Question

Sketch the frequency spectrum of he following signal and indicate the complex magnitude of each. All amplitudes must be positive

Determine fundamental frequency and period

*all phases need to be written in terms of pi between -pi and pi.

Sketch the frequency spectrum of the following signal and indicate the complex magnitude of each frequency component Manipulate phase as necessary to plot all components of the spectrum with positive amplitudes: x(t) = 14 cos( 1 60mt-π/4) + 5 cos(280mt-2π/3) -cos(600πt + π/s) f (Hz)

Answer 1

Ans)

Given signal

using $-cos\theta =cos(\pi -\theta )$

$x(t)=14cos(160\pi t-\frac{\pi }{4})+5cos(280\pi t-\frac{2\pi }{3})+cos(\pi -(600\pi t+\frac{\pi }{5}))$

$x(t)=14cos(160\pi t-\frac{\pi }{4})+5cos(280\pi t-\frac{2\pi }{3})+cos(-600\pi t+\frac{4\pi }{5})$

using $cos(-\theta )=cos\theta$

$x(t)=14cos(160\pi t-\frac{\pi }{4})+5cos(280\pi t-\frac{2\pi }{3})+cos(600\pi t-\frac{4\pi }{5})$

Now above signals has all amplitudes positive

using $cos\theta =\frac{1}{2}(e^{i\theta }+e^{-i\theta })$

$x(t)=14(\frac{1}{2}(e^{i(160\pi t-\frac{\pi}{4} )}+e^{-i(160\pi t-\frac{\pi}{4} )}))+5(\frac{1}{2}(e^{i(280\pi t-\frac{2\pi}{3} )}+e^{-i(280\pi t-\frac{2\pi}{3} )}))+(\frac{1}{2}(e^{i(600\pi t-\frac{4\pi}{5} )}+e^{-i(600\pi t-\frac{4\pi}{5} )}))$

$x(t)=7e^{-\frac{i\pi }{4}}e^{i(2\pi *80t )}+7e^{\frac{i\pi }{4}}e^{-i(2\pi 80t)})+2.5e^{-\frac{i2\pi}{3}}e^{i(2\pi 140t )}+2.5e^{\frac{i2\pi}{3}}e^{-i(2\pi 140t )}+0.5e^{-\frac{i4\pi}{5}}e^{i(2\pi 300t )}+0.5e^{\frac{i4\pi}{5}}e^{-i(2\pi300 t)}$ From the above signal frequencies and its magnitudes are

$80\: \: Hz\rightarrow complex\: \: magnitude\rightarrow 7e^{-\frac{i\pi }{4}}\rightarrow phase\rightarrow -\frac{\pi }{4}$

$-80\: \: Hz\rightarrow complex\: \: magnitude\rightarrow 7e^{\frac{i\pi }{4}}\rightarrow phase\rightarrow \frac{\pi }{4}$

$140\: \: Hz\rightarrow complex\: \: magnitude\rightarrow 2.5e^{-\frac{i2\pi }{3}}\rightarrow phase\rightarrow -\frac{2\pi }{3}$

$-140\: \: Hz\rightarrow complex\: \: magnitude\rightarrow 2.5e^{\frac{i2\pi }{3}}\rightarrow phase\rightarrow \frac{2\pi }{3}$

$300\: \: Hz\rightarrow complex\: \: magnitude\rightarrow 0.5e^{-\frac{i4\pi }{5}}\rightarrow phase\rightarrow -\frac{4\pi }{5}$

$-300\: \: Hz\rightarrow complex\: \: magnitude\rightarrow 0.5e^{\frac{i4\pi }{5}}\rightarrow phase\rightarrow \frac{4\pi }{5}$

The frequency spectrum is

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Fundamental frequency is the lowest frequency of periodic wave form ,here lowest frequency from the above signal is

$w_{o}=160\pi \: \: rad/s\: \: ,f_{o}=80\: \: Hz$

$T_{o}=\frac{1}{f_{o}}=\frac{1}{80}=0.0125\: \: s$