EE3302 Communication Theory Tutorial Assignment 3 PDF

SOUTH EASTERN UNIVERSITY OF SRI LANKA FACULTY OF ENGINEERING 2020/2021 (E20) BATCH DECEMBER 2024 EE3302 COMMUNICATION THEORY...

SOUTH EASTERN UNIVERSITY OF SRI LANKA FACULTY OF ENGINEERING 2020/2021 (E20) BATCH DECEMBER 2024 EE3302 COMMUNICATION THEORY TUTORIAL ASSIGNMENT – 3 Part I – Problem solving (Total Marks:100, 10 Marks for each question ) (1) You are asked to design a DSB-SC modulator to generate a modulated signal km(t) cos (ωct + θ), where m(t) is a signal band-limited to W Hz. Figure below shows a DSB-SC modulator available in the stockroom. The carrier generator available generates not cos (ωct), but cos3 (ωct). Explain whether you would be able to generate the desired signal using only this equipment. Your may use any kind of filter you like. (a) What kind of filter is required? (b) Determine the signal spectra at point (B) and (C), and indicate the frequency bands occupied by these spectra. (c) What is the minimum usable value of ωc ? (d) Would this scheme work if the carrier generator output were sin3 ωct ? Explain. (e) Would this scheme work if the carrier generator output were cosn ωct for any integer n≥2? m(t) km(t) cos ωct Filter (A) (B) (C) cos3 ωct (2) A DSB-SC wave is demodulated by applying it to a coherent detector. (a) Evaluate the effect of a frequency error Δf in the local carrier frequency of the detector, measured with respect to the carrier frequency of the incoming DSB-SC wave. (b) For the case of a sinusoidal modulating wave, show that because of this frequency error, the demodulated wave exhibits beats at the error frequency. Illustrate your answer with 1  a sketch of this demodulated wave. (A beat refers to a signal whose frequency is the difference between the frequencies of two input signals.) (3) In full AM with carrier (known as full AM, or AM) a message signal m(t) with peak value mp is first amplitude shifted and DSBSC modulated on the carrier 𝐴𝑐 cos(𝜔𝑐 𝑡). The resulting AM signal can be expressed as 𝑠(𝑡) = 𝐴𝑐 (1 + 𝜇 𝑚𝑛 (𝑡)) cos(𝜔𝑐 𝑡) where μ is the modulation index and 𝑚𝑛 (𝑡) is the normalized form of m(t) such that the peak is 1. (a) Write and expression for μ in terms of mp and 𝐴𝑐. (b) Comment on the value of 𝐴𝑐 with respect to mp required to avoid cross over in the AM waveform. (4) Consider a modulating wave that consists of a single tone or frequency component; that is, 𝑚(𝑡) = 𝐴𝑚 cos(2𝜋𝑓𝑚 𝑡). The corresponding AM signal is 𝑠(𝑡) = 𝐴𝑐 [1 + 𝜇 cos(2𝜋𝑓𝑚 𝑡)] cos(2𝜋𝑓𝑐 𝑡). Figure below shows the modulating, carrier, and AM signal waveforms. (a) What can you conclude about the value of 𝜇 here? (b) Write an expressioin for Amax /Amin in terms 𝜇 of and hence an expression for 𝜇 in terms of Amax and Amin. (c) Using appropriate triogonometric identity, rewrite s(t) as a sum of sinusoidal waves. (d) Write an expression for the Fourier transform S( f ) of the waveform s(t). (e) Sketch the Fourier transforms M( f ), C( f ), and S( f ). (f) Write expressions for the carrier power, upper side frequency power, the lower side frequency power, and the ratio of total sideband power to the total power. (g) Comment on the maximum value of the power ratio. 2 (5) Consider a message signal defined as: 𝑚(𝑡) = 𝐴𝑚 cos(2𝜋𝑓𝑎 𝑡)𝑐𝑜𝑠(2𝜋𝑓𝑏 𝑡) (a) Find the phase modulated signal. (b) Find the frequency modulated signal. (c) Find the narrowband FM signal using the signal obtained in Part (b). (6) Consider a modulating wave that increases linearly with time t, starting at t = 0 as shown by 𝑎𝑡, 𝑡 ≥ 0 𝑚(𝑡) = {. 0, 𝑡 < 0 where a is the slope parameter. For a = 1volt/second and a carrier frequeny fc = ¼ Hz, (a) Sketch m(t), the phase modulated waveform sp(t) with phase-sensitivity factor kp = π/2 radiance/volt and the frequency modulated waveform sf (t) with frequency-sensitivity factor, kf = 1 Hz/volt. Assume carrier amplitude Ac = 1 volt. (b) Find expressions for zero crossing times in the case of PM and FM. (7) Draw the PM and FM waves produced by the sawtooth wave shown in the figure below as the source of modulation. Write down any assumptions and derivations used in drawing the waveforms. (8) A carrier wave of frequency 100 MHz is frequency-modulated by a sinusoidal wave of amplitude 20 V and frequency 100 kHz. The frequency sensitivity of the modulator is 25 kHz/V. (a) Determine the approximate bandwidth of the FM wave, using Carson’s rule. (b) Determine the bandwidth obtained by transmitting only those side-frequencies with amplitudes that exceed one percent of the unmodulated carrier amplitude. Use the universal curve of the figure below for this calculation. 3  (c) Repeat your calculations, assuming that the amplitude of the modulating wave is doubled. (d) Repeat your calculations, assuming that the modulation frequency is doubled. (9) Design an Armstrong indirect FM modulator to generate an FM signal with carrier frequency 97.3 MHz and Δf = 10.24 kHz. A Narrowband FM generator of fc1 = 20 kHz and Δf = 5 Hz is provided. Only frequency doublers ( x2 ) can be used as multipliers. Additionally, a LO (local oscillator) with ajustable frequency between 400 and 500 kHz is available for frequency mixing. (10) Figure below shows the frequency-determining network of a voltage-controlled oscillator (VCO). Frequency modulation is produced by applying the modulating signal 𝐴𝑚 sin(2𝜋𝑓𝑚 𝑡) plus a bias Vb to a pair of varactor diodes connected across the parallel combination of a 200 μH inductor and 100 pF capacitor. The capacitor of each varactor diode is related to the voltage V (in volts) applied across its electrodes by C = 100V-1/2 pF. The unmodulated frequency of oscillation is 1 MHz. The VCO outpur is applied to a frequency multiplier to produce an FM signal with a carrier frequency of 64 MHz and a modulation index of 5. Determine (a) the magnitude of bias voltage Vb and the amplitude Am of the modulating wave, given that fm = 10 kHz. 4 Part II – Computational Exercise (Marked separately out of 100) AM: study sinusoidal modulation based on the following parameters: Carrier amplitude, 𝐴𝑐 = 1 Carrier frequency, 𝑓𝑐 = 0.4 Hz Modulation frequency, 𝑓𝑚 = 0.05 Hz Display and analyze 5 full cycles of the modulated wave, corresponding to a total duration of 100 seconds. Sample the modulted wave at the rate 𝑓𝑠 = 10 Hz obtaining a total of 100 x 100 × 𝑓𝑠 = 1000 data points. The frequency band occupied by the modulated wave is -5 Hz < f < +5 Hz. Since the separation between the carrier frequency and either side frequency is equal to the modulation frequency 𝑓𝑚 = 0.05 Hz, a frequency resolution of 𝑓𝑟 = 0.005 Hz is appropriate. To achieve this frequency resolution, it is recommended that the number of frequency samples satisfies the condition: 𝑓𝑠 5 𝑀≥ = = 1000 𝑓𝑟 0.005 Therefore choose M = 1000. To approximate the Fourier transform of the modulated wave, use a 1000-point FFT algorithm. Investigate three different situations with respect to the only variable parameter, the modulation factor μ: μ = 0.5, corresponding to undermodulation μ = 1.0, corresponding to 100 percent modulation μ = 2.0, corresponding to overmodulation In each case, plot the modulted wave in time domain (amplitude versus time) and frequency domain (normalized FFT versus frequency in Hz). Discuss the results in each case and draw conclusions. 5