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December 4, 2013

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Digital Communications, Electrical Engineering, Signal processing

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Passband and Baseband signal representation

I think this is worth bringing up because I never covered it in any signal processing class, but I certainly went over several times in different wireless/digital communication classes at UT. I had to use this in an acoustics application at work once on somebody else’s project, and it occurred to me this may not be common knowledge (my friend certainly didn’t know about it at the time). So let’s talk about it.

So let’s say we have a band-limited passband signal, say, an ideal noiseless radio signal going from a wireless radio to some other wireless radio. We can represent any signal of this type mathematically as

s(t) = A(t) \cos( \omega t + \theta (t) )

Here, the amplitude of our signal is represented by A(t) as a function of time, \omega is the carrier frequency (in radians) that our passband signal is centered at, and \theta is a phase component that is a function of time. So right off the bat, we can see that amplitude modulation and frequency modulation can be easily put into this form because of the general way we are expressing amplitude and phase of the signal.

We can express our passband signal in a slightly different way if we say that x(t) = A(t) \cos( \theta (t) ), y(t) = A(t) \sin( \theta(t) ), and with some trigonometry identities, we can get the following representation of our signal.

s(t) = x(t) cos( \omega t ) - y(t) sin( \omega t )

With not too much imagination, you can see that A(t) = \sqrt{x^{2}(t) + y^{2}(t)}, \theta (t) = \tan^{-1}\bigg( \frac{y(t)}{x(t)} \bigg)

Finally, we can get our baseband representation of our signal by taking the x and y components (the inphase and quadrature parts, respectively), and getting the following.

s_{bb}(t) = x(t) + jy(t)

where j is our imaginary number because I’m an electrical engineer and that’s what I’m used to using.

For the application I had at work, my friend had a baseband signal acoustic signal. Clearly it could not be played as is because it had real and imaginary parts to it, so we multiplied the real component by cosine, imaginary component (without the imaginary part) by negative sine, and we got what sounded like an underwater ping with reverberation from the ocean floor.

So why did I need to learn this for wireless communications? We know that radio signals are at some high frequency thanks to packaging materials on electronics. However, working with signals while they are at said frequency is needlessly difficulty and costly, especially when the actual bandwidth of our signal is much smaller. So we can work with a signal at baseband, spin it up to radio frequency, and receive it by spinning it down to baseband again and then decoding it. Essentially, everything we want to do, we can do it at baseband, and just spin it up to radio frequency when we’re done.

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