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Understanding Fourier Analysis: Waves, Modulation, and Signals

This introduction to Fourier Analysis covers the definition and analysis of periodic and aperiodic waves, digitization, time-frequency uncertainty, Fourier series, sound spectrum, types of waveforms, modulation, harmonic series, compound signals, and Fourier analysis of aperiodic signals. The text explores various concepts such as amplitude modulation, frequency modulation, harmonic frequencies, compound signals, and time-frequency uncertainty in sound spectrum analysis.

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Understanding Fourier Analysis: Waves, Modulation, and Signals

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  1. Intro to Fourier Analysis • Definition • Analysis of periodic waves • Analysis of aperiodic waves • Digitization • Time-frequency uncertainty

  2. The Fourier series • Any continuous waveform can be partitioned into a sum of sinusoidal waves • P(t) = Po + SPncos (2pfnt + Fn) • Po is the ambient pressure • Pn is the pressure of the nth cosine wave • fn is the frequency of the nth cosine wave • Fn is the phase of the nth cosine wave

  3. Sound spectrum Time domain Frequency domain Frequency spectrum

  4. Types of periodic waveforms • Amplitude varies in a repeating manner - amplitude modulation • Frequency varies in a repeating manner - frequency modulation • The shape of the waveform varies in a repeating manner - nonsinusoidal periodic wave

  5. Amplitude modulation (AM) Carrier frequency plus side bands

  6. Frequency modulation (FM) Modulation determines the number of side bands

  7. Periodic nonsinusoidal signals Harmonic series

  8. Harmonic series • Harmonic frequencies are integer multiples of the fundamental frequency, i.e. w, 2w, 3w, 4w … • Dirichlet’s rule states that the energy in higher harmonics falls off exponentially with the frequency of the harmonic • Note, however, that some animals alter the amplitude of harmonics by selective filtering during sound production

  9. Compound signals • Nonsinusoidal modulation of a sine wave • Sinusoidal modulation of a nonsinusoidal carrier wave • Nonsinusoidal modulation of a nonsinusoidal carrier wave

  10. Pulsed sine wave (frog or insect)

  11. Fourier analysis of aperiodic signals • Most natural signals have a short, not infinite, duration • The more aperiodic a signal is, the more frequency components are needed to describe the signal with a Fourier series • In the limit, an infinitely short signal has constant amplitude at all frequencies, a delta pulse

  12. Finite sounds and Fourier ‘lobes’

  13. Fourier “windows” Bartlett Hamming

  14. Digitizing sound

  15. Aliasing Nyquist frequency = 1/2 the sampling frequency

  16. Fourier window size and bandwidth

  17. Time-frequency uncertainty

  18. Sound spectrum “waterfall” Song sparrow

  19. Sonagram Song sparrow Narrow bandwidth analysis is good for frequency measurements, but not accurate for time measurements

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