Complex Signals & Fourier's Theorem Flashcards
Describe how waves are added together.
- 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
Who was Jean Baptiste Joseph Fourier?
-French mathematician and physicist
What is Fourier’s Theorem?
- 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
What is Fourier Analysis?
-The process of decomposing a complex waveform into its sinusoidal components
What is Fourier Series?
-The sinusoidal components of a complex periodic waveform
What is Fourier Synthesis?
-The process of reconstructing the waveform from its sinusoidal components
What is Fourier Transform?
- 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)
What are the processes of Fourier Transform? (4)
- Continuous Fourier Transform (CFT)- the theoretical transform going from negative to positive infinity in frequency
- Discrete Fourier Transform (DFT)- the same as CFT but with discrete/defined frequencies
- Fast Fourier Transform (FFT)- most used because it is efficient and possible
- Inverse Fourier Transform (IFT)- reverse of DFT or IFT
What are we interested in frequency information/analysis?
- 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
What is a complex tone?
- Sound with multiple frequencies
- Period is that of the lowest frequency wave
What is a harmonic complex tone?
- Sound with multiple frequencies that are multiples of some fundamental frequency
- F0 might not always be present (missing fundamental)
What are harmonics?
-Multiples of the fundamental frequency
What is an overtone?
- Anything over the fundamental
- EX: 2nd harmonic = 1st overtone
What is an inharmonic complex tone?
-Sound with multiple frequencies that are not multiples of some fundamental frequency
What are partials?
-Components in an inharmonic complex tone
What is a harmonic series?
- All components of a harmonic complex tone are exact integer multiples of the same fundamental frequency
- Each component is a harmonic
Describe a sawtooth waveform.
- -> A(n) = A(1) /n
- All components have a phase of -90 degrees (relative to cosine)
- Approximation of vocal fold movement
Describe a square waveform.
- 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
What is noise?
- Random signal with a continuous spectrum and random phases
- An example of an aperiodic complex wave
What are aperiodic waves?
- Waves that lack periodicity, where vibratory motion is random
- Sometimes called a random time function
What is white noise?
- 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
What is pink noise?
- Spectrum level decreases with increasing frequency (1/f)
- Double frequency –> half power
- -3 dB/octave or -10 dB/decade
What is red noise?
- Also known as Brown/Brownian noise
- Random walk or Brownian motion noise (1/f^2)
- -6 db/octave
What is grey noise?
- Psychoacoustics
- Applied to equal loudness contour A-weighting
What is Speech-Shaped Noise (SSN)?
- Average spectrum of long term speech
- Made by:
- Average a bunch of speech
- Use a known spectrum to create desired signal
What are some general rules to complex waves?
- Periodicity harmonics
- Shortening duration causes spectral spreading splatter
- Modulations add sidebands (extra spectral components)
What are condensation clicks?
- Positive amplitude
- Phase is +90 degrees
What are clicks?
- Transient signal
- Broadband sounds
- Zeroes on amplitude spectra at 1/D
- If duration decreases by half, the zero will be at double the Fc
What are envelopes?
-Slowly moving overall amplitude
What is fine structure?
-Fast phase changes
What are click trains?
- Basic psychoacoustic stimulus
- Combination of periodic and aperiodic
- Important for patients with cochlear implants
Describe Amplitude Modulation (AM).
- -> 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
Describe Frequency Modulation (FM).
x(t) = A * sin[2* pi * Fc * t + phi(t)] x(t) = A * sin[2* pi * Fc * t + B * sin(2 * pi * Fm * t)]
