Complex Form of Fourier Series
The complex form of Fourier series represents a periodic function as a sum of complex exponentials. It is a powerful tool in engineering and technology for analyzing and synthesizing periodic signals. By using complex exponentials, the complex form of Fourier series simplifies the mathematical representation of periodic functions and provides a convenient way to work with phase and magnitude information.
7 Key excerpts on "Complex Form of Fourier Series"
eBook - ePubDigital Signal Processing
Concepts and Applications
- Bernard Mulgrew, Peter Grant, John Thompson(Authors)
- 2002(Publication Date)
- Bloomsbury Academic(Publisher)
...There are two equivalent relationships based on the trigonometric and complex forms of the Fourier series respectively: Calculating the power in a signal is also a good example of how two alternative descriptions of a signal can be used to do the same thing. The power can either be calculated directly from the time-domain description using equation (1.5) or from the Fourier coefficients using either (1.14) or (1.15). Self Assessment Question 1.4: Evaluate the complex Fourier coefficients of the periodic signal x (t) which has a period of 1. The signal is defined as: x (t) = 4 t ; 0 ≤ t < 1. 1.4. The Fourier transform Although the Fourier series is a powerful and elegant concept, it suffers from one disadvantage, i.e. it is only applicable to periodic signals. Most signals of practical importance are not periodic. To see how a Fourier representation for a non-periodic signal might be developed, consider a periodic signal and examine what happens to the Fourier series as the period gets longer. Figure 1.7 illustrates the complex Fourier series of a pulse with increasing period. The waveform is similar to that illustrated in Figure 1.6. However this time, rather than plotting spectrum as a function of harmonic number n, it is plotted as a function of frequency ω. The frequency of the n th harmonic is now n ω 0. As the period T is increased from 1/4 of a second to 1/2 of a second to 1 second, the harmonics (or Fourier components) get closer and closer together in frequency (radiancy) − the spectrum gets denser. In fact every time the period is doubled, the frequency spacing between the harmonics is halved. Note however that the general outline (shape or envelope) of the spectrum stays the same. It would not be unreasonable to conclude that the outline is a property of the pulse itself and not the period between pulses. Thus a Fourier representation of a single pulse might be obtained by letting the period T get very large...
eBook - ePubSignal Processing for Neuroscientists
An Introduction to the Analysis of Physiological Signals
- Wim van Drongelen(Author)
- 2006(Publication Date)
- Academic Press(Publisher)
...5 Real and Complex Fourier Series 5.1 INTRODUCTION This chapter introduces the Fourier series in the real and the complex form. First we develop the Fourier series as a technique to represent arbitrary functions as a summation of sine and cosine waves. Subsequently we show that the complex version of the Fourier series is simply an alternative notation. At the end of this chapter, we apply the Fourier series technique to decompose periodic functions into their cosine and sine components. Because the underlying principle is to represent waveforms as a summation of periodic cosine and sine waves with different frequencies, one can interpret Fourier analysis as a technique for examining signals in the frequency domain. At first sight, the term frequency domain may appear to be a novel or unusual concept. However, in daily language we do use frequency domain descriptions; for instance, we use a frequency domain specification to describe the power line source as a 120-V, 60-Hz signal. Also, the decomposition of signals into underlying frequency components is familiar to most; examples are the color spectrum obtained from decomposing white light with a prism (Fig. 5.1), or decomposing sound into pure tone components. Figure 5.1 A prism performs spectral decomposition of white light in bands with different wavelengths that are perceived by us as different colors. An example showing an approximation of a square wave created from the sum of five sine waves is shown in Figure 5.2. This example can be reproduced with MATLAB script pr5_1.m. This example illustrates the basis of spectral analysis: a time domain signal (i.e., the (almost) square wave) can be decomposed into five sine waves, each with a different frequency and amplitude. The graph depicting these frequency and amplitude values in Figure 5.2 is a frequency domain representation of the (almost) square wave in the time domain...
eBook - ePubDigital Signal Processing
Fundamentals and Applications
- Li Tan(Author)
- 2007(Publication Date)
- Academic Press(Publisher)
...B Review of Analog Signal Processing Basics B.1 Fourier Series and Fourier Transform In electronics applications, we have been familiar with some periodic signals such as the square wave, rectangular wave, triangular wave, sinusoid, sawtooth wave, and so on. These periodic signals can be analyzed in frequency domain with the help of the Fourier series expansion. According to Fourier theory, a periodic signal can be represented by a Fourier series that contains the sum of a series of sine and/or cosine functions (harmonics) plus a direct-current (DC) term. There are three forms of Fourier series: (1) sine-cosine, (2) amplitude-phase, and (3) complex exponential. We will review each of them individually in the following text. Comprehensive treatments can be found in Ambardar (1999), Soliman and Srinath (1998), and Stanley (2003). B.1.1 Sine-Cosine Form The Fourier series expansion of a periodic signal x (t) with a period of T via the sine-cosine form is given by (B.1) whereas ω 0 = 2 π/T 0 is the fundamental angular frequency in radians per second, while the fundamental frequency in terms of Hz is f 0 = 1/ T 0. The Fourier coefficients of a 0, a n, and b n may be found according to the following integral equations: (B.2) (B.3) (B.4) Notice that the integral is performed over one period of the signal to be expanded. From Equation (B.1), the signal x (t) consists of a DC term and sums of sine and cosine functions with their corresponding harmonic frequencies. Again, note that nω 0 is the n th harmonic frequency. B.1.2 Amplitude-Phase Form From the sine-cosine form, we notice that there is a sum of two terms with the same frequency and that one from the first sum is a n cos (nω 0 t) and the other is b n sin (nω 0 t). We can combine these two terms and modify the sine-cosine form into the amplitude-phase form: (B.5) The DC term is the same as before; that is, (B.6) and the amplitude and phase are given by (B.7) (B.8) respectively...
eBook - ePubDigital Signal Processing
A Primer With MATLAB®
- Samir I. Abood(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...5 Discrete-Time Fourier Series (DTFS) The representation of periodic signals becomes the discrete-time Fourier series (DTFS), and for aperiodic signals, it becomes the discrete-time Fourier transform(DTFT). The motivation for representing discrete-time signals as a linear combination of complex exponentials is identical in both continuous-time and discrete-time. The complex exponentials are eigenfunctions of linear, time-invariant systems, and consequently, the effect of an LTI system on each of these basic signals is simply the amplitude change. An LTI system is completely considered by a spectrum applies at each frequency. In representing discrete-time periodic signals through the Fourier series, use harmonically related complex exponentials with fundamental frequencies. In this chapter will discuss the discrete-time Fourier transform and its application in digital signal processing. 5.1 DTFS Coefficients of Periodic Discrete Signals The discrete-time signal x (n) is periodic if for a positive value of N, x (n) = x n + N (5.1) Let us look at a process in which we want to estimate the spectrum of a periodic digital signal x (n), sampled at a rate of f s Hz with the fundamental period T 0 = NT, as shown in Figure 5.1, where there are N samples within the duration of the fundamental period, and T = 1/ f s is the sampling period. According to Fourier series analysis, the coefficients of the Fourier series expansion of a continuous periodic signal x (t) in a complex form is given. by: c k = 1 T 0 ∫ T 0 x (t) e − j k ω 0 t d t − ∞ < k < ∞ (5.2) where k is the number of harmonics corresponding to the harmonic frequency of kf 0, and ω 0 = 2 π / T 0 and f 0 = 1 / T 0 are the fundamental frequency in radians per second and in Hz, respectively. To apply this to Equation (5.2), we substitute T 0 = NT with ω 0 = 2 π / T 0 and approximate the integration over one period using a summation by replacing dt = T and t = nT...
eBook - ePub- David M. Howard, Jamie Angus(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
...from: a 0 = 1 T 0 ∫ − T 0 2 T 0 2 f (t) d t (A1.5a) a n = 2 T 0 ∫ − T 0 2 T 0 2 f (t) cos (n ω 0 t) d t (A1.5b) b n = 2 T 0 ∫ − T 0 2 T 0 2 f (t) sin (n ω 0 t) d t (A1.5c) A1.5 The Complex Fourier Series Like many things in. acoustics, dealing with the Fourier series in trigonometric form is messy and inconvenient. For example, calculating the Fourier series coefficients using the equations in A1.5 requires that we do three integrals! However, by using complex number theory, we can combine the sine and cosine to form a complex exponential. e j θ = cos θ + j sin θ (A1.6) where θ is in radians. This allows us to express the Fourier Series as a complex exponential as follows: f (t) = ∑ n = − ∞ ∞ C n e j n ω 0 t (A1.7) where the (now in general complex) coefficients C n are calculated. by: C n = 1 T 0 ∫ − T 0 2 T 0 2 f (t) e − j n ω 0 t d t (A1.8) Although in general the coefficients C n are complex if the waveform has odd symmetry, then resulting coefficients C n will be purely imaginary (the sine bit), whereas for even symmetry they will be purely real (the cosine bit). Equations A1.7 and A1.8 can be used instead of equations A1.1 and A1.5. They also have a further advantage in that when we do a Fourier analysis using equation A1.4 on a periodic signal that is neither odd nor even, we end up with two sets of coefficients, the a n s and the b n s. But when we listen to a periodic sound, we only hear one frequency spectrum. Therefore, to work out this spectrum we have to combine the a n ′s and the b n ′s to get the total contribution at each frequency, as shown in Equation A1.4. However, the complex exponential form of the Fourier series automatically gives us a single, complex value for the coefficients, and by finding their absolute values, or modulus, we get the magnitude of the spectrum, which is usually more perceptually relevant to the listener’s appreciation of timbre...
- Patrick F. Dunn(Author)
- 2019(Publication Date)
- CRC Press(Publisher)
...9 The Fourier Transform CONTENTS 9.1 Chapter Overview 9.2 Fourier Series of a Periodic Signal 9.3 Complex Numbers and Waves 9.4 Exponential Fourier Series 9.5 Spectral Representations 9.6 Continuous Fourier Transform 9.7 Continuous Fourier Transform Properties* 9.8 Discrete Fourier Transform 9.9 Fast Fourier Transform 9.10 Problems A single number has more genuine and permanent value than an expensive library full of hypotheses. Robert J. Mayer, c. 1840. 9.1 Chapter Overview Fourier analysis and synthesis are introduced in this chapter and used to find the amplitude, frequency, and power content of signals. These tools are applied to continuous signals; first to some classic periodic signals and then to aperiodic signals. In Chapter 10, these methods are extended to digital signal analysis. 9.2 Fourier Series of a Periodic Signal Before considering the Fourier series, the definition of orthogonality must be examined. The inner product (dot product), (x,y), of two real-valued functions x(t) and y(t) over the interval a ≤ t ≤ b is defined as (x, y) = ∫ a b x (t) y (t) d t. (9.1) If (x, y) = 0 over that interval, then the functions x and y are orthogonal in the interval. If each distinct pair of functions in a set of functions is orthogonal, then the set of functions is mutually orthogonal. For example, the set of functions sin(27 πmt/T) and cos(27πmt/T), m = 1,2,..., form one distinct pair and are mutually orthogonal because ∫ − T / 2 T / 2 sin (2 π m t / T) cos (2 π n t / T) d t = 0 for all m, n. (9.2) Also, these functions satisfy the other orthogonality. relations y (t) = ∫ − T / 2 T / 2 cos (2 π m t / T) cos (2 π n t / T) d t = { 0 m ≠ n T m = n (9.3) and y (t) = ∫ − T / 2 T / 2 sin (2 π m t / T) sin (2 π n t / T) d t =[--=PLGO-SEPAR...
eBook - ePub- Nassir H. Sabah(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...The different frequencies do not interact in an LTI circuit, so that each of these responses can be obtained using phasor analysis, as was done in Chapter 5. Circuit responses to periodic signals are of considerable practical interest because these signals are very common. The steady state of any linear or nonlinear circuit, other than the dc steady state, is periodic. Thus, the outputs of free-running oscillators, the time bases of TV and computer displays, and continuous vibrations of all kinds are periodic. The Fourier series expansion of periodic functions can be generalized to nonperiodic functions by means of the Fourier transform (Chapter 16). Learning Objectives To be familiar with: The Fourier series expansion of some commonly encountered periodic functions To understand: The derivation and basic properties of Fourier series expansions, including symmetry considerations The interpretation of amplitude and phase spectra How the Fourier series expansion of a given function may be obtained from those of other functions through addition, subtraction, multiplication, differentiation, or integration How the response of an LTI circuit to a periodic signal can be derived Power relations of periodic inputs, including rms values 9.1 Fourier Series The defining property of a periodic function f (t) of period T is that f (t) = f (t + nT), where n is any integer. That is, the function repeats every period T. The sinusoidal function discussed in Chapter 5 is a common example of a periodic function, so that the same definitions of cycle, frequency, and angular frequency for a sinusoidal function (Equation 5.1.2) apply to periodic functions in general. Strictly speaking, a periodic time function extends over all time, from − ∞ to + ∞. In practice, a function can be assumed periodic if it has been in a steady state for a time interval that is large compared to the period. Periodic functions need not be time functions...
