$1.$ Consider transmission of binary data using the following linear modulation scheme. Note that the symbol values are real$-$valued, and therefore this is not QPSK$.$

$-$ binary sequence $00$ is encoded as symbol $0$

$-$ binary sequence $01$ is encoded as symbol $1$

$-$ binary sequence $10$ is encoded as symbol $2$

$-$ binary sequence $11$ is encoded as symbol $3$

Assume the communication system operates with a bit rate of $1000$ bits per second. Assume the pulse shape $\backslash ($ p$($t$)\backslash )$ is rectangular with value $1$ over the entire symbol interval.

$($a$)$ Give a signal constellation for the modulation scheme. Label each symbol value with its corresponding bit sequence.

$($b$)$ Sketch the complex envelope $\backslash (\backslash$tilde$\{$s$\}($t$)\backslash )$ for the binary sequence $00101101.$ Give your answer as plots of the baseband I and Q waveforms.

$($c$)$ Sketch the corresponding passband communication signal $\backslash ($ s$($t$)\backslash )$ assuming a carrier frequency of $150$ kHz $.$