SCIENCE CHINA Information Sciences, Volume 62, Issue 2: 022302(2019) https://doi.org/10.1007/s11432-017-9322-2

## Quantitative SNR analysis of QFM signals in the LPFT domain with Gaussian windows

• ReceivedApr 7, 2017
• AcceptedJan 8, 2018
• PublishedOct 15, 2018
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### Abstract

The purpose of this paper is to present a quantitative SNR analysis of quadratic frequency modulated (QFM) signals. This analysis is located in the continuous-time local polynomial Fourier transform (LPFT) domain using a Gaussian window function based on the definition of 3 dB signal-to-noise ratio (SNR). First, the maximum value of the local polynomial periodogram (LPP), and the 3 dB bandwidth in the LPFT domain for a QFM signal is derived, respectively. Then, based on these results, the 3 dB SNR of a QFM signal with Gaussian window function is given in the LPFT domain with one novel idea highlighted: the relationship among standard SNR, parameters of QFM signals and Gaussian window function is clear, and the potential application is demonstrated in the parameter estimation of a QFM signal using the LPFT. Moreover, the 3 dB SNR in the LPFT domain is compared with that in the linear canonical transform (LCT) domain. The validity of theoretical derivations is confirmed via simulation results. It is shown that, in terms of SNR, QFM signals in the LPFT domain can achieve a significantly better performance than those in the LCT domain.

### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61671063), and also by Foundation for Innovative Research Groups of National Natural Science Foundation of China (Grant No. 61421001).

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• Figure 1

Geometry of the problem. Points $P_1$ and $P_2$ determine the limits of the rectangular area enclosing the baseline of the 3 dB surface and $P_0$ is the position of the maximum.

• Figure 3

(Color online) (a) The relation between ${\rm~SNR}_{\rm~LPFT}^{3~{\rm~dB}}$ and LPFT parameter $w_1$; (b) the relation between ${\rm~SNR}_{\rm~LPFT}^{3~{\rm~dB}}$ and Gaussian window parameter $\alpha$; (c) the relation between ${\rm~SNR}_{\rm~LPFT}^{3~{\rm~dB}}$ and $w_2-a_2$.

• Figure 4

(Color online) The ratio between ${\rm~SNR}_{\rm~LPFT}^{3~{\rm~dB}}$ and ${\rm~SNR}_{\rm~LCT}^{3~{\rm~dB}}$ versus $a/b$

• Figure 5

(Color online) LPP of a QFM signal in the noise environment $y(n)$ in (a) $w-w_1$ plane under optimal $w_2=18$, (b) $w-w_2$ plane under optimal $w_1=45$, and (c) $w_1-w_2$ plane under optimal $w=12$.

• Figure 6

(Color online) LCT spectrum of a QFM signal in the noise environment $y(n)$ in $u-k$ plane.

• Figure 7

(Color online) LCT of a QFM signal in the noise environment $y(n)$ projected in (a) LCT-frequency $u$ axis, (b) parameter-ratio $k$ axis.

• Figure 8

(Color online) MSEs of the parameter estimations via LPFT-based method. (a) Constant frequency $a_0$; (b) first chirp rate $a_1$; (c) second chirp rate $a_2$.

• Table 1   Numerical results for ${\rm~LPP}_x^3(t,\bar~w)\mid_{w_2=a_2}^{3~{\rm~dB}}$ examples in Figure
 Example $t$ $A$ $a_0$ $a_1$ $a_2$ $\alpha$ $P_0$ $P_1$ $P_2$ Average Avg/$P_0$ I $0$ $1$ $0.5$ $-2$ $-1$ $1$ $3.54$ $1.77$ $1.77$ $2.34$ $0.66$ II $2$ $2$ $1$ $4$ $-2$ $2$ $10.03$ $5.01$ $5.01$ $6.63$ $0.66$ III $10$ $4$ $3.5$ $1$ $-1.5$ $1.5$ $46.31$ $23.16$ $23.16$ $30.62$ $0.66$ IV $-100$ $8$ $-1$ $0.5$ $2$ $3$ $130.99$ $65.49$ $65.49$ $86.60$ $0.66$

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