### How It Works: Generalized Receiver

The process of transmitting data, $m(t)$, via a wireless signal, $v(t)$, is shown below: As can be seen from the figure above, in order to transmit $m(t)$, one must first process the data using a baseband processor (usually accomplished with software). As a result of this process the original data will be split into two signals, $x(t)$ and $y(t)$, and shifted to a frequency $f_c$ (and $-f_c$ because of the notion of a complex conjugate). Note that the original wireless communication data that you want to send is in the form known as baseband, which consists of frequencies near D.C. (or $f_c = 0$). When you actually send the communication data, however, you want to send it via much higher frequency (one which is inaudible to humans) and this creates a bandpass signal. This concept is illustrated below: How is the data split and shifted you ask? Mathematically speaking, in the world of "Communication Systems" a signal to be transmitted can be written as $\displaystyle v(t) = Re[g(t) \cdot e^{j(\omega_c t)}]$,

where $g(t) = x(t) + jy(t)$ is the signal to be sent and $e^{j\omega_{c}t} = \cos{(\omega_c t)} + j\sin{(\omega_c t)}$ (Euler's identity) shifts the signal in the frequency domain by $\omega_{c} = 2 \pi f_{c}$.
The above formula can then be rewritten to obtain the following relationship: $\displaystyle v(t) = Re[g(t) \cdot e^{j\omega_{c}t}] = Re[(x(t) + jy(t)) \cdot (\cos{(\omega_c t)} + j\sin{(\omega_c t)})] = x(t)\cos{(\omega_c t)} - y(t)\sin{(\omega_c t)}$.

As you can also see from the above relationship, in order to obtain the appropriate output signal $v(t)$ one must multiply $x(t)$ by a cosine function and $y(t)$ by a negative sine function and then sum the results (this is illustrated in the figure above). In simplified terms and details, this is essentially how the data $m(t)$ is transmitted.

Receiving the original data from a wireless signal is very similar to that of transmitting the data (as described before). The process of receiving data, $m(t)$, via a wireless signal, $v(t)$, is shown below: In short, the data that we are interested in ( $m(t)$) is embedded within the carrying signal ( $v(t)$) which is at a frequency inaudible to human ears. Thus, in order to get the original communication data it is necessary to somehow bring the bandpass signal back down to its original baseband form -- and this is what the second quarter of electronics is all about. If you have not already discovered, to recover the original data $x(t)$ and $y(t)$ from the signal signal $v(t)$ we must perform a task known as "mixing" and then run the results through low pass filters. Simply put, we need to multiply $v(t)$ by $2 \cdot \cos{(\omega_c t)}$ and $-2 \cdot \sin{(\omega_c t)}$ (the " $2$" is not necessary, but it makes the math a little nicer in the end). To see this mathematically, recall that $v(t) = x(t)\cos{(\omega_c t)} - y(t)\sin{(\omega_c t)}$. Therefore, $2\cos{(\omega_c t)} \cdot v(t) = 2cos{(\omega_c t)} \cdot [x(t)\cos{(\omega_c t)} - y(t)\sin{(\omega_c t)}] = 2x(t)\cos^{2}{(\omega_c t)} - 2y(t)\cos{(\omega_c t)}\sin{(\omega_c t)}$

Using the trigonometric identities $\cos^{2}{\theta} = \frac{1+\cos{2 \theta}}{2}$

and $\cos{\theta}\sin{\theta} = \frac{1}{2}\sin{\theta}$,

we can see that $\displaystyle 2x(t)\cos^{2}{(\omega_c t)} - 2y(t)\cos{(\omega_c t)}\sin{(\omega_c t)} = x(t) + x(t)\cos{(2 \omega_c t)} - y(t)\sin{(2 \omega_c t)}$.

But when this result is ran through a low pass filter it is fairly obvious that $x(t)\cos{(2 \omega_c t)} - y(t)\sin{(2 \omega_c t)}$ is filtered out (because of the $2 \omega_c$), but $x(t)$ is unaltered by the filter (because it is independent of frequency); therefore, we are left with $x(t)$ as desired! (Similarly, $y(t)$ can be obtained.) This is essentially how a generalized receiver functions: given a bandpass signal, $v(t)$, and we can use a mixer and low pass filters to obtain an original baseband signal, $m(t)$ (in the form of $x(t)$ and $y(t)$, of course).

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