6.003: Signals and SystemsFourier SeriesNovember 3, 2009Fourier RepresentationsRepresentations based on sinusoids.systemsignalinsignaloutTo date, we have focused primarily on time-domain techniques,especially with transient signals (e.g., impulse response).The primary focus for the next few weeks will be frequency-domaintechniques (e.g., frequency response) which concern eternal signals.Fourier SeriesToday: Fourier series represent signals in terms of sinusoids.This new representation for signals leads to a new representation forsystems asfilters.HarmonicsRepresenting signals by the amplitudes and phases of harmonic com-ponents.ωω02ω03ω04ω05ω06ω00 1 2 345 6← harmonic #DC →fundamental →second harmonic →third harmonic →fourth harmonic →fifth harmonic →sixth harmonic →Musical InstrumentsHarmonic content is natural way to describe some kinds of signals.Ex: musical instruments (http://theremin.music.uiowa.edu/MIS)pianotcellotbassoontoboethorntaltosaxtviolintbassoont1252secondsMusical InstrumentsHarmonic content is natural way to describe some kinds of signals.Ex: musical instruments (http://theremin.music.uiowa.edu/MIS)pianokcellokbassoonkoboekhornkaltosaxkviolinkMusical InstrumentsHarmonic content is natural way to describe some kinds of signals.Ex: musical instruments (http://theremin.music.uiowa.edu/MIS)pianotpianokviolintviolinkbassoontbassoonkHarmonicsHarmonic structure determines consonance and dissonance.octave (D+D’) fifth (D+A) D+E[time(periods of "D")harmonics0 1 2 3 4 5 6 7 8 9 1011120 1 2 3 4 5 6 7 8 9 1011120 1 2 3 4 5 6 7 8 9 101112–101DD'DADE0 1 2 3 4 5 6 7 8 90 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 90 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 90 1 2 3 4 5 6 7Harmonic RepresentationsWhat signals can be represented by sums of harmonic components?ωω02ω03ω04ω05ω06ω0T =2πω0tT =2πω0tOnly periodic signals: all harmonics of ω0are periodic in T = 2π/ω0.Harmonic RepresentationsIs it possible to represent ALL periodic signals with harmonics?What about discontinuous signals?2πω0t2πω0tFourier claimed YES — even though all harmonics are continuous!Lagrange ridiculed the idea that a discontinuous signal could bewritten as a sum of continuous signals.We will assume the answer is YES and see if the answer makes sense.Separating harmonic componentsUnderlying properties.1. Multiplying two harmonics produces a new harmonic with thesame fundamental frequency:ejkω0t× ejlω0t= ej(k+l)ω0t.2. The integral of a harmonic over any time interval with lengthequal to a period T is zero unless the harmonic is at DC:Zt0+Tt0ejkω0tdt ≡ZTejkω0tdt =0, k 6= 0T, k = 0= T δ[k]Separating harmonic componentsAssume that x(t) is periodic in T and is composed of a weighted sumof harmonics of ω0= 2π/T .x(t) = x(t + T) =∞Xk=−∞akejω0ktThenZTx(t)e−jlω0tdt =ZT∞Xk=−∞akejω0kte−jω0ltdt=∞Xk=−∞akZTejω0(k−l)tdt=∞Xk=−∞akT δ[k − l] = T alThereforeak=1TZTx(t)e−jω0ktdt =1TZTx(t)e−j2πTktdtFourier SeriesDetermining harmonic components of a periodic signal.ak=1TZTx(t)e−j2πTktdt (“analysis” equation)x(t)= x(t + T) =∞Xk=−∞akej2πTkt(“synthesis” equation)Check YourselfLet akrepresent the Fourier series coefficients of the followingsquare wave.t12−120 1How many of the following statements are true?1. ak= 0 if k is even2. akis real-valued3. |ak| decreases with k24. there are an infinite number of non-zero ak5. all of the aboveCheck YourselfLet akrepresent the Fourier series coefficients of the following squarewave.t12−120 1ak=ZTx(t)e−j2πTktdt = −12Z0−12e−j2πktdt +12Z120e−j2πktdt=1j4πk2 − ejπk− e−jπk=(1jπk; if k is odd0 ; otherwiseCheck YourselfLet akrepresent the Fourier series coefficients of the following squarewave.ak=(1jπk; if k is odd0 ; otherwiseHow many of the following statements are true?1. ak= 0 if k is even√2. akis real-valued X3. |ak| decreases with k2X4. there are an infinite number of non-zero ak√5. all of the above XCheck YourselfLet akrepresent the Fourier series coefficients of the followingsquare wave.t12−120 1How many of the following statements are true? 21. ak= 0 if k is even√2. akis real-valued X3. |ak| decreases with k2X4. there are an infinite number of non-zero ak√5. all of the above XFourier Series PropertiesIf a signal is differentiated in time, its Fourier coefficients are multi-plied by j2πTk.Proof: Letx(t) = x(t + T) =∞Xk=−∞akej2πTktthen˙x(t) = ˙x(t + T) =∞Xk=−∞j2πTk akej2πTktCheck YourselfLet bkrepresent the Fourier series coefficients of the followingtriangle wave.t18−1801How many of the following statements are true?1. bk= 0 if k is even2. bkis real-valued3. |bk| decreases with k24. there are an infinite number of non-zero bk5. all of the aboveCheck YourselfThe triangle waveform is the integral of the square wave.t12−120 1t18−1801Therefore the Fourier coefficients of the triangle waveform are1j2πktimes those of the square wave.bk=1jkπ×1j2πk=−12k2π2; k oddCheck YourselfLet bkrepresent the Fourier series coefficients of the following tri-angle wave.bk=−12k2π2; k oddHow many of the following statements are true?1. bk= 0 if k is even√2. bkis real-valued√3. |bk| decreases with k2√4. there are an infinite number of non-zero bk√5. all of the above√Check YourselfLet bkrepresent the Fourier series coefficients of the followingtriangle wave.t18−1801How many of the following statements are true? 51. bk= 0 if k is even√2. bkis real-valued√3. |bk| decreases with k2√4. there are an infinite number of non-zero bk√5. all of the above√Fourier SeriesOne can visualize convergence of the Fourier Series by incrementallyadding terms.Example: triangle waveformt0Xk = −0k odd−12k2π2ej2πkt18−1801Fourier SeriesOne can visualize convergence of the Fourier Series by incrementallyadding terms.Example: triangle waveformt1Xk = −1k odd−12k2π2ej2πkt18−1801Fourier SeriesOne can visualize convergence of the Fourier Series by incrementallyadding terms.Example: triangle waveformt3Xk = −3k odd−12k2π2ej2πkt18−1801Fourier SeriesOne can visualize convergence of the Fourier Series by incrementallyadding terms.Example: triangle waveformt5Xk = −5k odd−12k2π2ej2πkt18−1801Fourier SeriesOne can visualize convergence of the Fourier Series by incrementallyadding terms.Example: triangle waveformt7Xk = −7k odd−12k2π2ej2πkt18−1801Fourier SeriesOne can visualize convergence of the Fourier Series by
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