# Baseband Signals

The difference between the *baseband* signal and the *passband* signal in communications is really quite a simple one. The *baseband* signal refers to *any* signal that has not modulated a carrier waveform.

**NOTE: ** The use of the verb “modulated” there may made you think twice. If so, you are not alone. I always used to think that the carrier waveform was the object that acted on our baseband signal. This is the wrong way to think of it. The process of modulation is actually the process of how our baseband signal modifies the carrier waveform to create a *modulated*, *passband* signal! Thus we actually say that the baseband signal modulates the carrier waveform.

The important thing to realize is that a *baseband* signal can be an *analog* signal, a pulse code modulated signal (also actually analog really), or it can be *digital information*. Provided that the signal has not been used to modulate a high frequency carrier waveform, it is still considered to be *baseband*. Let’s go back to our model of a wireless communications system to understand *where* we may find *baseband* signals:

## Properties of Baseband Signals

The term B*aseband* is used due to the fact the signal has a frequency component that starts close to 0 Hz relative to the carrier wave’s frequency. *Baseband* signals have a defined *bandwidth* starting at a frequency greater than or equal to 0 Hz and ending at the highest *non-negligible* frequency component of the signal. And now that I have said that, and it made some sense, let me immediately seem to contradict myself.

### Bandwidth

It is important to note that there is no such thing as a *practical* time domain signal with a *finite bandwidth *as I just described in the previous sentence. If you were to see a *practical* time-domain signal with a *finite* bandwidth, it would have to continue on and on and on forever! These kind of signals exist in theory only.

The truth is that every *practical* time-domain signal we deal with has an infinite number of frequency components because it *has to be* limited in time for us to capture it! You *will* see time-domain signals with *most* of their power concentrated in a certain set of frequencies and negligible power in higher or lower bands. But they will always have some power in higher frequenciy component. You can attenuate the contribution of these frequency components with minimal error, but the total bandwidth of the signal will never be truly *finite*.

The picture below showing the benefits of a wide-band voice codec, actually illustrates the point very nicely (even though they stop counting at only 7kHz or so which makes it less than awe-inspiring). You can see how the energy in the *voice signal* *above* 4kHz is much less than that *below* 4 kHz. We can also apply the narrow-band filter of the digital telephone (shown in blue) to attenuate frequencies higher than 3400Hz and lower than 300Hz so that their contribution becomes *negligible and *we can *minimize* any errors that could be caused by sampling at 8 kHz. However, even the filtered signal (shown in blue) still does not have a truly finite bandwidth, its higher frequency components are just small enough for us to *ignore*.* *

The fact that all *practical* time-domain signals have infinite bandwidth has some critical implications for sampling time domain signals (read here).

### Frequency *Response*

*Response*

Baseband signals have a specific *frequency response* that describes the magnitude of each frequency component. Generally the value of each frequency component can be positive (add a sinusoidal wave of a given amplitude at some frequency) or negative (subtract a sinusoidal wave of given amplitude at some frequency).

### Negative Frequencies

Baseband signals don’t only have components with *positive-frequencies*. They also have components that operate at *negative-frequencies*. When I said that a baseband signal “*starts close to 0 Hz relative to the carrier wave’s frequency*” I omitted to mention that the baseband signal’s frequency-domain representation is in fact *centered around* the 0 frequency mark.

If you want to understand *where* these negative frequencies come from, I would suggest reading more about the Fourier Transform, Euler’s identity, and the complex sinusoid. Effectively what we need to understand is that when we look at the frequency domain representation of any *real numbered*, time-domain signal, we will end up with negative frequency components that have the exact same values as their corresponding positive frequency components. i.e the frequency response of a *real*, time-domain signal is symmetrical around the 0 Hz mark.

Here are some *real*, time-domain, *baseband* signals and their frequency-domain representations.

### Real vs Complex time-domain signals

Most engineering problems deal with *real* time-domain signals only. *Real* time-domain signals have *real* frequency components that are symmetrical around 0Hz mark and the imaginary frequency contributions on the positive and negative frequency bands cancel each other out! The above pictures show only the *real* frequency components.

*Complex* signals by comparison, (i.e time-domain signals that have both real and imaginary number components) have real and imaginary frequency components that are *not symmetrical* around the 0Hz mark and do not neatly cancel each other out. If you really want to read about it, go here.

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That’s all for now!

Rob