Frequency, Bandwidth, and Information
Our lives often depend on one interesting property of a signal: frequency. It is because the frequency is a very fundamental and important property of a signal. One of the ways to describe the characteristics of a signal is by analyzing its frequency components. With that knowledge, we can determine how fast information is transmitted with the signal, and also assess the integrity of the information contained within the signal. We are going to see how important frequency means to a signal.
Periodic and aperiodic signals
To deeper understand signals, first, we need to know about one of the properties of signal: periodicity. Periodicity is a property of a signal which describes its tendency to repeat itself after a finite period; while the value of the signal may vary over time, after a certain period, it may repeat the exact previous values. Based on this property, we can classify that there are two kinds of signals: periodic and aperiodic signals. A periodic signal is a signal which repeats the exact values after a certain finite period. Conversely, an aperiodic signal is a signal which does not show repetition. Most real signals are aperiodic. This should be intuitive because, for instance, we don’t speak repetitively. But to simplify things, let’s take a look at the periodic signal first.
A periodic signal has three main properties: waveform, period, and frequency. The waveform is the geometrical form of the signal values plotted over time. The period is a finite duration of time after which the signal will repeat the same value over and over again. Frequency is the exact inverse of the period; it describes how many repetitions a signal has for a certain duration. In physical terms, the period is measured in units of time (e.g. second), and frequency has a unit of Hertz, which is the inverse of a second.
One of the simplest periodic signals is a sine wave. Mathematically, its values follow sine function with a certain frequency, period, and amplitude (peak value). There are also other common periodic signals, such as square wave, triangle wave, and sawtooth. If we look closely, they kind of look similar, and we may ask, do those signals have any relation with sine wave?
The answer is yes, they do.
The basic idea of Fourier series
Those signals can be constructed as a sum of infinitely many sine waves with different amplitude, frequency, and phase. Mainly, we can view the magnitude and phase compositions in the frequency domain. Different waveforms have different frequency-domain compositions.
The summation of those sine waves is called the Fourier series. It was discovered by French mathematician Joseph Fourier. With the Fourier series, a periodic signal can be represented as a sum of many sine waves with discrete frequency steps, as illustrated above.
Let’s consider a square wave. What is the effect in the frequency domain, if the frequency of the signal is increased?
Intuitively, we can think that in the frequency domain, there will be more components in the higher frequencies. That is true. If we compare the frequency-domain composition between square waves with different frequencies, we can see that the signals with higher frequency have a ‘wider’ composition than the ones with lower frequency. Or in other words, the shorter a signal in the time domain, the wider it is in the frequency domain.
Fourier transform
We have seen quite a lot, but our journey is not complete yet. Most signals are aperiodic. Signals tend to stop or fade away. This is intuitive: we can’t speak a word forever. Pure periodic signals need to be generated with an infinite amount of energy.
To facilitate the analysis of real, aperiodic signals, we need to extend our idea of the Fourier series. Let’s consider a square wave as we’ve encountered before. If we increase the time separation between waves and see its Fourier series, we will see that the frequency components in the frequency domain will become denser as the separation between the waves increases.
How does that phenomenon relate to aperiodic signals? We can approximate an aperiodic signal as a periodic signal with an infinite delay between its pulses. As the time separation approaches infinity, we will see that its frequency composition in frequency domain gets closer, which will eventually form a continuous graph. The graph is called the Fourier transform of the signal.
In the case of a square wave with infinite delay, it becomes a single rectangular pulse. In the following picture, we can see that the shorter the rectangular pulse, the wider the frequency components.
Now we can see the difference between periodic and aperiodic signals. In the frequency domain, periodic signals have discrete frequency components, and aperiodic signals have continuous frequency components.
Real use-cases of Fourier transform
Fourier transform is very useful, especially in telecommunications. We can use it to analyze radio waves captured by an antenna. As in the air, many signals are traveling from different devices with different frequencies containing different information, we need Fourier transform to see the frequency components of the captured signal. Then, we can filter out the signal such that we can acquire the signal at our frequency of interest. Such a device is called filter, which passes signals in a particular frequency range and filters out the others. Filters are particularly useful, so useful that we can find it on every wireless device.
From the above example, we are assuming that communications signals use a finite band of frequency. This is true because signals with infinite frequency range are not very useful. They are impractical to transmit without loss of information because real communications channels and devices usually work well only on a particular frequency band. They can also interfere with other signals at different frequencies. Because of those impracticalities, wireless devices need to be assigned their frequency bands, and the signals they transmitted must use a finite frequency band.
Bandwidth
We often encounter the term ‘bandwidth,’ and we associate them with data speed. Although those terms are different, they are closely related. Wireless signals carrying digital information use a certain frequency band, with lower and upper boundaries. The width of that boundary is called ‘bandwidth,’ which explains the width of the band used by the signal. Its unit is in units of frequency (Hertz). Data speed or data rate is defined by how many bits a signal carries in a certain period. Its unit is bits per second (bps). Bits are carried in digital signals with a particular period. This period defines the data rate; the higher the data rate, the shorter the signal period. Let’s remember that a decrease in period means an increase in frequency, therefore the signal occupies a wider frequency band, and consequently, larger bandwidth.
We can also associate bandwidth with signal quality. Let’s see the example in the transmission of audio. Audio waves occupy frequencies from 20 Hz to 20 kHz. If we listen to sound directly from its source, for instance, direct conversation, we can hear the sound clearly without degradation of quality. But if the sound is transmitted through communication channels, for example, telephone, we can hear that the sound is distorted or muffled; its quality degrades. This happens as telephone filters signals.
Because telephone lines have many users, telephone providers need to manage the bandwidth capacity. To reduce bandwidth usage, the telephone only passes audio signals with a frequency range of 300 Hz — 3.4 kHz. The band is more limited than audio frequency (20 Hz — 20 kHz) or a typical human voice (80 Hz — 14 kHz). This means that there is a loss of information on the telephone. Any communications means which uses a narrower frequency band than the frequency band of original information it carries will impose information loss.
Frequency is an interesting property of a signal. Having appropriate knowledge of frequency is important for us to understand how information travels from one point to another.
