Time and Frequency Domain

Periodic signals can be examined from two points of view, or domains. These two domains are the time domain and the frequency domain. For periodic signals, time and frequency are the inverse of each other. Specifically, a periodic signal can be quantified by its period which is how long it takes for the signal to repeat itself or by its frequency which is how many times the signal repeats itself in a given time.:

period = \frac{1}{frequency}

and

frequency = \frac{1}{period}

Since period and frequency are inverses of each other, time domain analysis and frequency domain analysis are, in a way, inversely related as well.

Time Domain

The time domain refers to a description of the signal with respect to time. The basic tool for analyzing signals in the time domain is called an oscilloscope. An oscilloscope (or ‘scope) displays a two-dimensional graph of a signal’s magnitude in the y-axis, and time in the x-axis (for more info, read the “Features and Uses” section of wikipedia’s oscilloscope entry at http://en.wikipedia.org/wiki/Oscilloscope ). While the basic use of a scope is to determine the magnitude of the signal as time is changing, it can also be used to indirectly measure the frequency of a signal if that signal is periodic. To do this, simply configure the scope to show at least one period of the signal, then measure the time of that period.  The frequency of the signal is then   \frac{1}{signal period}.

The following picture shows two sine waves represented in the time domain on an oscilloscope (note that one of the sine waves is distorted at the point where the signal crosses 0V). The vertical scale shows 500mV per division and the horizontal scale shows 250uS per division.

Sine Waves in Time Domain
Sine Waves in Time Domain

Frequency Domain

When you do a measurement of the signal’s frequency, then you are said to be analyzing the signal in the frequency domain. While some oscilloscopes can be used for analyzing a (periodic) signal in the frequency domain (they need special functionality), a better tool for doing this is called a spectrum analyzer. A spectrum analyzer displays a two dimensional graph of a signal’s power in the y-axis, and the signal’s frequency in the x-axis.  Such a graph is called the frequency spectrum of a signal because it shows how strong a signal is at all frequencies.

The simplest example of a signal in the frequency domain, is the perfectly periodic signal the sine wave. The frequency spectrum of a 100 Hz sine wave consists of only one frequency (100 Hz), and so the frequency spectrum will look like this:

Frequency Spectrum
Frequency Spectrum

Other periodic, non-sinusoidal signals can also be represented in a frequency spectrum and these non-sinusoidal signals will be shown as signals made up of multiple frequencies as you can see in this picture (which is the frequency spectrum of the distorted sine wave in the earlier picture).

Frequency Spectrum of Distorted Sine Wave
Frequency Spectrum of Distorted Sine Wave

The next section explains why non-sinusoidal signals have multiple frequencies.

Fourier Analysis

As mentioned above, time domain and frequency domain are inversely related.In fact, if you know the mathematical description of the signal in one domain, it is possible to perform an operation on the signal to see what it looks like in the other domain. This operation is called the Fourier Transform.

This section is not going to get into the mechanics of Fourier Transforms (sigh of relief), but what you do need to know is the basis of Fourier Transforms. In 1826, the French mathematician Baron Jean Fourier proved that any periodic waveform can be broken down into a series of sinusoids. This series consists of a sinusoid at a frequency equal to the repetition rate of the waveform (i.e., the frequency of the waveform or  \frac{1}{period of waveform}) – this is called the fundamental frequency – plus a series of sinusoids at frequencies which are integer multiples of the fundamental frequency.  These sinusoids are called harmonics.  The fundamental frequency is the first harmonic.  The second harmonic is a sine wave two times the frequency of the fundamental frequency, the third harmonic is a sine wave three times the frequency of the fundamental etc.  So a periodic waveform can be expressed like this:

waveform = fundamental + 2nd harmonic + 3rd harmonic + 4th harmonic + …

Let’s look at an example for clarification.

Square Wave Example

A square wave is a periodic signal that alternates between two different levels, ideally with 0 time of transition between one level and the next. In the real world this is impossible to achieve, but it is possible to get a close approximation to an ideal square wave by adding a sine wave at the fundamental frequency of the desired square wave plus harmonics of the fundamental frequency. The more harmonics that you add, the closer the approximation will be. The following figure shows closer and closer approximations to a square wave by adding more and more harmonics.

Square Wave Approximations
Square Wave Approximations (Source Don Johnson CC-BY-1.0 via: http://cnx.org/content/m0041/latest/)

In this figure, K is the number of harmonics that are used in each approximation. The top graph shows a square wave with a sine wave of the same (fundamental) frequency superimposed on top of it. The second graph shows a square wave with an approximation that is equal to:

Asin(\alpha) + A_2sin(2\alpha) + A_3sin(3\alpha)+A_4sin(4\alpha) + A_5sin(5\alpha)

 

Where

  • \alpha is the fundamental frequency
  • A_2 and A_4 are equal to 0
  • A_3 is 1/3
  • A_5 is 1/5

The third graph includes even more more harmonics, and so has an even closer approximation, and the last graph has the most harmonics, and has a very close approximation to a square wave.

What is the bottom line here? We can approximate any periodic signal by adding sine waves of increasing frequency together. However as we add more and more sine waves together to get a better and better approximation, the frequency of those sine waves increase. In order to get a perfect representation of a periodic signal (such as a square wave) requires that we add sine waves of frequency approaching infinity. So for a perfect square wave, we say that it has infinite bandwidth.