Overview
Week 6 at a Glance
Fourier seriesDFT and FFTSpectral leakageWindow functionsLinear least squaresGoodness of fit (R², RMSE)
Why it matters in practice. The FFT is the most widely used signal processing tool in engineering. Least-squares fitting is how every calibration curve and sensor model is extracted from data. You will use both in virtually every lab and measurement task for the rest of your career.
Objectives
What You Will Be Able to Do
Course objectives (CO) define program-level skills. Module objectives (MO) define specific weekly targets that build toward them.
Course Objectives (CO)
CO5: Compute and interpret the FFT of a measured signal; apply linear and nonlinear curve fitting and assess goodness of fit.
Module Objectives (MO) — Week 6
Compute the DFT of a sampled signal and interpret the resulting frequency spectrum in terms of frequency resolution and aliasing.
CO5
CO5
Explain spectral leakage and apply Hanning or flat-top window functions to reduce it.
CO5
CO5
Set up and solve the linear normal equations to find the least-squares best-fit line through calibration data.
CO5
CO5
Compute R² and RMSE to assess goodness of fit and identify model inadequacy from residual patterns.
CO5
CO5
Extend least squares to polynomial and linearized nonlinear models (power law, exponential).
CO5
CO5
Review these objectives before you start each assignment. They map directly to what is assessed on the quiz, homework, and exams.
Suggested Learning Path
How to Work Through This Week
Follow this sequence. Each step prepares you for the next. Do not attempt graded work before completing the instructional material it depends on.
1
Read Alciatore Ch. 7 (FFT) and Ch. 9 Sec. 9.5 (Curve Fitting)
Read both chapters before lecture. The FFT material builds on the sampling concepts from Week 5. The curve fitting section introduces the normal equations — work through Example 9.5 by hand.
2
Attend Lecture
Lecture 1 covers the Fourier series, DFT algorithm, FFT, and windowing. Lecture 2 covers least-squares normal equations, R², RMSE, and polynomial fitting. Each lecture includes a problem session.
3
Lab: FFT and Curve Fitting on Measured Data
Use your Arduino to collect a time-domain signal, compute its FFT in MATLAB/Python, apply a window function, and fit a calibration curve to the amplitude spectrum.
4
Submit Module 3 Homework — due this week
Module 3 Homework covers Week 5 (filters, DAQ) and Week 6 (FFT, curve fitting) content. All problems must be submitted by the end-of-week deadline in Canvas.
Instructional Material
Required Readings, Videos, and Resources
Complete all required items before moving to graded activities. The Aligns to column maps each resource to the module objectives it directly supports.
| Resource | What You Will Gain | Aligns to | Est. Time |
|---|---|---|---|
| Read Alciatore Ch. 7 (Frequency Analysis) and Ch. 9 Sec. 9.5 (Curve Fitting) |
DFT derivation, FFT algorithm efficiency, spectral leakage, window functions, linear normal equations, R², RMSE, and polynomial fitting. | MO1-MO5 | 75 min |
| Lab Lab: FFT Analysis with Arduino and MATLAB/Python |
Capture a time-domain waveform on your Arduino, export to MATLAB or Python, compute the FFT, apply a Hanning window, and interpret the spectrum. | MO1, MO2 | ~2 hr lab |
| Watch Micro-lecture: Why the Hanning Window Works |
5-minute animated explanation of spectral leakage and the mechanism by which window functions suppress sidelobe energy. | MO2 | 5 min |
Graded Learning Activities
Assignments and Due Dates
All graded work is submitted through Canvas. Complete the listed prerequisites before attempting each assignment.
| Assignment | Prerequisites | What Is Assessed | Aligns to | Points |
|---|---|---|---|---|
| Module 3 Homework: Filters, DAQ, and FFT End of Week 6 |
Complete Ch. 6, Ch. 7, and Ch. 9 Sec. 9.5 readings. Attend both Week 5 and Week 6 lectures before attempting FFT and curve fitting problems. | Anti-aliasing specification, ADC quantization, FFT interpretation, window function selection, and least-squares calibration curve with R² and RMSE. | MO1-MO5 | 50 pts |
Academic integrity. FFT and curve fitting problems require your own measured time-domain data from the Arduino lab. Your reported spectrum and fit coefficients must correspond to your actual hardware measurements.