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Periodic signals can be written as a sum of sine and cosine waves: u(t) u(t) = a + n= (a ncosπnft+b n sinπnft) T = + T/3 T/ T +.65sin(πFt) -.6sin(πFt) +.6sin(πFt) + -.3cos(πFt) + T/ Fundamental Period: the smallest T > for which u(t + T) = u(t). Fundamental Frequency: F = T. The nth harmonic is at frequency nf. Some waveforms need infinitely many harmonics (countable infinity). E. (5-6) : / 3
Real versus Complex Fourier Series All the algebra is much easier if we use e iωt instead of cosωt and sinωt u(t) = a + n= (a ncosπnft+b n sinπnft) Substitute: cosωt = eiωt + e iωt sinωt = i u(t) = a + ( n= (a n ib n )e iπnft + (a n +ib n )e iπnft) = n= U ne iπnft U +n = (a n ib n ) and U n = (a n +ib n ). U +n and U n are complex conjugates. eiωt + i e iωt U +n is half the equivalent phasor in Analysis of Circuits. u(t) T U.5 Plot the magnitude spectrum and phase spectrum: U pi -pi -F -3F -F -F F F 3F F -F -3F -F -F F F 3F F E. (5-6) : 3 / 3
Fourier Series versus Periodic signals Fourier Series Discrete spectrum u(t) T u(t) = n= U ne iπnft -F -3F -F -F F F 3F F pi U U.5 -pi -F -3F -F -F F F 3F F Aperiodic signals Continuous Spectrum u(t) a=.5-5 5 Time (s) u(t) = f= U(f)eiπft df.5..3.. -5 5 Frequency (Hz) E. (5-6) : / 3 U(f) U(f) (rad) pi/ -pi/ -5 5 Frequence (Hz) Both types of spectrum are conjugate symmetric. If u(t) is periodic, its Fourier transform consists of Dirac δ functions with amplitudes {U n }.
Fourier Series: u(t) = n= U ne iπnft = how do you work out the Fourier coefficients, U n? Key idea: { e iωt if ω = = cosωt+isinωt = Orthogonality: e iπnft e iπmft = otherwise { for m = n for m n So, to find a particular coefficient, U m, we work out u(t)e iπmft = ( n= U ne iπnft) e iπmft = n= U n e iπnft e iπmft = U m [since all other terms are zero] Calculate the average by integrating over any integer number of periods U m = u(t)e iπmft = T T t= u(t)e iπmft dt Notice the negative sign in Fourier analysis: in order to extract the term in the series containing e +iπmft we need to multiply by e iπmft. E. (5-6) : 5 / 3
Power Conservation Fourier Series: u(t) = n= U ne iπnft Average power in u(t): P u u(t) = T T u (t)dt Average power in Fourier component n: Un e iπnft = U n e iπnft = U n Power conservation (Parseval s Theorem): P u = u(t) = n= U n The average power in u(t) is equal to the sum of the average powers in all the Fourier components. This is a consequence of orthogonality: u(t) = ( n= U ne iπnft)( m= U me iπmft) [u(t) real] = n= m= U nume iπnft e iπmft = n= m= U num e iπnft e iπmft = n= U n E. (5-6) : 6 / 3
Truncated Fourier Series: u N (t) = N n= N U ne iπnft Approximation error: e N (t) = u N (t) u(t) Average error power P en = n >N U n. P en monotonically as N. Gibbs phenomenon If u(t ) has a discontinuity of height h then: u N (t ) the midpoint of the discontinuity as N..5.5.5.5 max(u )=.5 N= 5 5 max(u )=.37 N= 5 5 max(u 3 )=. N=3 5 5 max(u 5 )=.9 N=5 5 5 u N (t) overshoots by ±9% h at t t ± T N+. For large N, the overshoots move closer to the discontinuity but do not decrease in size..5.5 max(u )=.89 N= 5 5 max(u )=.89 - -.5.5 [Enlarged View: u (t)] E. (5-6) : 7 / 3
Fourier Series: u(t) = n= U ne iπnft Integration: v(t) = t u(τ)dτ V n = provided U = V =. iπnf U n Differentiation: w(t) = du(t) dt W n = iπnf U n provided w(t) satisfies the Dirichlet conditions. : u(t) has a discontinuity U n is O ( n) for large n d k u(t) dt k is the lowest derivative with a discontinuity U n is O ( n k+ ) for large n If the coefficients, U n, decrease rapidly then only a few terms are needed for a good approximation. E. (5-6) : 8 / 3
If u(t) is only defined over a finite range, [, B], we can make it periodic by defining u(t±b) = u(t). Coefficients are given by U n = B B u(t)e iπnft dt. Example: u(t) = t for t < N= - N=3 - - Symmetric extension: To avoid a discontinuity at t = T, we can instead make the period B and define u( t) = u(+t). N= N=3 N=6 N=6 - - - Symmetry around t = means coefficients are real-valued and symmetric (U n = U n = U n ). Still have a first-derivative discontinuity at t = B but now we have no Gibbs phenomenon and coefficients n instead of n so approximation error power decreases more quickly. E. (5-6) : 9 / 3
Dirac Delta Function δ(x) is the limiting case as w of a pulse w wide and w high It is an infinitely thin, infinitely high pulse at x = with unit area. δ. (x) -3 - - 3 x Area: δ(x)dx = Scaling: δ(cx) = c δ(x).5 δ(x) -3 - - 3 x Shifting: δ(x a) is a pulse at x = a and is zero everywhere else Multiplication: f(x) δ(x a) = f(a) δ(x a) Integration: f(x) δ(x a)dx = f(a) : u(t) = δ(t) U(f) = We plot hδ(x) as a pulse of height h (instead of its true height of ) E. (5-6) : / 3
: u(t) = U(f)eiπft df U(f) = u(t)e iπft dt An Energy Signal has finite energy E u = u(t) dt < Complex-valued spectrum, U(f), decays to zero as f ± Energy Conservation: E u = E U where E U = U(f) df u(t) a=.5-5 5 Time (s) U(f).5..3.. -5 5 Frequency (Hz) Periodic Signals Dirac δ functions at harmonics. Same complex-valued amplitudes as U n from Fourier Series u(t) T E u = but ave power is P u = U.5 u(t) = n= U n -F -3F -F -F F F 3F F E. (5-6) : / 3
: w(t) = u(t) v(t) w(t) = u(τ)v(t τ)dτ [In the integral, the arguments of u( ) and v( ) add up to t] acts algebraically like : Commutative, Associative, Distributive over +. Identity element is δ(t): u(t) δ(t) = u(t) Multiplication in either the time or frequency domain is equivalent to convolution in the other domain: w(t) = u(t) v(t) W(f) = U(f)V(f) y(t) = u(t)v(t) Y(f) = U(f) V(f) Example application: Impulse Response: [ y(t) for x(t) = δ(t)] h(t) = RC e t RC for t Frequency Response: H(f) = +iπf RC : y(t) = h(t) x(t) Multiplication: Y(f) = H(f)X(f) E. (5-6) : / 3
Cross-correlation: w(t) = u(t) v(t) w(t) = u (τ t)v(τ)dτ [In the integral, the arguments of u ( ) and v( ) differ by t] is not commutative or associative (unlike ) Cauchy-Schwartz Inequality Bound on w(t) For all values of t: w(t) E u E v u(t t ) is an exact multiple of v(t) w(t ) = E u E v Normalized cross-correlation: w(t) Eu E v has a maximum absolute value of Cross-correlation is used to find the time shift, t, at which two signals match and also how well they match. Auto-correlation is the cross-correlation of a signal with itself: used to find the period of a signal (i.e. the time shift where it matches itself). E. (5-6) : 3 / 3