Fourier Series for CT & DT Signals

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1. Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC. The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (17681830), who made important contributions to the study of trigonometric series. Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 and publishing Analytical theory of heat in 1822. 2. The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine. This superposition or linear combination is called the Fourier series. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems. 3. Determining the Fourier Series Representation of a Continuous Time Signal tjntjk k k tjn eeaetx 000 )( 1)( )/2(0 tTjk k k tjk k k eaeatx A periodic CT signal can be expressed as a linear combination of harmonically related complex exponentials of the form :- Multiplying both sides with , we get :- tjn e 0 Now if we integrate both sides from 0 to T, we have :- dteeadtetx T tjntjk k k T tjn 00 000 )( 4. Interchanging the order of integration and summation :- 2)( 00 000 T tjntjk k k T tjn dteeadtetx Applying Eulers formula to the bracketed Integral :- TTT tnkj tdtnkjtdtnkdte 0 0 0 0 0 )( )sin()cos(0 T tnkj dte 0 )( 0 T, if k = n 0, if k n{ In the R.H.S. side of the integral, for k not equal to n, both the integrals will be zero. Fr k=n, the integrand equals 1 and thus, the integral equals T. 5. T tnkj dte 0 )( 0 T, if k = n 0, if k n{ dte T tnkj 0)( Integrating from 0 to T is same as integrating over any interval of length T because we are only concerned with integral number of periods of cosine and sine function in the previous equation. Now the R.H.S. of equation (2) reduces to Tan . Therefore :- dtetx T a T tjn n 0 0 )( 1 Consequently, we can write dtetx T a T tjn n 0 )( 1 6. Hence the Fourier series of x(t) can be expressed as :- 4)( 1 )( 1 3)( )/2( )/2( 0 0 dtetx T dtetx T a eaeatx T tTjk T tjk k tTjk k k tjk k k Here equation (3) is the analysis equation and equation (4) is the synthesis equation. Coefficients ak are called Fourier Series Coefficients or Spectral Coefficients of x(t). Here the DC or constant component of x(t) is :- dttx T a T 0 0 )( 1 7. 7 Example :- Periodic square wave defined over one period as 2/tT0 t1 1 1 T T tx 1 tx -T -T/2 T1 0 T1 T/2 T t Defining 10 1 sin 2 Tkc T T ak k Tk ak 10sin When k 0 At k = 0 T T dttx T a T T 1 2/ /2- 0 21 x x xc sin sin 8. Fourier Series of some Common Signals 9. Conditions for convergence of CT Fourier series Every function (x) of period 2 satisfying following conditions known as DIRICHLETS CONDITIONS, can be expressed in the form of Fourier series. 1. Over any period, x(t) must be absolutely integrable :- it guarantees that each coefficient ak will be finite 2. In a single period, x(t) should have finite number of max and min 3. In any finite interval of time, there are only a finite number of discontinuities. Each discontinuity should be finite. dttx T 2 )( ka 10. Properties of CT Fourier Series 1. Linearity z(t) = Ax(t) + By(t) ck = Aak + Bbk 2. Time Shifting z(t) = x(t-t0) ck = In time shifting magnitude of Fourier series coefficient remains the same |ck| = |ak| 3. Time Reversal z(t) = x(-t) ck = a-k If x(t) is even, ak = a-k If x(t) is odd, ak = -a-k x(t) & y(t) are two periodic signals with period T and Fourier coefficients ak & bk respectively 00tjk k ea 11. 4. Time Scaling z(t) = x(t) ck = ak But, the fundamental period is (T/) 5. Multiplication z(t) = x(t)y(t) ck = (DT convolution between coefficients) 6. Conjugation & Conjugate Symmetry Real x(t) a-k = a*k (conjugate symmetric) Real & Even x(t) ak = a*k (real & even ak) Real & Odd x(t) ak = -a*k (purely imaginary & odd ak ; a0 = 0) z(t) = Even part of x(t) ck = Real{ak} z(t) = Odd part of x(t) ck = jImaginary{ak} l lklba 12. 7. Periodic Convolution 8. Parsevals Relation Total average power = sum of average power in all harmonic components Energy in time domain equals to energy in frequency domain dtyxtytx T )()()(*)( Takbk k k T tjk k k T k tjk k T adtea T dtea T dttx T 222 2 2 0 0 1 1 )( 1 13. Determining the Fourier Series Representation of a Discrete Time Signal 1][ )/2(0 Nk nNjk k Nk njk k eaeanx A periodic DT signal can be expressed as set of N linear equations for N unknown coeffficients ak as k ranges over a set of N successive integers :- Nn nNjk e )/2( N, if k = 0,+N,+2N 0, otherwise { According to the fact that :- 14. Nk nNrkj k nNjr eaenx )/2)(()/2( ][ Multiplying both sides with , we get :- nNjr e )/2( Now if summing over N terms, we have :- Nn Nk nNrkj k Nn nNjr eaenx )/2)(()/2( ][ Interchanging the order of summation, we have :- Nk Nn nNrkj k Nn nNjr eaenx )/2)(()/2( ][ 15. Nn nNrkj e )/2)(( N, if k = 0,+N,+2N 0, otherwise { According to the fact stated before, we can conclude that :- So the R.H.S. of the equation reduces to Nar Nn nNjr r enx N a )/2( ][ 1 Hence the Fourier series of x[n] can be expressed as below where equation (3) is the analysis equation and equation (4) is the synthesis equation 4][ 1 ][ 1 3][ )/2( )/2( 0 0 Nn nNjk Nn njk k Nk nNjk k Nk njk k enx N enx N a eaeanx 16. Properties of DT Fourier Series 1. Linearity z[n] = Ax[n] + By[t] ck = Aak + Bbk 2. Time Shifting z[t] = x[n-n0] ck = In time shifting magnitude of Fourier series coefficient remains the same |ck| = |ak| 3. Time Reversal z[t] = x[-n] ck = a-k If x[n] is even, ak = a-k If x[n] is odd, ak = -a-k x[n] & y[n] are two periodic signals with period N and Fourier coefficients ak & bk respectively periodic with period N 0)/2( nNjk k ea 17. 4. Multiplication z[n] = x[n]y[n] ck = 5. Conjugation & Conjugate Symmetry Real x[n] a-k = a*k (conjugate symmetric) Real & Even x[n] ak = a*k (real & even ak) Real & Odd x[n] ak = -a*k (purely imaginary & odd ak ) z[n] = Even part of x[n] ck = Real{ak} z[n] = Odd part of x[n] ck = jImaginary{ak} 6. Periodic Convolution Nl lklba Nr rnyrxnynx ][][][*][ Nakbk 18. 7. Parsevals Relation Total average power = sum of average power in all harmonic components Energy in time domain equals to energy in frequency domain 22 ][ 1 Nk k Nn anx N 19. FOURIER SERIES, which is an infinite series representation of such functions in terms of sine and cosine terms, is useful here. Thus, FOURIER SERIES, are in certain sense, more UNIVERSAL than TAYLORs SERIES as it applies to all continuous, periodic functions and also to the functions which are discontinuous in their values and derivatives. FOURIER SERIES a very powerful method to solve ordinary and partial differential equation.. As we know that TAYLOR SERIES representation of functions are valid only for those functions which are continuous and differentiable. But there are many discontinuous periodic function which requires to express in terms of an infinite series containing sine and cosine terms. Advantages of using Fourier Series 20. Consider a mass-spring system as before, where we have a mass m on a spring with spring constant k, with damping c, and a force F(t) applied to the mass. Suppose the forcing function F(t) is 2L-periodic for some L > 0. The equation that governs this particular setup is The general solution consists of the complementary solution xc, which solves the associated homogeneous equation mx + cx + kx = 0, and a particular solution of (1) we call xp. mx(t) + cx(t) + kx(t) = F(t) Applications of using Fourier Series 1. Forced Oscillation 21. For c > 0, the complementary solution xc will decay as time goes by. Therefore, we are mostly interested in a particular solution xp that does not decay and is periodic with the same period as F(t). We call this particular solution the steady periodic solution and we write it as xsp as before. What will be new in this section is that we consider an arbitrary forcing function F(t) instead of a simple cosine. For simplicity, let us suppose that c = 0. The problem with c > 0 is very similar. The equation mx + kx = 0 has the general solution, x(t) = A cos(t) + B sin(t); Where, 22. Any solution to mx(t) + kx(t) = F(t) is of the form A cos(t) + B sin(t) + xsp. The steady periodic solution xsp has the same period as F(t). In the spirit of the last section and the idea of undetermined coefficients we first write, Then we write a proposed steady periodic solution x as, where an and bn are unknowns. We plug x into the deferential equation and solve for an and bn in terms of cn and dn. 23. It turns out that (almost) any kind of a wave can be written as a sum of sines and cosines. So for example, if a voice is recorded for one second saying something, I can find its Fourier series which may look something like this for example and this interactive module shows you how when you add sines and/or cosines the graph of cosines and sines becomes closer and closer to the original graph we are trying to approximate. The really cool thing about Fourier series is that first, almost any kind of a wave can be approximated. Second, when Fourier series converge, they converge very fast. So one of many applications is compression. Everyone's favorite MP3 format uses this for audio compression. You take a sound, expand its Fourier se