A bound involving n-dimensional instantaneous frequency

The author states and proves an integral inequality that bounds the absolute difference ε(x)=|m(x) -mˆ(x)| where m(x) is the response of a modulated n-dimensional real linear filter w to a complex exponential signal with n-dimensional instantaneous frequency Δu(x) and m(x)=|W[Δu(x)-u ₀]| where W is the Fourier transform of w . The quantity of ε(x) provides an estimate of the error incurred by using mˆ(x) as an estimate of m(x), e.g., if Δu(x) is unknown. Such estimates may be applied to the problem of measuring the n-dimensional instantaneous frequency of certain nonstationary phase-modulated signals