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How can headphone cables act as Antennas for built-in radios in smartphones ? Is it possible because of different frequency channels or what is the theory/mathematics behind this?

On a side note, what would be a hardware implementation for this application?

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    \$\begingroup\$ The "different frequencies" guess is basically right - audio is extremely low frequency below 20 KHz while broadcast radio is in the hundreds of KHz or 100 MHz range. So a matching network that creates different connections in different frequency ranges solves the problem of audio frequency and radio frequency signals co-existing on the same conductor. \$\endgroup\$ – Chris Stratton Jul 24 '14 at 20:47
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    \$\begingroup\$ That, and any RF that it picks up will be common-mode, while the audio signal is differential mode. \$\endgroup\$ – Phil Frost Jul 24 '14 at 20:51
  • \$\begingroup\$ I have no clue what hardware is used, but a headphone cable seems about right length for an FM antenna. For FM radio, a 1/4 wavelength would be roughly 0.75 meters (2.5 feet). \$\endgroup\$ – curtis Jul 25 '14 at 17:59
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If you think about it, any antenna picks up virtually anything from a few kHz to over a GHz - all these competing signals arrive on the input stages of a receiver but the receiver does what it does best - it focusses on the frequency band it is interested in and largely rejects everything else.

Clearly a big signal just outside what it focusses on can have an effect and these are known phenomena to radio guys.

Using a headphone audio wire is no different - the audio is never really anything like close to the frequency band of interest for the radio so it barely breaks a sweat rejecting it even though it might be a few volts peak-to-peak whilst the real signal received might be a few micro volts.

A single-order 1 MHz high pass filter will provide 120 dB rejection to a 1Hz signal. A 2nd order 1 MHz high pass filter will provide 120 dB of rejection to a 1000 Hz signal. A third order will reject 10 KHz at 120 dB and so on and so on. The filter in a radio is probably equivalent to a 6th order at least.

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They use capacitor to couple RF input stage to Audio OutPut Wire (+). This method is used too in old radio-clock where the RF input stage is connected in Line AC wire with capacitor. The capacitor blocks low frequencies.

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