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F. CHAPEAU-BLONDEAU and D. ROUSSEAU. Nonlinear SNR amplification of harmonic signal in noise. Electronics Letters, vol. 41:618–619, 2005. 105/197
Nonlinear SNR amplification of harmonic signal in noise F. Chapeau-Blondeau and D. Rousseau The SNR of a harmonic signal in additive white noise is computed after transformation by an arbitrary memoryless nonlinearity. With a simple saturating nonlinearity having direct electronic implementation, an amplification of the SNR can be obtained, an outcome which is inaccessible with linear devices. Assessing the presence of a harmonic signal hidden in additive noise is a very common problem in many areas of experimental sciences and technologies. This type of signal–noise mixture has a very characteristic signature in the frequency domain: its power spectrum is formed by a sharp spectral line at the harmonic frequency n s, emerging out of a broadband background contributed by the noise. A signal-to-noise ratio (SNR) R is conveniently defined as the ratio of the power contained in the spectral line at n s divided by the power contained in the noise background in a small reference frequency band DB around n s. This SNR quantifies how well the spectral line at ns emerges out of the noise background. A narrowband filter at n s used to extract the harmonic component, will have an efficacy directly increasing with this SNR [1]. As a preprocessing, it is known that no linear filter is able to improve (increase) such an SNR R. This is because a linear filter multiplies both the spectral line and the noise background at n s by the same factor (the squared modulus of its transfer function at n s), and therefore leaves the SNR R unchanged [1]. On the contrary, we will show that very simple nonlinear devices can act as an SNR amplifier providing an enhancement of R. We consider the signal–noise mixture x(t) ¼ s(t) þ x(t), with the harmonic component s(t) ¼ A cos(2pn st þ j), and x(t) a stationary white noise with probability density function f x(u). This signal x(t) is fed into a memoryless (nonlinear) system [2] with input–output characteristic g(.) producing the output yðtÞ ¼g½sðtÞþxðtÞŠ ð1Þ In this case, both x(t) and y(t) are cyclostationary random signals with period Ts ¼ 1=ns, both showing a power spectrum with a sharp spectral line at ns emerging out of a broadband noise background. The SNR, as defined above, for the output y(t) can be expressed as [3] Rout ¼ jhE½yðtÞŠ expð i2pt=TsÞij2 hvar½yðtÞŠiDtDB In (2), a time average is defined as h...i¼ 1 ...dt ð3Þ Ts 0 E[y(t)] and var[y(t)] ¼ E[y 2 (t)] E 2 [y(t)) represent the expectation and variance of y(t) at a fixed time t; and Dt is the time resolution of the measurement (i.e. the signal sampling period in a discrete time implementation). The white noise assumption here models a broadband physical noise with a correlation duration much shorther than the other 2 relevant time scales, i.e. Ts and Dt, and finite variance sx [3]. From (1), one has and E½yðtÞŠ ¼ E½y 2 ðtÞŠ ¼ ð þ1 1 ð þ1 1 ð Ts In a similar way, the SNR for the input x(t) is ð2Þ gðuÞf x½u sðtÞŠdu ð4Þ g 2 ðuÞf x½u sðtÞŠdu ð5Þ R in ¼ A2 =4 s 2 x DtDB We then consider for g(.) a very simple nonlinearity, easily implementable with an operational amplifier, the linear-limiting saturation 8 < l for u l gðuÞ ¼ : u l for for u l 0. With f x(u) a zero-mean Gaussian density associated to the cumulative distribution function F x(u), (4) and (5) give and E½ yðtÞŠ ¼ l þð l sðtÞÞFxð l sðtÞÞ ðl sðtÞÞ Fxðl sðtÞÞ þ s 2 x½ fxð l sðtÞÞ fxðl sðtÞÞŠ ð8Þ E½ y 2 ðtÞŠ ¼ l 2 þðl 2 s 2 ðtÞ s 2 xÞ½F xð l sðtÞÞ F xðl sðtÞÞŠ þ s 2 x½ð l sðtÞÞ f xðl sðtÞÞ ðl sðtÞÞ f xð l sðtÞÞŠ ð9Þ In these conditions, we can analyse the behaviour of the input–ouput SNR gain G ¼R out=R in. It turns out that there is a broad range of values for l, both >A or 1. Power-law nonlinearities tested in [4] exhibit a similar property of G >1 but with a more complex physical implementation. Additionally, application of the present treatment shows that hard-threshold nonlinearities, like signum or Heaviside functions for g(.), do not allow G >1 with Gaussian noise. Other common nonlinearities encountered, for instance in semiconductor devices, could also be tested for SNR amplification. Such simple nonlinear operators offer a useful complement to linear techniques for signal processing and sensors. # IEE 2005 23 March 2005 Electronics Letters online no: 20051065 doi: 10.1049/el:20051065 F. Chapeau-Blondeau and D. Rousseau (Laboratoire d’Ingénierie des Systèmes Automatisés (LISA), Université d’Angers, 62 avenue Notre Dame du Lac, 49000 Angers, France) E-mail: chapeau@univ-angers.fr 106/197
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