Introduction to Fourier Analysisscience.umd.edu/faculty/wilkinson/bsci338/ACL3_Fourier.pdf ·...
Transcript of Introduction to Fourier Analysisscience.umd.edu/faculty/wilkinson/bsci338/ACL3_Fourier.pdf ·...
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Introduction to Fourier Analysis
• Definition of Fourier analysis • Analysis of periodic waves • Analysis of aperiodic waves • Time-frequency uncertainty • Digitization and spectrograms • Reading focus:
– pp. 41-68 – boxes 3.2 and 3.3 but not 3.1 – Raven manual, appendices A and B
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The Fourier series
• Any continuous waveform can be partitioned into a sum of sinusoidal waves (Fourier is a type of infinite series)
• P(t) = Po + ΣPn sin (2πfnt + Φn) • Po is the ambient pressure • Pn is the pressure of the nth sine wave • fn is the frequency of the nth sine wave • Φn is the phase of the nth sine wave
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Sound spectrum
Frequency spectrum
Time domain
Frequency domain
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Frequency domain of a complex wave
Frequency spectrum Phase spectrum
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Types of periodic waveforms
• Amplitude varies in a repeating manner - amplitude modulation
• Frequency varies in a repeating manner - frequency modulation
• The repeating shape of the waveform is not a sine wave - nonsinusoidal periodic wave
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Amplitude modulation (AM)
Carrier frequency plus side bands
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Frequency modulation (FM)
Modulation determines the number of side bands
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Periodic nonsinusoidal signals
Harmonic series
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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
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Harmonic filtering example
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Harmonic Filtering
Frequency modulated sounds and periodic nonsinusoidal sounds can have identical spectra, but will differ in phase
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Compound signals
• Nonsinusoidal modulation of a sine wave
• Sinusoidal modulation of a nonsinusoidal
carrier wave
• Nonsinusoidal modulation of a
nonsinusoidal carrier wave
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Pulsed sine wave (frog or insect)
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Fourier analysis of aperiodic signals
• The more aperiodic a signal is, the more frequency components are needed to describe the signal with a Fourier series
• A - D have increasing intervals, between pulses, each of which has carrier frequency f. D is a single pulse
Most natural signals have a short duration, not infinite
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Delta signals
• In the limit, an infinitely short signal has constant amplitude at all frequencies, a delta pulse
• White noise also has equal amplitude at all frequencies, but this results from random waveform
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Finite sounds and Fourier ‘lobes’
Lobes are artifacts of short sample
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Fourier “windows”
Bartlett
Hamming
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Digitizing sound
Typical sampling rates are 22 kHz and 44 kHz
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Aliasing
Nyquist frequency = 1/2 the sampling frequency
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Aliasing consequences
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Bandwidth = minimum frequency difference that can be discerned
Fourier window size and bandwidth
Narrow Bandwidth (135 Hz, 46 ms)
Wide Bandwidth (540 Hz, 11.5 ms)
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Narrow band vs wide band spectrogram
• Narrow bandwidth analysis is good for frequency • Wide bandwidth is good for time
1 kHz signal sinusoidally amplitude modulated at 70 Hz
Narrow bandwidth (50 Hz) Wide bandwidth (300 Hz)
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Time-frequency uncertainty
Narrow Bandwidth (176 Hz, 23 ms)
Wide Bandwidth (1412 Hz, 2.9 ms)
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Problems
• If you want to measure the time delay between an echolocation pulse and the returning echo, should you use a 64 or 1024 point window?
• If you want to measure the frequency of a constant frequency echolocation pulse accurately, should you use a 64 or 1024 point window?
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Sound spectrum “waterfall”
Song sparrow
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Sonagram = sound spectrogram Song sparrow
Narrow bandwidth analysis is good for frequency measurements, but not accurate for time measurements