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e.g. a radio station at 100 mhz on FM vs a 50 mhz radio station. Wouldn't the latter have a lower bandwidth therefore lower quality audio/bit rate? If this is true, this must also mean that 2.4ghz router has less bandwidth capacity than a 5ghz router?

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    \$\begingroup\$ Neither 100 nor 50 milli-hertz carrier frequency is anywhere remotely sufficient for audio. \$\endgroup\$ – Olin Lathrop Jul 12 '15 at 13:30
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There is no fundamental difference between 50 and 100 MHz in this respect.

Stereo FM signals have spectral components up to 57kHz (including RDS), modulated onto carriers with a peak deviation of 75 kHz, which effectively leads to a 200kHz channel spacing, and that would apply equally whether the carrier was 100MHz, 50 MHz, .... or 1 MHz.

The latter poses political difficulties, as it would impose a limit of 5 stations in the available space on the "Medium Wave" band (aka "AM" in the USA) or one station on the Long Wave band.

So the AM bands operate with a much closer channel spacing (9 kHz worldwide, except 10kHz mainly N. America), and it's this reduced channel spacing, not the carrier frequency, that restricts quality on the AM bands.

Likewise, if a 5GHz router offers higher bandwidth, it's because the channel allocated to it is wider than the (now rather crowded) 2.4 GHz band.

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Nope. The bandwidth of the audio shows up as the modulation bandwidth, and as long as this is significantly less than the transmission frequency there is no real difference.

In other words, transmitting a 20 kHz modulated signal at 50 MHz gives no noticeable difference compared with a 20 kHz signal transmitted at 100 MHz.

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Public spectrums are a political issue, the band allocated to a given channel in FM radio is set by the stations FCC license, the carrier frequency is largely irrelevant for transmission bandwidth in this context.

From a technical perspective considering a signal modulated to 50/100MHz with an identical modulation/demodulation scheme, you will likely be able to squeeze more (signal) bandwidth at higher carrier frequency, however radio transmissions of all kinds are highly regulated and a station has to conform to a codified transmission scheme. Additionally as WhatRoughBeast pointed out, audio signal frequency spectrum is small enough that this would give no meaningful performance benefit for radio, from a pure transmission standpoint you would be able to transmit more data (more streams or higher objective signal quality) at 100MHz.

When it comes to Wifi, the 5GHz mode was specifically chosen to allow a larger number of channels. The physical (environment) bandwidth limitations are roughly the same at 5ghz and 2.4ghz, so the channel width cannot be decreased signficantly, however by jumping up to 5GHz, you can fit more channels in the legally allowed spectrum for the 5GHz ISM band.

The information here applies to the U.S.A where the FCC is the regulating agency.

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Bandwidth is literally the width of a frequency band, which is a measure other than the absolute frequency of the carrier. Example: a station at 50.1 MHz is assigned the 200 kHz band from 50.0 MHz to 50.2 MHz. Another station at 100.1 MHz is assigned the range 100.0 to 100.2. Again, a 200 kHz band. The width of the band (thus bandwidth) is the same.

It is the width of the band which is related to how much information can be carried, which is relevant to data communication, and the reason why "bandwidth" has become an informal synonym for "data rate". The capacity of a channel is given by the Shannon-Heartley theorem as:

$$C = B \log_2 \left( 1+\frac{S}{N} \right)$$

where \$C\$ is bits per second, \$B\$ is the bandwidth in Hertz, and \$S/N\$ is the signal to noise ratio. This tells us that the "digital bandwidth" is directly related to the "analog bandwidth" in the presence of some fixed quantity of noise.

About 2.4 GHz versus 5 GHz in Wi-Fi, that is apples versus oranges, so to speak. Wi-Fi in the neighborhood of 2.4 GHz uses IEEE 802.11g or the older 802.11b, whereas 5 GHz Wi-Fi is 802.11n.

802.11n can operate in 2.4 GHz and can squeeze more throughput even out of one 20 Mhz channel (as used by 802.11g). It performs better in 5 GHz because that band is much less congested.

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