Obtain and plot the first 12 values of the amplitude spectrum and the phase spectrum for a symmetric rectangular pulse train (see Table 16.2) having pulse duration τ=250μs , period T=1 ms and amplitude V0=500mV

Question

Obtain and plot the first 12 values of the amplitude spectrum and the phase spectrum for a symmetric rectangular pulse train (see Table 16.2) having pulse duration [latex] au=250 \mu \mathrm{s} [/latex] , period [latex] T=1 \mathrm{~ms} [/latex] and amplitude [latex] V_0=500 \mathrm{mV} [/latex].

Accepted Answer

From Table 16.2, the Fourier coefficients for the specified pulse train are given by

[latex] X_k=\frac{V_0 au}{T} \operatorname{sa}\left(\frac{k \pi au}{T} ight) [/latex] (16.45)

The amplitude and phase spectra are given by

[latex] A_k=\left\{\begin{array}{ll} \left|X_0 ight|, & k=0 \\ 2\left|X_k ight|, & k=1,2,3, \cdots \end{array} ight\}, [/latex] (16.46)

[latex] heta_k=∡ X_k, \quad k=0,1,2, \cdots [/latex]

Table 16.6 gives the values of the Fourier coefficients, the peak amplitude, and the relative phase of the first 12 components, where [latex] f=k f_0=k T^{-1} [/latex]. The values were calculated on a computer and rounded to three significant figures.

Figures 16.13 and 16.14 show the graphs of the amplitude spectrum and the phase spectrum, respectively. Conventionally, lines are drawn from the spectral values to the horizontal axis, presumably to make such graphs resemble what one would see on the screen of a spectrum analyzer. Partly for this reason, such spectra are called line spectra. But only the spectral values are significant. The lines have no mathematical meaning, but are visually helpful.

Question 16.13: Obtain and plot the first 12 values of the amplitude spectru...

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