Fourier Lab

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Run the Fourier Series Applet (http://falstad.com/fourier) by Paul Falstad. A workstation computer is more convenient than a tablet for this exercize, because of its mouse pointer and larger display. (Portrait orientation of a tablet is recommended on tablets to see its vertically stacked displays.) The full screen version (accessible via the button below the interface) is also recommended. Click Sound (below of the column of buttons) to hear the signal. Adjust the volume to a comfortable level. Avoid prolonged listening to sinusoidal (Sine or Cosine, etc.) waves (for more than a few minutes at a time) as it might be harmful for your ears.

Get to know the applet:
Click on the Sine button (top right) to hear a pure tone. Try another waveform, including one of the complex ones (Triangle, Sawtooth, or Square). The underlying white line in the display is the ideal signal; the red one is the approximation based on the sum of a finite number of components (set by the “Number of Terms” slider). Notice how decreasing the number of terms truncates the harmonic series and degrades the approximation to the target function. Enable Options > Show Full Expansion to show mathematical expression. Answer to your own satisfaction (no explicit answers required yet on this worksheet) these orientation questions. You may consult with other students if you like.

• What do the three displays show (from top to bottom)?
• Hover (place the cursor) over a frequency component stalk. A corresponding sinusoid (colored yellow) appears among the composite waveform of all the sinusoids added together at the top. What is the default frequency of the Sine function? (Hint: See the attribute of the fundamental frequency, adjusted by the “Playing Frequency” slider on the right control panel.)
  Shift the display from Sin[e]/Cos[ine] (rectangular coordinates) to polar coordinates Mag[nitude]/Phase. (Recall that x = r cos(θ), y = r sin(θ); r = sqrt(x²+y²), θ = arg(x,y) which is like arctan(y/x).)
• Use the cursor to drag the phase slider for that frequency up and down. How can a sin and cos waves be converted to each other?
• Most people are capable of both analytic and synthetic hearing. Analytic hearing decomposes sound into its component parts: a particular partial of a given periodic complex tone might be heard individually as a pure, sinusoidal tone, ringing separately from the whole. Synthetic hearing is listening to a sound as a unified whole. Making changes to the sound can drive listeners involuntarily into analytic listening, by focusing attention for example on a partial that has just been changed.
  Use the Mute and Select (Solo) controls, in the m and s rows at the bottom, corresponding to each harmonic. These can be used to exclude (mute) a selected set of harmonics from a complex tone (such as a sawtooth wave), or to “focus on” (select) some subset of its harmonics. Push single digits 1 through 0 to Mute corresponding harmonics; or Shift + single digit to Solo them.

• Combination of sinusoids of same frequency yields another sinusoid. In rectangular display mode (Sines/Cosines), assert two (out-of-phase) sinusoids with unity amplitude by letting sin(x) = cos(x) = 1. What is the resultant magnitude and phase? (The Mag/Phase representation normally has a maximum magnitude of 1, but combination of cos and sin components can push it [via Pythagorean theorem] beyond unity.)
• What is the difference between the frequency spectra of the sin and cosine waveforms? (How to transform between them?) Hint: In polar coordinates (Mag/Phase view), use phase shift (or just successively press the Phase Shift button).
• How many terms are needed to represent an uncentered pure tone (with non-zero average)?

• The DC offset of a signal is its bias or average (a.k.a. mean or expectation). (DC, direct current, is bias, constant offset in a signal, reflected in a change in its mean.) The 1^st harmonic merely adds such a constant to the waveform, and can be considered to be the zero frequency term. Since it is degenerately even (symetric about the amplitude axis), it is the cos component of the 0-order harmonic. How many terms are needed to represent a pure tone as a centered signal with no DC component (average = 0)?

• What does changing the lowest (0-order) frequency (the so-called “DC”) component do to the signal and the sound?
• Phase Shift advances a waveform (by angular intervals of π/10 radians). What is the effect of time-shifting (phase-shifting) a periodic signal on its waveform and its sound?
• Describe the spectra (amplitude or magnitude of the frequency components) for the Square, Triangle, and Sawtooth waves. How is the noise spectrum different from these? Triangle excites more harmonics than a pure tone. Log View (only for Mag/Phase) causes the stalk height to be the logarithm of the amplitude, as in expressions of signal level, to more easily see smaller magnitudes. Only odd harmonics f, 3f, 5f, ... are non-zero. Emphasis of odd harmonics is characteristic of clarinet-like musical instruments. (This can be contrasted to flutes, which play both even and odd harmonics of the overtone series.) Click on Sawtooth and note the greater amplitude of the higher harmonics, and the fact that even as well as odd harmonics are included; a sawtooth wave requires more Fourier frequencies to reproduce well.
  Enabling the Mag/Phase option to convert from rectangular coordinates to polar. To see the quantitative attributes of a harmonic, one can hover over its stalk (which turns yellow) to see the amplitude (a.k.a. magnitude) and phase. (For example, 1^st harmonic of sawtooth is “0.63662 cos(x - 1.56773)”, meaning its magnitude ≅ 0.6 and its phase delay is π/2 radians. The second harmonic is “0.31831 cos(2x + 1.57693)”. (Note the coefficient on the independent variable x indicating the harmonic number.) To calculate fall-off across this 8^ve interval (1 → 2), one can calculate 10 log₁₀ (x₂/x₁)² = 20 log₁₀ (x₂/x₁), which for this pair of values would be about 20 log 0.6/0.3 = 6.02 [dB]. Such calculation can be done with Wolfram Alpha, expressed as 20 log(10, 0.6/0.3) to override default natural logarithm (base e ≅ 2.7) with decimal base (since bels and decibels use base 10). Alternatively, enable the Log View for magnitude/phase display to see levels expressed directly, and subtract them to get the level difference: 1^st harmonic (fundamental freq.): -3.92 dB 2^nd harmonic (^ve up): -9.94 dB Level difference, then, is difference: -9.94 - -3.92 = -6 [dB/8^ve]. One can confirm that this attenuation (not “attention”) applies to any harmonics separated by an 8^ve (harmonic number n → 2n). Of course only even harmonics are an 8^ve above any other harmonics, so for waves with only odd harmonics (triangle, square, etc.), optically interpolate (estimating visually) a stalk along the decaying “harmonic envelope” (adjust a “fake” or "false" harmonic [which would otherwise be missing] to fit the curve) and use the displayed magnitude to estimate the 8^ve attenuation.
  Complete this table:

  Waveform	Harmonics (even, odd, both, etc.)	Attenuation	Sound	RMS
  Noise				⅓½ ≈ 0.58 ≈ 0.6
  Sin	(just one, so no attenuation)
  Triangle		-12 dB/8ve	Like a clarinet	⅓½
  Sawtooth		-6 dB/8ve	Like strings or brass
  Square		-6 dB/8ve	Hollow

• If you start with a Sine function, then invoke Rectify, what happens to the wave and the spectrum?
• Explore the Full Rectify and Clip functions as well. Which of these (half-Rectify, Full Rectify, or Clip) does not introduce a “DC” or “A₀” term (leftmost of the frequency components)? Explain this.
• What does the (half-)Rectify function do?
• What happens when a square wave is half- or full-rectified?
• Full Rectify a triangle wave, which subsequent half-amplitude triangle wave has every 4^th harmonic, as the original odd harmonics all get doubled. The original fundamental disappears, and a DC offset is introduced, which can be zeroed to iterate that operation...).
• What does the clipping function do? How does repeatedly clipping a non-square wave affect its higher-order harmonics? (Hint: amplification up to maximum value)
• Using the Beats set-up (http://falstad.com/fourier/e-beats.html), add two adjacent frequencies to see and hear the “beating” (or “throbbing” or “pulsing”). (Hint: Lower the fundamental frequency [Playing Frequency] to a low value (like around 2 Hz), and assert two adjacent high-order harmonics (such as #49 & #50). Can phase be heard from a complex tone? (Does adjusting phase affect the sound, except for crackling noise during adjustment?)

Humans can distinguish, with training, up to about 10 harmonics of a sound. Choose a broadband signal like a Square wave and make sure there are sufficient terms (~20). Listen to the sound and try to identify individual components of the sound.

• Can you “hear out” the respective frequency components by toggling its Select (Solo) attribute on the bottom row?
• Can you distinguish between the signal with and without a component by toggling the respective "Mute" attribute, in the second row from the bottom?
• How many harmonics can you “hear out” (using, for instance, the Mute operation on separate components) in a triangle wave?
• Select the Sawtooth wave. Now select (solo) the 4^th, 5^th, & 6^th harmonics (and, optionally, the 8^th, doubling the 4^th). What do you hear?

• How does the Resample button affect a waveform and its spectrum?  What does this simulate? (Increase number of terms so that enough high-order terms are present to observe a kind of aliasing.)
• How is the effect of the Quantize button different from Resample? (Hint: One down-samples in amplitude, the other in time.)
• In PCM (pulse code modulation), the amplitude of a signal is represented by a binary number. How many steps are in 8-bit sound?  16-bit sound? (Hint: Each bit doubles the dynamic range, the number of possible values.)
• “Draw” your own function in the top display. It's easiest in the middle third: Click and drag to draw in the display. Draw a straight line with constant amplitude. What frequencies are expected? What frequencies are actually produced by your hand-drawing? (Push Clear to null the signal.) Draw a constant slope line? What frequencies are expected? What frequencies are actually produced? Draw a sinusoid (with or without tracing over a sin or cos). The application analyses or decomposes the drawn function. When it looks like a sine or cosine, observe the frequency components below. How many have appreciable magnitude? What does that say about your drawing ability? How are errors in your drawing reflected in the harmonics? Log view (when in magnitude/phase) makes even small distortion visually apparent, but probably it might still be difficult to hear (using mute & solo). Can such residual harmonics be heard? Suppress the fundamental by resetting its amplitude. (Or listen to the signal with and without the respective harmonics by using Mute, or listen to the fundamental with and without the overtones by using Select (Solo). Unless you are a very talented artist, you can probably hear the “buzziness” representing the error in your drawing.)

• Assert a single impulse (at time 0). What is the spectrum?