Chapter 10.6, Problem 45PS

Solution

To show: The resultant signal strength of two frequencies on touch −tone phone.

Given data: The signals frequencies 480 Hertz and 620 Hertz are $y_1=\cos 960\pi t$ and $y_2=\cos 1240\pi t$ .

Method/Formula used:

The sum of cosines formula is $\cos A+\cos B=2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)$

Calculation:

The two signals on touch −tone phone are $y_1=\cos 960\pi t$ and $y_2=\cos 1240\pi t$ . The resultant of two signal y is given by

  $y=y_1+y_2$

Substitute $\cos 960\pi t$ for $y_1$ and $\cos 1240\pi t$ for $y_2$ ,

  $y=\cos 960\pi t+\cos 1240\pi t$

Use ‘sum of cosines formula’

  $\begin{array}{c}y=\cos 960\pi t+\cos 1240\pi t\\ =2\cos \left(\frac{960\pi t+1240\pi t}{2}\right)\cos \left(\frac{960\pi t-1240\pi t}{2}\right)\\ =2\cos \left(1100\pi t\right)\cos \left(-140\pi t\right)\leftarrow \cos \left(-\theta \right)=\cos \left(\theta \right)\\ =2\cos \left(1100\pi t\right)\cos \left(140\pi t\right)\end{array}$

Hence, the result.