# Fundamentals of Signals Processing

## Eric Johnston June 11, 2012

### Submitted as coursework for PH250, Stanford University, Spring 2012

To many people wireless communication seems to be somewhat of a mystery, how can "electromagnetic radiation" manage to carry a phone call, eventually to another cell phone? This question cannot be answered easily since there are many different components required for the transmission of a signal. The microphone is an electromechanical device, the circuitry required to modulate and transmit a high frequency signal requires expertise in circuit design and impedance matching. When a signal eventually reaches another phone, the process needs to be reversed: the signal is demodulated and converted back to a sound wave. Every process will not be explored in this article, but rather the mathematical foundations of wireless communication will be laid out. The following questions will be answered: How can signals be represented in the frequency domain (this is the domain of bandwidth). What is the mathematical meaning of signal modulation? Demodulation? Finally, how is the correct signal extracted (you don't want to listen to your neighbors phone call)? Once again, specifics such as circuitry will not be laid out here, but the overall process from a mathematical point of view will be explained.

## Describing a Signal by its Frequencies

It turns out that we can express any repeating time signal by a sum of sines and cosines that are multiples of the fundamental frequency - that is, the frequency that the time signal repeats.  For example, a repeating signal can be written in the form of:

A + C1 cos(x ) + C2 sin(x) + C3 cos(2x) + C4 sin(2x) + C5 sin(3x) + ...

Where in this case the fundamental frequency is 2π. Writing out a repeating signal in this form is often called a fourier series. Why is it useful? Well now instead of drawing the signal as some repeating function of time, we can instead represent it as some function of its frequencies. The lowest frequency is the fundamental frequency (or possibly the constant frequency A), and the magnitude at this frequency is related to the constant coefficients. This means that for each signal we can describe its bandwidth as the region between a signal's lowest frequency and its highest frequency. So when bandwidth is allotted to a radio station or a cell phone company, what is being allocated is a region of frequencies that one can transmit. Signals that are composed of combinations of frequencies within this bandwidth can be sent.

This seems useful, but there are two problems. First, cell phone conversations are not infinitely repeating signals. They have a start and an end, and this means they cannot be expressed as a simple sum of sines and cosines. Second, it's great that different bandwidths can be allocated, but conversations all more or less use the same frequencies: the frequencies of the human voice. Moreover, these frequencies cannot simply be converted to electromagnetic radiation.

Both problems can be solved using mathematical tricks. First, it turns out that if one takes a non-repeating signal, and repeats that signal with a fundamental frequency f, a fourier series is generated. In the limit that the frequency f goes to infinity, the discrete sums of sines and cosines become continuous. Transforming a signal into a continuous frequency representation is known as the fourier transform of a signal. Modern computer hardware can take the fourier transform of signals to extract their frequency components incredibly fast and efficiently.

The second problem, how to make voice frequencies have unique high frequency bandwidths, can be solved using frequency modulation.

## Modulating a Signal

The objective here is to transform a signal that has a low frequency into another signal that has a high frequency which can be transmitted. Referring to the previous equation for for series, what would happen if this signal were multiplied by a cosine with some frequency fc? The details can be explored on one's own, but conceptually speaking this multiplication would yield two copies of the original signal (when viewed in the frequency domain). These copies would occur at plus or minus the carrier frequency fc. Well this is very useful! By simply adjusting the carrier frequency, the original signal can be changed to a signal that looks the same (from a frequency point of view), except it is centered about the new carrier frequency. This means that if different electromagnetic signals are all emitted, but with different carrier frequencies, and each signal has its own bandwidth, many different signals can be sent and received at once.

However, once again there are a couple problems. First, a signal can be modulated to operate around a carrier frequency, but it is not immediately apparent how to demodulate this signal, or in other words how to go from a signal at some carrier frequency back to a signal that can be listened to. Second, how can one eliminate any noise in a demodulated signal?

## Demodulating a Signal

A signal is received that is centered around plus and minus the carrier frequency. In order to bring this signal back to one with the bandwidth of the human voice, the signal can simply be multiplied by the carrier frequency again. Two copies of the carrier frequency signal return to the original voice signal. This signal can be played with a speaker. 

Throughout this process lots of noise and other signals often need to be removed prior to or after a modulation or demodulation. Frequencies outside a given bandwidth can be removed using what is called a bandpass filter. Filters do exactly as their name says. Bandpass filters allow frequencies within a given band to remain unchanged while frequencies outside the range are suppressed. Mathematically, this can be done by multiplying a signal in the frequency domain by a square function that is constant at the desired bandwidth and zero everywhere else. In practice, a perfect filter such as this is impossible to make, although the theory of filter design is a well-studied area. Filters with smooth, rounded edges (such as a Parks-McClellan) instead of square edges are used. 

In practice there are many different modulation and signal transmission techniques, since many of the details were glazed over here. However, the basic concepts, that is, understanding a signal in terms of its frequency representation, modulation, demodulation, and filtering, are integral parts of any signal transmission.

© Eric Johnston. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

## References

 A. V. Oppenheim and A. S. Willsky, Signals and Systems, 2nd Ed. (Prentice Hall, 1996).

 B. P. Lathi and Z. Ding, Modern Digital and Analog Communications Systems, 4th ed. (Oxford, 2009).

 A. V. Oppenheim and R. W. Schafer, Discrete Time Signal Processing, 2nd Ed. (Prentice Hall, 1999).