# Performance Analysis of Joint Time Delay and Doppler-Stretch Estimation with Random Stepped-Frequency Signals

## Abstract

This paper investigates the performance of joint time delay and Doppler-stretch estimation with the random stepped-frequency (RSF) signal. Applying the ambiguity function (AF) to implement the estimation, we derive the compact expressions of the theoretical mean square errors (MSEs) under high signal-to-noise ratios (SNRs). The obtained MSEs are shown consistent with the corresponding Cramer-Rao lower bounds (CRLBs), implying that the AF-based estimation is approximately efficient. Waveform parameters including higher carrier frequencies, wider bandwidth covered by the carrier frequencies, and frequency shifting codewords with larger variance are expected for a better estimation performance. As a synthetic wideband signal, the RSF signal achieves the same estimation performance as the OFDM signal within an identical bandwidth. Due to its instantaneous narrowband character, requirement for the bandwidth of the receiver is much reduced.

Random stepped-frequency, time delay, Doppler-stretch, ambiguity function, MSE.

## 1 Introduction

The stepped-frequency (SF) signal has been widely adopted in modern wideband radar and sonar systems. Compared to the conventional narrowband signal, the SF signal achieves higher range resolution, and the multiple scatterers of the target can be thoroughly distinguished. Since its energy is dispersed to the whole bandwidth covered by the carrier frequencies, the SF signal attains a lower probability of interception [1]. Meanwhile, classified as one of the synthetic wideband signals, the SF signal only takes up a narrow bandwidth at any time instant, while the whole carrier frequency bandwidth can be occupied if an instantaneous wideband signal (e.g. the OFDM signal) is employed. From this scope, the SF signal could largely reduce the requirements for the bandwidth of the receiver [2].

The linear stepped-frequency (LSF) signal uses a fixed frequency shifting step, which introduces a “ridge” in its ambiguity function (AF). This causes a coupling problem between the range and Doppler dimensions [4]. For the random stepped-frequency (RSF) signal, however, with the carrier frequencies of the pulses randomly distributed over a given bandwidth, its AF appears in a thumbtack alike shape, where the range and Doppler dimensions are completely decoupled [3]. The resolutions in both dimensions thus meet further improvements, and the range ambiguity is efficiently suppressed. Besides, as the frequency shifting codeword of the RSF signal is usually highly self-correlated and hard to track, the interference between adjacent radars can be largely reduced, whereas the electronic counter-countermeasures (ECCM) capabilities can be also acquired [4]. As a result, parameter estimation with the RSF signal is of great practical significance and is gaining increasing research interests.

The joint estimation of time delay and Doppler-stretch is a fundamental problem that facilitates target tracking and localization in radar and sonar systems, upon which the location and velocity information of the target is managed to be attained [5]. One of the standard methods for joint delay-Doppler estimation is to adopt the AF [6, 7]. By locating the peak of the AF, the joint estimation was primarily implemented [8].
So as to evaluate the performance of the estimation, the Cramer-Rao lower bound (CRLB) is commonly employed, for the reason that it is regarded as a theoretical lower bound for the variance of any unbiased estimation and is usually easy to calculate. Based on a wideband signal model, the CRLBs of time delay and Doppler-stretch were derived, under the assumption that the scattering coefficient of the target was known a priori [8]. Derivation for a more realistic case was performed by [9], where the scattering coefficient was supposed to be unavailable at the receiver. The estimation problem of an extended target with multiple scatterers was even considered in [10].
Nevertheless, the CRLB is only reliable for accurately presenting the estimation performance when it is approached by the corresponding mean square error (MSE), i.e. when the estimation is (asymptotically) *efficient* [11]. By directly calculating the theoretical MSE, the estimation performance can be straightforwardly revealed. However, in most cases it is difficult to evaluate the MSE in a theoretical manner. Among the limited number of related works, the MSEs of the AF-based joint estimation were calculated in [8], whereas a correction followed in [12]. However, the calculating results are not accurate since too many approximations were made in both works.

All the approaches [8, 12, 9, 10] were built on a general wideband signal model, which failed to reveal the waveform parameters that influence the estimation performance with any specific signal. Based on the LSF signal model, the CRLBs of time delay and Doppler-stretch were derived in [13], whereas those of the high resolution range (HRR) profiles of an extended target were provided by [14]. Similar range and Doppler estimation problems with the OFDM signal were also considered in [15, 16, 17]. However, performance analysis on the parameter estimation with the RSF signal is quite limited in previous works.

In this paper, we investigate the performance of the AF-based joint delay-Dppler estimation with the RSF signal. Under high signal-to-noise ratio (SNR) assumption, compact expressions of the MSEs are obtained through a novel and strict derivation. The MSEs are shown consistent with their corresponding CRLBs, revealing that the AF-based estimation is approximately efficient. As illustrated by the derivations and simulations, three waveform parameters of the RSF signal, namely, the central carrier frequency, the bandwidth covered by the carrier frequencies, and the variance of the frequency shifting codeword mainly influence the estimation performance. In specific, by increasing either the bandwidth for the carrier frequencies or the variance of the frequency shifting codeword the performance of delay estimation can be improved, while better performance of Doppler-stretch estimation calls for higher central carrier frequency. As one of the synthetic wideband signals, the RSF signal only takes up a narrow bandwidth at any time instant, whereas it achieves the similar estimation performance as the OFDM signal does, contributing to a much reduced requirement for the bandwidth of the receiver.

The rest of the paper is organized as follows. Section II describes the model of the RSF signal and makes necessary preliminaries. Section III gives the main results of this paper, while Section IV provides the derivations for the main results. Section V verifies the main results by numerical examples, followed by a conclusion in Section VI.

## 2 Modeling and Preliminaries

Consider an RSF signal with pulses. Let denote the envelope of each pulse. The transmitted signal is then modeled as

(1) |

where denotes the carrier frequency of the -th pulse, and is the pulse repetition interval (PRI). Assume that the carrier frequency remains constant within each pulse and shifts randomly over the pulses within a given bandwidth. Then the carrier frequencies of the pulses can be represented as

(2) |

where denotes the central carrier frequency, is the minimum frequency shifting step, is the frequency shifting codeword, in which for each pulse is randomly selected from , . Suppose that a target is moving along the line of sight (LOS) with a constant radial velocity relative to the sensor (e.g. a radar or a sonar). The echo reflected from the moving target is then given by [8]

(3) |

where is the scattering coefficient of the target, accounting for the attenuation and reflection, and represent the time delay and Doppler-stretch, respectively, with the wave propagation velocity denoted as . The signal received by the sensor is contaminated by a white Gaussian noise (WGN) with power spectral density . The received signal is then expressed as

(4) |

Sampled at the rate of , the received signal (4) turns into

(5) |

, where is the total number of sampling points, is distributed as , and [11].

### 2.1 Ambiguity Function

As mentioned in Section I, the AF

(6) |

is generally applied for implementation of the joint delay-Doppler estimation [8, 11]. Denoted as

(7) |

the AF-based estimation is implemented by locating the peak of the AF [8, 12], where and respectively denote the estimations of time delay and Doppler-stretch, and regulates the searching area for the estimations of the parameters. Note that both and are continuous random variables that vary with . In addition, we also assume .

### 2.2 Preliminaries

For the ease of problem statement and derivations in the following sections, we make the necessary assumptions and definitions.

###### Assumption 2.1

The envelope of each pulse is time-limited within , .

Assumption 2.1 guarantees that each pulse does not overlap with any other ones in time domain.

###### Assumption 2.2

, i.e. has continuous derivatives up to order 2 inclusive.

###### Assumption 2.3

(8) | ||||

###### Assumption 2.4

has the unique maximizer , which holds almost surely if the sampling rate is greater than the bandwidth of .

###### Definition 2.5

For any sequences and , define

(9) |

###### Assumption 2.6

For RSF signals,

(10) |

###### Definition 2.7

,

(11) | ||||

where , and .

## 3 Main Results

In this section, the main results of the paper are provided.

Under high SNRs, i.e. when is sufficiently small compared to the amplitude of the echo, the unbiasedness of the AF-based joint estimation is shown as:

###### Theorem 3.1

For each ,

indicating that

(12) |

as , where , and “” denotes that converges to in probability.

See Appendix .1.

The estimations of time delay and Doppler-stretch respectively converge to their true values as the noise level gets weaker, implying that the AF-based joint estimation is asymptotically unbiased when SNR is sufficiently large [18].

Building on Theorem 3.1, we then evaluate the MSEs of the AF-based estimation. Define

(13) |

The theoretical MSEs of time delay and Doppler-stretch are formally given by:

###### Theorem 3.2

In the AF-based estimation, the theoretical MSEs of time delay and Doppler-stretch satisfy

(14a) | ||||

(14b) |

where the approximate in (14b) holds if .

See Section 4.3.

In Theorem 3.2, (14a) precisely describes the MSEs of the AF-based estimation under high SNRs. If the additional condition is involved, we obtain the approximated, but much simplified forms of the MSEs, which are presented by (14b). The condition suggests that the envelope is a symmetric function with an axis of symmetry . This is yielded by most of the radar signals. In this sense, Theorem 3.2 reliably presents the MSEs of most cases. Moreover, note that the MSEs of time delay and Doppler-stretch are represented with integrations, corresponding to the case where the sampling rate is sufficiently large, i.e. , such that the summations can be replaced by integrations.

The effectiveness of the AF-based estimation is judged by evaluating the gaps between the MSEs and their corresponding CRLBs, since the latters represent the minimum achievable variances of any unbiased estimation. In [8] and [9], the CRLBs for joint delay-Doppler estimation with known and unknown scattering coefficient were respectively derived. In the AF-based joint estimation as introduced in Section II-A, only the magnitude information of the AF is utilized. However, the phase information contained in the scattering coefficient is ignored. Therefore, regarding as one of the unknown parameters and according to the calculation of CRLBs in [9, 10], we have

###### Theorem 3.3

^{1}

The CRLBs of time delay and Doppler-stretch are given by

(15) |

Clearly, the MSEs in (14b) are confirmed to be consistent with the corresponding CRLBs. By Theorems 3.1–3.3, we describe the AF-based estimation as an approximately efficient estimation [18].

We should also notice that Theorem 3.2 and 3.3 in fact apply to estimations with arbitrary wideband signals modeled by (3) (including the RSF signals). However, the specific waveform parameters determining the estimation performance are not able to be revealed by (14) or (15). So as to explore the relationship between the estimation performance and the waveform parameters of the RSF signal, we substitute the signal model (1) into (14b) and obtain the compact expressions of the MSEs:

###### Theorem 3.4

For an RSF signal, the theoretical MSEs of joint delay-Doppler estimation under high SNRs are specifically expressed as

(16a) | ||||

(16b) |

See Section 4.4.

As revealed by (16), the estimation performance with an RSF signal is mainly dominated by three factors, namely, the central carrier frequency, the bandwidth covered by the carrier frequencies, and the frequency shifting pattern. Specifically, i) In both the expressions of MSEs above, there exists a component . Since , . Therefore, both the performances of time delay and Doppler-stretch estimations can be improved if the central carrier frequency increases. ii) The component in (16a) can be rewritten as , where determines the available bandwidth for the carrier frequencies, while is related to the frequency shifting pattern.

Beyond the revealings by the expressions of the MSEs in Theorem 3.2, simulation results in Section V show that the performance of delay estimation is only slightly improved as the central carrier frequency increases. This indicates that the performance of delay estimation is mainly influenced by the covered bandwidth of the carrier frequencies and the variance of the frequency shifting codeword.

With the analyses above, the theoretical MSEs given by Theorem 3.4 could serve as a guidance for waveform design, which aims at properly configuring the waveform parameters and improving the estimation performance. By increasing the central carrier frequency the performance of Doppler-stretch estimation can be significantly improved, while the performance of delay estimation can be improved by increasing the bandwidth covered by carrier frequencies and by adopting the frequency shifting codewords with large .

### 3.1 Comparison with Other Waveforms

The estimation performance with the RSF signal is then fairly compared with those with a monotone signal and an OFDM signal, where the three signals are comprised of the same number of pulses with the same amount of energy. The monotone signal fixes its carrier frequency to for all pulses, whereas the OFDM signal simultaneously uses orthogonal subcarriers within each pulse.

i) The estimation performance with the monotone signal is directly known from (16), since the signal can be considered as a special case of RSF signal with . The MSEs of time delay and Doppler-stretch then readily reduce to

(17a) | |||

(17b) |

with and . Recalling that mainly determines the MSE of time delay, and that always holds in (16a), we discover that the RSF signal leads to a better performance of delay estimation than the monotone signal. While due to the reason that holds in (16b), the MSEs of Doppler-stretch with the two signals are approximately the same.

ii) The OFDM signal can be considered as the sum of “monotone signals” regulated by , which is denoted as

(18) |

where [16, 17]. All the available carrier frequencies within the given bandwidth for the RSF signal are simultaneously used by the OFDM signal as its subcarriers. Substituting (18) into (14b) and following the derivation in Section 4.4, we obtain

(19a) | ||||

(19b) |

By comparing (19) with (16), it is obvious that the estimation performances with the OFDM and RSF signals are the same under high SNRs. Nevertheless, as one of the synthetic wideband signals, the RSF signal only takes up a narrow instantaneous bandwidth, while on the contrary, the whole bandwidth is occupied by all the subcarriers when transmitting an OFDM signal. From this perspective, the requirements for the receiver can be much reduced if the RSF signal is employed for parameter estimation.

Summarily, as for the performance of delay estimation, the RSF and OFDM signals have the identical performance under high SNRs. Due to their wideband character, they both outperform the narrowband monotone signal. While for the performance of Doppler-stretch estimation, which is mostly dependent on the central carrier frequency, the three signals have approximately the same performance.

## 4 Derivations and Proofs

In this section, we provide the detailed derivations of the main results.

Since the AF-based joint estimation can be interpreted as searching for the maximum point of the AF, we start our derivation by focusing on the properties of at its maximum point. Obviously, the partial derivatives of with respect to and both reach zero at [8, 12], i.e.,

(20) | ||||

Letting

(21) |

we have . Therefore, (20) can be reorganized as

(22a) | |||

(22b) |

where we define

(23) | ||||

As shown by Theorem 3.1, and respectively converge to their true values as . This indicates that under high SNRs, is distributed in a neighborhood of . Thus we expand the right hand sides (RHSs) of (22) in Taylor series around , respectively,

(24a) | |||

(24b) |

In (24) we respectively define

(25) | ||||

where , , , and , , . Since reaches its unique maximum at according to Assumption 2.4, both and equal zero and thus we have directly excluded them from (24). Then we convert (24) into the following forms:

(26a) | |||

(26b) |

where

(27) |

With MSE employed to evaluate the performance of estimation, (26) readily turns into

(28a) | |||

(28b) |

For each equation in (28), we next calculate the limits of both sides as , so as to investigate the MSEs of the joint estimation. Based on (28a), we focus on the further derivations for the MSE of time delay, while the MSE of Doppler-stretch can be obtained following the similar technique.

### 4.1 Calculations for LHS of (28a)

With the definitions in (23), we expand the LHS of (28a) into the following form:

(29) | ||||

Hence the value of under high SNR is evaluated by successively calculating the limit of each resultant expectation on the RHS of (29) as . Before starting the calculations, it is worth mentioning that since follows , the random variable is thus distributed as , which is independent of .

We firstly calculate the limit of as . For each , we have

(30) |

where denotes the indicator function [19]. To calculate the first term on the RHS of (30), we know from Theorem 3.1 that

(31) |

as , and

(32) | ||||

where

(33) |

and

(34) |

For (32), also note that . Therefore, according to Lebesgue’s dominated convergence theorem [19], we obtain

(35) |

For the second term on the RHS of (30), on one hand,

(36) | ||||

Letting and with (35), we have

(37) |

On the other hand, it can be also derived that

(38) |

Combining (37) and (38), we obtain

(39) | ||||

It is clear that if exists, the existence of can be also confirmed. With Theorem 3.1 and (32),

(40) |

as . Again, following Lebesgue’s dominated convergence theorem,

(41) |

Recall that . As a result, the random variable in (41) follows . Then (41) can be calculated as

(42) |

where we respectively define , and . The indices collected by in fact corresponds to all the non-zero samples of . Totally, together with (39) and (4.1), we conclude that exists and

(43) |

The calculations for the rest terms in (29) can be performed similarly. Due to limited space, we eliminate the tedious and repetive calculating processes as presented above, whereas the corresponding results can be found in Appendix C.

### 4.2 Calculations for RHS of (28a)

As for the RHS of (28a), given , we have