MEGR 3171 — Frequency Domain Analysis & Curve Fitting

Learning Objectives

Represent periodic signals as Fourier series and explain the DFT as the discrete-time equivalent.
Interpret FFT output: amplitude spectrum, phase spectrum, frequency resolution, and record length.
Apply windowing functions (Hanning, Hamming) to reduce spectral leakage.
Fit linear and polynomial models to experimental data using the method of least squares.
Apply the Gauss-Newton or Levenberg-Marquardt algorithm concept to nonlinear curve fitting.
Assess goodness of fit using R-squared and RMSE; analyze residuals for model validity.

1. Fourier Series and the DFT

Any periodic signal can be expressed as a sum of sinusoids at the fundamental frequency and its harmonics. The Fourier series provides the theoretical foundation; the Discrete Fourier Transform (DFT) extends this to sampled, finite-length signals.

Key FFT Parameters

Frequency resolution: Δf = f_s / N  (Hz per bin)
  where N = number of samples, f_s = sampling rate

Record length:  T = N / f_s  (seconds)

Useful frequency range: 0 to f_s/2  (Nyquist limit)

Amplitude scaling: divide magnitude by N/2 for single-sided spectrum

The Fast Fourier Transform (FFT, Cooley-Tukey algorithm) computes the DFT in O(N log N) operations instead of O(N²), making it practical for real-time use. FFT works most efficiently when N is a power of 2.

2. Spectral Leakage and Windowing

The DFT assumes the signal is periodic within the record. If the signal does not complete an exact integer number of cycles, the abrupt truncation at the record boundaries introduces artificial frequency components called spectral leakage. A windowing function tapers the signal to zero at the record edges, reducing leakage at the cost of slightly reduced frequency resolution.

Rectangular Window

No tapering. Best frequency resolution but worst leakage. Acceptable only when signal frequency aligns exactly with an FFT bin.

Hanning Window

Good leakage reduction, moderate resolution reduction. Most common choice for general-purpose spectral analysis of continuous signals.

Hamming Window

Similar to Hanning but with slightly different coefficients. Better for some transient signals. Widely used in audio and vibration analysis.

Flat-Top Window

Best amplitude accuracy. Used when precise amplitude measurement is critical (calibration). Poor frequency resolution.

3. Linear Curve Fitting (Least Squares)

Given data pairs (x_i, y_i), find the best-fit line y = a₀ + a₁x that minimizes the sum of squared residuals.

Linear Regression Formulas

a_1 = [N∑(x_i y_i) - ∑x_i ∑y_i] / [N∑(x_i²) - (∑x_i)²]

a_0 = (1/N)(∑y_i - a_1 ∑x_i)

R² = 1 - SS_res / SS_tot
SS_res = ∑(y_i - ŷ_i)²   SS_tot = ∑(y_i - ȳ)²

RMSE = √(SS_res / (N-2))

R² = 1.0 means perfect fit. R² = 0.99 means 99% of variance in y is explained by the model. For sensor calibration curves, R² > 0.9999 is typically expected.

Practice Problems

Problem 1 — Frequency Resolution An FFT is computed from 1024 samples collected at 8192 Hz. What is the frequency resolution? What is the highest frequency representable?

Δf = f_s / N = 8192 / 1024 = 8 Hz per bin

f_max = f_s / 2 = 8192 / 2 = 4096 Hz (Nyquist)

Frequency resolution = 8 Hz; Maximum frequency = 4096 Hz

Problem 2 — Spectral Leakage A 75 Hz sine wave is sampled at 1000 Hz for 0.1 seconds (100 samples). Will spectral leakage occur? Explain.

Frequency resolution = 1000/100 = 10 Hz. The FFT bins are at 0, 10, 20, ..., 70, 80, ... Hz. The 75 Hz signal falls exactly between the 70 Hz and 80 Hz bins. Since no bin aligns with 75 Hz, the record does not contain an integer number of cycles (75 Hz × 0.1 s = 7.5 cycles). Yes, significant leakage will occur. Apply a Hanning window to reduce it.

Yes, leakage will occur. 75 Hz falls between bins. Apply Hanning window.