Signals and Systems/Fourier Series

The Fourier Series

The Fourier Series is a specialized tool that allows for any periodic signal to be decomposed into an infinite sum of everlasting sinusoids. This may seem counterintuitive to some people, but it is a subject that is demonstrable both mathematically and graphically. Practically, this allows the user of the Fourier Series to understand the signal as the sum of various frequency components. Th

Rectangular Series

The type of sinusoids that a periodic signal can be decomposed into depends solely on the qualities of the periodic signal.

Calculations

If we have a function f(x), we can decompose it into a sum of sine and cosine functions as such:

$f(x)=\frac{1}{2}{a}_0+\sum _{n=1}^{\mathrm{\infty }}[a_n\cos (nx)+b_n\sin (nx)]$

The coefficients, a and b can be found using the following integrals:

$a_n=\frac{1}{\pi }\underset{-\pi }{\overset{\pi }{\int }}f(x)\cos (nx)dx$

$b_n=\frac{1}{\pi }\underset{-\pi }{\overset{\pi }{\int }}f(x)\sin (nx)dx$

"n" is an integer variable. It can assume positive integer numbers (1, 2, 3, etc...). Each value of n corresponds to values for A and B. The sinusoids with magnitudes A and B are called harmonics. Using Fourier representation, a harmonic is an atomic (indivisible) component of the signal, and is said to be orthogonal.

When we set n = 1, the resulting sinusoidal frequency value from the above equations is known as the fundamental frequency. The fundamental frequency of a given signal is the most powerful sinusoidal component of a signal, and is the most important to transmit faithfully. Since n takes on integer values, all other frequency components of the signal are integer multiples of the fundamental frequency.

Signal Properties

Various signal properties translate into specific properties of the Fourier series. If we can identify these properties before hand, we can save ourselves from doing unnecessary calculations.

Odd and Even

If the signal is even, it is composed of cosine waves. If the signal is odd, it is composed out of sine waves. If the signal is neither even nor odd, it is composed out of both sine and cosine waves.

DC Offset

If the periodic signal has a DC offset, then the Fourier Series of the signal will include a zero frequency component, known as the DC component. If the signal does not have a DC offset, the DC component has a magnitude of 0.

Discontinuities

If the periodic signal has no discontinuities, then the signal is composed of of sinusoids of all harmonics. If a signal has discontinuities, then it is composed of sinusoids lying on only the odd harmonics (1, 3, 5, etc...). This is important because a signal with discontinuities will require twice as much bandwidth to transmit the same number of harmonics as a signal with no discontinuities

Half-Wave Symmetry

If the periodic signal has halfwave symmetry, the magnitudes of each progressive harmonic N will fall off as a factor of N².

Summary

By convention, the coefficients of the cosine components are labeled "A", and the coefficients of the sine components are labeled with a "B". A few important facts can then be mentioned:

• If the function has a DC offset, A₀ will be non-zero. There is no B₀ term.
• If the signal is even, all the B terms are 0 (no sine components).
• If the signal is odd, all the A terms are 0 (no cosine components).
• If the function has discontinuities, then all the even coefficients (A and B) are zero.
• The Fourier series of a sine or cosine wave contains a single harmonic because a sine or cosine wave cannot be decomposed into other sine or cosine waves.

Polar Series

The Fourier Series can also be represented in a polar form which is more compact and easier to manipulate.

If we have the coefficients of the rectangular Fourier Series, A and B we can define a coefficient X, and a phase angle φ that can be calculated in the following manner:

$X_0=A_0$

$X_n=\sqrt{A_n^2+B_n^2}$

${\varphi }_n={\tan }^{-1}\left(\frac{B_n}{A_n}\right)$

We can then define f(x) in terms of our new Fourier representation, by using a cosine basis function:

$f(x)=X_0+\sum _{n=1}^{\mathrm{\infty }}{X}_n\cos (n\omega +{\varphi }_n)$

The use of a cosine basis instead of a sine basis is an arbitrary distinction, but is important nonetheless. If we wanted to use a sine basis instead of a cosine basis, we would have to modify our equation for φ, above.

Exponential Series

Using Eulers Equation, and a little trickery, we can convert the standard Rectangular Fourier Series into an exponential form. Even though complex numbers are a little more complicated to comprehend, we use this form for a series of reasons:

1. Only need to perform one integration
2. A single exponential can be manipulated more easily then a sum of sinusoids
3. It provides a logical transition into a further discussion of the Fourier Transform.

Negative Frequency

The Exponential form of the Fourier series does something that is very interesting in comparison to the rectangular and polar forms of the series: it allows for negative frequency components. To this effect, the Exponential series is often known as the "Bi-Sided Fourier Series", because the spectrum has both a positive and negative side. This, of course, prods the question, "What is a negative Frequency?"

Negative frequencies seem counterintuitive, and many people would be quick to dismiss them as being nonsense. However, a further study of electrical engineering (unfortunately, outside the scope of this particular book) will provide many examples of where negative frequencies play a very important part in modeling and understanding certain systems. While it may not make much sense initially, negative frequencies need to be taken into account when studying the Fourier Domain.

Negative frequencies follow the important rule of symmetry: For real signals, negative frequency components are always mirror-images of the positive frequency components. Once this rule is learned, drawing the negative side of the spectrum is a trivial matter once the positive side has been drawn.

However, when looking at a bi-sided spectrum, the effect of negative frequencies needs to be taken into account. If the negative frequencies are mirror-images of the positive frequencies, and if a negative frequency is analogous to a positive frequency, then the effect of adding the negative components into a signal is the same as doubling the positive components. This is a major reason why the exponential Fourier series coefficients are multiplied by one-half in the calculation: because half the coefficient is at the negative frequency.

Bandwidth

Bandwidth is the name for the frequency range that a signal requires for transmission, and is also a name for the frequency capacity of a particular transmission medium. For example, if a given signal has a bandwidth of 10kHz, it requires a transmission medium with a bandwidth of at least 10kHz to transmit without attenuation.

Bandwidth can be measured in either Hertz or Radians per Second. Bandwidth is only a measurement of the positive frequency components. All real signals have negative frequency components, but since they are only mirror images of the positive frequency components, they are not included in bandwidth calculations.

Bandwidth Concerns

It's important to note that most periodic signals are composed of an infinite sum of sinusoids, and therefore require an infinite bandwidth to be transmitted without distortion. Unfortunately, no available communication medium (wire, fiber optic, wireless) have an infinite bandwidth available. This means that certain harmonics will pass through the medium, while other harmonics of the signal will be attenuated.

Engineering is all about trade-offs. The question here is "How many harmonics do I need to transmit, and how many can I safely get rid of?" Using fewer harmonics leads to reduced bandwidth requirements, but also results in increased signal distortion. These subjects will all be considered in more detail in the future.

Pulse Width

Using our relationship between period and frequency, we can see an important fact:

$f_0=\frac{1}{T}$

As the period of the signal decreases, the fundamental frequency increases. This means that each additional harmonic will be spaced further apart, and transmitting the same number of harmonics will now require more bandwidth! In general, there is a rule that must be followed when considering periodic signals: Shorter periods in the time domain require more bandwidth in the frequency domain. Signals that use less bandwidth in the frequency domain will require longer periods in the time domain.

Further Reading

Wikipedia has an article on the Fourier Series, although the article is very mathematically rigorous.