Fourier Transform and Complex Numbers

The Fourier Transform and Complex Numbers

• The Fourier transform decomposes a complex sound into its constituent sine waves.
• For each frequency, the Fourier transform extracts a magnitude and a phase.
• The magnitude represents the amplitude of the sine wave at that frequency.
• The phase represents the shift of the sine wave relative to the original signal.
• The magnitude and phase can be represented as the polar coordinates of a complex number.
• The complex Fourier transform coefficient encapsulates both magnitude and phase.
• The complex Fourier transform coefficient is represented by the formula: C = |C| * exp(i * gamma), where:
  • C is the complex Fourier transform coefficient.
  • |C| is the magnitude of the coefficient.
  • gamma is the phase of the coefficient.
  • i is the imaginary unit.
• This formula maps the magnitude and phase of the Fourier transform coefficient to the absolute value and angle of a complex number, respectively.
• The complex Fourier transform coefficient provides a compact and elegant way to represent the frequency components of a signal.

Fourier Transform: A Visual Explanation

• The square root of 2 is used for normalization in the Fourier transform, but doesn't alter the result.
• Gamma is a function of phi: gamma = 2 * pi * phi.
• Fourier transform coefficients are plotted as points in the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis.
• The distance from the origin represents the magnitude, and the angle from the positive real axis represents the phase.
• The minus sign in the gamma function causes clockwise rotation as phase increases, unlike the counterclockwise rotation in the standard complex number definition.

Continuous Audio Signal and the Complex Fourier Transform

• A continuous audio signal is represented as a function g(t) that maps time (t) to sound pressure intensity.
• The complex Fourier transform, denoted as ĝ(f), maps frequency (f) to a complex number – the Fourier transform coefficient.
• Every frequency has a corresponding Fourier transform coefficient that provides information about the signal at that frequency.

Visualizing the Fourier Transform

• The exponential term in the complex Fourier transform formula traces a unit circle in the complex plane, moving clockwise due to the negative sign.
• The speed of the circle's traversal is determined by the frequency 'f'; higher frequencies result in faster rotations.
• Multiplying the original signal with a pure tone (sine wave) at a specific frequency visually "wraps" the signal around the complex plane. The resulting shape reflects the interaction of the original signal and the pure tone.
• When the pure tone frequency matches a frequency component in the original signal, the resulting shape is more stable. Mismatched frequencies lead to more erratic and symmetric shapes, ultimately centering around the origin.

Understanding the Integral and Summation

• The complex Fourier transform integral calculates the sum of the product of the signal and a pure tone, across all times.
• The "center of gravity" method provides a visual representation of the integral's outcome.
• While the center of gravity represents the average, the true integral sums all values along the path.
• The sum of the values is the actual Fourier transform coefficient, equal to the center of gravity multiplied by the number of time steps.

Interpretation of the Fourier Transform

• Fourier transform coefficients are points in the complex plane, where:
  • Magnitude is the distance from the origin.
  • Phase is the angle from the positive real axis.
• The coefficient's location represents the contribution of that frequency to the original signal.
  • High magnitude implies a strong frequency presence.
  • Zero magnitude means the frequency is absent.

The Fourier Transform: A Visual Explanation

• The center of gravity is multiplied by the number of time steps.
• More time steps yield a more accurate Fourier Transform analysis.
• The Fourier Transform breaks down a signal into its component frequencies.
• A signal with a frequency of 1.1 and a symmetrical shape will have a Fourier Transform concentrated around the center.
• Fourier Transform calculations involve summing values for different time steps, potentially causing cancellations.
• The Fourier Transform converts a signal from the time domain to the frequency domain.
• The Fourier Transform is a summation or integral, depending on the signal type (discrete or continuous).
• A continuous signal has infinitely many points, while a discrete signal has a finite number.
• Integration is used instead of summation for continuous signals.
• The Fourier Transform coefficient is a complex number.
• Euler's Formula allows decomposing the complex coefficient into real and imaginary parts.
• The magnitude is the absolute value of the complex Fourier Transform coefficient.
• The phase is determined by dividing the angle at the frequency by 2π.
• The Inverse Fourier Transform converts from the frequency domain back to the time domain.
• The Inverse Fourier Transform reconstructs the original signal by combining frequency components with their magnitudes and phases.
• The Fourier Transform and its inverse are integral operations, one over time, the other over frequency.

Discrete Fourier Transform

• The Fourier Transform works with continuous (analog) signals.
• Digital signals are discrete, created by sampling analog signals.
• In practice, the Discrete Fourier Transform (DFT) is used to analyze digital signals.
• The DFT is an adaptation of the continuous Fourier Transform for discrete signals.