# Compute the signal-to-noise ratio (SNR) of a signal Compute the SNR for a given signal and noise power spectral density. ```r snr(x, psd, two.sided = FALSE) ``` ## Arguments - `x`: the signal waveform, a time series (`ts`) object. - `psd`: the noise power spectral density. May be a vector of appropriate length (`length(x)/2+1`) or a function of frequency. - `two.sided`: a `logical` flag indicating whether the `psd` argument is to be interpreted as a one-sided or a two-sided spectrum. ## Details For a signal $s(t)$, the complex-valued discrete Fourier transform $s(f)$ is computed along the Fourier frequencies $f[j]=j/(N*delta_t) | j=0,...,N/2+1$, where $N$ is the sample size, and $delta_t$ is the sampling interval. The SNR, as a measure of "signal strength" relative to the noise, then is given by $$ \varrho=\sqrt{\sum_{j=0}^{N/2+1}\frac{\bigl|\tilde{s(f_j)}\bigr|^2}{\frac{N}{4\Delta_t} S_1(f_j)}},rho=sqrt(sum(abs(s(f))^2 / ((N/(4*delta_t)) * S1(f))), $$ where $S1(f)$ is the noise's one-sided power spectral density. For more on its interpretation, see e.g. Sec. II.C.4 in the reference below. ## Returns The SNR $rho$. ## References Roever, C. A Student-t based filter for robust signal detection. **Physical Review D**, 84 (12):122004, 2011. tools:::Rd_expr_doi("10.1103/PhysRevD.84.122004") . See also [arXiv preprint 1109.0442](https://arxiv.org/abs/1109.0442). ## Author(s) Christian Roever, christian.roever@med.uni-goettingen.de ## See Also `matchedfilter`, `studenttfilter` ## Examples ```r # sample size and sampling resolution: N <- 1000 deltaT <- 0.001 # For the coloured noise, use some AR(1) process; # AR noise process parameters: sigmaAR <- 1.0 phiAR <- 0.9 # generate non-white noise # (autoregressive AR(1) low-frequency noise): noiseSample <- rnorm(10*N) for (i in 2:length(noiseSample)) noiseSample[i] <- phiAR*noiseSample[i-1] + noiseSample[i] noiseSample <- ts(noiseSample, deltat=deltaT) # estimate the noise spectrum: PSDestimate <- welchPSD(noiseSample, seglength=1, windowingPsdCorrection=FALSE) # generate a (sine-Gaussian) signal: t0 <- 0.6 phase <- 1.0 t <- ts((0:(N-1))*deltaT, deltat=deltaT, start=0) signal <- exp(-(t-t0)^2/(2*0.01^2)) * sin(2*pi*150*(t-t0)+phase) plot(signal) # compute the signal's SNR: snr(signal, psd=PSDestimate$power) ```

snr function