# About sampling and wireless networks

Wireless networks operate on a high frequency, say some several gigahertz. Are the signals really sampled at that high frequencies? If I got it correctly there's some modulating/demodulating in signal transmit, does that effect the situation somehow?

What you describe is a direct-sampling architecture. Those are practically available for radio amateurs using frequencies up to 30MHz. They are also actively researched to open up their usage for much higher frequencies. Many interesting topics (e.g. using the aliasing effect to your advantage and undersample the signal) are investigated here and the possibilities of these receivers are large. However, it will probably take a long time until those are practically available for the WiFi frequency range.

However, your WiFi transceiver is most likely a direct-conversion architecture. In the digital domain of it, the data-carrying signal is generated. In case of 802.11n it is an OFDM modulated signal with up to 20/40MHz bandwidth. This signal is centered around 0Hz and hence cannot be transmitted as it is. This signal is called the baseband signal.

After Digital/Analog-conversion, in a stage called a mixer, this signal is then upconverted to the final carrier frequency around 2.4GHz. In the spectrum, the signal is literally shifted from a center frequency of 0Hz to a center frequency of 2.4GHz. At this point, the signal is called a passband (or bandpass) signal. After amplification, this signal is then transmitted via the antenna.

On the receiver side, the same happens the other way around. There is another mixer that downconverts the passband signal from 2.4GHz center frequency to a baseband signal at 0Hz center frequency. At this point, the signal has just a bandwidth of 20/40MHz and can easily be sampled in an A/D converter.

What follows is demodulation, error correction and other digital processing to eventually show you pictures of cute kittens in your browser.

As you suggest, most wireless systems incorporate modulation at the transmitter side and demodulation at the receiver side. Modulation shifts the baseband signal $x(t)$ with (two-sided) bandwidth $B$ to the desired carrier frequency $\omega_0$: $$y(t) = x(t)cos(\omega_0t)$$ Directly sampling the received signal $y(t)$ would require a sampling frequency $f_s \geq \omega_0/(2\pi) + B/2$ which is impractically high in many systems where the carrier frequency is in the Gigahertz range. Therefore demodulation is applied before sampling the receive signal: $$r(t)=y(t)cos(\omega_0t) = \frac{1}{2}x(t) + \frac{1}{2}cos(2\omega_0t)x(t)$$ Note that $r(t)$ has to be lowpass filtered before sampling to remove the radio frequency part and to avoid aliasing. The resulting signal has a bandwidth of $B/2$ and can be sampled at a rate $f_s \geq B$.

As the capabilities of digital-to-analog and analog-to-digital converters continously improve, the analog modulation and demodulation is replaced by a digital creation of $y(t)$ at tx side and sampling at a high rate at rx side in some systems. Especially so-called software defined radio (SDR) systems use this technique.

Analog mod/demod introduces some further requirements: the carrier frequency $\omega_0$ has to be recovered at transmitter side as the two oscillators in tx and rx will never run at the exact same frequency. In phase modulated systems also a phase recovery is required. Of course, the same is true for the sampling frequency in SDR systems.