determining whether the filter output, which is a compressed pulse, exceeds a specific threshold. Multiple targets will produce multiple pulses with varying magnitudes. The magnitude of a compressed pulse depends on the duration time of the receiving pulse (T_p), the reflectivity of target (σ), and the distance from radar to target (D). The effective width of a compressed pulse is determined by the duration time and frequency bandwidth (B) of the receiving pulse, or the time–bandwidth (T_pB) product. In the following discussion, a target is assumed to be an ideal target with reflectivity σ = 1, with no signal attenuation occurring between radar and target.

Given an input signal s(t) and its Fourier transform S(f), the matched filter h(t) and its Fourier transform H(f) are defined as

and

Let the output of a matched filter be f(t); then its Fourier transform satisfies the relation

Therefore

Since h(t) = s*(−t), it follows that

which is the definition of autocorrelation of s(t). In other words, the matched filtering of s(t) is equivalent to the autocorrelation of s(t) with itself.

Let the echo signal from a single target be the pulsed nonsymmetric LFM signal described in Eq. (5.11b):

With A₀ = 1 and h(t) = p*(−t), the matched filter output becomes

Let x = αtT_p = Bt; then the absolute value of f(t) becomes

The function |f(x)| has a maximum value at x = 0, and its magnitude equals T_p. From the discussion in Chapter 2, the first x value of sinc(x) = 0 occurs at x = Bt = 1, or t = 1/B. It is also true that the 3-dB mainlobe width of sinc(x) equals the first x value of sinc(x) = 0. Therefore, the 3-dB mainlobe width or the compressed pulsewidth is t = 1/B, where B = αT_p is the bandwidth of an LFM signal with time duration T_p. The pulse compression ratio is therefore

Notice that T_pB is the time–bandwidth product.

Since the time-domain convolution of two signals is equivalent to the multiplication of two signals in the frequency domain, the matched filter can be implemented by transforming both the received signal and the matched filter function into a frequency domain using FFT. The two frequency-domain functions are then multiplied together and followed by IFFT to convert back to the time domain. Because the matched filter function is normally shorter than the received signal (due to multiple targets at different ranges), zero padding on the matched filter function is required when computing the FFT on the matched filter function.

Figure 5.17 illustrates the process of DFT-based pulse compression (or matched filtering). Here the reference chirp signal is the matched filter function h(t), which equals p*(−t), with p(t) as the transmitting signal.

Examples 5.1 and 5.2 (below) illustrate the implementation of pulse compression in two approaches; one through direct convolution in time domain and the other through DFT in the frequency domain. The first example is based on a symmetric pulsed LFM with a single target, while the second example is based on a nonsymmetric pulsed LFM with dual targets. The signal and target related parameters are listed here:

Frequency chirp rate α = 2 × 10¹²

Pulse duration time T_p = 4 × 10⁻⁶ s

Frequency bandwidth of pulsed LFM B = α T_p = 8MHz

Sampling frequency f_s = 20 MHz

Number of samples in T_p = 80 (81 was used in simulation)

Time bandwidth product T_pB = 32

Delay time between Tx pulse and Rx signal:

Example 5.1: 5 × 10⁻⁶ s (or 100-sample delay)

Example 5.2: First target: 5 × 10⁻⁶ s (or 100-sample delay)

Second target: 7.5 × 10⁻⁶ s (or 150-samples delay)

FIGURE 5.17 DFT-based processing of chirp signal.

Example 5.1 Symmetric Pulsed LFM with Single-Target Response Figure 5.18 displays the waveforms of both the transmitter (Tx) signal and the matched filter (MF) function. Since the Tx signal is symmetric, the real part of the MF function is identical to that of the time reversed Tx signal, while the imaginary part of the MF function becomes negative of that of the time reversed Tx signal.

FIGURE 5.18 Waveforms of Tx signal and matched filter function.

Figure 5.19 displays the relationship between the transmitter signal and the received signal. The Rx and Tx waveforms are identical except that the former is delayed by 100 samples relative to the Tx. Here an ideal target is assumed, and the received signal has the same magnitude as the transmitter.

The frequency spectrum of a transmitter signal is shown in Fig. 5.20. Figures 5.20a and 5.20b display the real and imaginary parts of the spectrum, while Figs. 5.20c and 5.20d display the spectrum in terms of magnitude and phase. A 256-point FFT is used to compute the frequency spectrum with bin 128 as the center or origin of the spectrum. As can be seen, the frequency spectrum is symmetric around the origin (bin 128). Since the sampling frequency is 20 MHz, which corresponds to 256 frequency bins, the bandwidth is ~8 MHz (from bins ~80 to 180).

FIGURE 5.19 Waveforms of Tx signal and Rx signal.

The frequency spectrum of a matched filter is shown in Fig. 5.21. Figures 5.21a and 5.21b display the real and imaginary parts of the spectrum; Figs. 5.21c and 5.21d display the spectrum in terms of magnitude and phase. When comparing the spectrum of a matched filter with that of the Tx signal, one can see that the real part is identical to that of the Tx signal, and the imaginary part is negative of that of the Tx signal. The magnitude is the same, but the phase is negative of that of the Tx signal. Again, a 256-point FFT is used to compute the frequency spectrum with bin 128 as the origin of the spectrum.

Figure 5.22 displays the frequency spectrum of the received signal Rx. Although the received signal is a delayed version of the transmitter signal, the real and imaginary parts of the frequency spectrum are quite different from those of the Tx signal. The phase of the Rx spectrum also differs from that of Tx, but the magnitude is identical to that of Tx.

Figure 5.23 displays the compressed receiving signal, or matched filter output. Figures 5.23a and 5.23b show results based on time-domain processing; this is the direct convolution of received signal Rx with the MF function. Figures 5.23c and 5.23d show results based on frequency-domain processing, where the received signal and the MF function are first transformed into the frequency domain. The two frequency spectra are then multiplied together, followed by inverse DFT to obtain the final result as shown in Figs. 5.23c and 5.23d. Both time-domain and frequency-domain processing of pulse compression yield the same results.

FIGURE 5.20 Frequency spectrum of Tx signal.

Notice that the magnitude of the compressed pulse is 80, which equals the sample number of the LFM pulse. The peak of the compressed pulse occurs at time cell 180. This number corresponds to the sum of the Rx signal sample delay (100 in this example) and the sample number of LFM pulse (80 in this example).

The sample number in the compressed pulse, which equals the number between the peak of the pulse and the first zero crossing, is about 2.5 samples. Compared with the sample number in Tx pulse duration time, which equals 80, the pulse compression ratio is 80/2.5 = 32. This is the same number as the time–bandwidth product, or T_pB = 32.

Example 5.2 Nonsymmetric LFM Pulse with Dual-Target Response Figure 5.24 displays the time-domain waveforms of a pulsed LFM transmitter signal and the matched filter (MF). Figures 5.24a and 5.24b show the real and imaginary parts of the Tx signal. The waveform is nonsymmetric, and there are 80 samples within the pulse duration time T_p. Figures 5.24c and 5.24d show the real and imaginary parts of the MF. Both the real and imaginary part of MF waveforms are time-reversed relative to those of the Tx signal. In addition, the magnitude is negated for the imaginary part of the MF. Similar to the symmetric pulsed LFM matched filter, the nonsymmetric LFM MF is flipped around the origin (time cell 1), and appears from time cells 178 to 256 and cell 1 for a 256-point DFT computation.

FIGURE 5.21 Frequency spectrum of matched filter (MF) function.

Figure 5.25 displays the frequency spectra of a pulsed LFM Tx signal and the MF signal. Figures 5.25a and 5.25b show the magnitude and phase parts of the Tx signal, while Figs. 5.25c and 5.25d show the magnitude and phase parts of the MF signal. As can be seen, the magnitude of the MF spectrum is identical to that of Tx, while the phase of the MF spectrum is the negated version of the Tx spectrum. Given the sampling frequency f_s = 20 MHz and the 256-point DFT used, the Tx bandwidth is approximately 8 MHz (from about bins ~130 to 230).

The received signal consists of two target responses. One occurs at time cell 100 and the other, at time cell 150. These two target-reflected signals are added together and shown in Figs. 5.26a and 5.26b. The magnitude and phase of the frequency spectrum corresponding to the Rx signal are shown in Figs. 5.26c and 5.26d. A 512-point DFT is used to compute the spectrum of the received signal. Given the sampling frequency f_s = 20 MHz, the bandwidth of Tx is approximately 8 MHz (from about bins ~260 to 460).

FIGURE 5.22 Frequency spectrum of Rx signal.

Figure 5.27 displays the received signal after compression; Figs. 5.27a and 5.27b show results based on convolution of the Rx signal with the MF function, while Figs. 5.27c and 5.27d show results based on frequency-domain processing. Both time-domain and frequency-domain processes yield the same results.

Similar to the case of symmetric LFM, the magnitude of the compressed pulse is 80. There are two pulses appearing at time cells 180 and 230, respectively. Time cells 180 and 230 correspond with respect to the sum of the sample numbers of the LFM pulse (80) and the sample delay of the first Rx target (100); and also to the sum of the sample delay of the second target (150) and the sample number of the LFM pulse (80).

Similar to the case in Example 5.1, the sample number in the compressed pulse is ~2.5 samples and the pulse compression ratio is 80/2.5 = 32.

5.4.3 Target Detection Techniques of LFM Radar

A pulsed LFM signal with duration T_p can be represented as follows from Eq. (5.8):

p (t) = A₀ exp [j (2π f_ct + π αt₂)].

FIGURE 5.23 Comparison of pulse compression based on convolution and DFT.

The frequency spectrum of the signal is bounded as

and its frequency bandwidth is B = αT_p.

Assuming ideal targets and A₀ = 1, the received signal can then be expressed as

where t_n = 2R_n/c is the round-trip echo delay from the nth target.

To dechirp the received signal, the complex conjugate of the echo signal s(t) is multiplied with a locally generated chirp signal, which is the same as the transmitting chirp signal, except that it is a continuous LFM signal with a much greater bandwidth:

FIGURE 5.24 Waveforms of Tx signal and MF function.

bandwidth:

The dechirped signal r(t) has a frequency component αt_n, which carries information on the round-trip delay t_n, and is generally called the beat frequency f_bn. As an example, Figure 5.28 shows the time–frequency relationship of both the local chirp signal and the echo signals from two targets. Figure 5.28a shows the instantaneous frequency versus time, while Fig. 5.28b show beat frequencies of two targets versus time.

As can be seen in Fig. 5.28, the transmitted chirp signal with pulse duration T_p and period T has two echoes returned from targets at t₁ and t₂. After dechirp processing, the beat frequencies of the two targets become f_b1 and f_b2, respectively. The target range R_n can then be derived from the beat frequency f_bn and the delay time t_n as follows:

FIGURE 5.25 Frequency spectra of Tx signal and MF function.

Therefore

The minimum beat frequency can be computed, from Fig. 5.28, as f_bmin = f_max − f_c = αT_p, which corresponds to the minimum detectable range. The maximum beat frequency is f _bmax = α(T − T_p), which corresponds to the maximum detectable range. The bandwidth of the reference chirp signal is then αT. The block diagram of dechirp processing is shown in Fig. 5.29.

FIGURE 5.26 Time- and frequency-domain waveforms of Rx signal.

Since the maximum beat frequency is α(T − T_p), the Nyquist sampling frequency for the A/D converter is therefore 2α(T − T_p), which could be much smaller than the 2(f_c + αT_p) required for direct sampling on received signals. The output of the A/D converter is then processed through FFT to obtain the beat frequencies f_bn, and the target range can be computed using Eq. (5.20).

The pulsed LFM radar has a cost-effective way to detect the target range, yet it has a relatively lengthy pulsewidth and shares the same antenna for transmitting and receiving; therefore, it has poor minimum/blind range detection capability.

The CWLFM radar uses separate antennas for transmitting and receiving. It consumes less power than that of the pulsed LFM radar, yet still maintains the same signal-to-noise ratio as that of the pulsed LFM radar. It operates at a lower sampling frequency and therefore at a lower system cost. In addition, it can also detect target velocity, described next.

Consider the CWLFM waveform shown in Fig. 5.12 with T_p = T₀/2. Let p(t) be the transmitted signal of CWLFM radar:

FIGURE 5.27 Comparison of pulse compression based on convolution and DFT.

The returned echo signal from a single target with a time delay τ will have the following form:

By multiplying the transmitted signal with the complex conjugate of the echo signal, one obtains

The first item is a constant phase that depends on the delay time only, while the second item is a time-varying signal with a beat frequency

FIGURE 5.28 Time–frequency relationship of Tx, reference, and echo signals.

The maximum beat frequency occurs at τ = T_p and equals to f_bmax = αT_p = αT₀/2. The dechirped signal r(t) can then be sampled by an A/D converter with sampling frequency f_s ≥ 2 f_bmax = αT₀. Since τ = 2R/c, the target range R can then be computed as

Figure 5.30 shows an example of dechirping with two stationary targets; Fig. 5.30a shows the time–frequency relationship corresponding to the transmitting signal and two echo signals, while Fig. 5.30b shows the time–frequency relationship of the two dechirped or beat frequencies. The beat frequencies f_b1 and f_b2, corresponding to delay times t₁ and t₂, respectively, are positive values during the upsweep and the downsweep periods. However, during the crossover between the Tx signal and the echo signal, the echo's beat frequency value reduces to zero first and then increases to the original beat frequency. This is graphically represented in Fig. 5.30. Since T₀ = 2T_p, the beat frequency values in the upsweep and downsweep periods are identical.

FIGURE 5.29 Block diagram of dechirp processing.

FIGURE 5.30 Time–frequency relationship of Tx and echo signals from two stationary targets.

As discussed in Chapter 4, when a target approaches the radar, the received signal frequency is increased by an amount of the Doppler frequency f_D. This increment in f_D reduces the beat frequency in the upsweep region and increases the beat frequency in the downsweep region, that is

where f_bu and f_bd are the observed or measured frequencies. The true range-related beat frequency and Doppler frequency could then be computed as follows:

A similar situation applies if the target is moving away from the radar. Once the beat frequency f_b and the Doppler frequency f_D are known, the target range and velocity can be computed accordingly.

As an example, Fig. 5.31 displays the time–frequency relationship of Tx and echo signals from two moving targets. Figure 5.31a displays three triangular-shape plots, corresponding to the time–frequency relationship of the transmitting signal, the echo signal with target approaching, and the echo signal with target receding. The lighter (dotted) line shows the target approaching the radar while the darker (solid) line shows the target moving away from the radar. Figure 5.31b shows the beat frequencies corresponding to the targets both approaching and receding. Again, the dotted line refers to the approaching target and the solid line refers to the receding target.

FIGURE 5.31 Time–frequency relationship of Tx and echo signals from two moving targets.

5.4.4 Target Detection Techniques of SFM Radar

The SFM-based radar is used primarily for short-range measurement. This is because its detection range is limited to R ≤ (cT/2), where T is the time interval between each stepped frequency and T is normally a small number. Consider a target with distance R from a PSFM radar with stepped frequency f_k, where k = 0, 1, ..., N – 1. The transmitted PSFM signal p_k(2πf_kt) and the returned echo signal s_k(2πf_kt, τ), with time delay τ = 2R/c, can be expressed as follows:

The returned signal s_k(2πf_kt, τ) is then coherently downconverted by multiplying it by a portion of the transmitted signal, follows by a lowpass filter:

FIGURE 5.32 Baseband echo response from PSFM signal.

Let C_k be the lowpass filter output, which can be expressed as

Figure 5.32 shows an example of the digitized and normalized baseband echo signal C_k.

For simplicity, we let A_kB_k = 1 in the following discussion. The frequency-domain baseband signal C_k = exp(−j2πf_kτ) is then transformed into the time domain, using IFFT, as a range profile; that is

where n = 0, 1, ..., N–1. With f_k = f₀ + kΔf and τ = 2R/c, y_n becomes

Let a = (n/N) −(2Δf R/c); then

The magnitude response of the IFFT becomes

Note that the magnitude response of the IFFT, called the synthetic pulse, consists of N components and is repeated for each N time interval. The peaks of the mainlobes of Eq. (5.31) occur at a = ± l, or n_p = 2NRΔf/c + lN, l = 0, 1, 2, ....

The range of target R, which corresponds to the peak of Eq. (5.31), can then be computed as follows:

As an example, Fig. 5.33 shows the simulated plot of |y_n|, where N = 64, Δf = 20 × 10³ Hz, and R = 1.5 × 10³ m were used. According to this plot, the peak of |y_n| appears between time cells 13 and 14, which corresponds to n 12.8 = 13. (Notice that n is an integer and n = 0, 1, ..., N–1.) The range R can be calculated from Eq. (5.32), which is 1.52 × 10³ m. This result is very close to the data R used to plot Fig. 5.33.

The range resolution can be computed as follows, with n_p = 1 in Eq. (5.32):

FIGURE 5.33 A single-target range profile based on PSFM signal.

It is clear that the range resolution of step frequency depends on both the frequency step size and the number of steps that are used. This is different from the range resolution of single-frequency pulse radar with pulse duration T_p, which is

If a number of closely spaced targets are present, but the distance between each target is greater than ΔR, then each target will have its own unique frequency-dependent phase and baseband signal C_i. The final target function (or range profile) is the superposition of the synthetic pulses obtained from each target.

Figure 5.34 displays N-section stepped frequency radar pulses, together with three echo signals corresponding to each frequency pulse. The group consists of N stepped frequencies, starting from f₀ and ending at f_N−1, with each frequency separated by time interval T, shown in the top row of the figure. For each stepped frequency f_i, there exist three echo responses, C_i,1, C_i,2, and C_i,3. The first subscript i of echo response C_i,j denotes the ith stepped frequency. The second subscript j refers to the jth target. The individual frequency component f_i and its associated target echoes are shown under the top row. They were displayed for a period of T without being synchronized with the transmitted pulses.

FIGURE 5.34 Stepped frequency pulse train and echoes returned in one pulse period.

FIGURE 5.35 A multiple-target range profile based on PSFM.

Figure 5.35 displays the synthesized range profile, which consists of three targets. Figure 5.35a shows the target 1 range profile |y_n,1|, which is generated by taking IFFT on {C_k,1} for k = 0, 1, ..., N–1. Figure 5.35b shows the target 2 range profile |y_n,2|, and Fig. 5.35c shows the target 3 range profile |y_n,3|. Both |y_n,2| and |y_n,3| are obtained by taking IFFT on {C_k,2} and {C_k,3}, respectively, for k = 0, 1, ..., N–1. The three range profiles are then superimposed to become the complete range profile |y_n|, which is shown in Fig. 5.35d.

In summary, the following steps are used by the stepped frequency radar to obtain a high-resolution synthetic range profile (SRP) based on stationary targets:

1. Transmit a burst of N pulses (or sections), with each pulse shifted in frequency by a fixed step size Δf. (Assume that N is a number the power of 2.)
2. Collect the in-phase (I) and quadrature-phase (Q) complex value samples of the target's baseband echo response in each coarse range bin for every transmitted pulse. These samples are considered as the frequency-domain measurements of the target's spectral profile.
3. Apply IFFT to the N-complex samples of each coarse range bin to obtain an N-element SRP of the target in the range bin.
4. Repeat steps 1–3 for each multiple target.
5. The final range profile is the superposition of the SRP from each target.

In case the target is moving at speed v_t(n), n = 1, 2, 3, the target will be at a slightly different position with displacement Δr_i on a pulse-to-pulse basis. This position shift will generate an extra phase factor on the frequency response signal C_i; that is

C_i (2π f_i) ⇒ C_i (2π f_i) exp (− j2π f_i Δt_i),

where Δt_i = 2Δr_i/c and Δr_i = r_i – r₀ is the target displacement at the stepped frequency f_i.

The target radial velocity ν_t(n), n = 1, 2, 3, can be computed as follows. By applying M bursts of N-section SFM signal on the three targets described above, let r_1N(n) and r_MN (n) be the range profiles corresponding to the first and the Mth bursts of the N-section signal, and let Δr_MN(n), n = 1, 2, 3 be the range difference corresponding to targets 1, 2, and 3. Then

and

ν_t(n) = Δr_MN(n) /[(M − 1) NT ] for n = 1, 2, 3.