Beats — Question 9

Question 9

Finite recording time broadens a spectral line

A complex sinusoid $e^{2\pi i f_0t}$ is observed only for $-T/2\le t\le T/2$. Its normalized Fourier amplitude at trial frequency $f$ is $A(f)=T^{-1}\int_{-T/2}^{T/2}e^{2\pi i(f_0-f)t}\,dt$.

1. Evaluate the integral and find its nearest zeros to $f_0$.

2. Two tones are $2.0\,\mathrm{Hz}$ apart. What recording duration puts one at the first spectral zero of the other? Relate this to the beat period without claiming it is an absolute limit on all frequency estimation methods.

Question 9 - Solution

Finite recording time broadens a spectral line

(a) Method and calculation

For $\Delta=f_0-f\ne0$,$$A(f)=\frac{e^{i\pi\Delta T}-e^{-i\pi\Delta T}}{2\pi i\Delta T}=\frac{\sin(\pi\Delta T)}{\pi\Delta T}.$$The continuous limit at $\Delta=0$ is one. The closest zeros satisfy $|f-f_0|=1/T$.

(b) Interpretation and further calculation

Setting $1/T=2.0\,\mathrm{Hz}$ gives$$T=0.50\,\mathrm s=1/\Delta f.$$This equals one beat period. It is a useful resolution scale for the specified rectangular-window spectrum. Other windows, model-based fitting, noise levels and prior information can alter practical estimation performance; this calculation does not establish a universal hard bound.

Physical check

The product frequency offset times duration is dimensionless, and the zero spacing narrows as observation duration increases.