Christman GeneralizedReceiver

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How It Works: Generalized Receiver

The process of transmitting data, m(t), via a wireless signal, v(t), is shown below:

HW2 GenTran.jpg

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:

HW2 BandpassBaseband.png

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 \scriptstyle 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:

HW2 GenRec.jpg

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}


\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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