Sampling

For infinite precision sampling of a band-limited signal at the Nyquist Frequency, the signal-to-noise ratio after $N_q$ samples is

$${\rm SNR} = {\left\langle{r_\infty}\right\rangle{}\over\sigm... ...2}\sqrt{1+\rho^2}} = {\rho\over\sqrt{1+\rho^2}} \sqrt{N_q}\,,$$ (1)

where $\rho$ is the normalized cross-correlation Coefficient

$$\rho\equiv {\left\langle{x(t)}\right\rangle{}\left\langle{y(... ...e{x^2(t)}\right\rangle{}\left\langle{y^2(t)}\right\rangle{}}}.$$ (2)

For $\rho\ll 1$,

$${\rm SNR} \approx \rho\sqrt{N_q}\,.$$ (3)

The identical result is obtained for oversampling. For undersampling, the SNR decreases (Thompson et al. 1986).

See also Nyquist Sampling, Oversampling, Quantization Efficiency, Sampling Function, Shannon Sampling Theorem, Sinc Function

References

Feuer, A. Sampling in Digital Signal Processing and Control. Boston, MA: Birkhäuser, 1996.

Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. Interferometry and Synthesis in Radio Astronomy. New York: Wiley, pp. 214-216, 1986.