Geometric Effects Due to Quadrature Phase-Sensitive Detection: Artifact Minimization and Removal Schemes for Spectroscopic Lineshape Measurement

An integral operator is defined to simulate the quadrature phase-sensitive detection (PSD) of a time-independent characteristic function (e.g., spectroscopic lineshapes in magnetic resonance), with its dependent variable modulated and scanned during the detection. This integral operation is equivalent to a convolution, with its kernel determined by the modulation function and the demodulation order. The convolution kernel results from the corresponding PSD as applied to the Dirac delta function. The convolution form not only facilitates our understanding of the geometric effects due to the quadrature and the nonquadrature PSD but also indicates hardware schemes that are capable of accurately approximating or directly measuring the functional form that is being probed. These schemes involve no a priori assumptions such as those required in conventional post-PSD data manipulation to approximate true lineshapes. In the case of sinusoidal modulation, the significance of simultaneous multichannel demodulations at all even or all odd harmonics within the frequency bandwidth is presented. Unconventional PSD schemes using square-wave and sawtooth-wave modulations are also discussed. As an aside for mathematics, the PSD convolution kernel corresponds to a transform between two compact-support functions; it generates an infinite set of basis-function sets, with the set of the Chebyshev polynomials being a special case.