This section introduces several simulations to examine how the CRLB performs when the parameters are changed. The discussions above are compared herein.

#### 6.1. CRLB Comparison with Different Waveforms

According to the deviation of the time delay and Doppler stretch model, we generate the CRLB by various waveforms in different parameter configurations for comparison. These waveforms include LFM

${}_{1}$ and LFM

${}_{2}$ with changing pulse width and total bandwidth, LSF

${}_{1}$ and LSF

${}_{2}$ with changing pulse number and total bandwidth, two SSF waveforms with changing envelope, namely SSF

${}_{\mathrm{chirp}}$ (

13) and SSF

${}_{\mathrm{rect}}$ (

14). In each trial, the value of SNR is set from 0 to

$40\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$. Other basic parameters are

${\mathrm{T}}_{\mathrm{w}}=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$,

$\mathrm{T}=2\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$,

$M=100$,

$N=50$,

$\Delta f=50\phantom{\rule{3.33333pt}{0ex}}\mathrm{kHz}$,

${f}_{0}=10\phantom{\rule{3.33333pt}{0ex}}\mathrm{MHz}$. The theoretical CRLB for time delay and Doppler stretch estimation are respectively illustrated in

Figure 1a,b. For the legends [A,B,C] in the figure, three parts represent the total number of pulses in a burst, pulse width for each pulse, and equivalent coverage bandwidth. “

M in order” represents the using frequencies from 0 to

$M-1$ multiples of

$\Delta f$ in sequence. “Random

N from

M” means

N frequencies combination is generated as the rule of SSF from frequency span

$M\Delta f$.

As shown in the figure, the performance trend of the estimation from multiple waveforms is consistent along with the increasing SNR under the same basic parameters. However, the difference exists between time estimation and Doppler estimation. For time delay estimation, the CRLB of SF-based waveforms are obviously not as good as that of LFM signal, but have better performance in Doppler estimation. When the bandwidth range and pulse width change, it has slight influence on the estimation for LFM. With the increasing bandwidth in LSF, the performance improves significantly. By comparison, the CRLB of SSF with N sparse frequencies is used between N pulses and M pulses in LSF signal. Moreover, the envelope plays an insignificant role.

#### 6.2. Comparison of Different CRLBs

For the deviations in the time delay and Doppler stretch model, we use the CRLB of SSF-chirp signal as an example. The obtained statistical rules are also applied to the SSF-rect and LSF signals based on a reasonable definition of variables.

For each SNR value from 0 to 40 dB, we calculate the theoretical

$\mathrm{var}{\left(\widehat{\tau}\right)}_{ship}$ of 500 independent Monte Carlo simulations. In each trial, the frequency combination is generated randomly from the frequency span. The basic parameters are

${T}_{w}=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$,

$T=5\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$,

$M=100$,

$N=50$,

$\Delta f=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{KHz}$,

${f}_{0}=5\phantom{\rule{3.33333pt}{0ex}}\mathrm{MHz}$. The estimation performances with different frequency steps are illustrated in

Figure 2a. The basic parameters are set as the above setting while

$\Delta f$ is set to 0 kHz, 50 kHz, 100 kHz, and 200 kHz for comparison. Similarly,

${f}_{0}$ is set to 5 MHz, 15 MHz, 25 MHz and 35 MHz in

Figure 2b; frequency number

N is set to 10, 40, 70, and 100 in

Figure 2c. Any four groups of sparse frequency combination are selected from all the tests to compare in

Figure 2d.

As shown in the figure, the changes in $\Delta f$, ${f}_{0}$, and N exhibit corresponding effects on the range of the final CRLB. However, only $\Delta f$ exhibits uniform variations, the other two variables exhibit differentiations under equidistant values. The effect of random combination is minimal when the other conditions remain unchanged.

The CRLB of the joint multiple parameter estimation is the same as in (

4). Every normalized

${\mathrm{CRLB}}_{\mathrm{i}}$ as the

${i}^{th}$ independent parameter can also be calculated individually. For every SNR value from 0 to

$50\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$, the theoretical CRLB is calculated. In each trial, the frequency combination is generated randomly within the frequency span. The other basic parameters are the same as that in the first simulation. The target information to be estimated is set as

$R=5\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}$,

$V=2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}$,

$A=0.5+0.8j$.

The estimation performance of every

$\mathrm{CRL}{\mathrm{B}}_{i}$ is illustrated in

Figure 3a. It shows the contribution of each separated value to the overall CRLB by changing the SNR. If the best estimation performance is synthesized, the parameters related to the single coefficient to be estimated can be adjusted appropriately.

To observe the trend of performance caused by various parameters more intuitively, a statistical analysis is presented in

Figure 2b. The differences of the effects from various factors on the estimation performance are shown clearly. For each group of parameters, 500 independent Monte Carlo simulations are performed to obtain a statistical trend. The primary parameters of the variables are shown in

Table 1.

${\left(\xb7\right)}^{\left(0\right)}$ represents the initial value for any variables; ${\left(\xb7\right)}^{\left(i\right)}$ represents the value in the ${i}^{th}$ group. When testing a variable, other variables are defined as the initial value. For the parameters of every group, the logarithmic proportional function is used to characterize the trend, that is $P\left(i\right)=log{\displaystyle \frac{\mathrm{var}{\left(\xb7\right)}^{\left(i\right)}}{\mathrm{var}{\left(\xb7\right)}^{\left(0\right)}}}$. The testing variables are the frequency combination coefficient $\mathbf{C}$, frequency number N, environment parameter ${\sigma}^{2}$, PRI T, frequency step $\Delta f$ and initial carrier frequency ${f}_{0}$. As shown in the figure, the changes in $\Delta f$, ${f}_{0}$, and N exhibit corresponding effects on the range of the final CRLB. However, only $\Delta f$ exhibits uniform variations; the other two variables exhibit differentiations under equidistant values. The effect of random combination is minimal when the other conditions remain unchanged.

The related parameters are divided broadly into the following categories.

Envelope of signal: When $\gamma =0$, $\mathrm{var}{\left(\xb7\right)}_{chirp}=\mathrm{var}{\left(\xb7\right)}_{rect}$. The value of $\gamma $ has little effect. Because the value is sufficiently small, and the most of this coefficient concentrate in the items of n sequence with ${f}_{n}>>n>\gamma $, it implies that the envelope is not the dominant factor.

SNR, velocity, and amplitude: The real coefficient containing $\pi $, c, and ${\sigma}^{2}$ will directly affect the calculation results. The three items ${N}_{0}$, $\sigma $, $\left|a\right|$ are related to the SNR, velocity, and amplitude, respectively. The SNR can be obtained from the power and PSD, namely, $SNR=E/{N}_{0}$. Specifically, CRLB is proportional to $\sigma $, and inversely proportional to ${N}_{0}$.

Sparse frequency combination, frequency number N: When the value of N is fixed, the using signal bandwidth and the synthetic bandwidth are constant for each sparse frequency combination. Thus, the combination of frequencies does not have a significant effect. However, the maximum interval between two frequencies and the dispersion of every combination will impose higher requirements on the recovery method. Different methods involve particular constraints on the relationship between N and M, as well as special requirements for the rule of frequency hopping. We will not discuss this in detail; only the performance comparison will be emphasized herein.

Initial carrier frequency, pulse width, frequency step, and PRI: These parameters are determined by the signal, and are not changed by the environment and target scene. The trends of the three parameters is are synchronized. The value of ${f}_{0}$ is typically large, reaching MHz or even GHz, which is significantly larger than any other parameters. Therefore, it is the most significant factor. Conversely, T and ${T}_{w}$ are negligible owing to their small magnitudes.

#### 6.3. CRLBs Using Different Estimators

This simulation uses the bound comparison produced by different estimators. The symbols C1, C2, and C3 are used to represent the time delay and Doppler stretch estimator, joint multiple parameter estimator, and sparse based estimator, respectively. The comparison results are shown in

Figure 4.

The general parameters are set with

${T}_{w}=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$,

$T=4\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$,

$M=50$,

$N=80$,

$\Delta f=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{kHz}$,

${R}_{0}=15\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}$,

${V}_{0}=2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}$,

${A}_{0}=-0.5+0.5j$, and the SNR is changed from 0 to 30 dB. In

Figure 4a, the sampling rule includes the uniform

M frequencies in

M pulses, uniform

N frequencies in

N pulses, and random

M sequences from

N frequencies with

M pulses. The time delay

$\tau $ and Doppler stretch

$\sigma $ estimation are tested. In

Figure 4b, the primary target information is measured: target range

R, velocity

V, and amplitude

A under different sampling models. Jointly, the obtained overall CRLB

$C2-All$ is shown in the same figure. In

Figure 4c, the sparse estimation is considered with the method based on grid partition. Therefore, it depends on the numbers of range grids

$GR$ and velocity grids

$GV$. We use three groups of different grid numbers as an example.

As shown in the figure, the CRLB index is changed owing to the different samplings. In the uniform samplings, more samplings produce better estimation performance. However, the transmitting frequencies chosen randomly are somewhere in between. Therefore, we conclude from the statistical estimation, part of the estimation can be consistent with the theoretical value, while the estimation performance remains unchanged when the SNR is large.

#### 6.4. RD Comparison between Different Methods

A practical example is presented to demonstrate the effectiveness of the target location using different estimation methods. We use the IDFT method, correlation function method [

18], and CS method to estimate the range and velocity as a practical example. The contrasting figures illustrate the effect of resolution improvement and sidelobe suppression in the range domain. An SSF transmitting signal is designed using all available frequencies. The system parameters are set with carrier frequency

${f}_{0}=7.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{MHz}$, sample frequency

${f}_{s}=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{kHz}$, pulse width

${T}_{w}=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$, PRI

$T=4\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms}$, frequency span is

$500\phantom{\rule{3.33333pt}{0ex}}\mathrm{kHz}$, pulse number

$M=250$, step size

$\Delta f=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{kHz}$. Six targets are considered with the parameter configuration as (

$\#1$)

${R}_{1}=258\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}$,

${V}_{1}=3\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}$, (

$\#2$)

${R}_{2}=258\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}$,

${V}_{2}=6\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}$, (

$\#3$)

${R}_{3}=251\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}$,

${V}_{3}=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}$, (

$\#4$)

${R}_{4}=250\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}$,

${V}_{4}=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}$, (

$\#5$)

${R}_{5}=244\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}$,

${V}_{5}=-1\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}$, (

$\#6$)

${R}_{6}=240\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}$,

${V}_{6}=-5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}$. The SNR for every target is set to the same value

$20\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{dB}$. Thus, the frequency utilization rate is only 46%. We set multiple moving targets in the line-of-sight with the range and velocity values shown in

Figure 5d.

Targets

$\#3$ and

$\#4$ are located at a close proximity with the same (zero) velocity. For the convenience of observation, the targets are shown with the same amplitude and SNR. Two-dimensional signal processing for the echo signals is performed from 50 burst pulses transmitted continuously. Processing methods are used to estimate the information of the targets. The comparison results of the three methods are shown in

Figure 5a–c.

It is apparent that the IDFT method cannot determine the location of the targets. The range and velocity information of the targets can be obtained by the correlation function method. However, owing to the effect of the high-range sidelobe level, this method cannot distinguish the exact location of the targets. If we do not consider the ambiguity caused by the radial velocity, and only focus on the sidelobe suppression in the range domain, the CS method yields a better performance in improving the precision of the range measurement. The selected observation range of the axis is −20 to 0 dB. As shown, the sidelobe below −20 dB have little effect on the targets; additionally, it is easy to distinguish two closely spaced targets.

Next we will give a statistical result for IDFT, CORR and CS method. In

Figure 6, by adding the range, velocity and the amplitude estimated error together, the statistical values of the minimum mean square error (MMSE) are tested by Monte Carlo simulations. The times of Monte Carlo is 500, SNR from 0 to 30 dB. For CS, the traditional orthogonal matching pursuit method is chosen here. Three different methods are drawn separately because of the difference in error value level.

Figure 6a illustrates the MMSE of IDFT and CORR method. The results of comparison between CRLB and MMSE of CS method are shown in

Figure 6b.

As shown in the figure, with the increase of SNR, the trend of MMSE is decreased before stable in a certain value until it reaches a large SNR. CORR method has the fastest speed to achieve stable accuracy, and CS method has the best robustness in this test. The results of theoretical CRLB and MMSE are consistent with CS method. When SNR exceeds 20 dB, the limitation of grids will lead to a same error without further reduction.