COURSE NOTES
Sound Digitization and Fourier Analysis

Sound Digitization

• Analog-to-Digital (A/D) Conversion - Conversion of a continuous-time function to a discrete-time function

• Digital-to-Analog (D/A) Conversion - Conversion of a discrete-time function to a continuous-time function

Analog to Digital Conversion [HANDOUT]

Digital to Analog Conversion [HANDOUT]

The Sine Function (in the time domain)

• f(t) = A sin (2 pi f₀ t + o)
  A is the amplitude
  f₀ is the frequency (Hz)
  t is the time (sec)
  o is the phase angle offset (radians)
  pi is 3.141592...

•  A|       *
    |    *     *
    |  *         *
  --+-*-----------*---------------- t
    |*             *         *
    |                *     *
    |                   *

The Sine Function (in the frequency domain)

• f(t) = A sin (2 pi f₀ t + o)
  A is the amplitude
  f₀ is the frequency (Hz)
  t is the time (sec)
  o is the phase angle offset (radians)
  pi is 3.141592...

•   |    * A                                |
    |    *                         A/2 *    |    * A/2
    |    *                             *    |    *
  --+----|----------- f             ---|----+----|----  f
    |    f0                            -f0       f0

Periodic Waveforms

• All periodic waveforms can be constructed from the sum of harmonically related sinusoids.

• In general, if f is a periodic function, then
  f(t) = A₀ + A₁sin(2 pi f t + o₁) + A₂sin(2 pi 2 f t + o₂) + A₃sin(2 pi 3 f t + o₃) + ...

About Periodic Waveforms

• The frequency f is often called the fundamental frequency and denoted f₀.

• All sinusoidal components of a periodic waveform have a frequency which is an integral multiple of the fundamental frequency. There frequencies are said to be harmonically-related to f₀.

Building a Sawtooth Wave [HANDOUT]

Building a Square Wave [HANDOUT]

Fourier Analysis

• Algorithms used to convert the time-domain representation of a waveform to its equivalent frequency-domain representation

• Discrete Fourier Transform (DFT)

• Fast Fourier Transform (FFT)

Fourier Transforms

• The DFT of one period of a periodic waveform gives the frequency spectrum of that waveform.

• The IDFT of the frequency spectrum of a periodic waveform gives one period of a periodic waveform.

Discrete Fourier Transform (DFT)

• Given N time samples of a digital signal x[n], n=0, 1, 2, ..., N-1, the DFT of x[n], denoted X[k] is given by
  X[k] = (1/N) SUM_n=0,...,N-1(x[n] e^{-2 pi i k n/N}) k = 0, 1, 2, ... , N-1
  where i² = -1 and e^{-2 pi i k n/N} = cos (2 pi k n/N) - i sin (2 pi k n/N).

• Example: Use the equation for a square wave approximated using the first three terms (k=1,3,5). Substitute t = n/N and f = 1 to form an equation for x[n] from f(t) and evaluate the equation for n=0,1,...,N-1 to generate N equally-spaced samples of one period of this waveform for the DFT.

Inverse Discrete Fourier Transform (IDFT)

• Given N frequency samples of a digital signal X[k], k=0, 1, 2, ..., N-1, the IDFT of X[k], denoted x[n] is given by
  x[n] = SUM_k=0,...,N-1(X[k] e^{2 pi i k n/N}) n = 0, 1, 2, ... , N-1
  where i² = -1 and e^{2 pi i k n/N} = cos (2 pi k n/N) + i sin (2 pi k n/N).

DFT vs. FFT

• DFT - O(n²) algorithm

• FFT - O(n lg n) algorithm

The Sampling Theorem

• To represent a signal digitally that contains frequency components up to X Hz, it is necessary to use a sampling rate R of at least 2X samples/second.

• Example: If an analog waveform has frequency components at 1000, 2000 and 5000 Hz, we must sample at a minimum rate of 10000 Hz.

Aliasing [HANDOUT]

• Aliasing occurs when sampling is performed at a rate less than twice the maximum component frequency of the sound wave.

• If the sampling rate is R, then a sine wave component at frequency f > R/2 will be mapped to a sine wave component at frequency R/2 - (f - R/2) = R - f.

Frequencies in the DFT

• The frequency represented by frequency index k in the DFT X[k] is given by F = kR/N, where R is the sampling rate in samples/sec and N is the number of samples analyzed.

• Example: For the square wave frequency spectrum, if the number of samples taken for one period is 16, and the sampling rate is 32000 samples/sec, the fundamental frequency (k=1) of the square wave is 2000 Hz. The third harmonic (k=3) is at 6000 Hz. (NOTE: The second harmonic is silent.) etc.

Computer Storage Requirements

• Example: How many bytes of storage are required to sample a 120-second stereophonic audio wave that has frequencies up to 20 kHz with each sample quantized to 16 bits?

• Answer: 19,200,000 bytes (nearly 20 megabytes!)

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