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ELEC 324 Lab 3: Fourier Series, Fourier Transform and Signal Spectra

A comprehensive MATLAB implementation exploring discrete-time Fourier transforms, signal analysis, and spectrogram visualization techniques.

Overview

This lab demonstrates fundamental signal processing concepts including:

• Discrete-Time Fourier Series (DTFS) and Discrete Fourier Transform (DFT)
• Periodic and aperiodic signal analysis
• Frequency-domain representation of signals
• Time-frequency analysis using spectrograms
• Audio signal processing and visualization

Lab Structure

Section 2: Periodic Signals

• Analysis of discrete-time periodic signals
• Creation of DTPS from continuous-time signals
• Period identification and signal classification

Section 3: Discrete-Time Fourier Transform

3.2 DFT of Sinusoidal Signals

• Basic DFT computation and visualization
• Effects of transform size on frequency resolution
• Windowing effects and spectral leakage
• Zero-padding and frequency interpolation

3.3 DFT of Aperiodic Signals

• Linear frequency modulation (chirp signals)
• Effects of signal length and DFT size
• Aliasing and Nyquist limit violations

Section 4: Spectrum Analysis

4.0 Time and Frequency Analysis

• FM signal generation and audio playback
• Time-reversal properties in frequency domain
• Phase and magnitude spectrum comparison

4.1 Waterfall Spectrogram

• 3D time-frequency representation
• Chunk-based DFT computation

4.2 Two-Dimensional Spectrogram

• Color-coded frequency visualization
• Overlapping window implementation
• Comparison with MATLAB's built-in specgram
• Real-world audio file analysis

Key Concepts Demonstrated

Signal Properties

• Periodicity: Identification of periodic vs. aperiodic signals
• Spectral Leakage: Effects of non-integer period windows
• Time-Frequency Trade-off: Chunk size vs. resolution

DFT Properties

• Frequency Resolution: Δf = Fs/N
• Time Reversal: Magnitude preservation, phase conjugation
• Windowing: Rectangular window effects (sinc convolution)

Spectrogram Parameters

• Chunk Size: Frequency resolution (smaller = better time resolution)
• Overlap: Temporal smoothness (typically 50%)
• Colormap: Dynamic range visualization

Prerequisites

• MATLAB R2016a or later
• Signal Processing Toolbox (optional, for comparison functions)
• Audio files: newvoice.wav (provided or user-supplied)

Usage

Running Individual Sections

Each section can be run independently. Example:

% Section 3.2.1: Basic DFT
clear; close all; clc;
L = 100;
period = 20;
n = 0:L-1;
x = sin(2*pi*n/period);

X = fft(x, L);
x_reconstructed = ifft(X, L);

figure;
subplot(3,1,1);
stem(n, x, 'filled');
title('Original Signal');
% ... (see full code in lab report)

Complete Spectrogram Implementation

% Generate spectrogram with overlapping chunks
Fs = 2000;
[x, ~] = audioread('newvoice.wav');
chunk_size = 256;
overlap = 0.5;

% ... (see Section 4.2.3 for complete implementation)

Important Parameters

Parameter	Typical Value	Purpose
Fs	2000-8000 Hz	Sampling frequency
chunk_size	66-256 samples	DFT window size
overlap	0.5 (50%)	Window overlap factor
colormap	jet(256)	Visualization colors

Key Findings

1. DFT accurately reconstructs periodic signals when N = L and the signal completes integer periods within the window

2. Spectral leakage occurs when the analysis window contains non-integer periods, spreading energy across adjacent frequency bins

3. Overlapping chunks improve temporal resolution in spectrograms at the cost of increased computation

4. Multiplying signals by large constants (e.g., 1000×) expands the color range in spectrograms, revealing low-magnitude components

5. The DFT shows average frequency content but doesn't reveal when frequencies occur—spectrograms solve this limitation

Files Description

• Lab report with theoretical background and all answers
• MATLAB code snippets for each section
• Generated plots and spectrograms
• Audio analysis results

Learning Outcomes

After completing this lab, you will understand:

• How to compute and interpret DFT coefficients
• The relationship between DTFS, DFT, and DTFT
• Time-frequency analysis trade-offs
• Practical spectrogram implementation
• Real-world audio signal analysis

Common Issues and Solutions

Issue: IDFT produces small imaginary parts
Solution: This is normal due to floating-point precision. Use real(ifft(X)) for real signals

Issue: Spectrogram appears too dark
Solution: Multiply signal by 1000 or normalize DFT magnitudes before display

Issue: Frequency axis doesn't match expectations
Solution: Ensure proper axis scaling: freq_axis = linspace(0, Fs/2, half_spectrum)

Extensions and Improvements

Consider implementing:

• Hamming or Hann windows to reduce spectral leakage
• Logarithmic magnitude scaling (dB) for better dynamic range
• Variable overlap percentages (75%, 87.5%)
• Comparison with Short-Time Fourier Transform (STFT)
• Real-time spectrogram for live audio input

References

• Lab manual: "ELEC 324 Lab 3 – Fourier Series, Fourier Transform and Signal Spectra" © G. Chan, S. Blostein
• Course textbook (consult posted syllabus)
• MATLAB Documentation: Signal Processing Toolbox

Author

Daniil Nistribenko
Queen's University
ELEC 324
Date: November 13, 2025

License

This lab is for educational purposes as part of ELEC 324 at Queen's University. For licensing questions regarding distribution of this code, please see the repository license file.

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Note: This README accompanies the full lab report which contains detailed explanations, prelab answers, and comprehensive code listings for all sections.