Consider a phase modulated (PM) signal of the form xc(t) = Accos(2πfct + kpAmcos(2πfmt)). This modulated signal is applied to an ideal BPF whose characteristics are given below: Figure 1: The Bandpass Filter. (a) Find the Fourier series expansion of the signal xc(t). Note that xc(t) = Acℜ{ej(2πfct + kpAmcos(2πfmt)}, where ℜ{. } denotes real part of a complex number. (b) Determine the envelope, phase and instantaneous frequency of the modulated signal at the filter output as functions of time.
![Problem 4: Fourier Series [24 points] (a) (6 points) Obtain the Fourier series of the signal x(t) given in Figure 3. Using numerical solvers (like Wolfram alpha), sketch the synthesis equation of the Fourier series and see if it gives the same sketch as in Figure 3. Figure 3: Figure for Question 4 part (a) (b) (6 points) Consider a signal g(t) = t2 over the interval (−1, 1) which repeats its self with a period T = 4. Find the trigonometric Fourier Series to represent g(t). Sketch the Fourier Series over frequency. (c)(6 points) Verify Parseval's Theorem for this case given that ∑n = 1∞1 n4 = π490 (d) (6 points) Determine the periodic signal x(t) of period T = 4 whose Fourier series coefficients are given below cn = {jn|n| < 30 otherwise](https://www.doubtrix.com/uploads/editor/0258520695NeIiMMsiju.png)
![Bandlimited Baseband Modulation (18 points) Consider an M-ary PAM baseband communication system. The communication channel is an ideal lowpass filter with a cutoff frequency (or bandwidth) of W = 8 kHz. Let us denote the signal component at the output of the receiving filter as x(t) and its Fourier transform as X(f). Suppose the system is designed such that X(f) is a raised cosine spectrum: X(f) = {T0 ≤ |f| ≤ 1−α2 TT2[1+cos(πTα(|f|−1−α2 T))]1−α2 T ≤ |f| ≤ 1+α2 T0|f| > 1+α2 T where 0 ≤ α ≤ 1, and the corresponding x(t) is x(t) = sinc(tT)cos(παt/T)1−4 α2 t2 /T2 Note: sinc(x) = sin(πx) πx (a) Suppose there is no inter-symbol interference (ISI), then what is the maximum symbol rate supported by the channel, and what is the symbol rate when a "true" (α = 1) raised cosine spectrum is used? (b) Determine and plot the shape of x(t) when the maximum symbol rate is used, identify the zero crossings, and explain why there is no ISI.](https://www.doubtrix.com/uploads/editor/9781206924knTKkTxhts.jpg)
