Ring modulation

<h2 id="simplified-operation">Simplified operation</h2>
Denoting the carrier signal by 
 
 
 
 c
 (
 t
 )
 
 
 {\displaystyle c(t)}
 
, the modulator signal by 
 
 
 
 x
 (
 t
 )
 
 
 {\displaystyle x(t)}
 
 and the output signal by 
 
 
 
 y
 (
 t
 )
 
 
 {\displaystyle y(t)}
 
 (where 
 
 
 
 t
 
 
 {\displaystyle t}
 
 denotes time), ring modulation approximates <a href="/facts/Multiplication/VtcUjpNv">multiplication</a>:

y
        (
        t
        )
        =
        x
        (
        t
        )
        
        c
        (
        t
        )
        .
      
    
    {\displaystyle y(t)=x(t)\;c(t).}

If 
 
 
 
 c
 (
 t
 )
 
 
 {\displaystyle c(t)}
 
 and 
 
 
 
 x
 (
 t
 )
 
 
 {\displaystyle x(t)}
 
 are sine waves with frequencies 
 
 
 
 
 f
 
 c
 
 
 
 
 {\displaystyle f_{c}}
 
 and 
 
 
 
 
 f
 
 x
 
 
 
 
 {\displaystyle f_{x}}
 
, respectively, then 
 
 
 
 y
 (
 t
 )
 
 
 {\displaystyle y(t)}
 
 is the sum of two (<a href="/facts/Phase_shift/cpGtuVll">phase-shifted</a>) sine waves, one of frequency 
 
 
 
 
 f
 
 c
 
 
 +
 
 f
 
 x
 
 
 
 
 {\displaystyle f_{c}+f_{x}}
 
 and the other of frequency 
 
 
 
 
 f
 
 c
 
 
 −
 
 f
 
 x
 
 
 
 
 {\displaystyle f_{c}-f_{x}}
 
. This is a consequence of the <a href="/facts/Trigonometric_identity/Xw9Jx8Vz">trigonometric identity</a>:

sin
        ⁡
        (
        u
        )
        
        sin
        ⁡
        (
        v
        )
        =
        
          
            1
            2
          
        
        
          (
          
            cos
            ⁡
            (
            u
            −
            v
            )
            −
            cos
            ⁡
            (
            u
            +
            v
            )
          
          )
        
        .
      
    
    {\displaystyle \sin(u)\,\sin(v)={\frac {1}{2}}\left(\cos(u-v)-\cos(u+v)\right).}

Alternatively, one can use the fact that multiplication in the <a href="/facts/Time_domain/1w6mP2MU">time domain</a> is the same as <a href="/facts/Convolution/PVPrdz9J">convolution</a> in the <a href="/facts/Frequency_domain/pyTVrUSq">frequency domain</a>.
Ring modulators thus output the <a href="/facts/Combination_tone/LqDqDfFo">sum and difference</a> of the frequencies present in each waveform. This process of ring modulation produces a signal rich in <a href="/facts/Partial_(music)/PdElIrxL">partials</a>. Neither the carrier nor the incoming signal are prominent in the output, and ideally, not present at all.
Two oscillators, whose frequencies were harmonically related and ring modulated against each other, produce sounds that still adhere to the harmonic partials of the notes but contain a very different spectral makeup. When the oscillators' frequencies are not harmonically related, ring modulation creates <a href="/facts/Inharmonic/xDSTCe95">inharmonics</a>, often producing bell-like or otherwise metallic sounds.
If the carrier signal is a <a href="/facts/Square_wave_(waveform)/h1cWR1KM">square wave</a> of frequency 
 
 
 
 
 f
 
 c
 
 
 
 
 {\displaystyle f_{c}}
 
, whose <a href="/facts/Fourier_expansion/s3fsxPAT">Fourier expansion</a> contains the <a href="/facts/Fundamental_frequency/TJBJKHow">fundamental</a> and a series of reducing-amplitude odd <a href="/facts/Harmonic/9U9REzeR">harmonics</a>:

c
        (
        t
        )
        =
        sin
        ⁡
        
          f
          
            c
          
        
        t
        +
        
          
            1
            3
          
        
        sin
        ⁡
        3
        
          f
          
            c
          
        
        t
        +
        
          
            1
            5
          
        
        sin
        ⁡
        5
        
          f
          
            c
          
        
        t
        +
        
          
            1
            7
          
        
        sin
        ⁡
        7
        
          f
          
            c
          
        
        t
        +
        …
      
    
    {\displaystyle c(t)=\sin f_{c}t+{\frac {1}{3}}\sin 3f_{c}t+{\frac {1}{5}}\sin 5f_{c}t+{\frac {1}{7}}\sin 7f_{c}t+\ldots }

and the carrier frequency 
 
 
 
 
 f
 
 c
 
 
 
 
 {\displaystyle f_{c}}
 
 is at least twice the maximum frequency of the modulating signal 
 
 
 
 x
 (
 t
 )
 
 
 {\displaystyle x(t)}
 
, then the resulting output is a series of duplicates of 
 
 
 
 x
 (
 t
 )
 
 
 {\displaystyle x(t)}
 
 at increasing regions of the frequency spectrum.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> For example, let 
 
 
 
 x
 (
 t
 )
 
 
 {\displaystyle x(t)}
 
 represent a sine wave at 100 Hz, and the carrier 
 
 
 
 c
 (
 t
 )
 
 
 {\displaystyle c(t)}
 
 be an ideal square wave at 300 Hz. The output then includes sine waves at 100±300 Hz, 100±900 Hz, 100±1500 Hz, 100±2100 Hz, etc., at decreasing amplitudes according to the Fourier expansion of the carrier square wave. If the carrier frequency is less than twice the upper frequency of the signal then the resulting output signal contains spectral components from both the signal and the carrier that combine in the time domain. 
Because the output contains neither the individual modulator or carrier components, the ring modulator is said to be a double-balanced mixer,<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> where both input signals are suppressed (not present in the output)—the output is composed entirely of the sum of the products of the frequency components of the two inputs.

<h2 id="history">History</h2>
The ring modulator was invented by Frank A. Cowan in 1934 and patented in 1935<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> as an improvement on the invention of Clyde R. Keith at <a href="/facts/Bell_Labs/fKP5Gk0H">Bell Labs</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> The original application was in the field of <a href="/facts/Telephone/P7uEgWQX">analog telephony</a> for <a href="/facts/Frequency-division_multiplexing/lgj6TMU1">frequency-division multiplexing</a> for carrying multiple voice signals over telephone cables. It has since been applied to a wider range of uses, such as <a href="/facts/Voice_inversion/4Y8P099j">voice inversion</a>, radio <a href="/facts/Transceiver/S6HSzRPf">transceivers</a>, and <a href="/facts/Electronic_music/zfGNfUvR">electronic music</a>.
While the original Cowan patent describes a circuit with a ring of four diodes, later implementations used <a href="/facts/FET/TerYC6Nz">FETs</a> as the switching elements.

<h2 id="circuit-description">Circuit description</h2>
The ring modulator includes an input stage, a ring of four diodes excited by a carrier signal, and an output stage. The input and output stages typically include <a href="/facts/Transformer/EYYfXWrF">transformers</a> with center-taps towards the diode ring. While the diode ring has some similarities to a <a href="/facts/Bridge_rectifier/jN3TqC6M">bridge rectifier</a> the diodes in a ring modulator all point in the same clockwise or counterclockwise direction.
The carrier, which alternates between positive and negative current, at any given time makes one pair of diodes conduct, and <a href="/facts/P%E2%80%93n_junction/CaKIdLPg">reverse-biases</a> the other pair. The conducting pair carries the signal from the left transformer secondary to the primary of the transformer at the right. If the left carrier terminal is positive, the top and bottom diodes conduct. If that terminal is negative, then the side diodes conduct, but create a polarity inversion between the transformers. This action is much like that of a <a href="/facts/DPDT/vqbNDgYm">DPDT</a> (double pole, double throw) switch wired for reversing connections.
A particular elegance of the ring modulator is that it is bidirectional: the signal flow can be reversed allowing the same circuit with the same carrier to be used either as a <a href="/facts/Modulator/aAXPjINv">modulator</a> or <a href="/facts/Demodulator/GFLGoxFl">demodulator</a>, for example in low-cost radio transceivers.

<h2 id="integrated-circuit-methods-of-ring-modulation">Integrated circuit methods of ring modulation</h2>

Some modern ring modulators are implemented using <a href="/facts/Digital_signal_processing/PEG7azr8">digital signal processing</a> techniques by simply multiplying the time domain signals, producing a nearly-perfect signal output. <a href="/facts/Intermodulation/R9ex1SMn">Intermodulation</a> products can be generated by carefully selecting and changing the <a href="/facts/Frequency/QUSbPaW8">frequency</a> of the two input waveforms. If the signals are processed digitally, the frequency-domain convolution becomes <a href="/facts/Circular_convolution/krbXy3LH">circular convolution</a>. If the signals are <a href="/facts/Wideband/wv1yPnkF">wideband</a>, this causes <a href="/facts/Aliasing/Pzg84HL0">aliasing</a> distortion, so it is common to <a href="/facts/Oversample/9rka4Dpg">oversample</a> the operation or low-pass filter the signals prior to ring modulation.
The <a href="/facts/MOS_Technology_SID/vKQw3Jmt">SID</a> chip found in the <a href="/facts/Commodore_64/3TPcKDbb">Commodore 64</a> allows for <a href="/facts/Triangle_wave/72BUtbRl">triangle waves</a> to be ring modulated. Oscillator 1 gets modulated by oscillator 3's frequency, oscillator 2 by oscillator 1's frequency, and oscillator 3 by oscillator 2's frequency. Ring modulation is disabled unless the carrier oscillator is set to produce a triangle wave, but the modulating oscillator can be set to generate any of its available waveforms. However, no matter which waveform the modulating oscillator is using, the ring modulation always has the effect of modulating a triangle wave with a square wave.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
On an <a href="/facts/ARP_Odyssey/nTM1RAFP">ARP Odyssey</a> synthesizer (and a few others from that era as well) the ring modulator is an <a href="/facts/XOR_gate/WSfdqTcA">XOR function</a> (formed from <a href="/facts/XOR_gate/WSfdqTcA">four</a> <a href="/facts/NAND_gate/g6FUuhJT">NAND gates</a>) fed from the square wave outputs of the two oscillators. For the limited case of square or pulse wave signals, this is identical to true ring modulation.
Analog multiplier ICs (such as those made by Analog Devices) would work as ring modulators, of course with regard to such matters as their operating limits and scale factors. Use of multiplier ICs means that the modulation products are largely confined to sum and difference frequency of inputs (unless the circuit is overdriven), rather than the much more complicated products of the rectifier circuit.

<h2 id="limitations">Limitations</h2>
Any DC component of the carrier degrades the suppression of the carrier and thus in radio applications the carrier is typically transformer- or capacitor-coupled; in low frequency (e.g., audio) applications the carrier may or may not be desired in the output.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
Imperfections in the diodes and transformers introduce artifacts of the two input signals. In practical ring modulators, this leakage can be reduced by introducing opposing imbalances (e.g., variable resistors or capacitors).

<h2 id="applications">Applications</h2>
<h3>Radio communications</h3>
Ring modulation has also been extensively used in <a href="/facts/Radio_receiver/5olDTxkg">radio receivers</a>, for example, to <a href="/facts/Demodulate/GFLGoxFl">demodulate</a> an <a href="/facts/FM_broadcasting/TpKl3FJv">FM</a> stereo signal, and to heterodyne microwave signals in mobile telephone and wireless networking systems. In this case, the circuit is sometimes called a ring demodulator, one of many possible <a href="/facts/Chopper_(electronics)/tckYde1J">chopper circuits</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> A ring modulator can be used to generate a double-sideband suppressed-carrier (DSB-SC) wave used in radio transmission.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

<h3>Music and sound effects</h3>

One of the earliest musical instruments utilizing a ring modulator was the Melochord (1947) built by <a href="/facts/Harald_Bode/fuaSX4AT">Harald Bode</a>. It was a two-tone melody keyboard instrument with foot controllers and later added a second keyboard for timbre control, featuring a white-noise generator, envelope controller, formant filters and ring modulators for harmonics.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> The early Melochord was extensively used by <a href="/facts/Werner_Meyer-Eppler/tm8wby84">Werner Meyer-Eppler</a> in the early days of the electronic music studio at <a href="/facts/Bonn_University/PpaE8Xvf">Bonn University</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> Meyer-Eppler mentioned the musical application of ring modulator in his book Elektrische Klangerzeugung, published in 1949.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>
Meyer-Eppler's student <a href="/facts/Karlheinz_Stockhausen/XCrTaejh">Karlheinz Stockhausen</a> used ring modulation in 1956 for some sounds in <a href="/facts/Gesang_der_J%C3%BCnglinge/HflnXxgd">Gesang der Jünglinge</a> and his realization score for <a href="/facts/Telemusik/jLcCsNj3">Telemusik</a> (1966<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>) also calls for it. Indeed, several entire compositions by Stockhausen are based around it, such as <a href="/facts/Mixtur/8DaXPo9c">Mixtur</a> (1964), one of the first compositions for orchestra and live electronics; <a href="/facts/Mikrophonie_(Stockhausen)/1a12396g">Mikrophonie II</a> (1965), where the sounds of choral voices are modulated with a <a href="/facts/Hammond_organ/1Xr2Ia2c">Hammond organ</a>; <a href="/facts/Mantra_(Stockhausen)/Pf4nX8dS">Mantra</a> (1970),<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> where the sounds from two pianos are routed through ring modulators; and Licht-Bilder (2002) from <a href="/facts/Sonntag_aus_Licht/Umc3bgKx">Sonntag aus Licht</a> (2003),<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> which ring-modulates flute and trumpet.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a><a class="footnote-ref" id="fnref:19" href="#fn:19">19</a><a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> Other Stockhausen pieces employing ring modulation include <a href="/facts/Kontakte/9FpYV4xz">Kontakte</a> (1960),<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> Mikrophonie I (1964),<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> <a href="/facts/Hymnen/4ExG6tqr">Hymnen</a> (1969),<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> <a href="/facts/Prozession/Me6Glvok">Prozession</a> (1967),<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a> and <a href="/facts/Kurzwellen/5vtMMLrm">Kurzwellen</a> (1968).<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>
A ring-modulator was the major component used in <a href="/facts/Louis_and_Bebe_Barron/QUYXYBDw">Louis and Bebe Barron</a>'s music for the film <a href="/facts/Forbidden_Planet/FYboIEsp">Forbidden Planet</a> (1956). One of the best-known applications of the ring modulator may be its use by <a href="/facts/Brian_Hodgson/bFl3Fg1M">Brian Hodgson</a> of the <a href="/facts/BBC_Radiophonic_Workshop/RHv0b8BA">BBC Radiophonic Workshop</a> to produce the distinctive voice of the <a href="/facts/Dalek/drNMtTqF">Daleks</a> in the <a href="/facts/Television_series/eooTEhwW">television series</a> <a href="/facts/Doctor_Who/R7X9rdLc">Doctor Who</a>, starting in 1963.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a>
One of the first products dedicated for music was the Bode Ring Modulator developed in 1961 by <a href="/facts/Harald_Bode/fuaSX4AT">Harald Bode</a>. Also in 1964 he developed the Bode Frequency Shifter, which produced a clearer sound by eliminating a side band.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> These devices were designed to be controlled by voltage, compatible with modular synthesizer architecture also advocated by him,<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a> and these modules were licensed to <a href="/facts/R.A._Moog/w2jekxxq">R.A. Moog</a> for their <a href="/facts/Moog_modular_synthesizer/I6EkoW5J">Moog modular synthesizers</a> started in 1963–1964.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a> In 1963, <a href="/facts/Don_Buchla/EEpvGHPh">Don Buchla</a> included an optional ring modulator in his first modular synthesizer, the <a href="/facts/Buchla/Dm1dWJrw">Model 100</a>.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a> Also <a href="/facts/Tom_Oberheim/tn7A1qAR">Tom Oberheim</a> built a ring modulator unit for <a href="/facts/The_United_States_of_America_(band)/0uFY2pfp">his musician friend</a> in the late 1960s,<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a><a class="footnote-ref" id="fnref:32" href="#fn:32">32</a> and it became an origin of <a href="/facts/Oberheim_Electronics/1G9AyTBU">Oberheim Electronics</a> Music Modulator<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> and <a href="/facts/Gibson_Guitar_Corporation/0LDC3Qen">Maestro</a> Ring Modulator,<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a> one of the earliest ring modulator <a href="/facts/Effects_unit/XhTKhJQC">effect</a> products for guitarists. The <a href="/facts/Electronic_Music_Studios/mGz2zWPa">EMS</a> <a href="/facts/VCS3/TdG8zIMJ">VCS3</a>, <a href="/facts/Synthi_A/TdG8zIMJ">Synthi A</a>, <a href="/facts/ARP_2600/khHxUkM3">ARP 2600</a>, <a href="/facts/ARP_Odyssey/nTM1RAFP">Odyssey</a>, <a href="/facts/Rhodes_Chroma/kFc4qp0m">Rhodes Chroma</a> and <a href="/facts/Yamaha_CS-80/qXt8fm4N">Yamaha CS-80</a> synthesizers also featured built-in ring modulators.
<a href="/facts/John_McLaughlin_(musician)/3YrUa79f">John McLaughlin</a> employs the ring modulator heavily in the 1974 <a href="/facts/Mahavishnu_Orchestra/uG3wL4Qn">Mahavishnu Orchestra</a> album <a href="/facts/Visions_of_the_Emerald_Beyond/4csdDSBl">Visions of the Emerald Beyond</a>, especially on the track "On the Way Home to Earth". On <a href="/facts/Miles_Davis/sywS2Krc">Miles Davis</a>' 1975 live album <a href="/facts/Agharta_(album)/bdVgnbRx">Agharta</a>, guitarist <a href="/facts/Pete_Cosey/CqH7VeAu">Pete Cosey</a> ran the sounds he played through a ring modulator.<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a> <a href="/facts/Deep_Purple/oIpApBaM">Deep Purple</a>'s <a href="/facts/Jon_Lord/6qWqu2FA">Jon Lord</a> fed the signal from his Hammond through a Gibson Ring Modulator unit live on stage, which he described in 1989.<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a><a class="footnote-ref" id="fnref:37" href="#fn:37">37</a> Founding member of <a href="/facts/Hawkwind/1RT3YxvR">Hawkwind</a>, Dik Mik, a self-confessed non-musician, used a ring modulator as his main instrument during his time with the band (1969-1973).<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a>
<a href="/facts/Vangelis/WooCLYPf">Vangelis</a> used a ring modulator with his <a href="/facts/Yamaha_CS-80/qXt8fm4N">Yamaha CS-80</a> to improvise his 1978 avant-garde-experimental album <a href="/facts/Beaubourg_(album)/UDsqyhtn">Beaubourg</a>. The music on the album is often atonal, with the ring modulator converting the synthesizer's sound into complex metallic timbres.<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a> It remains the most experimental released work by the artist, with reviewers calling it "difficult listening at best".<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a>
Ring modulation is used in the piece Ofanim (1988/1997) by <a href="/facts/Luciano_Berio/tWJ2he4H">Luciano Berio</a>, and in the first section is applied to a child's voice and a <a href="/facts/Clarinet/57a45Ia7">clarinet</a>: "The transformation of the child voice into a clarinet was desired. For this purpose, a pitch detector computes the instantaneous frequency 
 
 
 
 
 f
 
 0
 
 
 (
 n
 )
 
 
 {\displaystyle f_{0}(n)}
 
 of the voice. Then the child voice passes through a ring modulator, where the frequency of the carrier 
 
 
 
 
 f
 
 c
 
 
 
 
 {\displaystyle f_{c}}
 
 is set to 
 
 
 
 
 f
 
 0
 
 
 (
 n
 )
 
 /
 
 2
 
 
 {\displaystyle f_{0}(n)/2}
 
. In this case odd harmonics prevail which is similar to the sound of a clarinet in the low register."<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a>

<h3>Analogue telephone systems</h3>
An early application of the ring modulator was for combining multiple analog telephone voice channels into a single wideband signal to be carried on a single cable using <a href="/facts/Frequency-division_multiplexing/lgj6TMU1">frequency-division multiplexing</a>. A ring modulator in combination with <a href="/facts/Carrier_wave/ae98pBlY">carrier wave</a> and filter was used to assign channels to different frequencies.
Early attempts at <a href="/facts/Privacy/C269YalS">securing</a> analog telephone channels used ring modulators to modify the spectrum of the audio speech signals. One application is spectral inversion, typically of speech; a carrier frequency is chosen to be above the highest speech frequencies (which are low-pass filtered at, say, 3 kHz, for a carrier of perhaps 3.3 kHz), and the sum frequencies from the modulator are removed by more low-pass filtering. The remaining difference frequencies have an inverted spectrum: high frequencies become low, and vice versa.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Theremin/qygjqB7S">Theremin</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li>Scott Lehman. <a href="https://web.archive.org/web/20051201090154/http://www.harmony-central.com/Effects/Articles/Ring_Modulation/">"Effects Explained: Ring Modulation"</a>. Harmony Central. Archived from <a href="http://www.harmony-central.com/Effects/Articles/Ring_Modulation/">the original</a> on 1 December 2005.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Richard Orton, "Ring Modulator", The New Grove Dictionary of Music and Musicians, page 429, volume 21, second edition, edited by Stanley Sadie and John Tyrrell (London: Macmillan Publishers, 2001) <a href="/wiki/Stanley_Sadie" target="_blank">/wiki/Stanley_Sadie</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Strange, Allen (1972). Electronic Music, p. 11. Wm. C. Brown Co. Publishers. ISBN 0-697-03612-X. <a href="/wiki/Allen_Strange" target="_blank">/wiki/Allen_Strange</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">"Multipliers vs. Modulators". <a href="http://www.analog.com/library/analogdialogue/archives/47-06/multipliers_modulators.html" target="_blank">http://www.analog.com/library/analogdialogue/archives/47-06/multipliers_modulators.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"Double Balanced Mixer – Theory; Circuit; Operation – Tutorial – Electronics Notes". <a href="https://www.electronics-notes.com/articles/radio/rf-mixer/double-balanced-mixer.php" target="_blank">https://www.electronics-notes.com/articles/radio/rf-mixer/double-balanced-mixer.php</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">U.S. patent 2,025,158 <a href="http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=2025158" target="_blank">http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=2025158</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">U.S. patent 1,855,576 <a href="http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=1855576" target="_blank">http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=1855576</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Commodore Programmer's Reference Guide, page 463 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">"Synth Secrets, Part 11: Amplitude Modulation". <a href="http://www.soundonsound.com/sos/mar00/articles/synthsecrets.htm" target="_blank">http://www.soundonsound.com/sos/mar00/articles/synthsecrets.htm</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Hamish Meikle (2008). Modern Radar Systems. Artech House. p. 336. ISBN 978-1-59693-243-2. <a href="978-1-59693-243-2" target="_blank">978-1-59693-243-2</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Abhishek Yadav (2008). Analog Communication System. Firewall Media. p. 83. ISBN 978-81-318-0319-6. <a href="978-81-318-0319-6" target="_blank">978-81-318-0319-6</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">T G Thomas S Chandra Sekhar (2005). Communication Theory. Tata McGraw-Hill Education. p. 37. ISBN 978-0-07-059091-5. <a href="978-0-07-059091-5" target="_blank">978-0-07-059091-5</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Rebekkah Palov (July 2011), "Harald Bode—A Short Biography", EContact!, 13 (4), Canadian Electroacoustic Community <a href="http://cec.sonus.ca/econtact/13_4/palov_bode_biography.html" target="_blank">http://cec.sonus.ca/econtact/13_4/palov_bode_biography.html</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
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<li id="fn:14">Werner Meyer-Eppler, Elektronische Klangerzeugung: Elektronische Musik und synthetische Sprache, (Bonn: Ferdinand Dümmlers, 1949) <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
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<li id="fn:19">Karlheinz Stockhausen, "Electroacoustic Performance Practice", translated by Jerome Kohl, Perspectives of New Music 34, no. 1 (Winter, 1996): 74–105. Citation on 89. <a href="/wiki/Jerome_Kohl" target="_blank">/wiki/Jerome_Kohl</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Karlheinz Stockhausen, "Einführung"/"Introduction", English translation by Suzanne Stephens, in booklet accompanying Karlheinz Stockhausen, Licht-Bilder (3. Szene vom SONNTAG aus LICHT), 2-CD set, Stockhausen Gesamtausgabe/Complete Edition 68A–B (Kürten: Stockhausen-Verlag, 2005): 10 & 51 <a href="/wiki/Suzanne_Stephens" target="_blank">/wiki/Suzanne_Stephens</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Curtis Roads (1996). The Computer Music Tutorial, pp. 220-221. MIT Press. ISBN 9780262680820. <a href="/wiki/Curtis_Roads" target="_blank">/wiki/Curtis_Roads</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Curtis Roads (1996). The Computer Music Tutorial, pp. 220-221. MIT Press. ISBN 9780262680820. <a href="/wiki/Curtis_Roads" target="_blank">/wiki/Curtis_Roads</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Curtis Roads (1996). The Computer Music Tutorial, pp. 220-221. MIT Press. ISBN 9780262680820. <a href="/wiki/Curtis_Roads" target="_blank">/wiki/Curtis_Roads</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Curtis Roads (1996). The Computer Music Tutorial, pp. 220-221. MIT Press. ISBN 9780262680820. <a href="/wiki/Curtis_Roads" target="_blank">/wiki/Curtis_Roads</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Curtis Roads (1996). The Computer Music Tutorial, pp. 220-221. MIT Press. ISBN 9780262680820. <a href="/wiki/Curtis_Roads" target="_blank">/wiki/Curtis_Roads</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Jeremy Bentham (1986), Doctor Who: The Early Years, (London: W. H. Allen & Co., 1986), p. 127, ISBN 0-491-03612-4 <a href="0-491-03612-4" target="_blank">0-491-03612-4</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
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<li id="fn:28">Harald Bode, "European Electronic Music Instrument Design", Journal of the Audio Engineering Society 9 (1961): 267. <a href="/wiki/Audio_Engineering_Society" target="_blank">/wiki/Audio_Engineering_Society</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
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<li id="fn:31">Thomas E. Oberheim, "A Ring Modulator Device for the Performing Musician", AES Convention 38 (May 1970), No. 708 (G-4).
 <a href="https://secure.aes.org/forum/pubs/conventions/?elib=1304" target="_blank">https://secure.aes.org/forum/pubs/conventions/?elib=1304</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">"Session Transcript: Tom Oberheim", Red Bull Music Academy Barcelona 2008 <a href="http://www.redbullmusicacademy.com/lectures/tom-oberheim-polyphonic-one-love?template=RBMA_Lecture%2Ftranscript" target="_blank">http://www.redbullmusicacademy.com/lectures/tom-oberheim-polyphonic-one-love?template=RBMA_Lecture%2Ftranscript</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
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<li id="fn:34">Maestro RM-1A Ring Modulator, DiscoFreq's Effects Database <a href="http://www.effectsdatabase.com/model/maestro/rm1/a" target="_blank">http://www.effectsdatabase.com/model/maestro/rm1/a</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
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<li id="fn:39">"Vangelis - Beaubourg". Synthtopia. 7 December 2003. Retrieved 26 August 2022. <a href="https://www.synthtopia.com/content/2003/12/07/vangelis-beaubourg/" target="_blank">https://www.synthtopia.com/content/2003/12/07/vangelis-beaubourg/</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">McDonald, Steven. "Vangelis - Beaubourg". AllMusic. Retrieved 26 August 2022. <a href="https://www.allmusic.com/album/r108871" target="_blank">https://www.allmusic.com/album/r108871</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Zölzer, Udo; ed. (2002). DAFX - Digital Audio Effects, p.76-7. John Wiley & Sons. ISBN 9780471490784. <a href="https://books.google.com/books?id=h90HIV0uwVsC&pg=PA75" target="_blank">https://books.google.com/books?id=h90HIV0uwVsC&pg=PA75</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
</ol>

Ring modulation open-in-new

Ring modulation