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Department of Electrical Engineering University of Arkansas EE 2000 SIGNALS AND SYSTEMS ELEG 3124 SYSTEMS AND SIGNALS Ch Fourier Series Dr Jingxian Wu wuj@uark.edu (These slides are taken from Dr Jingxian Wu, University of Arkansas, 2020.) OUTLINE • Introduction • Fourier series • Properties of Fourier series • Systems with periodic inputs INTRODUCTION: MOTIVATION • Motivation of Fourier series – Convolution is derived by decomposing the signal into the sum of a series of delta functions • Each delta function has its unique delay in time domain • Time domain decomposition + x(t ) = x( ) (t − )d = lim − →0 + x(n) (t − n) n = − x(t) t Illustration of integration INTRODUCTION: MOTIVATION • Can we decompose the signal into the sum of other functions – Such that the calculation can be simplified? – Yes We can decompose periodic signal as the sum of a sequence of complex exponential signals ➔ Fourier series j0t j 2f 0t f0 = e =e 2 – Why complex exponential signal? (what makes complex exponential signal so special?) • Each complex exponential signal has a unique frequency ➔ frequency decomposition • Complex exponential signals are periodic INTRODUCTION: REVIEW • Complex exponential signal e j 2ft = cos(2ft) + j sin( 2ft) – Complex exponential function has a one-to-one relationship with sinusoidal functions – Each sinusoidal function has a unique frequency: f • What is frequency? – Frequency is a measure of how fast the signal can change within a unit time • Higher frequency ➔ signal changes faster f = Hz f = Hz Department of Engineering Science Sonoma State University Sinusoidal at different frequencies f = Hz INTRODUCTION: ORTHONORMAL SIGNAL SET • Definition: orthogonal signal set – A set of signals, 0 (t ), 1 (t ), 2 (t ), , are said to be orthogonal over an interval (a, b) if b C , l = k * a l (t )k (t )dt = 0, l k • Example: k (t ) = e jk t k = 0,1,2, are – the signal set: orthogonal over the interval [0, T0 ] , where 2 0 = T0 OUTLINE • Introduction • Fourier series • Properties of Fourier series • Systems with periodic inputs FOURIER SERIES • Definition: – For any periodic signal with fundamental period T0 , it can be decomposed as the sum of a set of complex exponential signals as + c e x (t ) = n = − jn 0t n 0 = • cn , n = 0,1,2, , Fourier series coefficients cn = T0 • derivation of cn : T0 x (t )e − jn0t dt 2 T0 FOURIER SERIES • Fourier series x (t ) = + c e n = − jn 0t n – The periodic signal is decomposed into the weighted summation of a set of orthogonal complex exponential functions – The frequency of the n-th complex exponential function: n • The periods of the n-th complex exponential function: Tn = T0 n – The values of coefficients, cn , n = 0,1,2, , depend on x(t) • Different x(t) will result in different c n • There is a one-to-one relationship between x(t) and cn s(t ) ➔ [, c−2 , c−1,c0 , c1 , c2 ,] For a periodic signal, it can be either represented as s(t), or represented as cn 10 FOURIER SERIES • Example x(t) − K , − t x(t ) = K, t t -3 -2 -1 Rectangle pulses 24 PROPERTIES: PARSEVAL’S THEOREM • Review: power of periodic signal P= T T | x(t ) |2 dt • Parseval’s theorem if x(t ) ➔ n then T | x(t ) |2 dt = T + | | m m = − – Proof: The power of signal can be computed in frequency domain! 25 PROPERTIES: PARSEVAL’S THEOREM • Example – Use Parseval’s theorem find the power of x(t ) = Asin( 0t ) 26 OUTLINE • Introduction • Fourier series • Properties of Fourier series • Systems with periodic inputs 27 PERIODIC INPUTS: COMPLEX EXPONENTIAL INPUT • LTI system with complex exponential input x(t ) = e jt h(t ) y(t ) y(t ) = x(t ) h(t ) = h(t ) x(t ) + = h( ) x(t − )d − + = exp( jt ) h( ) exp( − j )d − • Transfer function + H () = h( ) exp( − j )d − – For LTI system with complex exponential input, the output is y(t ) = H () exp( jt ) – It tells us the system response at different frequencies 28 PERIODIC INPUT • Example: – For a system with impulse response h(t ) = (t − t0 ) find the transfer function 29 PERIODIC INPUT: • Example – Find the transfer function of the system shown in figure RL circuit 30 PERIODIC INPUTS • Example – Find the transfer function of the system shown in figure RC circuit 31 PERIODIC INPUTS: TRANSFER FUNCTION • Transfer function – For system described by differential equations n py i =0 i m (i ) (t ) = qi x ( i ) (t ) i =0 m H () = q ( j) i i i =0 n p ( j) i =0 i i 32 PERIODIC INPUTS • LTI system with periodic inputs + – Periodic inputs: x (t ) = c n = − e linear: + jn0t c e n = − n h(t ) e jn0t H (n0 ) + jn0t n exp( jn0t ) 2 0 = T h(t ) jn0t c e n H ( n ) n = − + x(t ) h(t ) jn0t c e n H ( n ) n = − For system with periodic inputs, the output is the weighted sum of the transfer function 33 PERIODIC INPUTS • Procedures: – To find the output of LTI system with periodic input • Find the Fourier series coefficients of periodic input x(t) T 2 n = x (t )e − jn0t dt = f = 0 T T • Find the transfer function of LTI system H () • The output of the system is y (t ) = + jn 0t c e n H (n0 ) n = − period of x(t) 34 PERIODIC INPUTS • Example – Find the response of the system when the input is x(t ) = cos(t ) − cos(2t ) RL Circuit 35 PERIODIC INPUTS • Example – Find the response of the system when the input is shown in figure x(t) t -3 RC circuit -2 -1 Square pulses 36 PERIODIC INPUTS: GIBBS PHENOMENON • The Gibbs Phenomenon – Most Fourier series has infinite number of elements→ unlimited bandwidth x (t ) = + jn 0t c e n n = − • What if we truncate the infinite series to finite number of elements? x N (t ) = +N jn 0t c e n n=− N – The truncated signal, xN (t ) , is an approximation of the original signal x(t) 37 PERIODIC INPUTS: GIBBS PHENOMENON x(t) t -3 -2 -1 2K , n odd, cn = j n 0, n even x N (t ) = +N jn 0t c e n n=− N Square pulses x3 (t ) x5 (t ) x19 (t ) 38 FOURIER SERIES • Analogy: Optical Prism – Each color is an Electromagnetic wave with a different frequency Optical prism ... open-source software Audacity 14 FOURIER SERIES • Example – Find the Fourier series of s(t ) = exp( j 0t ) 15 FOURIER SERIES • Example – Find the Fourier series of s(t ) = B + Acos(0t + ... exponential signals as + c e x (t ) = n = − jn 0t n 0 = • cn , n = 0,1,2, , Fourier series coefficients cn = T0 • derivation of cn : T0 x (t )e − jn0t dt 2 T0 FOURIER SERIES • Fourier series. .. T = 1, T = 15 17 FOURIER SERIES: DIRICHLET CONDITIONS • Can any periodic signal be decomposed into Fourier series? – Only signals satisfy Dirichlet conditions have Fourier series • Dirichlet
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