The Fourier Series and Fourier Transform

Module 3: The Fourier Transform

It is useful to describe any signal in terms of certain primitives. This may be thought of as a language or vocabulary to describe signals. There are several alternative methods of describing signals. One of the most common is the Fourier method. This method uses sinusoids to describe signals. However, given a non-periodic signal (like the EMG), we can suppose that it will repeat itself after infinite time.

The Fourier method of describing signals comprises two parts, one is obtaining the descriptors given an arbitrary signal, and the second is synthesizing the actual signal given the descriptors. The former is termed Fourier analysis and the latter Fourier synthesis.

In order to show how the Fourier method works, we can see how the combination of several sinusoids can produce almost any signal we want. Thus, the signal we want will be described by the set of sinusoids. Each sinusoid in this set will have a specific amplitude and relative phase. The "phase" refers to the "starting point" of the each sinusoid with respect to a reference point in time.

Synthesis of signals from sinusoidal primitives

1-Dimensional signals

1-D sinusoids can be represented on a flat screen (or paper) using one of the screen's dimensions for the independent variable and the other for the dependent variable. In the accompanying simulation of sinusoids (hyperlink at the end of this paragraph), the horizontal dimension represents time, and the vertical dimension represents the signal amplitude. Different sinusoids are shown in different colours. Upto thirty-one sinusoids can be generated simultaneously. (The first scroll bar labelled "0" is an unvarying signal of constant amplitude - i.e., zero frequency). These are all multiples of the basic sinusoid which cycles once per second. The addition of all the thirty-one sinusoids is shown in black in the lower panel below the sinusoid set.

The amplitude and the phase of each sinusoid can be adjusted by the user.

Adjust the scroll bars to see how different signals can be synthesised from this set of sinusoids.

Clicking on the "ECG" or "Aortic Pressure" buttons will set the amplitudes and phases to specific values so that the appropriate signal is synthesised.
Interactive Example: Fourier synthesis of time signals

Learning exercise I-a:
1. Adjust the amplitude of one sinusoid while the others are at zero amplitude. Does the adjustment of phase produce any noticeable effect?
2. Set the amplitude of two sinusoids non-zero and the rest zero. Adjust the phase of one of the sinusoids. How is the composite signal affected? Does it change shape? Is the notion of phase clear?
3. Adjust several sinusoids and see if you can produce a recognizable signal? If you can't, why not?
4. Click on the ECG button to have the amplitudes and phases set to generate an ECG signal. See how by adjusting any one of the amplitudes or phases, the entire ECG signal changes appearance. What do you think will happen if all the amplitudes are halved? What do you think will happen if all the phases are halved?
5. Click on the Aortic P button to have the amplitudes and phases set to generate an Aortic Pressure signal. Repeat the same operations as in question 4. What happens to the waveform when the phases of frequencies above "10" are adjusted? Do you notice that the amplitudes of the sinusoids above 10 are zero?
6. What signals are produced when you click on the "Random" button? Are these signals periodic? Can you notice that although they are usually not recognizable as any known signal, there is a pattern that is repeated in each signal?
Frequency Spectrum of 1-D signals

The amplitudes and phases of the sinusoids comprising any signal can be drawn pictorially in a graph with the amplitudes on the vertical axis and the frequency number on the horizontal axis. The result would look very much like the positions of the scrollbars in the preceding simulation. The following two figures show both the scrollbars in the ECG synthesis and the corresponding amplitude plotted against frequency. This plot is called the amplitude spectrum. As seen in the simulation the phase is also important in specifying the signal, and therefore, the spectrum should show both the amplitude and phase of the component sinusoids plotted against frequency. The power spectrum is a compact (but reduced) way of showing the Fourier components, where the square of the amplitude (which is related to the power) is plotted against frequency.

The following figure shows the positions of the amplitude scrollbars set for ECG synthesis.

The following figure shows the positions of the amplitude scrollbars in the above figures connected by straight lines. The horizontal axis shows the frequency (corresponding to the sinusoid controlled by that scrollbar) and the vertical axis shows the amplitude of that sinusoid (the position the scrollbar has been set at). This is the amplitude spectrum of the composite signal, i.e., the ECG.

2-Dimensional signals

2-D sinusoids can be represented on a flat screen using both the screen's dimensions for the independent variables and the dependent variable must be represented using different colours on the screen. The simplest scheme is to use different intensities of grey - black indicates the smallest amplitude and white the largest amplitude. A simple sinusoid in two dimensions is like a corrugated sheet with the length and breadth of the sheet representing the independent variables and the undulations representing the amplitude of the signal. If the undulations of the corrugated sheet are in the X-direction it is said to be a frequency on the X-axis, if they are in the Y-direction it's a frequency on the Y-axis. Undulations diagonal to the X-Y plane can be obtained from a sinusoid that is on the diagonal line between the X & Y axes. Similarly, any other orientation of the undulations will be obtained from sinusoids from other points between the X & Y axes. In the accompanying simulation of 2-D signals (hyperlink at the end of this paragraph) you have 64 adjustable sinusoids whose amplitudes and phases can be adjusted. One image of the brain synthesized from sinusoids is also included. However, in order to synthesise this image purely from sinusoids, about 400 sinusoids are required. Since, it is not possible to have 400 scrollbars for amplitude and another 400 for phase on the computer screen, only 64 have been provided. Therefore, the user can manipulate the image only to a very small extent.

Interactive Example: Fourier synthesis of images (Note: This needs Java 2 - update your JRE if it doesn't run)

Learning exercise I-b:
1. Adjust the amplitude of one sinusoid on the screen. What does the adjustment of phase do?
2. Set the amplitude of two sinusoids non-zero and the rest zero. Adjust the phase of one of the sinusoids. How is the composite signal affected? Does it change shape? Is the notion of 2-dimensional phase clear?
3. Adjust several sinusoids and see if you can produce a recognizable signal? If you can't, why not?
4. What images are produced when you click on the "Random" button?
Frequency Spectrum of 2-D signals

As in the case of 1-dimensional signals, the amplitudes and phases of the component sinusoids in a Fourier decomposition of a 2-dimensional signal can be plotted against the frequency.

Such Fourier spectra that have two independent variables and one dependent variable can be represented as an image (with frequency on the X-Y plane and amplitude as colour intensity). The following figure shows two different ways of representing 2-D Fourier amplitudes plotted against frequency (the horizontal plane represents frequencies in two directions). The figure on the right is a 3-dimensional surface plot diagram, and the figure on the left is a flat contour map diagram.

The Fourier Series and the Fourier Transform
- Any periodic function can be decomposed into a set of sinusoids. The frequencies of these sinusoids is a multiple of the fundamental frequency, f_o, which is the frequency of repetition of the periodic function.
- Each sinusoid has two parameters, namely its amplitude and its phase in relation to some reference point.
- Conversely, using a particular set of sinusoids we can synthesise a given periodic waveform of any arbitrary shape. This is Fourier Synthesis and it can be written in four main equivalent forms:
- The first form is convenient for graphical and physical explanation, while the last form, the complex exponential form, is convenient for calculation.
- For the complex exponential form, each of the coefficients, a_k is a complex number.
The interactive example shown earlier demonstrates the first of the above equations.
- The converse of Fourier Synthesis is Fourier Analysis whereby we decompose a periodic function into a set of sinusoids.
- Fourier Analysis provides a method of obtaining the magnitude and phase values for the set of sinusoids. (Note: T_o is the period of the periodic function, x(t), and T_o=1/f_o)
- An alternative convention is to use the notation, a[k] for the Fourier coefficent, a_k.
- This pair of equations, namely, the Fourier Synthesis and Analysis equations can be used to decompose any signal into sinusoids and also to reconstitute a signal from sinusoids.
The Fourier Transform
- The above definition of Fourier analysis and synthesis is only for periodic function, x(t). In the case of non-periodic functions we can assume the period to be infinite and extend the above equations:
Note that the coefficients of the Fourier series, a[k], have been replaced with a continuous function, X(w).

In summary:
- A periodic function, x(t), can be decomposed into a set of sinusoids whose frequencies are multiples of the basic periodicity of the original function.
- In the case of a non-periodic function, the period being infinite, the fundamental frequency is zero, and therefore, the Fourier series becomes a continuous function, namely, the Fourier Transform.
Calculation Examples

Consider the two signals shown below. One is a periodic square wave and the second is a non-periodic square/rectangular pulse.

The Fourier series calculation can be applied to the periodic function, and the Fourier transform calculation can be applied to the non-periodic one.

Euler’s relation has been used in the results in the final form given above.

[ Note: Euler’s relation is: e^j^x=cos(x)+jsin(x)

The magnitudes of Fourier series coefficients and the Fourier transform function are plotted in the figures below.

See the textbook for details of the above calculations. The above examples show the similarity and the difference between the Fourier series and the Fourier transform.

Finally, use the interactive example to compare the above Fourier decomposition with the square wave synthesis in the program. Note that the synthesis of the square wave using a limited number of sinusoids does not reproduce the square wave very well. As you increase the number of sinusoids (number of Fourier series terms), the square wave becomes better. But because of the discontinuity (sudden change) in the square wave, the reconstruction never becomes perfect, but always retains some oscillations at the edges- this behaviour was described in detail by Gibbs and is referred to as Gibbs’ phenomenon.

© Suresh Devasahayam

The Fourier Series and Fourier Transform

Module 3: The Fourier Transform

Synthesis of signals from sinusoidal primitives

1-Dimensional signals

Learning exercise I-a:

Frequency Spectrum of 1-D signals

2-Dimensional signals

Learning exercise I-b:

Frequency Spectrum of 2-D signals

The Fourier Series and the Fourier Transform

The Fourier Transform

Calculation Examples