Analysis and Problem Solving in Angle and Frequency Modulation: Assignment 4 ELG 3175

Assignment 4 ELG 3175

P1

5.1-1
Sketch ψPM(t)\psi_{PM}(t)ψPM(t) and ψFM(t)\psi_{FM}(t)ψFM(t) for the modulating signal m(t)m(t)m(t) shown in Fig. P5.1-1.
Given ac=106a_c = 10^6ac=106, kf=103k_f = 10^3kf=103, and kp=25k_p = 25kp=25.

P2

5.2-6
Estimate the bandwidth for ψFM(t)\psi_{FM}(t)ψFM(t) and ψPM(t)\psi_{PM}(t)ψPM(t) in Prob. 5.1-1.
Assume the bandwidth of m(t)m(t)m(t) in Fig. P5.1-1 to be the third-harmonic frequency of m(t)m(t)m(t).

P3

5.2-4
An angle-modulated signal with carrier frequency ωc=2π×106\omega_c = 2\pi \times 10^6ωc=2π×106 is described by the equation:

ψAM(t)=10cos⁡(ωct+0.1sin⁡2000πt)\psi_{AM}(t) = 10 \cos \left( \omega_c t + 0.1 \sin 2000\pi t \right)ψAM(t)=10cos(ωct+0.1sin2000πt)

(a) Find the power of the modulated signal.
(b) Find the frequency deviation Δf\Delta fΔf.
(c) Find the phase deviation Δϕ\Delta \phiΔϕ.
(d) Estimate the bandwidth of ψAM(t)\psi_{AM}(t)ψAM(t).
(e) Plot the spectrum of ψAM(t)\psi_{AM}(t)ψAM(t).
(f) Determine the power included in the bandwidth found in (d).

P4

Consider a superheterodyne FM receiver designed to receive the frequency band of 1 to 30 MHz with an IF frequency of 8 MHz. What is the range of frequencies generated by the local oscillator for this receiver?
An incoming signal with a carrier frequency of 10 MHz is received at the 10 MHz setting. At this setting of the receiver, we also get interference from a signal with the same carrier frequency if the receiver RF-stage bandpass filter has poor selectivity. What is the carrier frequency of the interfering signal?

Summary of Assessment Requirements and Academic Mentor’s Guiding Approach

Assessment Requirements

This assignment focuses on signal modulation concepts within Assignment 4 – ELG 3175. Students must work through four problem-based tasks involving phase modulation (PM), frequency modulation (FM), angle modulation, and superheterodyne receiver calculations.

The key requirements include:

1. P1 – Modulation Sketching

   • Sketch the PM and FM waveforms based on the given modulating signal.

   • Use the provided constants aca_cac, kfk_fkf, and kpk_pkp.

2. P2 – Bandwidth Estimation

   • Estimate FM and PM bandwidth using the characteristics of the modulating signal.

   • Refer to the third-harmonic frequency of m(t)m(t)m(t).

3. P3 – Angle-Modulated Signal Analysis

   • Determine power, frequency deviation, phase deviation, and bandwidth.

   • Plot the resulting spectrum.

   • Calculate power contained within the estimated bandwidth.

4. P4 – Superheterodyne Receiver Analysis

   • Determine local oscillator frequency range for a receiver covering 1–30 MHz.

   • Identify the interfering signal frequency when receiving a 10 MHz carrier.

The task evaluates students’ understanding of modulation theory, signal behaviour, bandwidth estimation, and RF system function.

Academic Mentor’s Step-by-Step Guidance Approach

The academic mentor supported the student using a structured, analytical, and concept-driven method, ensuring each question was approached logically and linked to course learning outcomes.

1. Understanding the Problem Context

The mentor first ensured the student clearly understood:

• The difference between PM, FM, and angle modulation.

• The relationship between modulating signal characteristics and waveform shape.

• How deviations and bandwidth formulas apply to each modulation type.

• The architecture and behaviour of superheterodyne receivers.

This foundational clarity helped the student approach each part confidently.

2. Breaking Down Each Section of the Assessment

P1: PM and FM Sketching

The mentor guided the student to:

• Observe the shape, slope, and amplitude of the given m(t)m(t)m(t).

• Identify where frequency changes (FM) and phase changes (PM) occur.

• Map portions of m(t)m(t)m(t) to corresponding increases/decreases in frequency or phase.

• Sketch idealised waveforms rather than mathematically exact plots.

The student learned how qualitative waveform interpretation forms the basis for modulation analysis.

P2 : Bandwidth Estimation

The mentor demonstrated:

• How to compute the third-harmonic frequency of m(t)m(t)m(t).

• How Carson’s Rule applies differently to FM and PM.

• How deviation constants relate to bandwidth.

The student was taught to justify decisions using signal properties + known FM/PM formulas rather than guessing.

P3: Angle Modulation Calculations

The mentor walked the student through:

• Calculating average power from amplitude.

• Identifying the modulation index β\betaβ from the equation.

• Determining frequency deviation and phase deviation.

• Estimating bandwidth using Carson’s Rule.

• Understanding how sidebands form the spectrum.

• Summing sideband power within the estimated bandwidth.

This section reinforced core skills in mathematical analysis, spectrum interpretation, and modulation index evaluation.

P4: Superheterodyne Receiver Analysis

The mentor explained:

• The concept of image frequency and local oscillator mixing.

• Why both fLO=fRF+IFf_{LO} = f_{RF} + IFfLO=fRF+IF and fLO=fRF−IFf_{LO} = f_{RF} - IFfLO=fRF−IF may be valid.

• How poor selectivity leads to image interference.

• How to compute the interfering carrier frequency that mixes to the same IF.

Here, the student practised RF systems reasoning, linking theory to practical receiver behaviour.

Ensuring Alignment with Learning Objectives

Throughout the guidance process, the mentor emphasised:

• Correct interpretation of modulation parameters.

• Clear step-by-step reasoning.

• Proper application of formulas and theoretical models.

• Logical justification for every numerical or conceptual answer.

• Accuracy in the representation of spectra, deviations, and bandwidth.

• Precision in explaining receiver interference.

Thus, the following learning outcomes were fully supported:

LO1: Applying modulation theory to PM, FM, and angle-modulated systems

LO2: Estimating bandwidth and analysing signal parameters

LO3: Understanding superheterodyne receiver operation and interference

LO4: Demonstrating technical reasoning and problem-solving skills

Final Outcome

By following this structured, mentor-guided approach, the student developed:

• A clear understanding of modulation behaviour and waveform representation.

• Accurate calculations for deviations, power, bandwidth, and spectrum.

• A step-by-step method for approaching complex modulation problems.

• Confidence in applying radio-frequency and signal-processing concepts.

The assessment was completed with clear reasoning, well-structured explanations, and accurate technical interpretation, fully meeting the task requirements and intended learning objectives.

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