Signals and Systems, their analysis and characterization is one of the most passionate tasks any Communications Engineer would want to get drawn into. That given, there has been a thorough understanding of the fundamentals. Otherwise, a scenario of disconnect as it is common with many of us would occur.

In this post, we try to understand one of the most fundamental concepts of Signals and Systems.

Frequency Spectrum of signals using the Fourier Analysis: Without a clear understanding of this fundamental concept, the communication student would end up doing just the Algebra with no application whatsoever.

**What are the ‘signals’ and ‘systems’ we are talking about?**

Signals are the simplest forms of energy carriers, and hence information carriers as well. Seismic waves, sound waves, Electro cardiogram information, galvanic current on the skin, microwaves, radio waves, flow of gases and liquids, vibrations in solids, cosmic radiation, pressure, voltage and temperature gradients,and the list would go on!

While signals are the carriers, Systems are, simply put, specifically designed (either by nature, or us humans) to be able to transmit, receive and analyze these signals. These systems have certain properties which makes our task of analyzing them easier, and consistent; this we might want to discuss in subsequent posts.

### **Periodicity of signals**

Another desirable feature in signals, while analyzing them is periodicity. When a signal repeats itself at regular intervals of time, it is said to be periodic in nature. This would in turn make our task of analyzing them a time-limited one, instead of having to probe them to eternity.

While there are many real life signals which are periodic in nature, making it easier for us to predict and analyses them , we also encounter aperiodic signals with aberrations and sporadic nature which are of prime import to us (ex, the seismic waves).

**Beauty of sinusoids**

Now, we look at one of the most beautiful formulas: Euler’s identity

e is Euler’s number, the base of natural logarithms,

i is the imaginary unit, which satisfies i^{2} = −1, and

π is pi, the ratio of the circumference of a circle to its diameter.

This is a special case of one other formula we would deeply be interested in – *The Euler’s Formula*

where cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians.

The usefulness of this formula will be understood once we have seen the Fourier expansion of a periodic signal. But until then, the point to be taken is that sinusoids are impeccably beautiful mathematical functions which are periodic, fit into Euler’s formula and make our computational lives easier and exciting.

**The Optical Prism:**

Firstly, let us look at a close analogy to the concept of Frequency Spectrum analysis, using the Light Scattering by an optical prism as an example.

What does an optical prism do to white light?

Yes, it does scatter the white light into its constituent wavelengths and gives us the VIBGYOR pattern. Such scattering allows us to study the contribution by each of the component frequencies and hence attribute specific traits and phenomena associated with a set of frequencies in a spectrum.

The usefulness of such abilities,i.e, to be able to look at and analyze constituent frequencies is immense. For instance, the Frequency spectrum analysis allows us to describe local conditions on a distant star, its composition and gives us details which otherwise would have been impossible to understand.

**The Mathematical Prism**

The central starting point of Fourier analysis is Fourier series. They are infinite series designed to represent general periodic functions in terms of simple ones, namely, cosines and sines.

So, thinking of Fourier Series as a Mathematical Prism, yielding the constituent frequency components of various real world signals would help us appreciate the idea better.

Fourier series are very important to the engineer and physicist because they allow the solution of ODEs (Ordinary Differential Equations) in connection with forced oscillations, and the approximation of periodic functions. We confine ourselves to the approximation of periodic functions in this article.

Fourier series are, in a certain sense, more universal than the familiar Taylor series in calculus because many discontinuous periodic functions that come up in applications can be developed in Fourier series but do not have Taylor series expansions. Fourier analysis can also be extended to non-periodic signals or in applications where we require the whole of x-axis, to arrive at integral forms of Fourier series called the Fourier Integrals/ Fourier Transforms.

Fourier analysis is a precious mathematical tool using which the component frequencies of any real world signal can be analyzed. When the signal to be analyzed in periodic with a period 2 π, its frequency approximation is given by the Fourier Series

The Fourier series expansion for any periodic signal with a period 2 π is given as,

Where a0, an and bn are the coefficients of the Fourier series expansion.

(Further mathematical elaboration Fourier Series can be obtained in any standard Mathematics book).

The above formula shows that any periodic signal can be decomposed into a DC content ( zero frequency) and weighted sines and cosines of varying frequency. To approximate signals with full fidelity, in ideal conditions the expansion should comprise of infinite terms.

As dealing with infinite terms in a summation is not pragmatic when trying to implement, in most of the cases an adequate number of terms are used to get the approximation. The following example from Wikipedia shows the impact of increasing the terms of the series, to yield better convergence.

The approximation tends to converge to the original wave (in this case a square waveform), and as we note with the increase of terms from one to four in the series the Fourier Series is able to approximate the Square wave better. As mentioned previously, for complete convergence infinite terms need to be used for the approximation.

Fourier approximation of square wave

Another example in this animation shows the approximation of a saw-tooth waveform by increasing the terms of the Fourier Series.Saw tooth waveform

Fourier approximantion of a saw tooth wave**Frequency Analysis**

Now that we have seen that any periodic signal can be decomposed into sines and cosines, this gives us an excellent opportunity to understand and interpret signals in another ‘domain’. From varying time domain, when signals are approximated into their Fourier Series, we can now analyze the behavior of these signals in the ‘varying frequency domain’.

When signals are transformed into Frequency domain, certain properties which are subtle and unnoticed in the Time domain get accentuated in the Frequency domain. For instance, the clumsy process of convolving two time domain sequences for a filtering application supposedly, will require us to perform multiplication of shifted instances of the original sequence with the second sequence.

The same operation in frequency domain would be a simple element wise multiplication of the Spectral components.

Another easy example of Frequency analysis is with respect to Image Processing.

Consider the Fourier Spectrum of a gray scale image of size MxN, and it is considered to be periodic with a discrete period of M and N in the two dimensions.

As seen before, the Fourier Series would comprise of the following:

- A DC component with zero frequency: With respect to the Frequency spectrum of an image this corresponds to the average gray scale level present in the image

- Low frequency sinusoids: The first few terms of the expansion would correspond to the gradual progressions in the image. For instance, a landscape, or a wall.

- High Frequency sinusoids: These are the higher order frequency components which indicate the sudden changes in the image regions. For, example edges of a wall, or boundaries in the image.

If there is an application to either vary the average gray scale level of the image, or to enhance the details (gradual progressions) or to enhance the edges, one only needs to perform appropriate filtering on the Frequency Spectrum of the image. For instance, a lowpass filtering of the image spectrum will yield a blurred image with all the edges removed (high frequencies removed) and quite the opposite with highpass filtering. The same operation in Time domain might end up being counter-intuitive unlike in Frequency domain.

Apart from this, operations such as building Filter Banks, advanced application based signal processing etc.. are rendered simple (in an implementation point of view).

Although, the very same results can be obtained by performing equivalent steps in Time domain, the ease and subsequent intuition like understanding that the Frequency Domain yields is austere in efforts required, yet grand in the results it yields!

References:

https://secure.wikimedia.org/wikipedia/en/wiki/Euler’s_formula

https://secure.wikimedia.org/wikipedia/en/wiki/Euler’s_identity

https://secure.wikimedia.org/wikipedia/en/wiki/Fourier_series

Advanced Engineering Mathematics by Erwin Kreyszig

Digital Image Processing by Richard Gonzales

## 1 Trackback or Pingback for this entry:

[…] https://eceforum.wordpress.com/2011/10/24/fourier-analysis-the-mathematical-prism/ […]