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McMaster University
Paul Faure

September 27 , 2013 Psych 3A03: Audition Filters Complex Sounds - SO far we have been discussing sinusoidal motion and sine waves - A sine wave is a very simple (homogenous) sound also known as a pure tone signal - Sine waves are the fundamental component of all other sound waves - Most acoustic sounds, including animal vocalizations, are complex signals - All complex sounds can be described as the sum of many simple sinusoidal signals that differ in their amplitude, frequency, or phase Jean Baptise Joseph Fourier - The fourier series has since transcended its originally intended purpose, spawning many new areas of study in mathematics and physics Fourier Analysis - When 2 or more sine waves are added together result is a complex wave Discrete Fourier Analysis (Periodic Signals) - All complex sounds can be described as the sum of many simple sinusoidal signals that differ in their amplitude, phase and frequency - If two signals are in phase and have the same frequency you end up with a simple sound - This is known as Fourier’s theorem - ( ) ∑ ( ), where  n = integers from 1 to ∞ that represent the different numbers of frequencies present in the complex signal - Remember that ω = 20f (or ω = 20/T) - By using known trigonometric identities the precious equation can be rewritten as: ( ) ∑ ( ( ) ), for periodic signals Time Domain Displays - Spectra varies as function of the phase - Looking at periodic vs. non periodic Frequency Domain Displays - Line or discrete spectrum  Why is this not continuous?  Fourier: the sum of sinusoids that describe a complex sound  Infinite signal which repeats; therefore, only the sum of the tree pure tones present - Continuous spectrum: pulse sound of discrete duration  Peak in the magnitude is at 1000Hz which is the dominant frequency  Attenuates  Power as a function of frequencies  The phase spectrum all start at 0 for the frequency  1000Hz is the dominant frequency but we have to also explain the occurrence of the pulse  To get the signal to be flat then pulsate at 1000Hz we need more signals present at different relative amplitudes in order for them to cancel and end up with a 20ms signal of 1000Hz Square Wave - Going to infinity in both directions - Oscillates around 5 - Amplitudes are falling exponentially with increasing frequency - Phase remains the same - All of the integer multiples are present The Making of a Square Wave - Sine waves which are added to infinity - Increasing frequencies and decreasing amplitudes Square Wave - 50% on 50% off: duty cycle - Oscillation around 1; f 0 1 - f(target) = 1 + 2 + f3+ fn Some Common Complex Waves - Square wave  Fundamental a
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