# Analysis of the spectrum transform for narrow-band signal passing through nonlinear section of a digital radar receiver

- Haijiang Wang
^{1}Email author, - Yangyang Deng
^{1}, - Bao Dong
^{1}and - Debin Su
^{1}

**2015**:265

https://doi.org/10.1186/s13638-015-0499-7

© Wang et al. 2015

**Received: **30 September 2015

**Accepted: **15 December 2015

**Published: **24 December 2015

## Abstract

In this paper, with the analytical method, the problem of unexpected bandwidth occupying is investigated firstly when a narrow-band signal passes through the inertialess nonlinear section of a digital radar receiver. Afterwards, the frequency aliasing that is caused by the digitalization is studied. To solve these two issues, the selection of the sampling frequency is of great importance. In this paper, a maximizing minimum arc-distance strategy is proposed, which is dependent on the frequency range of the signal. The arc-distance method is designed for optimizing the sampling frequency. Furthermore, with this method, the minimum arc distances between the frequency bands of the signal and the interferences can be calculated. The inter-band arc distances are important references for optimizing the transition bandwidth of the filter after the sampling and demodulation process.

## Keywords

## 1 Introduction

The dynamic range of the signal to be processed by some certain kinds of radar receivers is very large. For example, the RF signal’s dynamic range of weather radar can be as wide as about 94 to 100 dB [1]. To meet this wide dynamic range requirement, automatic gain control (AGC) in IF stage plays an indispensable role in analog circuitry. With the rapid development and popularization of digital electronic technology, in recent years, to improve performance and simplify system, hardware digitization has long been introduced in this area and more and more analog function circuits have been replaced with digital ones [2, 3]. Orthogonal demodulation now proceeds digitally with satisfactory performance and the function of amplification of IF stage is actually saved [4, 5]. As ADC is moved further towards radar’s analog front end by employing advanced under-sampling ADC chip, the signal added to AD converter has a large dynamic range because of the absence of AGC function which is originally in the analog circuit, and nonlinear effects caused by the strong signal become apparent. Especially, after the digitization of the signal which has been subjected to nonlinear distortion, the harmonic component of the signal will invade the signal frequency band because of aliasing effect, causing additional distortion.

Many researchers have investigated the nonlinearity in digital communication and radar systems. For example, Lin et al. analyzed the problem of fifth-order nonlinear distortion and spectral regrowth in a communication system [6, 7]. Wang et al. studied the nonlinearity of the mixer and VCO in the FMCW V-band radar receiver [8]. In [9], Harrington discussed the nonlinearity of the active array radar and gave a spectral analysis of third-order intermodulation clutter. In [10], Mark B. Yeary analyzed the intermodulation product caused by the power amplifier’s nonlinear response. In [11], A. W. Doerry et al. investigated spurious effects of analog-to-digital conversion nonlinearities on radar range-Doppler maps. And in [12], Grimm et al. focused on the analysis and digital mitigation of nonlinear distortion in software-define radio receivers; they derived a nonlinear distortion model in time and frequency domain and suggested an adaptive digital feed-forward linearization structure to mitigate the nonlinear distortion. But by now, few literatures were seen to take the nonlinear section and digitalization processing as a whole to analyze the spectrum transform of the input signal. Especially, in the existing work, the effect of sampling and the selection of sampling frequency were seldom investigated intensively.

Because the nonlinearity section has a suppression effect upon the strong signal, some people have the idea of trying to take advantage of the nonlinearity to realize automatic gain control function. It is considered that the signal frequency bands are duplicated and shifted to the position of each harmonic frequency of the carrier because of the nonlinearity, forming several harmonic frequency bands. So by choosing the sampling frequency carefully, it is expected that the fidelity of signal can be ensured and the effect of AGC can be achieved at the same time through digital filtering. In this paper, however, the research shows that the situation is not good as seems to be.

*s*(

*t*), that is

*V*

_{ m }> > max[|

*s*(

*t*)|]. If the dynamic range of

*s*(

*t*) is small enough, the output signal can be expressed as first-order Taylor expansion near the carrier signal:

*f*and

*f*' are both periodic functions of

*t*, they can be expanded into Fourier series [13]:

The first summation item on the right-hand side of the expression above represents the spectrum line of each harmonic wave caused by carrier. While the second summation item represents the duplication and shifting near each harmonic frequency of the modulated signal spectrum. In the above equation, \( {B}_0\left({V}_m\right)=\frac{1}{2\uppi}{\displaystyle \underset{-\uppi}{\overset{\uppi}{\int }}f\hbox{'}\left({V}_m \cos \varphi \right)}\kern0.1em d\varphi \) is the gain of the signal. If the gain decreases with the increase of carrier amplitude, it can realize the function of automatic gain control.

This is the theoretical basis of the idea of duplication and shifting. However, for the “deep” modulated signal, especially for the pulse-modulated signal, such assumptions will not hold.

*y*=

*f*(

*x*) can be expressed as the power series

We will discuss the property of the output signal’s spectrum *Y*(*f*) transformed from the output signal *y*(*t*) under the condition of knowing the spectrum *X*(*f*) transformed from the input signal *x*(*t*).

*x*(

*t*) is

*x*(

*t*) and

*y*(

*t*) can be expressed as

*X*

^{⊗ n }(

*f*) to denote the spectrum with

*n*th power of a signal, then the Laplace transform of

*X*

^{⊗ n }(

*f*) is exactly the

*n*th power \( {X}_L^n(s) \) of the Laplace transform of the signal’s spectrum. The following equations show the relationship between the symbols above clearly and intensively. That is to say, if

This result is greatly helpful for us to analyze the spectrum transform of a signal passing through the nonlinear section.

## 2 The spectrum transform of baseband signal passing through nonlinear section

*x*

_{ b }(

*t*) has uniform spectrum within a frequency band

*u*(

*f*) is the unit step function, and

*b*represents the single-side bandwidth. Its Laplace transform is easy to draw:

*s*-plane: Re(

*s*) > 0. Its

*n*th power is

*s*

^{− n }. The inverse transform of

*s*

^{− n }is

Bandwidth ratio of the first 10 power items

Power | Bandwidth ratio |
---|---|

1 | 1. |

2 | 1.8 |

3 | 2.2254 |

4 | 2.5264 |

5 | 2.8099 |

6 | 3.0716 |

7 | 3.3118 |

8 | 3.5357 |

9 | 3.7464 |

10 | 3.9459 |

*k*is approximately proportional to the square root of the harmonic order, that is

According to the criterion of minimum mean square error, the coefficient can be determined as *a* = 1.2544. For nonuniform signal spectrum, the value of *a* will be smaller.

## 3 The spectrum transform of modulated signal passing through nonlinear section

*n*th power item’s spectrum is

*n*th power item can be deduced:

*F*by the 2

*k*− 1 -th power item is

*y*(

*t*) around the fundamental frequency

*F*is obtained:

Thus, it can be seen that the modulated signal has been contaminated around the carrier frequency by the third and larger odd power items.

## 4 Filtering of the harmonic aliases after nonlinear transform

From the discussion in Section 3, it can be seen that the contamination put on the signal frequency band by the combinatorial frequency items is inevitable. In fact, the signal has experienced this kind of contamination before sampling. If the degree of contamination is mild, the signal is still usable and can be put into digital processing after sampling. However, the sampling will produce aliasing problem of the harmonic frequency band, and this will bring new challenges to filtering processing.

To show the aliasing effect more intuitively, the spectrum of the signal after digitization can be plotted onto a cylindrical surface which can indicate the periodic property of the digital spectrum and we call it “cylindrical spectrum”.

*V*(

*ω*) is the scaled analog spectrum. Firstly, let us only consider the first two terms

*V*(

*ω*) and

*V*(

*ω*+ 2

*π*) in Eq. (30). A length of magnitude spectrum |

*V*(

*ω*)| within the interval (−

*π*,

*π*] is drawn in solid line in Fig. 4a, meanwhile, the magnitude spectrum |

*V*(

*ω*+ 2

*π*)| within the interval (−

*π*,

*π*] is drawn in dotted line. Suppose that we cut down this strip of paper on which the scaled analog spectrum in the interval (−

*π*,

*π*] is drawn and stick it on a cylinder whose cross section is a unit circle, then a cylindrical spectrum is obtained, which is shown as Fig. 4b.

*f*

_{0}and − 3

*f*

_{0}, whose harmonic band completely covers the signal band. This is because according the equation below, the two digital frequencies corresponding to these two analog frequencies are both equal to 0.5

*π*, which is the digital frequency of the carrier:

*f*and

*ω*in Eq. (32) is drawn as Fig. 6. The bold lines denote the signal frequency range and the harmonic frequency ranges. The shaded areas denote the signal frequency band after sampling. Obviously, the contamination is very serious under this situation, because the signal frequency band is seriously overlapped by the other ones.

*f*

_{0}and the bandwidth is

*B*) and all the harmonic frequency bands generated by the nonlinear items whose center frequencies are not

*f*

_{0}:

*D*cB is the arc distance and is defined as

*D*

_{ c min}(

*x*) can be drawn by changing the value of

*x*gradually, which is shown in Fig. 7. The solid line represents the minimum arc distance between the signal frequency band and the harmonic frequency bands with center frequencies not equal to the carrier frequency, which are generated by nonlinearity when the first to fifth power are all exist.

*D*

_{ c min}(

*x*). The coordinate position of the peak corresponding to the largest

*x*is

*x*= 0.43,

*D*

_{ c min}(

*x*) = 0.0619, and the corresponding sample frequency is

*f*

_{ s }= 139.5349

*MHz*. The

*f*↔

*ω*graph in this case is shown as Fig. 8, and the cylindrical spectrum diagram is shown as Fig. 10b. Indeed, with the minimum arc distance method, by choosing the sampling frequency carefully, the harmonic frequency band caused by nonlinearity has been separated from the signal frequency band, so the contamination to the signal frequency band is greatly eliminated.

*f*(−

*x*) = −

*f*(

*x*), then all the even power items at the right side of equation (9) will disappear. In this case, the curve of

*D*

_{ c min}(

*x*) is shown as the dotted line in Fig. 7. On this line, the abscissas of the peaks increase greatly compared to the ones on the solid line. The sampling frequency corresponding to the peak with the maximum abscissa is

*f*

_{ s }= 68.1818

*MHz*, while the corresponding minimum arc distance is only 0.0336, making the filter design and implementation after sampling and digital demodulation very difficult. In order to get an enough large minimum arc distance, we can choose

*x*= 0.62;

*D*

_{ c min}(

*x*) = 0.1655, and the corresponding sampling frequency is

*f*

_{ s }= 96.7742

*MHz*. In this situation, the

*f*↔

*ω*relation is shown as Fig. 9, and its cylindrical spectrum diagram is shown as Fig. 10d.

## 5 Conclusion

Compared to the bandwidth of the baseband, the bandwidth of frequency band generated by the high power items of the baseband signal that has passed through nonlinear section of a digital radar receiver is wider. In theory, the bandwidth generated by *n*th power item is *n* times of the width for the baseband. Although due to the attenuation on band edge, the bandwidth calculated at a certain level increases slow along with *n* (approximately proportional to \( \sqrt{n} \)), but it is still much wider than that of the baseband. The signal of modulated narrow band will generate a series of harmonic frequency band after passing through the nonlinear section of a digital radar receiver. The harmonic frequency bands are shifted copy of the spectrum from respective power item of baseband signal according to the integer multiple of the carrier frequency, rather than the simple duplicated and shifted copy of the signal spectrum of the baseband signal. Each item of odd order power generates frequency component near the fundamental frequency, causing the unnecessary contaminant to the fundamental frequency. So, it is impossible to eliminate the contaminant by using the filtering method.

If the contaminant discussed above is minor, it is still acceptable to sample the signal and make the digital processing. However, the sampling frequency is suggested to be selected prudently. In this paper, a maxmin arc-distance strategy that is dependent on the frequency range of the signal is proposed. The arc-distance method is designed for optimizing the sampling frequency. Furthermore, with this method, the minimum arc distances between the frequency bands of the signal and the interferences can be calculated. The inter-band arc distances are important references for optimizing the transition bandwidth of the filter after the sampling and demodulation process.

## Declarations

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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