Wireless Communications Exercise I
Q1: Why do we use the modulation technique?
Q2: What is relationship between information frequency, $f_m$, and carrier signal frequency $f_c$ ?
Q3: Write the mathematical formula for signals of the following:
Carrier signal $\mathrm{c}(\mathrm{t})$
Amplitude Modulation ${\mathrm{M}}_{\mathrm{AM}}(\mathrm{t})$
Amplitude Modulation ${\mathrm{M}}_{\mathrm{FM}}(\mathrm{t})$
Q4: Sketch the modulated signals of the following digital modulation schemes for digitized information signal ${\mathrm{m}}_{\mathrm{d}}(\mathrm{t})$, and sketch the constellation diagram for each one?
$${\mathrm{m}}_{\mathrm{d}}(\mathrm{t})=(1,0,0,1,1,0,0)$$
Q5: explain the QAM modulation?
Q6: Explain in detail the OFDMA ?
Q7: Does $x_1(t)=\cos (2\pi ft)$, and $x_2(t)=\sin (2\pi ft)$ are orthogonal signals?

Step 1 ：A1: Modulation techniques are used to represent information and transmit it over wireless channels with improved performance and robustness.

Step 2 ：A2: The relationship between information frequency $f_m$ and carrier signal frequency $f_c$ is typically given by: $f_c>>f_m$.

Step 3 ：A3: Carrier signal: $c(t)=A_c\cos (2\pi f_{c}t)$; AM: $M_{AM}(t)=(A_c+m(t))\cos (2\pi f_{c}t)$; FM: $M_{FM}(t)=A_c\cos (2\pi f_{c}t+k_{f}m\int m(\tau )d\tau )$

Step 4 ：A4: Sketch not possible with text; see resources such as lecture notes or textbooks for example diagrams.

Step 5 ：A5: QAM (Quadrature Amplitude Modulation) combines two amplitude-modulated signals, with carriers in quadrature (90-degree phase shift), to transmit multiple bits per symbol.

Step 6 ：A6: OFDMA (Orthogonal Frequency-Division Multiple Access) is a technique that allocates subsets of OFDM subcarriers to multiple users simultaneously, enabling efficient and flexible use of the available frequency spectrum.

Step 7 ：A7: $x_1(t)$ and $x_2(t)$ are orthogonal signals if their integral product over a period is zero: ${\int }_0^T{x}_1(t)x_2(t)dt={\int }_0^{T}cos(2\pi ft)sin(2\pi ft)dt=0$