Determine the first four values of the modulation index

Question

DATA COMMUNICATION

(1) A carrier wave of frequency 100 MHz and amplitude A_c = 100V is frequency modulated by a sinusoidal signal of frequency f_m = 100 kHz and amplitude A_m = 20V. Assuming the frequency sensitivity of the modulator k_f= 25 kHz/V:

(a) Determine the approximate bandwidth of the FM signal

(b) Repeat your calculations assuming the modulating frequency is doubled.

(c) Sketch the amplitude spectrum of the FM signals in parts (a) and (b) [Hint: use Matlab function call Jn(β) = besselj(n,β) to calculate J_n(β)]. Consider 10 spectral components on either side of the carrier.

(2) A carrier wave is frequency modulated using a sinusoidal signal of frequency f_m and amplitude A_m.

(a) Determine the first four values of the modulation index β for which the carrier component is reduced to zero [Hint: use Matlab function call J_n(β) = besselJ(n,β) to calculate J₀(β)]

(b) In certain experiment conducted with f_m =1kHz and increasing Am (starting from zero), it is found that the carrier component of the FM signal is reduced to zero for the first time when A_m = 2V.

i. What is the frequency sensitivity of the modulator

ii. What is the value of Am for which the carrier component is reduced to zero for the second time.

(3) The signal:

is sampled at a 10 kHz rate.

s(t) = sin(400Πt) + 0.5cos(12000πt)

(a) What is the spectrum of the sampled signal?

(b) Use the following Matlab script to simulate the sampled spectrum. Explain the results.

(c) How does the spectrum change if the sampling rate was changed to 15 kHz?

% 6/3/2016

% ELEG390 HW3 Q3: Sampling

Fs = 10;             % Sampling Rate (kHz)

Ts = 1/Fs;           % Sample period (ms)

t = 0:Ts:100; % Observation period (ms)

s = sin(2*pi*0.2*t)+0.5*cos(2*pi*6*t);       % given signal

figure(1)

plot(t,s)

xlabel 'Time (ms)'

ylabel 's(t)'

FFTsize = 1024;

spec = fftshift(abs(fft(s,FFTsize)));           % spectrun

freq = -Fs/2:Fs/FFTsize:Fs/2;

freq = freq(1:end-1);

figure(2)

plot(freq,spec/length(freq))

xlabel 'Frequency (kHz)'

ylabel 'Amplitude Spectrum'

(4) A PAM telemetry system involves the multiplexing of four input signals: si(t), ?? = 1,2,3,4. Two of the signals s₁(t) and s₂(t) have a bandwidth of 80 Hz each, whereas the remaining two signals s₃(t) and s₄(t) have a bandwidth of 2 kHz each.

The signals s₃(t) and s₄(t) are sampled at the rate of 2400 sample per second. This sampling rate is divided by 2R (R is an integer) in order to derive the sampling rate for s₁(t) and s₂(t). Find the maximum value of R.

(5) Consider the following sequences of 1s and 0s:
(a) An alternating sequence of 1s and 0s,
(b) A long sequence of 1s followed by long sequence of 0s,
(c) Along sequence of 1s followed by a single 0 and then long sequence of 1s. Sketch the waveform for each of these sequences with:
(4.1) On-off signaling
(4.2) Bipolar return-to-zero signaling

(6) The figure below shows a PCM signal in which the amplitude levels of +1 vole and -1 volt are used to represent binary symbols1 and 0 respectively. The code word used consists of three bits. Find the sampled version of an analog signal from which this binary signal is derived.