52:
51:
56:
order intermodulation products (D3 and D4) grow by 3 dB in each frame. Higher-order intermodulation products (5th order, 7th order, 9th order) are visible at very high input power levels as the amplifier is driven past saturation. Near saturation, each additional dB of input power results in proportionally less output power going into the amplified carriers and proportionally more output power going into the unwanted intermodulation products. At and above saturation, additional input power results in a
2507:
intermodulation signal is typically many orders of magnitude lower than the power of the transmit signal, the power in the passive intermodulation signal is often times on the same order of magnitude (and possibly higher) than the power of the receive signal. Therefore, if a passive intermodulation finds its way to receive path, it cannot be filtered or separated from the receive signal. The receive signal would therefore be clobbered by the passive intermodulation signal.
40:
2647:. Hard to test are intermodulation signals in the GHz-range generated from passive devices (PIM: passive intermodulation). Manufacturers of these scalar PIM-instruments are Summitek and Rosenberger. The newest developments are PIM-instruments to measure also the distance to the PIM-source. Anritsu offers a radar-based solution with low accuracy and Heuermann offers a frequency converting vector network analyzer solution with high accuracy.
1399:
2948:
2621:(Audio IMD) include SMPTE standard RP120-1994 where two signals (at 60 Hz and 7 kHz, with 4:1 amplitude ratios) are used for the test; many other standards (such as DIN, CCIF) use other frequencies and amplitude ratios. Opinion varies over the ideal ratio of test frequencies (e.g. 3:4, or almost — but not exactly — 3:1 for example).
2495:
metal-oxide junctions, such as loose corroded connectors. The higher the signal amplitudes, the more pronounced the effect of the nonlinearities, and the more prominent the intermodulation that occurs — even though upon initial inspection, the system would appear to be linear and unable to generate intermodulation.
2577:
IEC 62037 is the international standard for passive intermodulation testing and gives specific details as to passive intermodulation measurement setups. The standard specifies the use of two +43 dBm (20 W) tones for the test signals for passive intermodulation testing. This power level
2494:
Passive intermodulation (PIM), however, occurs in passive devices (which may include cables, antennas etc.) that are subjected to two or more high power tones. The PIM product is the result of the two (or more) high power tones mixing at device nonlinearities such as junctions of dissimilar metals or
2522:
Passive intermodulation can also be generated in components with manufacturing or workmanship defects, such as cold or cracked solder joints or poorly made mechanical contacts. If these defects are exposed to high radio frequency currents, passive intermodulation can be generated. As a result, radio
2506:
Passive intermodulations are a major concern in modern communication systems in cases when a single antenna is used for both high power transmission signals as well as low power receive signals (or when a transmit antenna is in close proximity to a receive antenna). Although the power in the passive
55:
3rd order intermodulation products (D3 and D4) are the result of nonlinear behavior of an amplifier. The input power level into the amplifier is increased by 1 dB in each successive frame. The output power of the two carriers (M1 and M2) increases by about 1 dB in each frame, while the 3rd
2549:
Trapped dielectric materials (adhesives, foam, etc.), cracks or distortions at the end of the outer conductor of coaxial cables, often caused by overtightening the back nut during installation, solid inner conductors distorted in the preparation process, hollow inner conductors excessively enlarged
212:
oscillator-mixer circuits. However, while the intermodulation products of the received signal with the local oscillator signal are intended, superheterodyne mixers can, at the same time, also produce unwanted intermodulation effects from strong signals near in frequency to the desired signal that
654:
2110:
In many radio and audio applications, odd-order IMPs are of most interest, as they fall within the vicinity of the original frequency components, and may therefore interfere with the desired behaviour. For example, intermodulation distortion from the third order
2498:
The requirement for "two or more high power tones" need not be discrete tones. Passive intermodulation can also occur between different frequencies (i.e. different "tones") within a single broadband carrier. These PIMs would show up as
1528:
1149:
In general, each of these frequency components will have a different amplitude and phase, which depends on the specific non-linear function being used, and also on the amplitudes and phases of the original input components.
2952:
2529:
In the field, passive intermodulation can be caused by components that were damaged in transit to the cell site, installation workmanship issues and by external passive intermodulation sources. Some of these include:
1326:
132:
Practically all audio equipment has some non-linearity, so it will exhibit some amount of IMD, which however may be low enough to be imperceptible by humans. Due to the characteristics of the human
2609:(RMS) value of the various sum-and-difference signals as a percentage of the original signal's root mean square voltage, although it may be specified in terms of individual component strengths, in
355:
1043:
47:
plot showing intermodulation between two injected signals at 270 and 275 MHz (the large spikes). Visible intermodulation products are seen as small spurs at 280 MHz and 265 MHz.
456:
2515:
Ferromagnetic materials are the most common materials to avoid and include ferrites, nickel, (including nickel plating) and steels (including some stainless steels). These materials exhibit
1623:
822:
60:
in output power, with most of that additional input power getting dissipated as heat and increasing the level of the non-linear intermodulation products with respect to the two carriers.
1388:
1230:
225:
system cannot produce intermodulation. If the input of a linear time-invariant system is a signal of a single frequency, then the output is a signal of the same frequency; only the
2413:
2367:
2267:
2221:
1841:
1787:
1733:
1679:
2526:
Passive intermodulation can also be inherent in the design of a high power radio frequency component where radio frequency current is forced to narrow channels or restricted.
1402:
Distribution of third-order intermodulations: in blue the position of the fundamental carriers, in red the position of dominant IMPs, in green the position of specific IMPs.
2803:
2105:
2061:
2017:
1973:
1929:
1885:
360:
Intermodulation occurs when the input to a non-linear system is composed of two or more frequencies. Consider an input signal that contains three frequency components at
703:
271:
2861:
2491:
components, meaning that the components must be biased with an external power source which is not the input signal (i.e. the active components must be "turned on").
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916:
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418:
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2321:
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2171:
2144:
2932:
2523:
frequency equipment manufacturers perform factory PIM tests on components, to eliminate passive intermodulation caused by these design and manufacturing defects.
1430:
680:
1171:
758:
2699:
2578:
has been used by radio frequency equipment manufacturers for more than a decade to establish PASS / FAIL specifications for radio frequency components.
1438:
95:
or time variance in a system. The intermodulation between frequency components will form additional components at frequencies that are not just at
2562:
Nearby metallic objects in the direct beam and side lobes of the transmit antenna including rusty bolts, roof flashing, vent pipes, guy wires, etc.
3017:
2909:
2785:
2748:
110:, but also at the sum and difference frequencies of the original frequencies and at sums and differences of multiples of those frequencies.
2713:
2589:(SID) can produce intermodulation distortion (IMD) when the first signal is slewing (changing voltage) at the limit of the amplifier's
1238:
2963:
84:
2957:
2624:
After feeding the equipment under test with low distortion input sinewaves, the output distortion can be measured by using an
2636:, or when determining intermodulation effects in communications equipment, may be made using the receiver under test itself.
2847:
649:{\displaystyle \ x(t)=M_{a}\sin(2\pi f_{a}t+\phi _{a})+M_{b}\sin(2\pi f_{b}t+\phi _{b})+M_{c}\sin(2\pi f_{c}t+\phi _{c})}
276:
2666:
2644:
2597:
the second signal. If SID only occurs for a portion of the signal, it is called "transient" intermodulation distortion.
962:
148:
2672:
273:
then the output is a signal which includes a number of integer multiples of the input frequency signal; (i.e. some of
136:, the same percentage of IMD is perceived as more bothersome when compared to the same amount of harmonic distortion.
2678:
240:
in response to sinusoidal input, meaning that if the input of a non-linear system is a signal of a single frequency,
3027:
2701:
Wireless
Receiver Architectures and Design: Antennas, RF, Synthesizers, Mixed Signal, and Digital Signal Processing
2661:
2618:
1536:
766:
159:
2546:
Inconsistent metal-to-metal contact between radio frequency connector surfaces caused by any of the following:
1334:
1176:
190:
2773:
2586:
33:
2629:
103:
2372:
2326:
2226:
2180:
2537:
Loose mechanical junctions due to inadequate torque, poor alignment or poorly prepared contact surfaces.
1793:
1739:
1685:
1631:
2594:
2487:, intermodulation can only occur in non-linear systems. Non-linear systems are generally composed of
1232:, the output signal will contain a number of frequency components, each of which may be described by
162:(with a solitary sine wave input) and IMD (with more complex tones). In music, for instance, IMD is
80:
1533:
For example, in our original example above, third-order intermodulation products (IMPs) occur where
2900:
2516:
155:
107:
2067:
2023:
1979:
1935:
1891:
1847:
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953:
163:
44:
39:
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2777:
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222:
194:
140:
114:
685:
243:
147:. For radio transmissions this increases the occupied bandwidth, leading to adjacent channel
2812:
2765:
2656:
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2572:
2503:
in a telecommunication signal, which interfere with adjacent channels and impede reception.
1111:
1081:
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2445:
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2122:
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of a given intermodulation product is the sum of the absolute values of the coefficients,
186:
133:
122:
1412:
662:
3022:
2989:
2705:
1156:
743:
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in that the stimulus signal is different. The same nonlinear system will produce both
32:
This article is about signal processing. For the album by Bill Evans and Jim Hall, see
3011:
2766:
2556:
2555:
Passive intermodulation can also occur in connectors, or when conductors made of two
1523:{\displaystyle \ O=\left|k_{a}\right|+\left|k_{b}\right|+\cdots +\left|k_{N}\right|,}
230:
126:
2801:
Lui, P. L. (1990). "Passive
Intermodulation Interference in Communication Systems".
1138:
are arbitrary integers which can assume positive or negative values. These are the
2967:
193:) where signals to be modulated are presented to an intentional nonlinear element (
171:
2973:
2740:
2173:. When you cube the sum of these sine waves you will get sine waves at various
179:
118:
88:
17:
1398:
209:
198:
92:
2832:
2824:
139:
Intermodulation is also usually undesirable in radio, as it creates unwanted
2816:
2540:
Loose mechanical junctions caused during transportation, shock or vibration.
2174:
2116:
226:
144:
2990:"Intermodulation Performance and Measurement of Intermodulation Components"
2628:
to remove the original frequencies, or spectral analysis may be made using
2534:
Contaminated surfaces or contacts due to dirt, dust, moisture or oxidation.
2500:
2115:) of a circuit can be seen by looking at a signal that is made up of two
237:
96:
2519:
when exposed to reversing magnetic fields, resulting in PIM generation.
2610:
100:
1153:
More generally, given an input signal containing an arbitrary number
705:
are the amplitudes and phases of the three components, respectively.
2675:(TOI), a metric of an amplifier or system related to intermodulation
121:
outcome of these non-linearities can be calculated by generating a
2640:
1397:
205:
38:
2593:
product. This induces an effective reduction in gain, partially
2605:
Intermodulation distortion in audio is usually specified as the
2885:
151:, which can reduce audio clarity or increase spectrum usage.
2543:
Metal flakes or shavings inside radio frequency connections.
185:
IMD is also distinct from intentional modulation (such as a
178:
harmonics of the tones being played on the instrument. See
50:
2902:
Cohen 3-4 Ratio: A method of measuring distortion products.
1321:{\displaystyle k_{a}f_{a}+k_{b}f_{b}+\cdots +k_{N}f_{N},\,}
113:
Intermodulation is caused by non-linear behaviour of the
858:
will contain the three frequencies of the input signal,
117:(physical equipment or even algorithms) being used. The
2448:
2421:
2375:
2329:
2302:
2275:
2229:
2183:
2152:
2125:
2070:
2026:
1982:
1938:
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1539:
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1337:
1241:
1179:
1159:
1114:
1084:
1054:
965:
924:
894:
864:
832:
769:
746:
740:, by passing our input through a non-linear function
714:
688:
665:
459:
426:
396:
366:
279:
246:
2804:
350:{\displaystyle ~f_{a},2f_{a},3f_{a},4f_{a},\ldots }
2461:
2434:
2407:
2361:
2315:
2288:
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2215:
2165:
2138:
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2011:
1967:
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1727:
1673:
1617:
1522:
1424:
1382:
1320:
1224:
1165:
1130:
1100:
1070:
1038:{\displaystyle \ k_{a}f_{a}+k_{b}f_{b}+k_{c}f_{c}}
1037:
940:
910:
880:
850:
816:
752:
732:
697:
674:
648:
442:
412:
382:
349:
265:
125:of the characteristic, or more approximately by a
2643:applications, intermodulation may be measured as
2559:unmatched metals come in contact with each other.
956:of the fundamental frequencies, each in the form
2484:
8:
2550:or made oval during the preparation process.
1618:{\displaystyle \ |k_{a}|+|k_{b}|+|k_{c}|=3}
817:{\displaystyle \ y(t)=G\left(x(t)\right)\,}
2931:: CS1 maint: location missing publisher (
213:fall within the passband of the receiver.
2848:"Passive Intermodulation Characteristics"
2453:
2447:
2426:
2420:
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2386:
2374:
2353:
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2003:
1990:
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1543:
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1507:
1480:
1459:
1440:
1414:
1383:{\displaystyle k_{a},k_{b},\ldots ,k_{N}}
1374:
1355:
1342:
1336:
1317:
1308:
1298:
1279:
1269:
1256:
1246:
1240:
1225:{\displaystyle f_{a},f_{b},\ldots ,f_{N}}
1216:
1197:
1184:
1178:
1158:
1122:
1113:
1092:
1083:
1062:
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335:
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287:
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254:
245:
27:Non-linear effect in amplitude modulation
2735:Rumsey, Francis; Mccormick, Tim (2012).
2690:
2323:are large but very close together then
2924:
2582:Intermodulation in electronic circuits
2880:
2878:
952:frequencies), as well as a number of
166:to electric guitars using overdriven
7:
2737:Sound and Recording: An Introduction
2408:{\displaystyle 2\times f_{1}-f_{2}}
2362:{\displaystyle 2\times f_{2}-f_{1}}
2262:{\displaystyle 2\times f_{1}-f_{2}}
2216:{\displaystyle 2\times f_{2}-f_{1}}
2764:Davis, Gary; Jones, Ralph (1989).
2511:Sources of passive intermodulation
233:can differ from the input signal.
25:
1836:{\displaystyle f_{b}+f_{c}-f_{a}}
1782:{\displaystyle f_{a}+f_{c}-f_{b}}
1728:{\displaystyle f_{a}+f_{b}-f_{c}}
1674:{\displaystyle f_{a}+f_{b}+f_{c}}
87:containing two or more different
2951: This article incorporates
2946:
2899:Cohen, Graeme John (July 2008),
2886:"AES Pro Audio Reference for IM"
2768:The Sound Reinforcement Handbook
2964:General Services Administration
2915:from the original on 2022-04-07
2864:from the original on 2022-07-11
2567:Passive intermodulation testing
3018:Audio amplifier specifications
1605:
1590:
1582:
1567:
1559:
1544:
1390:are arbitrary integer values.
845:
839:
805:
799:
782:
776:
727:
721:
643:
608:
586:
551:
529:
494:
472:
466:
1:
2988:Butler, Lloyd (August 1997).
2473:Passive intermodulation (PIM)
708:We obtain our output signal,
2667:Second-order intercept point
2645:adjacent channel power ratio
2100:{\displaystyle 2f_{c}-f_{b}}
2056:{\displaystyle 2f_{c}-f_{a}}
2012:{\displaystyle 2f_{b}-f_{c}}
1968:{\displaystyle 2f_{b}-f_{a}}
1924:{\displaystyle 2f_{a}-f_{c}}
1880:{\displaystyle 2f_{a}-f_{b}}
450:; which may be expressed as
236:Non-linear systems generate
2673:Third-order intercept point
2632:in software or a dedicated
3044:
2846:Eron, Murat (2014-03-14).
2698:Rouphael, Tony J. (2014).
2570:
2476:
154:IMD is only distinct from
73:intermodulation distortion
31:
2772:(2nd ed.). Yamaha /
2662:Audio system measurements
2619:Audio system measurements
217:Causes of intermodulation
191:superheterodyne receivers
160:total harmonic distortion
1173:of frequency components
1140:intermodulation products
948:(which are known as the
174:to produce new tones at
2908:, Adelaide, Australia,
2774:Hal Leonard Corporation
2630:Fourier transformations
2587:Slew-induced distortion
1331:where the coefficients
698:{\displaystyle \ \phi }
266:{\displaystyle ~f_{a},}
143:, often in the form of
34:Intermodulation (album)
2959:Federal Standard 1037C
2953:public domain material
2679:Luxemburg–Gorky effect
2463:
2436:
2415:will be very close to
2409:
2363:
2317:
2290:
2263:
2217:
2167:
2140:
2101:
2057:
2013:
1969:
1925:
1881:
1837:
1783:
1729:
1675:
1619:
1524:
1426:
1403:
1384:
1322:
1226:
1167:
1132:
1131:{\displaystyle ~k_{c}}
1102:
1101:{\displaystyle ~k_{b}}
1072:
1071:{\displaystyle ~k_{a}}
1039:
942:
941:{\displaystyle ~f_{c}}
912:
911:{\displaystyle ~f_{b}}
882:
881:{\displaystyle ~f_{a}}
852:
851:{\displaystyle \ y(t)}
818:
754:
734:
733:{\displaystyle \ y(t)}
699:
676:
650:
444:
443:{\displaystyle ~f_{c}}
414:
413:{\displaystyle ~f_{b}}
384:
383:{\displaystyle ~f_{a}}
351:
267:
61:
48:
2972: (in support of
2817:10.1049/ecej:19900029
2464:
2462:{\displaystyle f_{2}}
2437:
2435:{\displaystyle f_{1}}
2410:
2364:
2318:
2316:{\displaystyle f_{2}}
2291:
2289:{\displaystyle f_{1}}
2264:
2218:
2168:
2166:{\displaystyle f_{2}}
2141:
2139:{\displaystyle f_{1}}
2102:
2058:
2014:
1970:
1926:
1882:
1838:
1784:
1730:
1676:
1620:
1525:
1427:
1401:
1394:Intermodulation order
1385:
1323:
1227:
1168:
1133:
1103:
1073:
1040:
943:
913:
883:
853:
819:
755:
735:
700:
677:
651:
445:
415:
385:
352:
268:
223:linear time-invariant
164:intentionally applied
54:
42:
2613:, as is common with
2595:amplitude-modulating
2446:
2419:
2373:
2327:
2300:
2273:
2227:
2181:
2150:
2123:
2068:
2024:
1980:
1936:
1892:
1848:
1794:
1740:
1686:
1632:
1537:
1439:
1413:
1335:
1239:
1177:
1157:
1112:
1082:
1052:
963:
922:
892:
862:
830:
767:
744:
712:
686:
663:
457:
424:
394:
364:
277:
244:
180:Power chord#Analysis
81:amplitude modulation
1425:{\displaystyle \ O}
954:linear combinations
675:{\displaystyle \ M}
156:harmonic distortion
108:harmonic distortion
2485:a previous section
2459:
2432:
2405:
2359:
2313:
2286:
2259:
2213:
2163:
2136:
2097:
2053:
2009:
1965:
1921:
1877:
1833:
1779:
1725:
1671:
1615:
1520:
1422:
1404:
1380:
1318:
1222:
1163:
1128:
1098:
1068:
1035:
938:
908:
878:
848:
814:
750:
730:
695:
672:
646:
440:
410:
380:
347:
263:
141:spurious emissions
106:) of either, like
62:
49:
45:frequency spectrum
3028:Radio electronics
2853:Microwave Journal
2787:978-0-88188-900-0
2750:978-1-136-12509-6
2634:spectrum analyzer
2626:electronic filter
2479:Rusty bolt effect
1542:
1444:
1418:
1166:{\displaystyle N}
1117:
1087:
1057:
968:
927:
897:
867:
835:
772:
753:{\displaystyle G}
717:
691:
668:
462:
429:
399:
369:
282:
249:
115:signal processing
16:(Redirected from
3035:
3004:
3002:
3001:
2977:
2971:
2966:. Archived from
2950:
2949:
2938:
2936:
2930:
2922:
2921:
2920:
2914:
2907:
2896:
2890:
2889:
2882:
2873:
2872:
2870:
2869:
2843:
2837:
2836:
2798:
2792:
2791:
2771:
2761:
2755:
2754:
2739:(5th ed.).
2732:
2726:
2725:
2723:
2722:
2715:978-0-12378641-8
2695:
2657:Beat (acoustics)
2607:root mean square
2573:Two-tone testing
2483:As explained in
2468:
2466:
2465:
2460:
2458:
2457:
2441:
2439:
2438:
2433:
2431:
2430:
2414:
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2314:
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2222:
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2211:
2199:
2198:
2172:
2170:
2169:
2164:
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2145:
2143:
2142:
2137:
2135:
2134:
2106:
2104:
2103:
2098:
2096:
2095:
2083:
2082:
2062:
2060:
2059:
2054:
2052:
2051:
2039:
2038:
2018:
2016:
2015:
2010:
2008:
2007:
1995:
1994:
1974:
1972:
1971:
1966:
1964:
1963:
1951:
1950:
1930:
1928:
1927:
1922:
1920:
1919:
1907:
1906:
1886:
1884:
1883:
1878:
1876:
1875:
1863:
1862:
1842:
1840:
1839:
1834:
1832:
1831:
1819:
1818:
1806:
1805:
1788:
1786:
1785:
1780:
1778:
1777:
1765:
1764:
1752:
1751:
1734:
1732:
1731:
1726:
1724:
1723:
1711:
1710:
1698:
1697:
1680:
1678:
1677:
1672:
1670:
1669:
1657:
1656:
1644:
1643:
1624:
1622:
1621:
1616:
1608:
1603:
1602:
1593:
1585:
1580:
1579:
1570:
1562:
1557:
1556:
1547:
1540:
1529:
1527:
1526:
1521:
1516:
1512:
1511:
1489:
1485:
1484:
1468:
1464:
1463:
1442:
1431:
1429:
1428:
1423:
1416:
1389:
1387:
1386:
1381:
1379:
1378:
1360:
1359:
1347:
1346:
1327:
1325:
1324:
1319:
1313:
1312:
1303:
1302:
1284:
1283:
1274:
1273:
1261:
1260:
1251:
1250:
1231:
1229:
1228:
1223:
1221:
1220:
1202:
1201:
1189:
1188:
1172:
1170:
1169:
1164:
1137:
1135:
1134:
1129:
1127:
1126:
1115:
1107:
1105:
1104:
1099:
1097:
1096:
1085:
1077:
1075:
1074:
1069:
1067:
1066:
1055:
1044:
1042:
1041:
1036:
1034:
1033:
1024:
1023:
1011:
1010:
1001:
1000:
988:
987:
978:
977:
966:
947:
945:
944:
939:
937:
936:
925:
917:
915:
914:
909:
907:
906:
895:
887:
885:
884:
879:
877:
876:
865:
857:
855:
854:
849:
833:
823:
821:
820:
815:
812:
808:
770:
759:
757:
756:
751:
739:
737:
736:
731:
715:
704:
702:
701:
696:
689:
681:
679:
678:
673:
666:
655:
653:
652:
647:
642:
641:
626:
625:
601:
600:
585:
584:
569:
568:
544:
543:
528:
527:
512:
511:
487:
486:
460:
449:
447:
446:
441:
439:
438:
427:
419:
417:
416:
411:
409:
408:
397:
389:
387:
386:
381:
379:
378:
367:
356:
354:
353:
348:
340:
339:
324:
323:
308:
307:
292:
291:
280:
272:
270:
269:
264:
259:
258:
247:
208:and even single-
21:
18:Cross modulation
3043:
3042:
3038:
3037:
3036:
3034:
3033:
3032:
3008:
3007:
2999:
2997:
2987:
2984:
2982:Further reading
2956:
2947:
2945:
2942:
2941:
2923:
2918:
2916:
2912:
2905:
2898:
2897:
2893:
2884:
2883:
2876:
2867:
2865:
2845:
2844:
2840:
2800:
2799:
2795:
2788:
2763:
2762:
2758:
2751:
2743:. p. 538.
2734:
2733:
2729:
2720:
2718:
2716:
2708:. p. 244.
2697:
2696:
2692:
2687:
2653:
2615:radio frequency
2603:
2591:power bandwidth
2584:
2575:
2569:
2513:
2481:
2475:
2449:
2444:
2443:
2422:
2417:
2416:
2395:
2382:
2371:
2370:
2349:
2336:
2325:
2324:
2303:
2298:
2297:
2276:
2271:
2270:
2249:
2236:
2225:
2224:
2203:
2190:
2179:
2178:
2153:
2148:
2147:
2126:
2121:
2120:
2087:
2074:
2066:
2065:
2043:
2030:
2022:
2021:
1999:
1986:
1978:
1977:
1955:
1942:
1934:
1933:
1911:
1898:
1890:
1889:
1867:
1854:
1846:
1845:
1823:
1810:
1797:
1792:
1791:
1769:
1756:
1743:
1738:
1737:
1715:
1702:
1689:
1684:
1683:
1661:
1648:
1635:
1630:
1629:
1594:
1571:
1548:
1535:
1534:
1503:
1499:
1476:
1472:
1455:
1451:
1437:
1436:
1411:
1410:
1396:
1370:
1351:
1338:
1333:
1332:
1304:
1294:
1275:
1265:
1252:
1242:
1237:
1236:
1212:
1193:
1180:
1175:
1174:
1155:
1154:
1118:
1110:
1109:
1088:
1080:
1079:
1058:
1050:
1049:
1025:
1015:
1002:
992:
979:
969:
961:
960:
928:
920:
919:
898:
890:
889:
868:
860:
859:
828:
827:
795:
791:
765:
764:
742:
741:
710:
709:
684:
683:
661:
660:
633:
617:
592:
576:
560:
535:
519:
503:
478:
455:
454:
430:
422:
421:
400:
392:
391:
370:
362:
361:
331:
315:
299:
283:
275:
274:
250:
242:
241:
219:
187:frequency mixer
134:auditory system
123:Volterra series
65:Intermodulation
37:
28:
23:
22:
15:
12:
11:
5:
3041:
3039:
3031:
3030:
3025:
3020:
3010:
3009:
3006:
3005:
2983:
2980:
2979:
2978:
2970:on 2022-01-22.
2940:
2939:
2891:
2874:
2838:
2811:(3): 109–118.
2793:
2786:
2756:
2749:
2727:
2714:
2706:Academic Press
2689:
2688:
2686:
2683:
2682:
2681:
2676:
2670:
2664:
2659:
2652:
2649:
2602:
2599:
2583:
2580:
2568:
2565:
2564:
2563:
2560:
2553:
2552:
2551:
2544:
2541:
2538:
2535:
2512:
2509:
2477:Main article:
2474:
2471:
2456:
2452:
2429:
2425:
2402:
2398:
2394:
2389:
2385:
2381:
2378:
2356:
2352:
2348:
2343:
2339:
2335:
2332:
2310:
2306:
2283:
2279:
2256:
2252:
2248:
2243:
2239:
2235:
2232:
2210:
2206:
2202:
2197:
2193:
2189:
2186:
2160:
2156:
2133:
2129:
2108:
2107:
2094:
2090:
2086:
2081:
2077:
2073:
2063:
2050:
2046:
2042:
2037:
2033:
2029:
2019:
2006:
2002:
1998:
1993:
1989:
1985:
1975:
1962:
1958:
1954:
1949:
1945:
1941:
1931:
1918:
1914:
1910:
1905:
1901:
1897:
1887:
1874:
1870:
1866:
1861:
1857:
1853:
1843:
1830:
1826:
1822:
1817:
1813:
1809:
1804:
1800:
1789:
1776:
1772:
1768:
1763:
1759:
1755:
1750:
1746:
1735:
1722:
1718:
1714:
1709:
1705:
1701:
1696:
1692:
1681:
1668:
1664:
1660:
1655:
1651:
1647:
1642:
1638:
1614:
1611:
1607:
1601:
1597:
1592:
1588:
1584:
1578:
1574:
1569:
1565:
1561:
1555:
1551:
1546:
1531:
1530:
1519:
1515:
1510:
1506:
1502:
1498:
1495:
1492:
1488:
1483:
1479:
1475:
1471:
1467:
1462:
1458:
1454:
1450:
1447:
1421:
1395:
1392:
1377:
1373:
1369:
1366:
1363:
1358:
1354:
1350:
1345:
1341:
1329:
1328:
1316:
1311:
1307:
1301:
1297:
1293:
1290:
1287:
1282:
1278:
1272:
1268:
1264:
1259:
1255:
1249:
1245:
1219:
1215:
1211:
1208:
1205:
1200:
1196:
1192:
1187:
1183:
1162:
1125:
1121:
1095:
1091:
1065:
1061:
1046:
1045:
1032:
1028:
1022:
1018:
1014:
1009:
1005:
999:
995:
991:
986:
982:
976:
972:
935:
931:
905:
901:
875:
871:
847:
844:
841:
838:
825:
824:
811:
807:
804:
801:
798:
794:
790:
787:
784:
781:
778:
775:
749:
729:
726:
723:
720:
694:
671:
657:
656:
645:
640:
636:
632:
629:
624:
620:
616:
613:
610:
607:
604:
599:
595:
591:
588:
583:
579:
575:
572:
567:
563:
559:
556:
553:
550:
547:
542:
538:
534:
531:
526:
522:
518:
515:
510:
506:
502:
499:
496:
493:
490:
485:
481:
477:
474:
471:
468:
465:
437:
433:
407:
403:
377:
373:
346:
343:
338:
334:
330:
327:
322:
318:
314:
311:
306:
302:
298:
295:
290:
286:
262:
257:
253:
218:
215:
204:such as mixer
172:effects pedals
93:nonlinearities
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3040:
3029:
3026:
3024:
3021:
3019:
3016:
3015:
3013:
2995:
2994:Amateur Radio
2991:
2986:
2985:
2981:
2975:
2969:
2965:
2961:
2960:
2954:
2944:
2943:
2934:
2928:
2911:
2904:
2903:
2895:
2892:
2887:
2881:
2879:
2875:
2863:
2859:
2855:
2854:
2849:
2842:
2839:
2834:
2830:
2826:
2822:
2818:
2814:
2810:
2806:
2805:
2797:
2794:
2789:
2783:
2779:
2775:
2770:
2769:
2760:
2757:
2752:
2746:
2742:
2738:
2731:
2728:
2717:
2711:
2707:
2703:
2702:
2694:
2691:
2684:
2680:
2677:
2674:
2671:
2668:
2665:
2663:
2660:
2658:
2655:
2654:
2650:
2648:
2646:
2642:
2637:
2635:
2631:
2627:
2622:
2620:
2616:
2612:
2608:
2600:
2598:
2596:
2592:
2588:
2581:
2579:
2574:
2566:
2561:
2558:
2554:
2548:
2547:
2545:
2542:
2539:
2536:
2533:
2532:
2531:
2527:
2524:
2520:
2518:
2510:
2508:
2504:
2502:
2496:
2492:
2490:
2486:
2480:
2472:
2470:
2454:
2450:
2427:
2423:
2400:
2396:
2392:
2387:
2383:
2379:
2376:
2354:
2350:
2346:
2341:
2337:
2333:
2330:
2308:
2304:
2281:
2277:
2254:
2250:
2246:
2241:
2237:
2233:
2230:
2208:
2204:
2200:
2195:
2191:
2187:
2184:
2176:
2158:
2154:
2131:
2127:
2118:
2114:
2092:
2088:
2084:
2079:
2075:
2071:
2064:
2048:
2044:
2040:
2035:
2031:
2027:
2020:
2004:
2000:
1996:
1991:
1987:
1983:
1976:
1960:
1956:
1952:
1947:
1943:
1939:
1932:
1916:
1912:
1908:
1903:
1899:
1895:
1888:
1872:
1868:
1864:
1859:
1855:
1851:
1844:
1828:
1824:
1820:
1815:
1811:
1807:
1802:
1798:
1790:
1774:
1770:
1766:
1761:
1757:
1753:
1748:
1744:
1736:
1720:
1716:
1712:
1707:
1703:
1699:
1694:
1690:
1682:
1666:
1662:
1658:
1653:
1649:
1645:
1640:
1636:
1628:
1627:
1626:
1612:
1609:
1599:
1595:
1586:
1576:
1572:
1563:
1553:
1549:
1517:
1513:
1508:
1504:
1500:
1496:
1493:
1490:
1486:
1481:
1477:
1473:
1469:
1465:
1460:
1456:
1452:
1448:
1445:
1435:
1434:
1433:
1419:
1409:
1400:
1393:
1391:
1375:
1371:
1367:
1364:
1361:
1356:
1352:
1348:
1343:
1339:
1314:
1309:
1305:
1299:
1295:
1291:
1288:
1285:
1280:
1276:
1270:
1266:
1262:
1257:
1253:
1247:
1243:
1235:
1234:
1233:
1217:
1213:
1209:
1206:
1203:
1198:
1194:
1190:
1185:
1181:
1160:
1151:
1147:
1145:
1141:
1123:
1119:
1093:
1089:
1063:
1059:
1030:
1026:
1020:
1016:
1012:
1007:
1003:
997:
993:
989:
984:
980:
974:
970:
959:
958:
957:
955:
951:
933:
929:
903:
899:
873:
869:
842:
836:
809:
802:
796:
792:
788:
785:
779:
773:
763:
762:
761:
747:
724:
718:
706:
692:
669:
638:
634:
630:
627:
622:
618:
614:
611:
605:
602:
597:
593:
589:
581:
577:
573:
570:
565:
561:
557:
554:
548:
545:
540:
536:
532:
524:
520:
516:
513:
508:
504:
500:
497:
491:
488:
483:
479:
475:
469:
463:
453:
452:
451:
435:
431:
405:
401:
375:
371:
358:
344:
341:
336:
332:
328:
325:
320:
316:
312:
309:
304:
300:
296:
293:
288:
284:
260:
255:
251:
239:
234:
232:
228:
224:
216:
214:
211:
207:
203:
200:
196:
192:
188:
183:
181:
177:
173:
169:
165:
161:
157:
152:
150:
146:
142:
137:
135:
130:
128:
127:Taylor series
124:
120:
116:
111:
109:
105:
102:
99:frequencies (
98:
94:
90:
86:
82:
78:
74:
70:
66:
59:
53:
46:
41:
35:
30:
19:
2998:. Retrieved
2993:
2968:the original
2958:
2917:, retrieved
2901:
2894:
2866:. Retrieved
2857:
2851:
2841:
2808:
2802:
2796:
2767:
2759:
2736:
2730:
2719:. Retrieved
2700:
2693:
2638:
2623:
2604:
2585:
2576:
2557:galvanically
2528:
2525:
2521:
2514:
2505:
2497:
2493:
2488:
2482:
2112:
2109:
1532:
1407:
1405:
1330:
1152:
1148:
1143:
1139:
1047:
949:
826:
707:
658:
359:
235:
220:
184:
175:
153:
149:interference
138:
131:
112:
91:, caused by
76:
72:
68:
64:
63:
57:
29:
2974:MIL-STD-188
2741:Focal Press
2601:Measurement
2175:frequencies
2146:and one at
950:fundamental
119:theoretical
89:frequencies
3012:Categories
3000:2012-01-30
2919:2022-07-01
2868:2022-07-11
2776:. p.
2721:2022-07-11
2685:References
2571:See also:
2517:hysteresis
2177:including
2117:sine waves
659:where the
210:transistor
199:non-linear
195:multiplied
168:amplifiers
2937:(5 pages)
2860:: 34–38.
2833:0954-0695
2825:2051-218X
2501:sidebands
2393:−
2380:×
2347:−
2334:×
2247:−
2234:×
2201:−
2188:×
2119:, one at
2085:−
2041:−
1997:−
1953:−
1909:−
1865:−
1821:−
1767:−
1713:−
1494:⋯
1365:…
1289:⋯
1207:…
693:ϕ
635:ϕ
615:π
606:
578:ϕ
558:π
549:
521:ϕ
501:π
492:
345:…
238:harmonics
227:amplitude
145:sidebands
104:multiples
79:) is the
2927:citation
2910:archived
2862:Archived
2651:See also
2611:decibels
197:). See
97:harmonic
58:decrease
2996:. VK5BR
101:integer
85:signals
2831:
2823:
2784:
2747:
2712:
2617:work.
2489:active
2269:. If
1541:
1443:
1417:
1116:
1108:, and
1086:
1056:
1048:where
967:
926:
918:, and
896:
866:
834:
771:
716:
690:
667:
461:
428:
420:, and
398:
368:
281:
248:
206:diodes
202:mixers
3023:Waves
2955:from
2913:(PDF)
2906:(PDF)
2821:eISSN
2669:(SOI)
2641:radio
1408:order
231:phase
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