Fourier Transform and its applications
Jatin Kumar
Murray State University

Abstract
It has been widely recognized that waveforms are an integral part of the various universe phenomenon. Waveforms can be used to represent almost everything in the world. Therefore it is understandable that concepts related to waveforms or signals are extremely important as their applications exist in a broad variety of fields. The processes and ideas related to waveforms play a vital role in different areas of science and technology such as communications, optics, quantum mechanics, aeronautics, image processing to name a few. Even though the physical nature of signals might be completely different in various disciplines, all waveforms follow one fundamental principle; they can be represented by functions of one or more independent variables.
This paper would focus on the concept of Fourier Transform, the technique through which signals can be deconstructed and represented as sum of various elementary signals. It briefly describes Linear Time Invariant systems and their response to superimposed signals. Fourier transform has many applications in physics and Engineering. This paper would also cover some of Fourier Transform applications in telecommunication and its impact on society.

Introduction

Some of the basic signals that exist in the world and are useful in various technology fields are continuous and discrete time signals. These signals depend on a single independent variable. Generally the independent variable is considered to be time (though it is not universally true).We will focus our discussion on the signals which are considered to be a function of a single independent variable. With the use of transformation techniques, a signal can be deconstructed. The continuous and discrete signals can be represented as composite signals with the help of Fourier transform. The result of this transformation provides an output which is easy to interpret, to be changed and analyzed
To understand Fourier transform with technical details, let us start with the definitions of continuous and discrete time signals. Later we present the mathematics related to signal representations, transformation and synthesis.
Signals
A continuous signal is a signal in which the independent variable is continuous and thus the signal is defined for a continuum of values of the independent variable [1, p. 9]
A discrete time signal is only defined at discrete times, and consequently for these signals the independent variable takes on only a discrete set of values [1, p. 9]
It is also useful to define periodic functions. These functions are used to define various periodic signals
A function f(t) is said to be periodic with period T if there exists a number T > 0 such that f(t + T) = f(t) for all values of t [2, p. 4]. If such a T exists, then the smallest T for which the equation holds is called the fundamental period of the function f. Every integer multiple of the fundamental period is also a considered to be a period of the function: f(t + nT) = f(t), n= 0,±1,±2,
Continuous Time complex Exponential and Sinusoidal Signals
The continuous time complex exponential signal is of the form [1, p. 17] x(t)= 〖Ce〗^at
Where C and a are complex numbers. Depending on the values of C and a, the exponential equation can exhibit various characteristics. If we restrict the value of a to be purely imaginary, then we get signals of the form x(t)= e^(jω_o t) Important property of such type of signals is that they are periodic. Such class of signals is used to derive the sinusoidal functions. x(t)=Acos(ω_o t+ φ) ω_ois called the fundamental frequency of the signal, and φ is the phase of the signal
Sinusoidal and other periodic exponential signals are used to describe the characteristics of various real world processes.
Similarly discrete time sinusoidal signals can be defined as x[n]=Acos(ω_o n+ φ)
Linear time Invariant systems
Linear time invariant systems can be used to model many physical processes. Their response to arbitrary input signal forms the basis for development of many real world systems which includes a variety of telecommunication systems as well. Several networking and routing hardware are built on the principles of LTI systems. LTI systems provide basis for understanding the basics of how signal’s decomposition and signal processing systems works. Hence it is important to understand their definition and properties as they are at the core of various telecommunication systems.
Properties of LTI systems
Linearity of LTI systems: The input and output of the system have a linear relationship.
If an arbitrary input x1 has an output y1 and an input x2 has output y2, then the scaled summation of input(ax1 + bx2) will produce a scaled summation of output response (ay1 + by2). Here a, and b are real numbers.
Time invariance of LTI systems: If the output response to an input x(t) at time t is y(t), then the output response to signal x(t-T) is y(t-T). That means that the output is independent of the instant at which the input signal starts; it is only a time shifted version of output at time t. Output does not depends upon the time at which input is applied
Hence, if input to an LTI system consists of a linear combination of signals x (t) = a_1 x_1 (t) + a_2 x_2 (t)+ …
Then by superposition the response is given by y (t)= a_1 y_1 (t) + a_2 y_2 (t)+…. [1, p. 69]
Hence it is possible to represent input signals to LTI systems in terms of basic signals and use the superposition property of the system to calculate the response.
Signals can be represented in terms of impulses by decomposing them into weighted sum of shifted impulses. Figure 1 show how a discrete time signal can be represented as a sum of shifted impulse response. [1, p. 71] (a) Shows the original signal and the other five are generated by multiplying the scaled time shifted impulses with the original signal.

Figure 1

Hence an arbitrary signal can be expressed as x[n]= ∑_(k= -∞)^(+∞)▒〖x[k]δ[n-k]〗
Where δ[n-k] is the time shifted impulse function, and x[k] is the amplitude (for scaling). If we know the impulse response h[n] of the system to input impulse δ [n], we can determine the output by applying h[n] to δ [n] and then superposing the output as a linear combination of outputs to individual shifted scaled impulse x[n]. Hence the response of a system to an input x[n] can be represented by [1, p. 72] y[n]= ∑_(k= -∞)^(+∞)▒〖x[k]h[n-k]〗
This result is the convolution sum or superposition sum
Figure 2 (a) [1, p.76]shows an input signal which is non zero for t = -1,0,1
Figure 2 (b) shows impulse response of the LTI system for δ [n-1], δ [n], δ [n+1] Figure 2

Figure 3, Output of LTI system to Input Impulses[1, p.77]
Similarly for continuous time signals, continuous time LTI system response can represent output as y(t)= ∫_(-∞)^(+∞)▒〖x(τ)h(t-τ)〗
Response of continuous time LTI systems to complex periodic exponentials
When a complex exponential input is applied to an LTI system, the output is the same complex exponential with different amplitude. [1, p. 166]
(e^s )^t=〖H(s)e〗^st
Where e^st is called the eigenfunction of the system and the amplitude factor H(s) is known as the eigenvalue. Just like any arbitrary signal, the response of an LTI system to a complex exponential signal can be represented as a superposition of individual responses.
Representation of periodic signals as Fourier series
As mentioned previously a periodic signal can be represented by the equation x(t)=e^(jω_o t)
Harmonically related complex exponentials with this signal can be represented as e^(jkω_o t) , Where k = 0, +-1, +-2, +-3…, where x(t) has a fundamental period of T after which it repeats itself. This set of harmonically related signals can be represented as [1, p. 169] x(t)=∑_(k=-∞)^(+∞)▒〖a_k e^(jkω_o t) 〗
This composite signal is also periodic with fundamental period T and frequency ω_o. Terms with k as +1 and -1 have fundamental frequency and period T. Terms with k as +2 and -2 have double the fundamental frequencies and are periodic with half the fundamental period (T/2). Each such component is also called harmonics (nth harmonic in general).
This representation of a periodic signal is known as the Fourier series representation of a periodic signal.

Figure 4,
Signal x(t)=∑_(-3)^(+3)▒〖a_k e^jk2πt 〗 represented as linear combination of harmonically related sinusoidal signals[1, p.170]

If a signal can be represented with Fourier series, its coefficients are determined by the formula a_k=1/To ∫_To▒〖x(t) e^(-jkω_o t) dt〗
Coefficients are called spectral coefficients or Fourier series coefficients.
So far we have seen that Fourier series can be used to represent periodic signals. But it was also determined by Fourier that it is possible to extend this principle to represent aperiodic signals. Fourier transform pair for aperiodic signals is represented by x(t)=1/2π ∫_(-∞)^(+∞)▒〖X(ω) e^jωt dω〗
〖X(ω)〗_ =∫_(-∞)^(+∞)▒〖x(t) e^(-jωt) dt〗
Applications of Fourier Transform in Telecommunications
Modulation: We can use one signal to scale or modulate the characteristics of another signal. This can be thought of as multiplication of two signals (convolution property). This is of particular importance in communication channels. Channels typically have a frequency range over which they are best suited to transmit the signals without significant loss. For example, our atmosphere adversely affects the audible frequency range signals (10Hz to 20 KHz) whereas it allows transmission of higher frequency signals over a wide range without significant loss. Hence all the audio signals like music and speech which are transmitted through an atmosphere communication channel are modulated by high frequency carrier waves. Sinusoidal amplitude modulation is a common modulation technique used for this purpose. A high frequency sine wave acts as the information carrier signal whose frequency is in the appropriate range, and whose amplitude is modulated by various sound signals.
Filtering: In many applications, it is required to change the amplitudes of frequency components of a signal or at times remove the whole frequency component itself, a process which is called filtering. Filtering is used in communication for various applications and industries. One application of filtering is in the audio systems. A filter is provided to either include the low energy (Bass) or high energy (Treble) frequency components. Filtering is also used in mobile communication to improve receiver quality even over interference limited channels.
Multiplexing: The concept of modulation can be used to transmit many signals over a channel. For example, normally microwave transmission links have bandwidth of several gigahertz which is a lot more than what is required for one voice channel. If many voice signals have overlapping frequency, their frequency content range is shifted so that they all belong to distinct frequency bands. This is achieved through amplitude modulation with a sinusoidal wave. After modulation, all the signals can be transmitted simultaneously on the same wideband channel. This concept is known as frequency division multiplexing.
Fourier Transform’s impact on society and economic value
Technology has changed the way humans see the world today. Even if we look at our recent past, we did not communicate over cell phones, communicate via video calls or fly on jet planes to different countries of the world and on space shuttles over to the moon! But we have come a long way thanks to developments in Technology. Fourier Transform has played a vital role in the development of innumerable technologies in various industries. It can be said to have had a huge impact in the way we see technology today. A lot of hardware and applications are built which utilize the principles of Fourier transform. There are many instruments in the Healthcare industry that work on the principles of FT using spectroscopy like an MRI machine. MRI machines have enabled doctors to identify various problems related to health and proper diagnosis of patients. . In the food industry, spectroscopic instruments are used to identify the adulteration of fats and edible oils
If we look at geology, FT principles are used to calculate the probability and intensity of earthquakes which can save a lot of human lives.
“Signals are ubiquitously present in manufacturing machines and systems. For example, metal removal is essential to many manufacturing processes, as seen in turning, milling, and drilling During such a process, interactions between the cutting edge of the tool and the work piece lead to removal of fragments of varying volumes, producing whereby time-varying or transient components in the vibration signals” [5]
Identification of such signals at initial stage and their removal can improve the machine performance and help reduce structural defects and machine downtime in the future. These signals are identified with the help of Fourier Transform. Hence it provides economic value to the manufacturers and machine buyers as it helps reduce manufacturing defects and costly downtime. In stock markets, various financial models for stock prices and market predictions are built using Fourier series. Looking into the future, Affine Fourier Transform (AFT-MC) can be used for transmission of data over aeronautical satellite channels. Using AFT-MC would provide higher date rate for communication when flying on airplanes. Hence we can see that Fourier Transform can be said to have played a vital role in the overall development of technology, human beings and thus the society. As we keep coming up with more innovative theories and ideas, we will find more applications of Fourier Transform and its derivatives in future.

Conclusion
We briefly described few types of signals in the paper. Then we talked about Linear Time Invariant systems and their importance in various fields of technology. We focused on the technical aspects of Fourier Transform and represented it mathematically. Later part of the paper briefly mentioned some applications of Fourier Transform in telecommunications and various other fields and also how it has had an impact on modern day technology and society.

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