Application Radar Automobile Radar Pulsed Vs Continuous

Pulse Doppler Radar

Michael Parker , in Digital Signal Processing 101 (Second Edition), 2017

Abstract

This chapter provides an overview of pulse Doppler radar. In pulse Doppler radar, the range is estimated by binning the returns of the individual pulses by their time of arrival, which is proportional to the range. The Doppler processing is done by coherently measuring the phase shifts across many pulses. In this way, both range and velocity of targets can be determined. In addition, moving targets can be easily detected against a nonmoving background clutter. Fast Fourier transforms are used for both range and Doppler processing. The rate of pulses, or pulse repetition frequency, is an important parameter. Aliasing can occur in radar systems, similar to sampled signals. These concepts are all covered in this chapter.

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Pulse Doppler Radar

Michael Parker , in Digital Signal Processing 101, 2010

Publisher Summary

This chapter throws light on the pulse Doppler radar. Doppler shifts are a key part of the detection and tracking of objects. For this reason, nearly all radar systems incorporate Doppler processing. It has been working since the ground returns are at the same range as the vehicle, the difference in velocity is the means of discrimination using Doppler measurements. It may be caused by a frequency shift called the Doppler Effect. Although we cannot sense this, the light waves are affected in the same way as the sound waves. The description has also revealed that the radar is ground-based and then all Doppler frequency shifts are due to the target object motion. If the radar is vehicle- or airborne-based, then the Doppler frequency shifts are due to the relative motion between the radar and target object. Target ranges, velocities, azimuths, and elevations may cross over each other. These changes need to be interpolated into a trajectory that can be matched to a specific target. Using radar digital signal processing followed by software-enabled target identification, tracking, monitoring, and classification, the radar system can automate all these functions.

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Radar and Inverse Scattering

Hsueh-Jyh Li , Yean-Woei Kiang , in The Electrical Engineering Handbook, 2005

10.11.2 Pulse Doppler Radar

A pulse Doppler radar uses the Doppler shift to discriminate moving targets from stationary clutter. A low PRF radar has a long unambiguous range but results in blind speeds. On the contrary, a high PRF radar can avoid blind speeds but experiences ambiguity in range. MTI usually refers to a radar in which the PRF is low enough to avoid ambiguity in range but results in blind speeds. The pulse Doppler radar operates with high PRF to avoid blind speeds but at the expense of range ambiguity.

The pulse Doppler radar usually uses filter banks, which are implemented with a discrete Fourier transform rather than delay cancellers to remove the clutter. The improvement factor is a function of the size of the Fourier transform and the window function used.

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RF MEMS for automotive radar

J. Oberhammer , ... Z. Baghchehsaraei , in Handbook of Mems for Wireless and Mobile Applications, 2013

Pulse radar

The pulse Doppler-radar has the advantage of being able to detect small amplitude moving target returns against a large amplitude clutter background. Pulse-delay ranging is based on the measurement of the time delay between the transmitted pulse and the received echo. ⁴

$R=c\frac{\text{Δ}T}{2}$

where c is the speed of light, ΔT is the time difference between transmitted and received pulse and echo. The velocity is related to the Doppler frequency shift between the transmitted pulse and received echo: ⁴

$v=\frac{f_{\text{d}}{\lambda }_{\text{o}}}{2}$

where f _d is the Doppler frequency shift and λ _o is the free space wavelength at the centre frequency. Pulse Doppler radars are half duplex, meaning that they either transmit or receive, which results in high isolation between the transmitter and receiver, thus increasing the dynamic range of the receiver and the range detection of the radar. The disadvantage of this system is the existence of a blind zone given by: ⁴

$R_{\text{b}}=c\frac{({\tau }_{\text{p}}+t_{\text{s}})}{2}$

where τ_p is the pulse width, and t _s is the switching time of the transmit and receive switch, if applicable. Pulse Doppler radar systems are therefore better suited for long-range detection, whereas frequency-modulated continuous wave (FMCW) radar is better suited for short-range detection. ⁴

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RF MEMS for automotive radar sensors

J. Oberhammer , ... Z. Baghchehsaraei , in MEMS for Automotive and Aerospace Applications, 2013

Pulse radar

The pulse Doppler radar has the advantage to detect small amplitude moving target returns against a large amplitude cluttered background. Pulse-delay ranging is based on the measurement of the time delay between the transmitted pulse and the received echo ( Koen and Van Caekenberghe, 2007):

[5.3] $R=c\frac{\Delta \mathit{T}}{2}$

where c is the speed of light and T is the time difference between transmitted and received pulse and echo. The velocity is related to the Doppler frequency shift between the transmitted pulse and received echo (Koen and Van Caekenberghe, 2007):

[5.4] $v=\frac{f_d{\lambda }_o}{2}$

where f_d is the Doppler frequency shift and λ_o is the free space wavelength at the center frequency. Pulse-Doppler radars are half duplex, meaning that they either transmit or receive, which results in high isolation between the transmitter and receiver, thus increasing the dynamic range of the receiver and the range detection of the radar. The disadvantage of this system is the existence of a blind zone given by (Koen and Van Caekenberghe, 2007):

[5.5] $R_b=c\frac{\left({\tau }_p+t_s\right)}{2}$

where τ _p is the pulse width and t_s is the switching time of the transmit and receive switch, if applicable. Pulse-Doppler radar systems are therefore better suited for long-range detection, whereas FMCW radar is better suited for short-range detection (Koen and Van Caekenberghe, 2007).

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Micro-Doppler Effect in Wideband Radar

Qun Zhang , ... Yong-an Chen , in Micro-Doppler Characteristics of Radar Targets, 2017

3.4.1 Linear Frequency Modulation Continuous Wave Signal

The transmission signal and receiving signal of pulse Doppler (PD) radar are separate on time, which accomplish the transmission process and receiving process by the converting of the transmit-receive switch. Since the transmission energy of PD radar gathers in a narrow pulse, its power of peak value is quite high. The PD radar has very high requirements for the sensor on volume, weight, and power. However, the LFMCW radar transmits the signal continuously. Consequently, the energy is distributed in the whole period. Besides, its transmission power is low, so the solid-state amplifier will meet the requirement, which leads to many advantages compared with PD radar, such as smaller volume, smaller weight, lower cost, and stronger concealment. At present, two kinds of LFMCW signal, the saw-tooth-wave frequency modulation and the triangle-wave frequency modulation, are applied popularly, which are shown in Fig. 3.17.

Figure 3.17. The schematic diagram of the frequency modulation form of LFMCW signal. (A) The saw-tooth-wave frequency modulation; (B) the triangle-wave frequency modulation.

The expression of the saw-tooth-wave frequency modulation is

(3.64) $p\left(t\right)=\mathrm{rect}\left(\frac{t}{T_p}\right)·\exp \left(\mathrm{j}2\mathrm{\pi }\left(f_{c}t+\frac{1}{2}\mu t^2\right)\right)$

where the expression of LFMCW signal is the same as the expression of LFM signal, the difference is pulse duration T _p equals the pulse repetition interval (PRI), and pulse duration is much larger compared with LFM signal, which is often at the millisecond.

The expression of the triangle-wave frequency modulation is

(3.65) $p\left(t\right)=\{\begin{array}{cc}\exp \left(\text{j}2\mathrm{\pi }\left(f_{c}t+\frac{1}{2}\mu t^2\right)\right)& t\in \left[0,\frac{T_p}{2}\right)\\ \exp \left(\text{j}2\mathrm{\pi }\left(f_{c}t-\frac{1}{2}\mu t^2\right)\right)& t\in \left[\frac{T_p}{2},T_p\right)\end{array}$

The saw-tooth-wave frequency modulation signal is applied in imaging radar in general. But the triangle-wave frequency modulation signal is applied on the aspects of target velocity estimation, motion-speed detection, etc. In this section, we mainly use the high resolution of LFMCW signal to analyze the micro-Doppler feature of the target, so the following LFMCW signal we referred to is the saw-tooth-wave frequency modulation signal.

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Out-of-seam dilution: Economic impacts and control strategies

Joseph C. Hirschi , Y. Paul Chugh , in Advances in Productive, Safe, and Responsible Coal Mining, 2019

9.4.2.5 Radar-based CID

This technology utilizes a single antenna, which transmits and receives Doppler radar pulses. A network analyzer controls signal frequency. Signals are attenuated as they pass through coal and bounce off the density interface of the confining rock, which is interpreted by the network analyzer to determine the distance to that interface. This system has reliable accuracy and operates well under most roof conditions. It is also suited for monitoring rib thickness between adjacent holes in highwall mining. Two disadvantages of this system are that it does not work well in coal seams with wave-dispersing properties and it requires the transmitter to be located within 4   in. (10   cm) of the coal [23].

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Radar

Merrill I. Skolnik , in Reference Data for Engineers (Ninth Edition), 2002

PULSE RADARS THAT EMPLOY DOPPLER

There are three types of pulse radars that extract the doppler frequency shift, or relative velocity, in addition to the range information. These are:

1.

A moving target indicator (MTI) radar in which the pulse repetition frequency (prf), denoted by f_p , is low enough to have no range ambiguities; i.e., its maximum unambiguous range R _unam = c/2 f_p . On the other hand, the MTI radar has ambiguities in doppler and many blind speeds (due to doppler frequency ambiguities) where moving targets are not detected.

2.

A high pulse repetition frequency pulse doppler radar in which the prf is large enough to have no blind speeds [as given by Eq. (32)] within the expected values of target doppler frequencies. There will be, however, many range ambiguities because of the high prf. Range ambiguities are resolved and the true range is found by transmitting three separate waveforms, each at a different prf. (In theory, only two different prfs are required to resolve range ambiguities, but in practice at least three are needed.) An advantage of a high-prf pulse doppler radar is that it can readily detect targets with a high relative-velocity since such echoes do not compete with the echoes from clutter that are at lower doppler frequencies. It has much poorer performance, however, against low relative-velocity targets which compete with the large clutter echoes seen by this type of radar in its antenna sidelobes because of the high prf.

3.

A medium prf pulse doppler radar has both range and doppler ambiguities. It will not detect high relative-velocity targets as well as can a high-prf pulse doppler radar, but it will detect low relative-velocity targets better because its low prf sees less clutter than does the high prf.

Most ground-based air-surveillance radars that must see aircraft in the midst of clutter are generally MTI radars. When flown in an aircraft for purposes of air-surveillance, these are called AMTI, or airborne moving target indicator radar. AMTI radars are very good for airborne air-surveillance radars at UHF, but result in too many blind speeds when employed at the higher microwave frequencies. The high-prf and the medium-prf pulse doppler radars can both be used for airborne air-surveillance purposes at the higher frequencies. (The Sband AWACS, or AN/APY-1, airborne air-surveillance radar utilizes a high-prf pulse doppler waveform.) An X-band fighter/attack radar in a modern military aircraft might employ on a time-shared basis three widely differing prfs, depending on the operational situation. When no clutter is present (as when the antenna beam is looking up above the surface of the earth), a low prf waveform without any doppler processing might be employed. (The low prf waveform will provide the greatest range, if no clutter is present.) When searching for a target in clutter, the high prf and the medium-prf pulse doppler waveforms might be interleaved. The high prf would be looking for high-speed approaching targets at long range and the medium prf would be looking for slower-speed targets at shorter ranges.

The doppler frequency shift is important for many purposes in radar. In addition to the MTI, AMTI, and pulse doppler radars, the doppler frequency is the key to such radars as

•

Doppler weather radars

Nexrad, whose output is regularly shown on TV weather broadcasts.

Terminal Doppler Weather Radar (TDWR). These are located in the vicinity of major airports to warn aircraft, that are landing or taking-off, of the presence of dangerous wind shear.

Wind profiler, that measures as a function of altitude the wind speed and direction for both weather prediction and the efficient routing of aircraft.

Airborne weather-avoidance doppler radar, that detects dangerous wind shear to warn the pilot of danger during take-off or landing.

•

Synthetic aperture radar (SAR), (for mapping a scene on the surface of the earth, and the inverse synthetic aperture radar (ISAR), for imaging a target well enough to recognize it from other similar targets. (SAR is more usually thought of as an antenna synthesized in a digital processor, but it was originally invented as a doppler radar and can be described as depending on the doppler effect.)

•

Doppler navigator, a multi-beam radar which can provide the vector velocity of an aircraft which carries it.

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HF over-the-horizon radar, which depends on the doppler frequency to detect aircraft and ships in the presence of large clutter echoes. The doppler spectrum of the sea echo obtained by an HF radar can also be used to extract the strength and direction of the winds over the ocean.

The continuous wave (CW) radar also depends on the doppler frequency shift to detect targets in clutter and/ or to measure relative velocity.

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MEMS for passenger safety in automotive vehicles

J. Swingler , in MEMS for Automotive and Aerospace Applications, 2013

1.6.3 Environment monitoring systems

The motor vehicle is expected to become more automated to enable more fuel efficiency, to reduce traffic congestion, and deliver higher safety for occupants and pedestrians. This will require more sensors and novel types of sensing technology, as well as novel systems. Currently, long-range distance sensors are used on adaptive cruise control to not only maintain a constant speed but to maintain a safe distance from the car in front. The adaptive cruise control slows the vehicle as it gets too close to one in front. These sensor types are already used for warning of a forward collision. Sensor types current used are 'pulse doppler radar', 'frequency-modulated/continuous-wave radar', 'monopulse radar' and 'laser radar' ( Fleming, 2008). The challenge for the future is to implement these, or novel versions, using MEMS technology to reduce the size of the sensing units.

Short-range distance sensors are used for parking assist and lane changing assist systems, to help the driver minimize collisions. Sensor types available are 'ultra-wideband radar', 'multibeam-forming radar', 'laser radar', 'camera vision' and 'ultrasonic sensors' (Fleming, 2008). Again the challenge for the future is to implement these, or novel versions, using MEMS technology, without the need for larger structures being attached to the micro-fabricated components. The ideal is to integrate as much as possible onto a single micro-fabricated substrate.

These long-range and short-range sensors are used to assist the driver to prevent any crash in the first instance. However, another area that is likely to expand is to adopt technology to anticipate a crash event in the moments before it actually occurs (pre-crash sensing). Then the on-board systems can prepare the vehicle and occupants with appropriate protection measures. The anticipation of a crash event gives more time for early triggering of multiple protection devices.

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Sparsity-based radar technique

Matthias Weiß , in Academic Press Library in Signal Processing, Volume 7, 2018

6.2.2 Sparse Sampling in Range and Doppler

For the second example we consider a pulse radar which emits a pulse train with a given pulse repetition time interval (PRI) or reciprocal pulse repetition frequency (PRF), respectively. The samples are then arranged into a matrix form with one index corresponding to the range (fast-time) and the other to the pulse index (slow-time). The Doppler frequency of the moving object can then be obtained by a FFT along the slow-time. As shown in the previous example sparse signal processing techniques allows us to implement a nonuniform or so-called co-prime sampling in range (fast-time). In the second example this technique is expanded to include also a co-prime sampling in slow-time to further reduce the amount of data which has to be processed. It will be shown that the estimation accuracy of the Doppler frequency of the moving target will not degrade.

Let an ordinary pulse Doppler radar emit M pulses with a pulse repetition interval (PRI) of τ _⊓. A single transmitted pulse is described by its baseband function x(t) and its continuous-time Fourier transformation $X(\omega )={\int }_{-\infty }^{\infty }x(t)e^{-j\omega t}dt$ . The frequency bandwidth of the pulse is B and we presume that X(ω) has almost no energy at frequencies outside of B. With these constraints the transmitted signal x _trans (t) can be written in a matrix form as:

(6.14) $\begin{array}{ll}\hfill {\mathit{X}}_{trans}(t)& =[{\mathit{x}}_1^T(t),\dots ,{\mathit{x}}_M^T(t)]\hfill \\ \hfill \text{with}{\mathit{x}}_m(t)& =x(t-(m-1){\tau }_{\sqcap }),(m-1)\le t\le m{\tau }_{\sqcap }.\hfill \end{array}$

The matrix X _trans consists of M columns which represent the equally spaced pulses x(t) with a pulse to pulse delay of τ _⊓. The range and Doppler resolution obtained by standard radar signal processing techniques is reciprocally proportional to the signal bandwidth Δr ∼ 1/b and to the coherent processing time Δf _D ∼ 1/Mτ _⊓, respectively.

For the sake of simplicity let us assume that a scene contains K constant moving point-like targets. According to that the transmitted pulses x(t − (m − 1) τ _⊓) are reflected by these objects and propagate back to the receiver. Therefore the received signal is described by:

(6.15) $\begin{array}{ll}\hfill {\mathit{Y}}_{received}(t)& =[{\mathit{y}}_1^T(t),\dots ,{\mathit{y}}_M^T(t)]\hfill \\ \hfill \text{with}{\mathit{y}}_m(t)& =\sum _{k=1}^K{s}_{k}x(t-{\tau }_k-(m-1){\tau }_{\sqcap })e^{-j{f}_{Dk}(m-1){\tau }_{\sqcap }},(m-1)\le t\le m{\tau }_{\sqcap },\hfill \end{array}$

where s _k is the complex amplitude corresponding to kth target radar cross section and the propagation attenuation, τ _k = 2 r _k /c ₀ the time delay, and f _Dk the Doppler radial frequency, proportional to the radial velocity of the target.

For a co-prime sampling in slow-time we introduce the variable $P\subset [1,\dots ,M]$ , where the index p denotes the pth pulse transmitted at time m _p τ _⊓:

(6.16) $\begin{array}{l}\hfill {\mathit{Y}}_p(t)=[{\mathit{y}}_1^T(t),\dots ,{\mathit{y}}_P^T(t)].\end{array}$

By introducing index p we extend the sparse recovering technique in such a way that it can work likewise with sparse sampling in slow-time (pulse repetition or Doppler-domain). The objective of the following steps are to estimate accurately the target range and Doppler frequency, actually time delay τ _l and Doppler shifts v _l , from the received signals y _p (t). It will be shown that again sparse recovering technique is able to recover this information with less samples in fast-time (range) and slow-time (quantity of transmitted pulses supp(P) < M) without degrading the accuracy of the estimation. However, this will only be true if the observation time Mτ _⊓ is equal.

At first we transform the time-domain representation of the aligned received pulses Y _received (t) from Eq. (6.15) into the Fourier domain with N discrete frequencies f _Dn ∈ (−f _D , …, f _D ):

(6.17) $\begin{array}{l}\hfill {\mathit{Y}}_p[n]=\frac{1}{{\tau }_{\sqcap }}X[n]\sum _{l=1}^L{s}_l\cdot e^{-j2\pi f_{Dn}{\tau }_l}\cdot e^{-j{v}_l(m_p-1){\tau }_{\sqcap }/\lambda }\end{array}$

with λ the wavelength of the center frequency of the transmitted modulated pulse and L the number of grid points in the time-delay/Doppler plane (s _l , v _l ). For the continuous case the sampling rate in fast-time and slow-time is determined by the pulse bandwidth (t _n ≥ 1/2 B) and by the demanded Doppler resolution (∼ PRF/Δf _D ), respectively. The unknown parameters of the targets (s _l , τ _l , v _l ) are contained in the Fourier coefficients Y _p [n].

The measurement matrix $\mathit{Y}\in {\mathbb{C}}^{N\times P}$ with its pth column defined by y _p [n] describes the measurement in fast- and slow-time in the Fourier domain and can be expressed by:

(6.18) $\begin{array}{l}\hfill \mathit{Y}=\mathit{X}{\mathit{F}}_{\tau }\mathit{S}{({\mathit{F}}_D)}^T\end{array}$

with X[n] the Fourier transformed pulse at N discrete frequencies (f _Dn = −f _D , …, f _D ), $\mathit{X}=\frac{1}{{\tau }_{\sqcap }}\text{diag}(X[n]))$ , F _τ the Fourier time shift matrix with its M columns and N selected Fourier coefficients $\left(F_{\tau }^K(m,n)=exp(-j2\pi f_{D}n{\tau }_m)\right)$ , and ${\mathit{F}}_D\in {\mathbb{C}}^{N\times P}$ the Fourier Doppler shift matrix with its P rows ( $F_T^P=exp(-j2\pi f_D(m_p-1)\tau )$ ). The matrix S of size M × N holds the values s _l at the searching grid points L = M N in the range/Doppler plane (τ _l , v _l ). As the scene contains only a limited number of targets the relation K ≪ L is true and consequently S is a sparse matrix with only K nonzero elements.

To estimate the coefficients of the sparse matrix S we first consider that

(6.19) $\begin{array}{l}\hfill \mathit{Z}={\mathit{X}}^{-1}\mathit{Y}\end{array}$

and then we obtain

(6.20) $\begin{array}{l}\hfill \mathit{Z}={\mathit{F}}_{\tau }\mathit{S}{({\mathit{F}}_D)}^T.\end{array}$

For a successful parameter estimation of the sparse matrix elements s _{n, p} it has been shown for the noiseless case that for K targets the minimal number of samples of 4K ² are needed, with N ≥ 2 K (fast-time) and P ≥ 2 K (slow-time), respectively [29, 30].

For the noise free environment we have to solve the following optimization problem to recover the nonzero elements of the sparse matrix S [31]:

(6.21) $\begin{array}{l}\hfill min\parallel \mathit{S}{\parallel }_1\text{subject to}{\mathit{F}}_{\tau }\mathit{S}{({\mathit{F}}_D)}^T=\mathit{Z},\end{array}$

where $\parallel S{\parallel }_1={\sum }_{i,j}|S_{ij}|$ is the ℓ ₁-norm of vec( S ), where vec( S ) vectorize the matrix S by stacking the columns into a vector.

If there is any noise present Eq. (6.21) has to be adapted in such a manner that a threshold is taken into account:

(6.22) $\begin{array}{l}\hfill \underset{\mathit{S}}{min}\left\{\frac{1}{2}{‖\mathit{Z}-{\mathit{F}}_{\tau }\mathit{S}{({\mathit{F}}_D)}^T‖}_2^2+\lambda \parallel \mathit{S}{\parallel }_1\right\},\end{array}$

where λ is a regularization parameter and $\parallel X{\parallel }_2={\sum }_{i,j}|X_{ij}{|}^2$ is the ℓ ₂-norm of vec( X )

To solve the underdetermined linear equation set defined in Eq. (6.22) several sparse recovering algorithms exist (comp. in [31]). An alternative procedure to estimate the coefficients of the sparse state matrix S is to transform Eq. (6.21) into a vector version by using the following relation:

(6.23) $\begin{array}{l}\hfill \text{vec}({\mathit{F}}_{\tau }\mathit{S}{\mathit{F}}_D)=({\mathit{F}}_D^T\otimes {\mathit{F}}_{\tau })\text{vec}(\mathit{S}),\end{array}$

where ⊗ is the Kronecker operator and vec( B ) is the operator which stacks the columns of a matrix B into a vector b .

Applying Eq. (6.23) to Eq. (6.21) yield:

(6.24) $\begin{array}{l}\hfill \mathit{y}=\mathit{C}\mathit{s},\end{array}$

where y is the stacked version of matrix Z , $\mathit{C}={\mathit{F}}_D^T\otimes {\mathit{F}}_{\tau }$ , and s = vec( S ). With this equation we can find the sparse solution of s by solving the ℓ ₁-norm minimization problem:

(6.25) $\begin{array}{l}\hfill min\parallel \mathit{s}{\parallel }_1\text{subject to}\mathit{C}\mathit{s}=\mathit{y}.\end{array}$

To confirm the improved performance of this discussed method let's assume that a standard pulse-Doppler radar transmits M chirped pulses with a bandwidth of B = 147   MHz and a PRI of τ _⊓ = 20   ms. For the first case we consider that P is equal M = 20 pulses and in the fast-time the Nyquist rate is fulfilled. Hence, all available data is used.

Fig. 6.2 shows the output of the simulation for the standard matched filter approach (left image) and the sparse matrix recovering method on the right.

Fig. 6.2. Result for a pulse-Doppler radar with a B = 147   MHz, τ _⊓ = 20   ms, uniform sampling in fast- and slow-time with M = 20 fulfilling Nyquist criteria. Left diagram shows the result of the matched filter and the right diagram obtained by sparse reconstruction technique for the case of four moving targets.

For the co-prime sampling case P = 10 pulses are randomly chosen from the M = 20 transmitted pulses. Also the sampling rate in fast-time (range) is reduced by a factor of 2. The result is shown in Fig. 6.3.

Fig. 6.3. Result for a pulse-Doppler radar with a B = 147   MHz, τ _⊓ = 20   ms, nonuniform sampling in fast- and slow-time with P = 10∈1, …, 20. Left diagram shows the result of the matched filter and the right diagram of the compressive sensing case for four moving targets.

The result obtained by the standard signal processing chain is shown in the left image. The direct comparison with the right image, which shows the solution obtained by the sparse reconstruction technique, confirms that the new approach is clearly able to estimate positions and velocity of all four moving targets with high accuracy. Furthermore, this method does not show any side-lobes.

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