UNIT II Fourier series & transform Representation of Continuous Time Signals Learning Objectives: To introduce the concept of various Fourier series To introduce the concept of convergence of Fourier series. To introduce the concept of representation of a periodic function by Fourier series To introduce the concepts of sampling of continuous time signlas. To introduce the concept of Fourier transform. Learning Outcomes: Students will be able to Apply symmetric conditions Represent the periodic function by Fourier series Obtain the relation between trigonometric and exponential Fourier series Perform transformations on signals Syllabus: Trigonometric and exponential Fourier series, relationship between trigonometric and exponential Fourier series, representation of a periodic function by the Fourier series over the entire interval, convergence of Fourier series, alternative form of trigonometric series, symmetry conditions: Even, Odd and Half-wave symmetry. Properties of Fourier series: linearity, time scaling, time shifting, time reversal, differentiation, integration, modulation, convolution and Parseval’s theorem. Complex Fourier transforms. Representation of an arbitrary function over the entire interval: Fourier transform, Existence of Fourier transform, Fourier transform of some useful functions, Fourier transform of periodic function, Properties of Fourier transform, Energy Density spectrum, Parseval’s Theorem. Introduction Many of the phenomena studied in engineering and science are periodic in nature eg. The current and voltage in an alternating current circuit. These periodic functions can be analysed into their constituent components (fundamentals and harmonics) by a process called Fourier analysis. Fourier series was developed by Joseph Fourier Definition . Fourier series is a mathematical tool which decomposes any periodic function or periodic signal into sum of set of sine and cosine terms. OR The representation of signals over a certain time interval interns of the linear composition of orthogonal functions is called Fourier series. It is sometimes called the Harmonic analysis. There are 3 important types of Fourier series. They are 1. Trigonometric form 2. Cosine form 3. Exponential form Trigonometric Fourier series The representation of signals over a certain time interval interns of the linear composition of trigonometric functions is called Trigonometric Fourier series. The trigonometric Fourier series representation of any periodic signal x (t) is defined as ∞ 푥(푡) = 푎표 + ∑푛=1[푎푛푐표푠푛푤표푡 + 푏푛푠푖푛푛푤표푡] (1) Where푎푛, 푏푛 are called constants and 푎표 is called dc component and is given by 1 푡표+푇 푎 = ∫ 푥(푡)푑푡 표 푇 푡표 2 푡표+푇 푎 = ∫ 푥(푡)푐표푠푛푤 푡 푑푡 푛 푇 표 푡표 2 푡표+푇 푏푛 = ∫ 푥(푡)푠푖푛푛푤표푡 푑푡 푇 푡표 Exponential Fourier series The representation of signals over a certain time interval interns of the linear composition of exponential functions is called Exponential Fourier series. The complex exponential form is more general and usually more convenient & more compact when compared to Trigonometric Fourier series. The exponential Fourier series of signal x (t) is given by ∞ 푗푛푤표푡 푥(푡) = ∑푛=−∞ 푐푛푒 OR −1 푗푛푤표푡 ∞ 푗푛푤표푡 푥(푡) = 푐0 + ∑푛=−∞ 푐푛푒 + ∑푛=1 푐푛푒 (2) Relationship between trigonometric and exponential Fourier series The complex exponential Fourier series is given by ∞ 푗푛푤표푡 푥(푡) = ∑푛=−∞ 푐푛푒 −1 푗푛푤표푡 ∞ 푗푛푤표푡 = 푐0 + ∑푛=−∞ 푐푛푒 + ∑푛=1 푐푛푒 ∞ −푗푛푤표푡 푗푛푤표푡 = 푐0 + ∑푛=1(푐−푛푒 + 푐푛푒 ) ∞ = 푐0 + ∑푛=1 푐−푛 (푐표푠푛푤표푡 − 푗푠푖푛푛푤표푡) + 푐푛(푐표푠푛푤표푡 + 푗푠푖푛푛푤표푡)) ∞ 푥(푡) = 푐0 + ∑푛=1(푐−푛 + 푐푛) 푐표푠푛푤표푡 + 푗(푐푛 − 푐−푛)푠푖푛푛푤표푡) (3) Compare eq(1) and eq(3) 푎0 = 푐0 푎푛 = 푐푛 + 푐−푛 푏푛 = 푗(푐푛 − 푐−푛) Similarly 푐0 = 푎0 1 푐 = (푎 − 푗푏 ) 푛 2 푛 푛 1 푐 = (푎 + 푗푏 ) −푛 2 푛 푛 Representation of a Periodic Function by the Fourier Series Over the Entire Interval\ So far we have considered the Fourier series representation of an arbitrary function over finite interval between 푡표푎푛푑 (푡표 + 푇). The function and its Fourier series may not be equal outside this interval. However, a periodic function which repeats after every T seconds is expected to have the same Fourier series for the entire interval (-∞, ∞). A periodic function has an identical Fourier series for the entire interval (-∞, ∞) as for the interval (푡표 + 푇) since same function is repeated for every T seconds. Thus for a periodic function x (t) we have ∞ 푗푛푤표푡 푥(푡) = ∑푛=−∞ 퐹푛푒 , (-∞‹t‹∞) 1 푡표+푇 퐹 = ∫ 푥(푡)푒−푗푛푤0푡푑푡 푛 푇 푡표 Dirichlets Conditions (Convergence of Fourier Series) For the Fourier series to exist for a periodic signal it must satisfy certain conditions and they are 1. Function x(t) must be a single valued function 2. Function x(t) has only finite number of maxima & minima. 3. Function x(t) has finite number of discontinuities. 푇| | 4. Function x(t) is absolutely integral over one period i,e ∫0 푥(푡) 푑푡 < ∞ Alternative form of Fourier series (Cosine Fourier series) The trigonometric Fourier series of x(t) contains “sine” and “cosine” terms of the same frequency ∞ 푥(푡) = 푎0 + ∑[푎푛푐표푠푛푤0푡 + 푏푛푠푖푛푛푤0푡] 푛=1 ∞ 푎 푏 2 2 푛 푛 푥(푡) = 푎0 + ∑ √푎푛 + 푏푛 [ 푐표푠푛푤0푡 + 푠푖푛푛푤0푡] 2 2 2 2 푛=1 √푎푛 + 푏푛 √푎푛 + 푏푛 2 2 −1 푏푛 푎푛 Substituting the values 푎0 = 퐴0 , 퐴푛 = √푎푛 + 푏푛 , 휃푛 = 1 tan , = 푐표푠휃푛 and 푎푛 2 2 √푎푛+푏푛 푏푛 = −푠푖푛휃푛 in the above equation of x(t) we get 2 2 √푎푛+푏푛 ∞ 푥(푡) = 퐴0 + ∑ 퐴푛 [푐표푠휃푛푐표푠푛푤0푡 − 푠푖푛휃푛푠푖푛푛푤0푡] 푛=1 ∞ 푥(푡) = 퐴0 + ∑푛=1 퐴푛 [cos (푛푤0푡 + 휃푛] The term 퐴0 is called the dc component. The term 퐴푛 represents the amplitude coefficients (or) harmonic amplitudes (or) spectral amplitudes of the Fourier series and 휃푛 represents the phase coefficients (or) phase angles of Fourier series. The cosine form is also called as “Harmonic Form” (or) “Polar Form” Fourier series. Symmetric conditions or Wave symmetry There are 4 types of symmetry of function x(t) can have 1. Even symmetry 2. Odd symmetry 3. Half wave symmetry Even symmetry A function x(t) is said to be even symmetry if 푥(−푡) = 푥(푡) These types of functions always symmetrical w.r.t the vertical axis 푥(푡) = 푥푒(푡) + 푥0(푡) If x(t) is even function then, 푥0(푡) = 0 and 푥푒(푡) = 푥(푡) . When even symmetry exists the trigonometric Fourier series coefficients are 2 푇/2 푎표 = ∫ 푥(푡)푑푡 푇 0 4 푇/2 푎 = ∫ 푥(푡)푐표푠푛푤 푡 푑푡 푛 푇 0 표 푏푛 = 0 Odd symmetry A function x(t) is said to be odd symmetry if 푥(−푡) = −푥(푡) These types of functions always antisymmetrical w.r.t the vertical axis 푥(푡) = 푥푒(푡) + 푥0(푡) If x(t) is odd function then, 푥푒(푡) = 0 and 푥(푡) = 푥0(푡) . When odd symmetry exists the trigonometric Fourier series coefficients are 푎표 = 0 푎푛 = 0 4 푇/2 푏 = ∫ 푥(푡)푠푖푛푛푤 푡 푑푡 푛 푇 0 표 Half wave symmetry A periodic signal x(t) which satisfies the condition 푥(푡) = −푥 (푡 ± 푇) is said to be half wave symmetry. 2 The Fourier series expansion of half wave symmetry signal contains odd harmonics only (When “n” is odd). 4 푇/2 푎 = ∫ 푥(푡)푐표푠푛푤 푡 푑푡 푛 푇 0 표 4 푇/2 푏 = ∫ 푥(푡)푠푖푛푛푤 푡 푑푡 푛 푇 0 표 When “n” is even 푎푛 = 0 푏푛 = 0 Properties of Fourier Series Let us consider 푥1(푡)푎푛푑 푥2(푡) are two periodic signals with period ‘T’ and with Fourier series coefficients 퐶푛& 퐷푛. 1. Linearity Property The linearity property states that, if 퐹푆{푥1(푡)} = 퐶푛 푎푛푑 퐹푆{푥2(푡)} = 퐷푛 Then 퐹푆 {퐴 푥1(푡) + 퐵푥2(푡)} = 퐴퐶푛 + 퐵 퐷푛 Proof: From the definition of Fourier series, we have 1 푡표+푇 퐹푆{퐴 푥 (푡) + 퐵푥 (푡)} = ∫ [퐴 푥 (푡) + 퐵푥 (푡)]푒−푗푛푤0푡푑푡 1 2 푇 1 2 푡표 1 푡표+푇 1 푡표+푇 퐹푆{퐴 푥 (푡) + 퐵푥 (푡)} = 퐴 ∫ [ 푥 (푡)]푒−푗푛푤0푡푑푡 + 퐵 ∫ [푥 (푡)]푒−푗푛푤0푡푑푡 1 2 푇 1 푇 2 푡표 푡표 퐹푆{퐴 푥1(푡) + 퐵푥2(푡)} = 퐴퐶푛 + 퐵 퐷푛 2. Time Reversal The time reversal property states that, if 퐹푠{푥(푡)} = 퐶푛 Then 퐹푠{푥(−푡)} = 퐶−푛 Proof: From the dentition of Fourier series, we have ∞ 푗푛푤표푡 푥(푡) = ∑ 푐푛푒 푛=−∞ ∞ 푗푛푤표(−푡) 푥(−푡) = ∑ 푐푛푒 푛=−∞ Substitute “n = -p” in the right hand side, we get −∞ ∞ 푗(−푝)푤표(−푡) 푗푝푤표푡 푥(−푡) = ∑ 푐−푝푒 = ∑ 푐−푝푒 푝=∞ 푝=−∞ Substitute “p = n”, we get ∞ 푗푛푤표푡 −1 푥(−푡) = ∑ 푐−푛푒 = 퐹푆 {푐−푛} 푛=−∞ Therefore 퐹푠{푥(−푡)} = 퐶−푛 3. Time Shifting The time shifting property states that, if 퐹푠{푥(푡)} = 퐶푛 −푗푛푤0푡0 Then 퐹푆 {푥(푡 − 푡0) = 푒 퐶푛 Proof: From the dentition of Fourier series, we have ∞ 푗푛푤표푡 푥(푡) = ∑ 푐푛푒 푛=−∞ ∞ 푗푛푤표(푡−푡0) 푥(푡 − 푡0) = ∑ 푐푛푒 푛=−∞ ∞ −푗푛푤표푡0 푗푛푤표푡 푥(푡 − 푡0) = ∑ [푐푛푒 ] 푒 푛=−∞ −1 −푗푛푤표푡0 푥(푡 − 푡0) = 퐹푆 [푐푛푒 ] −푗푛푤0푡0 Therefore 퐹푆 {푥(푡 − 푡0) = 퐶푛푒 4. Time Scaling: The time scaling property states that, if 퐹푠{푥(푡)} = 퐶푛 Then 퐹푠{푥(푎푡)} = 퐶푛 푤푖푡ℎ 휔0 → 푎휔0 Proof: From the definition of Fourier series, we have ∞ 푗푛푤표푡 푥(푡) = ∑ 푐푛푒 푛=−∞ ∞ 푗푛푤표푎푡 푥(푎푡) = ∑ 푐푛푒 푛=−∞ ∞ 푗푛(푎푤표)푡 푥(푎푡) = ∑ 푐푛푒 푛=−∞ −1 푥(푎푡) = 퐹푆 {푐푛} Where 휔0 → 푎휔0 Therefore 퐹푠{푥(푎푡)} = 퐶푛 푤푖푡ℎ 푓푢푛푑푎푚푒푛푡푎푙 푓푟푒푞푢푒푛푐푦 표푓 푎휔0 5. Convolution Property The convolution property states that, if 퐹푆{푥1(푡)} = 퐶푛 푎푛푑 퐹푆{푥2(푡)} = 퐷푛 Then 퐹푆{푥1(푡) ∗ 푥2(푡)} = 퐶푛퐷푛 6. Time Differentiation Property The time differentiation property states that, if 퐹푠{푥(푡)} = 퐶푛 ( ) Then 퐹푆 {푑푥 푡 } = 푗푛휔 퐶 푑푡 0 푛 7. Time Integration Property The time integration property states that, if 퐹푠{푥(푡)} = 퐶푛 푡 퐶푛 Then 퐹푆 {∫ 푥(휏)푑휏} = (if 퐶0 = 0) −∞ 푗푛휔0 8.