# Theory and Measurement of Signal-to-Noise Ratio in Continuous-Wave Noise Radar

^{*}

## Abstract

**:**

## 1. Introduction

^{2}, while the RCS of the ground is 1000 m

^{2}. Therefore, the dynamic range dependent on RCS changes is 33 dB. The dynamic range related to distance is based on the law that a reflected signal’s power reduces with the fourth power of distance. For example, for the shortest (10 m) to the longest (1000 m) distance ratio, the dynamic range is 80 dB. Therefore, it is necessary to ensure a very high reception path with a dynamic range equal to or exceeding 100 dB. An optimal receiver for this type of signal is a correlation receiver. A correlation receiver not only deals with the noise of the receiver’s electronic circuits and components, but also with the noise occurring as a correlation function of noise signals, which is the so-called noise floor. The RMS (root mean square) of the noise signal’s correlation function estimator is not reduced to zero with a higher distance. Therefore, it is transferred to all other distance cells. This results in the emergence of an additional (apart from the self-noise of components and circuits) interference noise source that limits the receiver sensitivity. Recently, a number of studies presenting research reducing clutter with algorithmic methods in the digital correlation processing of noise signals have been published [8,9,10,11,12,13]. This paper discusses the mutual signal power relations in an actual noise radar solution with a range-Doppler digital correlator. This research allows us to determine an achievable limit level of SNR (signal-to-noise ratio) at the output of the correlation detector depending on the SNR at its input for band-limited white noise signals without the use of algorithmic signal processing techniques to reduce the noise floor. The considerations presented in the study concern various noise signals with the same probabilistic characteristics. The signal coming from the analyzed target and the signals coming from other targets, leakage signals, and ground-reflected signals are noise signals with zero mean value and Gaussian distribution of probability density. For this reason, a classic approach to radar system parameters such as receiver sensitivity, SNR, or detection gain requires determination of the signal and the noise in a noise radar both at the input and at the output of the correlation detector. The specific nature of continuous-wave noise radars is that both the power of the noise signal received from the target and the power of the interfering signals depends on the radar transmitter signal power. One exception is the receiver self-noise. The studies which are presented in this paper focus on discovering the causes of the “noise floor” occurrence in the correlative detector output signal. The conducted measurements prove the theoretical considerations, which were sufficiently described in the work of Axelsson [14]. This paper analyzes a more complex case in which, while referring to Reference [14], an interfering signal σ

_{z}consists of a signal of internal leakage, antenna leakage, and other signals received by an antenna (e.g., signals reflected from the ground). There is also a reference signal σ

^{2}

_{ref}that is not required to be equal to a received signal σ

^{2}

_{x}as it is in Reference [14].

## 2. Signals in a Noise Radar

_{R}(t); and some other signals indicated with S

_{I}(t), which are interfering signals from the viewpoint of signal S

_{R}(t). The interference signals include two types of signals: a signal associated with the transmitter signal S

_{T}(t) and a signal independent of the transmitter, which is the receiver self-noise S

_{N}

_{,R}(t). These signals have the characteristics of random signals and limit the receiver sensitivity.

- leakage 1—internal leakage of a signal from transmitter circuits to the receiver circuits with the power P
_{LC}. - leakage 2—leakage of a signal between the transmitting and receiving antennas with the power P
_{LA}. - interfering signal—signal from Object 2 with the power P
_{R2}and signals reflected from the ground and other elements on its surface at different distances from the radar with the total power P_{C}a so-called clutter.

_{REF}(t) was reduced to two: S

_{R}(t), the signal from Object 1; and S

_{I}(t), the interfering signal. The power of the interfering noise signal can be defined as:

_{N}

_{,T}(t) are sources of correlation peaks for the delay times other than the delay time of the target in the process of correlation with the reference signal. However, they constitute a significant interference for the signal reflected from Object 1 due to the additive residue of the correlation function of these signals.

## 3. Noise Signal Correlation Function

#### 3.1. The Variance of the Random Signal Correlation Function Estimator

_{xy}(τ) regardless of the signal duration T. Therefore, the mean squared error of estimator ${\widehat{R}}_{xy}\left(\tau \right)$ is equal to its variance according to Equation (4) below [16]:

#### 3.2. Noise at the Correlation Detector Output

_{R}(t) with variance σ

_{R}

^{2}, reference signal S

_{REF}(t) with variance σ

_{REF}

^{2}, and interfering signal S

_{I}(t) with variance σ

_{I}

^{2}. As a result of the correlation processing in the noise radar receiver, these signals correlate with each other as follows: the reference signal with the signal received from the target results in the output signal S

_{OUT1}(ΔT), which is the estimate of the cross-correlation function R

_{xy}

_{,I}(ΔT) of these signals. The reference signal with the interfering signal results in the output signal S

_{OUT2}(ΔT), which is the estimate of the cross-correlation function R

_{xy}

_{,II}(ΔT) of the reference signal and the interfering signal. In fact, one correlation processing is performed, which results in the determination of the estimate of the cross-correlation function R

_{xy}

^{∑}(ΔT). This estimate is identical to the sum of the estimates.

_{O}− T

_{DL}, T

_{O}= 2D

_{O}/c, D

_{O}is the distance from radar to an object, and T

_{DL}is the time delay of the delay line.

_{xy}

^{∑}(ΔT) can be described with Equation (8) below:

_{OUT}

_{1}(ΔT) and S

_{OUT}

_{2}(ΔT). The relationship between the parameters of the input and output signals of the correlation detector are as follows: R

_{R}(0) = σ

_{R}

^{2}is the variance of the noise signal from the target equal to the power P

_{R}

_{,IN}; R

_{I}(0) = σ

_{I}

^{2}is the variance of the signal from the interference equal to the power P

_{I}; R

_{REF}(0) = σ

_{REF}

^{2}is the variance of the reference signal equal to the power P

_{REF}. All powers are measured at 1 Ω resistance. On the other hand, the voltage at the correlator output for ΔT = 0 is determined by Equation (9) below:

^{2}

_{xy}

_{,I}(ΔT ≠ 0) is only 10% of σ

_{R}

^{2}σ

_{REF}

^{2}. A similar situation is seen in the case of the second component of Equation (10). Therefore, for a sufficiently high value of BT, the values of factors R

_{xy}

_{,I}

^{2}(ΔT) and R

_{xy}

_{,II}

^{2}(ΔT) are negligible compared to the values of the corresponding products (σ

_{R}

^{2}σ

_{REF}

^{2}) and (σ

_{REF}

^{2}σ

_{I}

^{2}). Considering the above, Equation (10) takes the following form:

_{N,OUT}, or the noise floor, which is shown in Figure 2. The power of the correlator output noise is equal to the variance of the correlation function estimator for ΔT, which significantly differs from zero. It is also possible to determine the PNFR (peak-to-noise floor ratio) as a ratio of the power of the output signal S

_{OUT}

_{1}(ΔT) for ΔT = 0, to the variance of estimator ${\widehat{R}}_{xy}^{\Sigma}\left(\Delta T\right)$ for ΔT ≠ 0.

## 4. Experimental Measurements

_{T}= 10 dBm, the noise signal bandwidth B

_{RF}= 150 MHz, center frequency f

_{0}= 9.2 GHz, and integration time T = 1 ms. In real systems, the main measurement problem is the separation of the noise signal coming from the target from other noise signals such as internal leakage, antenna leakage, and other signals received by an antenna. This includes signals reflected from the ground. These signals determine the noise power at the output of the digital correlator.

#### 4.1. Measuring System

_{R}

_{,IN}(t), which reached the receiver signal input through a power combiner. Reference signal S

_{REF}(t) was transmitted to the receiver reference input through a power divider. Part of the signal from the noise transmitter was transferred to the combiner through the power divider and adjustable attenuator T2, and then, to the receiver signal input. This signal represents the sum of the following signals: internal leakage, leakage between antennas, and ground-reflected signals specified in Equation (1). At the receiver input, there was a known value of noise signal P

_{R}

_{,IN}coming from the target and the value of interference power P

_{I}. The lengths of the transmission lines connecting the noise generator to the correlation receiver in the reception and reference paths were negligible in relation to the length of the lines L1 and L2. The power of the receiver self-noise included in the power of interference noise according to Equation (1) was calculated on the basis receiver noise ratio measurement.

#### 4.2. Measurement Results

_{R}

_{,OUT}and P

_{N}

_{,OUT}of the correlation receiver output for ΔT = 0 as a function of input signal power P

_{R}

_{,IN}. These characteristics were plotted for three different values of interfering signal power P

_{I}. The change in the value of power P

_{R}

_{,IN}was implemented using an adjustable attenuator T1. The plot of the measured characteristics P

_{R}

_{,OUT}= f(P

_{R}

_{,IN}) was consistent with the equation below:

_{T}

_{1}and k

_{T}

_{3}are coefficients of the transmittance of the noise generator signal power for the receiving channel and the reference channel, respectively, and σ

^{2}

_{T}is the variance of the noise generator signal.

_{R}

_{,OUT}were plotted for different values of power P

_{I}starting at points (A), (B), and (C), marked in Figure 6, and partially overlapped with each other. These points correspond to the unit value of the SNR at the correlator output for P

_{I}

_{1}= P

_{LC}+P

_{N}

_{,R}, P

_{I}

_{2}= P

_{I}

_{1}+1nW, and P

_{I}

_{3}= P

_{I}

_{1}+ 0.1 μW. Figure 6 also includes P

_{N}

_{,OUT}measured under the same conditions, which partially overlapped as well. This is similar to the signal characteristics. In this case, output noise power P

_{N}

_{,OUT}is demonstrated by a broken curve described by the following Equation (14):

_{R}

_{,IN}is much less than the interfering signal power P

_{I}, then (14) takes the form below:

_{N}

_{,OUT}does not depend on the power of the input signal coming from the target. In Figure 6, this situation is represented by a flat part in the characteristics with the level depending only on the value of the interfering signal power P

_{I}. P

_{R}

_{,IN}≫ P

_{I}(15) is simplified to the form below:

_{RF}T. For P

_{R}

_{,IN}= const, the difference in the signal power levels and noise power levels at the correlator output, expressed in dB, is equal to value 10log(B

_{RF}T).

_{REF}on P

_{R}

_{,OUT}and P

_{N}

_{,OUT}at the correlation receiver output as a function of the input signal power.

_{REF}decided only the increase in the output power of the signals. However, SNR

_{OUT}for specified P

_{R}

_{,IN}remained unchanged.

_{I}, which corresponds to the sum of internal leakage power P

_{LC}and receiver self-noise power P

_{N}

_{,R}. According to Equation (12), SNR

_{OUT}ratio as a function of SNR

_{IN}is expressed in Equation (17) below:

_{IN}<< 1:

_{OUT}is directly proportional to the value of the SNR at the input with a factor of proportionality equal to B

_{RF}T. On the other hand, for SNR

_{IN}≫ 1, (17) takes the form below:

_{RF}T. The above analytical conclusions are consistent with the measurement results shown in Figure 8.

## 5. Conclusions

_{I}defined by Equation (1)). This reduction can be achieved through a hardware solution by minimizing powers P

_{LC}and P

_{LA}, as well as through signal processing.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Functional diagram of a noise radar together with the types of the signals occurring in real working conditions. LPF – Low Pass Filter.

**Figure 2.**Real plot of the correlation function module square for D

_{o}= 3.2 m, v = 0, B = 1 GHz, T = 160 μs.

**Figure 6.**The plot of signal power transmittance for various interfering signals for B

_{RF}= 150 MHz, T = 1 ms, f

_{0}= 9.2 GHz. SNR

_{OUT}: output signal-to-noise ratio.

**Figure 7.**The plot of the signals power transmittance for various reference signals for B

_{RF}= 150 MHz, T = 1 ms, f

_{0}= 9.2 GHz.

**Figure 8.**The plot of the SNR at the correlator output for P

_{REF}= −18 dBm, B

_{RF}= 150 MHz, T = 1 ms, f

_{0}= 9.2 GHz.

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**MDPI and ACS Style**

Stec, B.; Susek, W.
Theory and Measurement of Signal-to-Noise Ratio in Continuous-Wave Noise Radar. *Sensors* **2018**, *18*, 1445.
https://doi.org/10.3390/s18051445

**AMA Style**

Stec B, Susek W.
Theory and Measurement of Signal-to-Noise Ratio in Continuous-Wave Noise Radar. *Sensors*. 2018; 18(5):1445.
https://doi.org/10.3390/s18051445

**Chicago/Turabian Style**

Stec, Bronisław, and Waldemar Susek.
2018. "Theory and Measurement of Signal-to-Noise Ratio in Continuous-Wave Noise Radar" *Sensors* 18, no. 5: 1445.
https://doi.org/10.3390/s18051445