The Uncertainty Principle

The uncertainty principle (for Fourier transform pairs) follows immediately from the scaling theorem (§2.4.4). It may be loosely stated as

Time Duration Frequency Bandwidth c

where is some constant determined by the precise definitions of ``duration'' in the time domain and ``bandwidth'' in the frequency domain.

If duration and bandwidth are defined as the ``nonzero interval,'' then we obtain , which is not very useful. This conclusion follows immediately from the definition of the Fourier transform and its inverse in §2.2.

---

Subsections

• Second Moments
• Time-Limited Signals
• Time-Bandwidth Products are Unbounded Above

---

Order a Hardcopy of Spectral Audio Signal Processing

Previous: Sampling Theory
Next: Duration and Bandwidth as Second Moments

---

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.