Question

Screen Shot 2021-03-17 at 9.57.42 AM.png

Screen Shot 2021-03-17 at 9.57.49 AM.png

3 Decoding a Morse Code message

In this exercise you will decipher a Morse code message sent to Agent 008 by Agent \(007 .\) The last words of Agent 007 were "The future of technology lies in \(\ldots\) " at which point she produced a memory stick containing a MATLAB file ctftmod.mat. The file ctftmod \(.\) mat contains the following:

af, bf the denominator and numerator coefficients of a lowpass filter, whose frequency response can be plotted using the freqs command, i.e., freqs (bf, af).

- Modulation frequencies \(\mathrm{f} 1\) and \(\mathrm{f} 2\).

- Two prototype signals dot and dash.

- Signal \(x(t)\)

A time vector t.

The signal \(x(t)\) is of the form \(x(t)=m_{1}(t) \cos \left(2 \pi f_{1} t\right)+m_{2}(t) \sin \left(2 \pi f_{2} t\right)+m_{3}(t) \sin \left(2 \pi f_{1} t\right) .\) It is known that

\(m_{1}(t), m_{2}(t)\) and \(m_{3}(t)\) correspond to single letters of alphabet which has been encoded using International Morse code. Morse code encodes characters as standardized sequence of dots and dashes (signals dot and dash are given in the file). To read more about Morse Code and to find sequence of dots and dashes corresponding to each alphabet visit this link.

1. Using the signals dot and dash, construct the signal that corresponds to ' \(Z\) ' in Morse code and plot it against t. As an example, the letter \(\mathrm{C}\) can be constructed by, \(\mathrm{c}=\) [ dash dot dash dot ].

2. Plot the frequency response of the filter using freqs (bf, af).

3. The signals dot and dash are both composed of low frequency components, such that their Fourier transform lies within the passband of the filter. Verify this by using the \(1 \mathrm{sim}\).

ydash \(=1 \operatorname{sim}(b f, a f,\) dash, \(t(1:\) length \((\) dash \()))\)

\(y\) dot \(=1 \operatorname{sim}(b f, a f, \operatorname{dot}, t(1:\) length \((\) dot \()))\)

Plot ydash and ydot along with the original signals.

4. When the signal dash is modulated by \(\cos \left(2 \pi f_{1} t\right)\) most of the energy of its Fourier transform moves outside the passband of the filter. Create \(y(t)\) by executing \(\mathrm{y}=\) dash. \(* \cos (2 * \pi * \mathrm{f} 1 * \mathrm{t}(1:\) length \((\) dash \())\). Plot the signal \(y(t)\). Also plot the output yo \(=1\) sim(bf, af, \(y, t)\). Do you get a result that you expected?

5. Analytically determine the Fourier transform of each of the signals

$$ \begin{array}{l} \left.m(t) \cos \left(2 \pi f_{1} t\right) \cos \left(2 \pi f_{1} t\right)\right) \\ \left.m(t) \cos \left(2 \pi f_{1} t\right) \sin \left(2 \pi f_{1} t\right)\right) \end{array} $$

and

$$ \left.m(t) \cos \left(2 \pi f_{1} t\right) \cos \left(2 \pi f_{2} t\right)\right) $$

in terms of \(M(j \omega),\) the Fourier transform of \(m(t)\).

6. Using your result from part 5 and by examining the frequency response of the filter in part \(2,\) devise a plan to extract \(m_{1}(t)\) from \(x_{1}(t) .\) What alphabet does it correspond to?

7. Repeat above step to extract \(m_{2}(t)\) and \(m_{3}(t) .\) Where does the future of technology lie?

Thank you