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HESP600 > Complex Signals & Fourier's Theorem > Flashcards

Complex Signals & Fourier's Theorem Flashcards

1
Q

Describe how waves are added together.

A
  • Sinusoids are the building blocks of all sounds
  • The sine wave is the fundamental component of all other sound waves
  • All waves that aren’t sinusoids are complex waves
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2
Q

Who was Jean Baptiste Joseph Fourier?

A

-French mathematician and physicist

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3
Q

What is Fourier’s Theorem?

A
  • Any arbitrary waveform can be represented as a sum of sine/cosine tones
  • Thus, any complex wave consists of a series of simple sinusoids that can differ in amplitude, frequency, and phase
  • Called Fourier Series for periodic waveforms
  • Called Fourier Transform for aperiodic waveforms
  • Fourier Series or Transform can be derived from Fourier Analysis
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4
Q

What is Fourier Analysis?

A

-The process of decomposing a complex waveform into its sinusoidal components

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5
Q

What is Fourier Series?

A

-The sinusoidal components of a complex periodic waveform

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6
Q

What is Fourier Synthesis?

A

-The process of reconstructing the waveform from its sinusoidal components

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7
Q

What is Fourier Transform?

A
  • The frequency characterization of a complex aperiodic waveform
  • Treat the entire waveform as the period
  • Output of transform will contain N harmonics from 0 to N-1
  • High amplitude indicate a correlation between input and output (how much a harmonic contributes to the overall waveform)
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8
Q

What are the processes of Fourier Transform? (4)

A
  1. Continuous Fourier Transform (CFT)- the theoretical transform going from negative to positive infinity in frequency
  2. Discrete Fourier Transform (DFT)- the same as CFT but with discrete/defined frequencies
  3. Fast Fourier Transform (FFT)- most used because it is efficient and possible
  4. Inverse Fourier Transform (IFT)- reverse of DFT or IFT
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9
Q

What are we interested in frequency information/analysis?

A
  • Using a complex waveform to determine its sinusoidal components
  • Understanding a system
    • EX: input –> decompose into constituent sinusoids –> alter each sinusoid according to transfer function –> add together modified sinusoids –> output
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10
Q

What is a complex tone?

A
  • Sound with multiple frequencies

- Period is that of the lowest frequency wave

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11
Q

What is a harmonic complex tone?

A
  • Sound with multiple frequencies that are multiples of some fundamental frequency
  • F0 might not always be present (missing fundamental)
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12
Q

What are harmonics?

A

-Multiples of the fundamental frequency

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13
Q

What is an overtone?

A
  • Anything over the fundamental

- EX: 2nd harmonic = 1st overtone

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14
Q

What is an inharmonic complex tone?

A

-Sound with multiple frequencies that are not multiples of some fundamental frequency

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15
Q

What are partials?

A

-Components in an inharmonic complex tone

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16
Q

What is a harmonic series?

A
  • All components of a harmonic complex tone are exact integer multiples of the same fundamental frequency
  • Each component is a harmonic
17
Q

Describe a sawtooth waveform.

A
  • -> A(n) = A(1) /n
  • All components have a phase of -90 degrees (relative to cosine)
  • Approximation of vocal fold movement
18
Q

Describe a square waveform.

A
  • Progression summation of 4 components with identical starting phases
  • Each component is an odd integer multiple of F0
  • Amplitude still the same [A(n) = A(1)/n]
  • Same amplitude spectra as sawtooth waveform
19
Q

What is noise?

A
  • Random signal with a continuous spectrum and random phases

- An example of an aperiodic complex wave

20
Q

What are aperiodic waves?

A
  • Waves that lack periodicity, where vibratory motion is random
  • Sometimes called a random time function
21
Q

What is white noise?

A
  • An aperiodic waveform with equal energy in every frequency band 1 Hz wide
  • Called white noise because it’s analogous to white light that has equal energy in all light wavelengths
  • Also called Gaussian Noise
    - It’s a random time function described by a cumulative probability distribution
    - A plot of the changing slope of a cumulative probability distribution is called a probability density function
  • Flat power spectrum
22
Q

What is pink noise?

A
  • Spectrum level decreases with increasing frequency (1/f)
  • Double frequency –> half power
  • -3 dB/octave or -10 dB/decade
23
Q

What is red noise?

A
  • Also known as Brown/Brownian noise
  • Random walk or Brownian motion noise (1/f^2)
  • -6 db/octave
24
Q

What is grey noise?

A
  • Psychoacoustics

- Applied to equal loudness contour A-weighting

25
Q

What is Speech-Shaped Noise (SSN)?

A
  • Average spectrum of long term speech
  • Made by:
    - Average a bunch of speech
    - Use a known spectrum to create desired signal
26
Q

What are some general rules to complex waves?

A
  • Periodicity harmonics
  • Shortening duration causes spectral spreading splatter
  • Modulations add sidebands (extra spectral components)
27
Q

What are condensation clicks?

A
  • Positive amplitude

- Phase is +90 degrees

28
Q

What are clicks?

A
  • Transient signal
  • Broadband sounds
  • Zeroes on amplitude spectra at 1/D
  • If duration decreases by half, the zero will be at double the Fc
29
Q

What are envelopes?

A

-Slowly moving overall amplitude

30
Q

What is fine structure?

A

-Fast phase changes

31
Q

What are click trains?

A
  • Basic psychoacoustic stimulus
  • Combination of periodic and aperiodic
  • Important for patients with cochlear implants
32
Q

Describe Amplitude Modulation (AM).

A
  • -> x(t) = A(t) * sin(2 * pi * Fc * t)
  • -> A(t) = A * [1 + m * sin(2* pi * Fm * t)]
  • M: depth of modulation in %
  • Fc: carrier frequency
  • Fm: modulation frequency
  • Sidebands equal Fc-Fm and Fc+Fm at a height of A * m/2
33
Q

Describe Frequency Modulation (FM).

A 
x(t) = A * sin[2* pi * Fc * t + phi(t)]
x(t) = A * sin[2* pi * Fc * t + B * sin(2 * pi * Fm * t)]

HESP600 (4 decks)

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