iv) (2%) For sending a same signal, linear modulation typically requires a larger bandwidth than angle modulation.

2. Consider sending a message $m(t)$ whose power spectral density (PSD) is $S_m(f) = f^4$ for $|f| \le W$ and $S_m(f) = 0$ otherwise. The transmitted signal is given by $x(t) = m(t) \cos 2\pi f_c t$. The signal is corrupted by an additive white Gaussian noise with PSD $N_0/2$. i) (5%) Pass the received signal through an ideal bandpass filter centered at \( f_{c} \) with bandwidth \( B \geq 2W \). The noise then becomes \( n_{c}(t) \cos 2\pi f_{c}t - n_{s}(t) \sin 2\pi f_{c}t \) narrowband noise. Draw \( S_{n_c}(f) \) the PSD of \( n_{c}(t) \).

ii) (8%) The output signal now is \( y(t) = x(t) + n_c(t) \cos 2\pi f_c t - n_s(t) \sin 2\pi f_c t \). Use a coherent demodulator followed by a low-pass filter with bandwidth \( W \) to demodulate it. Derive the demodulation signal and the SNR at the demodulator output.

iii) (8%) Suppose the signal is further corrupted by an interference \( k(t) = \text{sinc}(wt) \) and the received signal after the bandpass filter becomes \( y_2(t) = (t) + k(t) \cos 2\pi(f_c + 2W)t + n_c(t) \cos 2\pi f_c t - n_s(t) \sin 2\pi f_c t \). Design a low-pass filter for the demodulator in part ii) such that the signal-to-interference-plus-noise ratio (signal power divided by the sum of interference and noise power) is maximized.

3. Let $\{a_k\}$ be a sequence of data which is multiplied with a pulse shaping function $h(t)$ and sent every $T_s$ seconds. The signal is given by $x(t) = \sum_{k=-\infty}^{\infty} a_k h(t - kT_s)$. We sample this signal every $T_s$ seconds to get the discrete signal $x[n] = \sum_{k=-\infty}^{\infty} a_k h[n - k]$ for every integer $n$. i) (5%) Derive the condition on \( h(t) \), in time domain, such that there will be no inter-symbol interference?

ii) (8%) Now, derive from part i) the condition on \( H(f) \), in frequency domain, which guarantees no inter-symbol interference.

iii) (8%) Provide an example of \( H(f) \) satisfying the condition in part ii). Show in time domain that it indeed will not result in any inter-symbol interference.

4. (10%) In wireless communication systems, Additive White Gaussian Noise (AWGN) channel is often assumed. What do "Additive", "White", and "Gaussian" mean?

5. (10%) Plot the 16QAM and 8PSK constellations with normalized power 1. How many bits can one 16QAM symbol and one 8PSK symbol carry, respectively?

6. (10%) A matrix \( H \) for a linear block code is defined such that each vector \( v \) is a valid codeword if and only if \( Hv^T = 0 \). Assume that $$H = \begin{bmatrix} 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 1 \end{bmatrix}$$ i) Is the vector \( v_{1} = [1\ 0\ 0\ 1\ 0\ 1] \) a valid codeword vector? Why or why not? ii) How many valid codewords can you find?

7. (20%) Given $\varphi_1(t) = 1$, $\varphi_2(t) = t$, $\varphi_3(t) = 3t^2 - 1$, i) Show that \(\varphi_1(t)\) and \(\varphi_2(t)\) are mutually orthogonal over the interval (-2, 2)

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