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Phase’s Impact on Sonics & Imaging
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<center><H2>Phase’s Impact on Sonics & Imaging</H2></center>
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Let’s start the concept of sonic imagery with examining a steady-state signal-generator tone in an anechoic chamber.&nbsp You would readily notice that the perceived size of the sonic image is proportional to wavelength.&nbsp I.E.: High-frequencies (HFs), as in treble (E.G.: <a href=10khz.wav>10kHz</a>), seem tiny & easily to pin-point due to their short wavelengths ( ¾"; 8.5mm ); low-frequencies (LFs), as in bass (E.G.: <a href=100hz.wav>100Hz</a>), seem huge & harder to track back to the source due to their long wavelengths ( 6'2"; 85cm ).
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N.B. #1:&nbsp These values are estimates as the speed of sound is dependent on temperature & humidity, which is why it also reduces with altitude.
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If we stayed longer to experiment with different tones, eventually, you'd notice that determining direction happens immediately.&nbsp We'll discuss part of the reasons later.
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N.B. #2:&nbsp Since we're done discussing frequency dependence, the rest of the various waveform signals on this page are all at 1kHz for comparison purposes.
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<center><img src=longitest3bis.gif width="490" height="360"></center>
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<u><b>Compression & rarefaction waves</b></u>.&nbsp While compression is self-explanatory, rarefaction isn’t.&nbsp Rarefaction waves are the opposite of compression waves.&nbsp You might want to say expansion waves, but no one does.&nbsp Instead it's named after the air becoming rarefied.&nbsp Illustrated above we see these propagating longitudinal sound waves shown graphically, as reciprocation alternates equally compression & rarefaction waves.&nbsp However, humans hear predominantly compression waves, whose magnitude is perceived logarithmically (EG: half-power is -3dB but people subjectively perceive -10dB as half-volume).&nbsp They are many cases of a sustained symmetrical signals that this distinction doesn't matter, but there remain many exceptions that are crucial.&nbsp Before we explore them we must define some things 1<small><b><small><sup>st</sup></small></b></small>.
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<u><b>Harmonics</b></u>.&nbsp Harmonics, or overtones, are multiples of the original <a href=sin.wav>sinusoidal signal frequency</a> which usually have proportional envelopes.&nbsp Thus, the terms “<a href=even.wav>even-order harmonics</a>” & “<a href=odd.wav>odd-order harmonics</a>” you may heard before in regard to tube amplifier total harmonic distortion (THD) for even & odd multiples, respectively; <i>good tube amplifiers produce more even order THD than solid-state but less odd order THD</i>.&nbsp The 1st harmonic is a multiple of 2, which means it’s an even harmonic.&nbsp This span also happens to correspond with a musical octave, as in one octave above the original signal.&nbsp Then there’s a sub-harmonic, which is one octave below the original signal.&nbsp Below is an example tabulated from a 1kHz original signal.
<center><table width=50%>
<tr align=middle>
  <th>frequency</th>
  <th>multiple</th>
  <th>harmonic</th>
  <th>octave</th>
<tr align=middle>
  <td>500Hz</td>
  <td>½</td>
  <td><i>(sub)</i></td>
  <td>-1</td>
</tr><tr align=middle>
  <td>1,000Hz</td>
  <td>1</td>
  <td> … </td>
  <td>0</td>
</tr><tr align=middle>
  <td>2,000Hz</td>
  <td>2</td>
  <td>1st <i>(even)</i></td>
  <td>1</td>
</tr><tr align=middle>
  <td>3,000Hz</td>
  <td>3</td>
  <td>2nd <i>(odd)</i></td>
  <td>1.585</td>
</tr><tr align=middle>
  <td>4,000Hz</td>
  <td>4</td>
  <td>3rd <i>(even)</i></td>
  <td>2</td>
</tr><tr align=middle>
  <td>5,000Hz</td>
  <td>5</td>
  <td>4th <i>(odd)</i></td>
  <td>2.322</td>
</tr><tr align=middle>
  <td>6,000Hz</td>
  <td>6</td>
  <td>5th <i>(even)</i></td>
  <td>2.585</td>
</tr><tr align=middle>
  <td>7,000Hz</td>
  <td>7</td>
  <td>6th <i>(odd)</i></td>
  <td>2.807</td>
</tr><tr align=middle>
  <td>8,000Hz</td>
  <td>8</td>
  <td>7th <i>(even)</i></td>
  <td>3</td>
</tr><tr align=middle>
  <td>9,000Hz</td>
  <td>9</td>
  <td>8th <i>(odd)</i></td>
  <td>3.170</td>
</tr>
</table></center>
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<u><b>Fourier series</b></u>.&nbsp This is a set of harmonic simple sinusoidal functions that comprise a complex waveform.&nbsp Say for a simplified academic exercise your instrument produces square waves.&nbsp This characteristic waveform determines the majority of an instrument’s unique voice.&nbsp Below, we show the construction of a square wave by consecutively adding increasing odd harmonics of ever reducing loudness.&nbsp Consequently, whereas a simple sinusoid images at it's own single Fourier frequency, a square-wave of the same frequency has more pinpoint image due to its associated Fourier harmonics (much higher frequencies; much shorter wavelengths).&nbsp 
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<center><img src=fourier-ani.gif width="700" height="206"></center>
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We can also show the analysis of these components via a plot of the magnitude vs frequency.&nbsp This is often referred to as a Fast-Fourier Transform (FFT).&nbsp However, those of you with equalizers may be more familiar with a similar plot called Real-Time Analysis (RTA).
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<center><img width=100% src=fft-1.gif>
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<i><b><big>f(t) = sin(ω•t) + sin(3•ω•t)/3 + sin(5•ω•t)/5 + sin(7•ω•t)/7 + sin(9•ω•t)/9 + …</big></b></i></center>
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<center><img width=100% src=fft-2.gif>
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<i><b><big>f(t) = sin(ω•t) + sin(2•ω•t)/2 + sin(3•ω•t)/3 + sin(4•ω•t)/4 + sin(5•ω•t)/5 + …</big></b></i></center> 
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<center><img width=100% src=fft-3.gif>
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<i><b><big>f(t) = 1 + cos(ω•t) + cos(2•ω•t) + cos(3•ω•t) + cos(4•ω•t) + cos(5•ω•t) + …</big></b></i></center>
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<b><u>Phasing</u></b>, or rate of phase is important as it can alter the characteristic voices in the music.&nbsp In extreme cases, it has been known to make a grand piano sound like a keyboard.&nbsp This is a <b><u>relative phase</u></b> phenomena, or how the sound changes due to the overtones phase shifting with respect to the fundamental, regardless of how off the absolute phase is.&nbsp EG: Take a 1khz sinuous tone & a smaller 2nd harmonic of 3kHz.&nbsp I can coalesce the two with a shifted phase & make something resembling a <img src=square.gif width="46" height="14"> <a href=square.wav>square-wave</a> resemble a <img src=triangle.gif width="46" height="14"> <a href=triangle.wav>triangular-wave</a>.
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<img src=superimpose1.gif width="381" height="220">
<img src=superimpose2.gif width="381" height="220">
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N.B. #3:&nbsp 
The following is the Fourier series of an ideal triangular-wave.&nbsp The negative sine-wave components mean an inversion from the square-wave components, or a 180º phase shift.
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<center><embed width=550 height=180 src="fourier2.mpg" autostart="True" loop=True></embed>
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<i><b>f(t) = sin(ω•t) - sin(3•ω•t)/3² + sin(5•ω•t)/5² - sin(7•ω•t)/7² + sin(9•ω•t)/9² - …</b></i></center>
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However, this case (@ 120º/octave) is an extreme example.&nbsp In my <a href=circuits/>crossover circuits</a> experience that includes the <a href=woofer/>drivers</a>, phasing of 30º/octave is virtually undetectable.&nbsp However, the absolute phase can accumulate at that phasing rate over several octaves & eventually invert the signal.&nbsp “That” is audible, as we will discuss.  
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<u><b>Envelope</b></u>.&nbsp Up until now we were discussing the characteristic waveform of the carrier signal.&nbsp Envelopes bound the magnitude of a carrier signal (shown below in green).&nbsp There can be periodic envelopes like beating, which imposes their own affect on sonic imagery, but here we're interested in an exponential decay envelope of a transient signal (shown below in yellow), which is phase crucial.&nbsp We can examine the following crude approximation (shown below at right).
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<img src=envelope1.gif width="381" height="221">
<img src=envelope2.gif width="380" height="221">
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We can approximate a step function by indexing a biased extremely long period square wave by simply adding unity & halving the sum.&nbsp Multiplying the steady-state signal by this step function models the semi-infinite initiation. 
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<center><i><b>f(t) = ½•{ 1 + sin(ω•t) + sin(3•ω•t)/3 + sin(5•ω•t)/5 + sin(7•ω•t)/7 + … }</b></i></center>
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Furthermore, we can superimpose an approximate exponential decay by indexing a reverse-sawtooth wave.&nbsp 
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<center><i><b>f(t) = ⅔•{ sin(ω•t) + sin(2•ω•t)/2 + sin(3•ω•t)/3 + sin(4•ω•t)/4 + … }</b></i></center>
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Take note of all the harmonic components & how every one is a positive sine wave.&nbsp That means, as we superimpose them to generate an approximate transient envelope that we can readily examine, at the crucial time when “t=0” each component is at maximum positive slope.&nbsp This means they are all about to compress air.&nbsp We hear every single harmonic component.&nbsp It’s these HF <b><u>cues</u></b> that are directly coupled to the signature waveform that help really pinpoint the sonic image.&nbsp It’s this aspect that is inaudible when polarity is inverted from <b><u>absolute phase</u></b>.
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Re-examining the transient envelope as a waveform, we can readily analyize the <img src=saw2.gif width="42" height="14"> <a href=revsaw.wav>reverse sawtooth waves</a>.&nbsp If polarity inverts, the resultant signal becomes standard <img src=saw1.gif width="42" height="14"> <a href=sawtooth.wav>saw-tooth waves</a>.&nbsp All the compressive harmonic components become rarefactive components.&nbsp Thus, these HF cues lose audibility.
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Consequently in application, inverting HF cues causes otherwise holographic images sculpted in vivid depth of imagery become magnified, panoramic & shallow.&nbsp A good example is a tympani drum, whose excitation transmits an initial rarefacted impulse.&nbsp That means the tympani transients are inverse that of a kick-drum, whose excitation transmits an initial compressive impulse.&nbsp That inversion renders a very ambient sonic signature as opposed to the concussive impact slam of a kick-drum.&nbsp That's also why reverse-polarity subwoofers may seem to rumble more as they lose impact.
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N.B. #4:&nbsp Other cues help too like valving on horns, fretting on guitars, plucking on a violin, the slap of a drum skin, the strike on a cymbal, etc..&nbsp However, we are ignoring the relationship because they have their own transient envelopes & sonic images that already happen to coincide with the principle signal.&nbsp So, for our academic purposes we’re treating them separately.
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<img src=polarity1.gif width="380" height="220">
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Finally, since the signal is decaying, the peak compression wave of the inverted signal is a tad quieter than the original signal.&nbsp This also gives one the impression that inverted signals are about 2dB quieter.&nbsp Of course this perceived discrepancy depends on the rate of decay of each signal, whose damping is unique to each & every instrument.
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