Overview
This lecture explains how to analyze non-periodic (time-bounded, real-world) signals using Fourier methods, culminating in the definition and interpretation of the Fourier Transform and its inverse.
Periodic vs. Non-Periodic Signals
- Periodic signals repeat every interval T₀ and are defined from minus to plus infinity in time.
- Most real signals are non-periodic, starting and ending at finite times, making them "energy signals."
- Energy signals, unlike periodic signals, have finite duration and finite energy.
Constructing a Periodic Signal from a Non-Periodic One
- To analyze a finite signal with Fourier series, artificially repeat it every T₀ to create a periodic extension.
- The original signal is then a special case as T₀ approaches infinity.
Fourier Series and the Transition to Fourier Transform
- Fourier series represent periodic signals as sums of harmonics with coefficients Dₙ.
- For the periodic extension, the fundamental frequency F₀ = 1/T₀.
- As T₀ → ∞, F₀ → 0, harmonics become densely packed and the sum approaches an integral.
- The Fourier coefficients Dₙ become samples of a new function, G(F), at discrete frequencies.
- When the frequency step ΔF becomes infinitesimal, the sum becomes the Fourier Transform integral.
The Fourier Transform Pair
- The Fourier Transform of signal g(t): G(f) = ∫ g(t) e^(–j2πft) dt (integrate over time from –∞ to ∞).
- The Inverse Fourier Transform: g(t) = ∫ G(f) e^(j2πft) df (integrate over frequency from –∞ to ∞).
- This pair allows transforming between time and frequency domains for non-periodic signals.
Interpretation and Properties of G(f)
- G(f) describes amplitude and phase for all frequencies, generalizing Fourier coefficients to a continuum.
- Individual G(f) values at specific frequencies carry no "energy," but integration over a frequency range gives meaningful results.
- Analogous to probability density: the probability at a single point is zero, but over a range is finite.
Key Terms & Definitions
- Periodic Signal — a signal that repeats at regular intervals over all time.
- Energy Signal — a time-limited signal with finite energy (does not extend to infinity).
- Fourier Series — represents a periodic signal as a sum of sinusoids with specific frequencies.
- Fourier Transform — generalizes the Fourier series to non-periodic signals using an integral over all frequencies.
- G(f) — the Fourier Transform of a signal, indicating its frequency content.
- Inverse Fourier Transform — reconstructs the time signal from its Fourier Transform.
Action Items / Next Steps
- Review the derivation and properties of the Fourier Transform and inverse.
- Prepare for discussion on the implications of the Fourier Transform in the next class.