3.1 Fourier Series The Fourier series are applicable to periodic signals. They were discovered by the famous French mathematician Joseph Fourier in 1807. By using the Fourier series, a periodic signal satisfying certain conditions can be expanded into an infinite sum of sine and cosine functions. Consider a real periodic signal of time,, with the period, that is Define the fundamental angular frequency as. By the Fourier theorem, under certain mild conditions, the following expansion for holds This is the real coefficient trigonometric form of the Fourier series. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 1
The required real number coefficients in the above formula are obtained from In these integrals plays the role of a parameter, hence the coefficients and are functions of, that is, and. Due to periodicity, it is sometimes more convenient to use the integration limits from to. Note that the coefficient gives the average value of in the interval of one period, that is The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 2
If the period is measured in seconds, denoted by, then the fundamental angular frequency is measured in radians per second ( ). It is well known from elementary electrical circuits and physics courses that, where is the frequency defined by and measured in Hertz ( ). Formulas for the Fourier series coefficients can be justified by using the following known integrals of trigonometric functions The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 3
Multiplying the signal Fourier series expansion formula by and integrating from to we obtain which implies formula for. Integrating the signal Fourier series expansion formula directly from to produces which implies the formula for. Similarly, multiplying the signal Fourier series expansion formula by and integrating from to, we obtain the formula for, that is The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 4
In the Fourier series expansion formula, defines the fundamental harmonic, gives the second harmonic and so on, a sinusoidal signal of angular frequency is called the th harmonic of the periodic signal of the fundamental angular frequency. Practical applications of the infinite sum of the Fourier series expansion formula comes from the fact that this infinite sum can be well approximated by a finite sum containing only a few first harmonics. In other words, for many real periodic signals coefficients and decay very quickly to zero as increases, hence periodic signals can be approximated by the truncated Fourier series where stands for the number of harmonics included in the approximation. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 5
Calculations of the Fourier series coefficients can be simplified if we use the notions of even and odd functions. Recall that a function is even if and odd if. For example, since and, we conclude that is an even function and is an odd function. According to the Fourier series expansion formula, periodic signals are expanded in terms of cosine and sine functions. Hence, the cosine terms represent the even part of and the sine terms represent the odd part of. This indicates that even periodic signals will have no sine terms in their Fourier series expansions, that is, for them the coefficients for every. Similarly, odd periodic signals will be represented only in terms of sine functions, that is, for odd periodic signals for every. The following example demonstrates this property of the coefficients of the Fourier series. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 6
Example 3.1: Consider an even periodic signal presented in Figure 3.1. E/2 x(t) -T -T/2 -T/4 0 T/4 T/2 T t -E/2 Figure 3.1: An even periodic signal For this signal the coefficients for every, which can be formally shown by performing integration The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 7
!! # " # # " # # # # The coefficients! are obtained from $% & % $ $% & % $ # # # Note that # represents the average value of the signal over the time interval of one period. Hence, in this example #. Replacing # by, we obtain ')(!+* ')(!,* The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 8
The original even periodic signal is represented by the Fourier series as follows - - - - Apparently, with more terms included in this infinite series, a better approximation for the signal is obtained. However, since the cosine function is bounded by. -, we see that can be well approximated by a truncated series due to the fact that as increases (theoretically as ). In Figure 3.2 we present the approximations of the signal for and. In Figure 3.3, we compare the higher order approximations for. Figures 3.2 and 3.3 are generated and plotted using MATLAB. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 9
0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.5 1 0.5 0 0.5 1 1.5 Figure 3.2: Signal approximation by the Fourier series: /10325476 dotted line, /98:2;4<6 dashed line, and />=?2;4<6 solid line The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 10
B @ Gibbs Phenomenon It is expected that the signal approximation defined by the truncated Fourier series expansion formula improves as increases; theoretically, as, the signal is perfectly approximated by the signal @. This is valid at all points where the signal is continuous. However, according to the theory of Fourier series, at the point where a periodic signal has a jump discontinuity ( A B B C ), @ the signal at the discontinuity point; that is, A B B C converges to the average value of B, converges to. Furthermore, in an arbitrary small neighborhood of the discontinuity point a small ripple always exists. This was first observed by Gibbs, hence this is known as the Gibbs phenomenon. No matter how many harmonics are included in the approximation @, the ripple that exists in a very close neighborhood of B can not be less than 9%, see Figure 3.3, approximation D)E. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 11
approximation x10 0.6 0.4 0.2 0 0.2 0.4 0.6 2 1 0 1 2 time approximation x20 0.6 0.4 0.2 0 0.2 0.4 0.6 2 1 0 1 2 time approximation x30 0.6 0.4 0.2 0 0.2 0.4 approximation x50 0.6 0.4 0.2 0 0.2 0.4 0.6 0.6 2 1 0 1 2 2 1 0 1 2 time time Figure 3.3: Higher order signal approximations by the Fourier series: signals FHGJILK3M>N)OPFRQSILK3MTN9OUFWVPIXK3M9N9O and FRYSIZK3M9N. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 12
d d e e d f e f f f f f f Another form of the Fourier series, known as the trigonometric Fourier series with complex coefficients, can be derived from the real coefficient trigonometric Fourier series expansion formula by using the Euler formula []\_^ \`^ ab\c^ \c^ ab\`^ These expressions for the sine and cosine signals with d used in the signal Fourier series expansion formula imply fhgi fhgpi \ f+jnk?l \ f+j,kml ab\ f+jnk?l fg i \ fqj k l r f ab\ f+jhk:l ab\ fqj k l \ f+jnk?l ab\ f+jnkol The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 13
t t t where the star denotes the complex conjugate and s s t t t t uzvlwyx z t tz t {} Note that t and t coefficients are functions of s. It follows that the complex coefficients are also functions of s, that is t t s. It can be observed that the following relationships hold s s t t t t The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 14
~ ~ ~ ~ The trigonometric form of the Fourier series with complex coefficients p +ˆn?Š qˆn?š ƒ_ X ƒ bƒœ L y bƒ qˆ Š +ˆ Š: ƒž L Œ bƒx L _ ~ h p The coefficients, in the above formula can be calculated from the real Fourier series coefficient formula as follows ~ bƒ +ˆ Š bƒ qˆn?š ~ ~ The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 15
Convergence Conditions for the Fourier Series (Dirichlet s Conditions) It is possible to find the Fourier series coefficients for a given periodic function, but the infinite Fourier series obtained does not converge to the given periodic function. In general, not all periodic signals can be expanded into convergent Fourier series. A periodic function has the convergent Fourier series if the following conditions are satisfied: (1) The single-valued function is bounded, and hence absolutely integrable over the finite period, that is (2) The function has a finite number of maxima and minima over the period. (3) The function has a finite number of discontinuity points over the period. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 16
š Under the Dirichlet conditions the Fourier series converges to at all points where is continuous and to the average of the left- and right- limits of at each point where is discontinuous. In practical applications, it is sufficient to check only condition (1) since it is almost impossible to find real physical signals that satisfy condition (1) and do not satisfy conditions (2) and (3). The third form of the Fourier series, very useful in the passage from the Fourier series to the Fourier transform, can be derived as follows. Observe first that œ œ ž +Ÿh : œ œ žž Ÿn? š The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 17
Using this result we have h h ª ª q«h p ªŽ ª q«+«h : «n? This form is called the exponential form of the Fourier series. Line Spectra Frequency plots of the magnitude and phase of the Fourier series coefficients ªŒ±L²y³Ž ª q«n µ define the line spectra. The line spectra can be plotted independently for the magnitude and for the phase. Hence, we distinguish between the magnitude and phase line spectra. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 18
From the complex Fourier series coefficient formula, we have and ¹ZºL»y¼X½;¹ +¾, PÀ Áb¹ŒºL»y¼Â½Ã¹ q¾n µà ¹_ºL» ¼ ½ Áb¹ +¾ À It follows that is an even function of and is an odd function of, that is It is customary in engineering to plot the amplitude (magnitude) and phase line spectra for all frequencies from to at discrete frequency points defined at. We can plot the line spectra only for positive frequencies. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 19
Ä È É Å Å Å É È Å Example 3.2: Consider the signal represented in Figure 3.4. x(t) E -T -T/2 0 T/2 T -E t Figure 3.4: Sawtooth waveform Let us find the complex coefficients Ä Å. Since this signal is odd it follows that Ä, which implies that Å. The real Fourier series coefficients Ä are obtained as ÆÇ ÆÇ ÆÇ ÆÇ ÆÇ The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 20
Ì Ê Ê Ì Ì Ê Ê Ì Ì Ì Ê Ê Ê Ê Substituting the corresponding integration limits, we have Ê Ë Ì Ê Í Î The coefficients are given by Hence, the magnitude and phase line spectra are represented by Ï Ê Ð Ê Ð Ë Note that. The corresponding line spectra are presented in Figures 3.5 and 3.6. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 21
X n (jn ω o ) E/π E/2π E/3π E/4π... 0 ω o 2ω o 3ω o 4ω o nω o Figure 3.5: The magnitude line spectrum for Example 3.2 X n(jn ) ω o π/2 0 π/2 ω... o 2ω o 3ω o 4ω o nω o... Figure 3.6: The phase line spectrum for Example 3.2 The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 22
In summary, we have introduced three forms of the Fourier series. The Fourier series trigonometric form with the real coefficients is used for calculating the Fourier series of a given signal. The Fourier series trigonometric form with complex coefficients is used for signal spectral analysis (frequency domain analysis) since the corresponding complex coefficients provide information about signal magnitude and phase in terms of frequency Ñ (signal line spectra). The trigonometric Fourier series formula with complex coefficients will be also used in Section 3.4 for finding the zero-state system response due to periodic and sinusoidal inputs. The third form of the Fourier series, the exponential form, is important for the development of the Fourier transform. The Fourier transform represents a generalization of the Fourier series that can be applied to both periodic and aperiodic signals. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 23
3.1.1 From Fourier Series to Fourier Transform Since in linear system analysis we deal with general signals and since the use of the Fourier series is limited to periodic signals only, we need to derive a more general transform, which will be applicable to nonperiodic signals as well. That transform is known as the Fourier transform. The Fourier transform can be derived from the Fourier series by using the following mathematical artifice: any function is periodic with a period, that is, by stating By doing this (assuming that it can be mathematically and practically justified) we are passing from a discrete frequency representation Ò Ò Ò Ò, with Ò used for the Fourier series, to a continuous frequency representation since for we have Ò (an infinitesimally small quantity) The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 24
ß Û and Ó (evenly spaced discrete frequencies Ó become a continuum of frequencies ). These limiting facts become more obvious from the line spectra diagrams (see the corresponding ones presented in the previous example and observe that for it follows that Ó ). Using these facts in the exponential form of the Fourier series and the corresponding coefficients, we obtain that is Ù Ú ÔÖÕØ ÙÚ Û ÙhÚ Ùqß à Õ ÙhÚ Û Û ÜÝ ÛbÞ`Ù+ßhàJá Þ`Ù+ßhà:â ÜÝ ÛbÞcÙqßnàmá Þ`Ù+ßhà:â The infinite sum in the limit becomes the integral, hence ÛbÞãß]á ÞãßWâ Û Û The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 25
å ä å ä ä The term in parentheses is a function of the angular frequency. It defines the Fourier transform of the signal, which we denote by ä åæcçhè where the symbol means by definition. The signal is obtained from its Fourier transform by the inverse operation å}é æêçwè We call this expression the inverse Fourier transform of. In other words, the signal is obtained by finding the inverse Fourier transform of. We say that the signals and form the Fourier transform pair and denote their relationship by. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 26
ì ì ë ë ë ë Some authors, especially in communication systems and signal processing, prefer to express the Fourier transform in terms of frequency rather than in terms of angular frequency. In that case, the corresponding formulas become ìbíœî)ïhðzñ ìóò íœî)ïhðxô In this textbook we will use the formulas for the Fourier transform and its inverse expressed in terms of as the definitions of the Fourier transform and its inverse. It is interesting to point out that the almost ridiculously simple mathematical trick (every signal is periodic with the period ), used almost two hundred years ago by the mathematician Fourier in an attempt to deal with infinity in a mathematical way, led to the development of the Fourier transform, which provided The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 27
ö õ õ ö õ õ ö õ ö õ õ a foundation for the development of practical communication and signal processing systems, such as telephone, television, computer networks, and so on. Existence Condition for the Fourier Transform Since the Fourier transform is defined by an integral, the main existence condition of the Fourier transform is in fact the existence condition of the corresponding integral, that is, the following integral must exist (must be convergent) öb ãøwù öb ãøwù öb ãøwù Using the fact that öb êøwù, the above condition leads to õ This absolute integrability condition is the main existence condition for the Fourier transform. Some additional mathematical restrictions have to be imposed also, similar to Dirichlet s conditions of the Fourier series (the signal must have a finite The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 28
number of minima and maxima and a finite number of discontinuities on any finite time interval). For the purpose of this course, it is enough to test only the absolute integrability condition in order to make a decision whether or not the corresponding Fourier transform exists in terms of ordinary functions. It should be pointed out that the absolute integrability condition states a sufficient condition only, which means that if it is satisfied than the Fourier transform exists (in terms of ordinary functions), but not the other way around: the existence of the Fourier transform does not imply that the absolute integrability holds. In the follow up sections, we will see that some standard signals do not satisfy the absolute integrability condition, but they do have Fourier transforms given in terms of generalized functions. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 3 29