EECS20N: Signals and Systems

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Week 10

• FS coefficients
• Negative freq.
• Transfer function
• Filtering
• Filtering sounds
• Filtering images
• Cascade
• Freq response
• Commutativity
• Feedback
• Freq response

Negative frequencies

The Fourier series expansion for a periodic signal x(t) is

This includes Fourier series coefficients for the constant term (when k = 0, exp(jkω ₀t) = 1), plus the fundamental and harmonics (k ≥ 1). But it also includes terms that seem to correspond to negative frequencies. When k ≤ − 1, the frequency kω ₀ is negative. These negative frequencies balance the positive ones so that the resulting sum is real valued. The following applet demonstrates this:

(Applet by Cory Sharp)

Rotating Phasors No JDK 1.1 support for applet!!

This applet supports exploration of a number of other features of complex exponentials. It includes an illustration of the first few Fourier series terms of the waveforms studied before.