Continuous-time Fourier analysis consists of the Fourier series, or Fourier expansion, and the Fourier transform, or Fourier integral. The former is discussed in this section. Continuous-time Fourier analysis will not be presented in depth but rather as a review.

Trigonometric expansion The famous French mathematician Jean Baptiste Joseph Fourier demonstrated that a periodic waveform, such as the one in , can be expanded into sinsoidal components having frequencies which are the multiples of the fundamental frequency of the waveform.

A periodic waveform of period T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaa@37A4@ Let’s begin with the time signal x(t) size 12{x \( t \) } {}, periodic at period T0 (sec) or angular frequency Ω0=2π/T0 size 12{ %OMEGA rSub { size 8{0} } =2π/T rSub { size 8{0} } } {} (rad/sec) or frequency F 0 =1/ T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaaGimaaqabaGccqGH9aqpdaWcgaqaaiaaigdaaeaacaWGubWaaSbaaSqaaiaaicdaaeqaaaaaaaa@3B2E@ (Hz) . The trigonometric expansion, or series, is x(t)= a 0 + ∑ n=1 ∞ a n cos⁡n Ω 0 t + ∑ n=1 ∞ b n sin⁡ Ω 0 t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpcaWGHbWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSYaaabCaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaci4yaiaac+gacaGGZbGaamOBaiabfM6axnaaBaaaleaacaaIWaaabeaakiaadshaaSqaaiaad6gacqGH9aqpcaaIXaaabaGaeyOhIukaniabggHiLdGccqGHRaWkdaaeWbqaaiaadkgadaWgaaWcbaGaamOBaaqabaGcciGGZbGaaiyAaiaac6gacqqHPoWvdaWgaaWcbaGaaGimaaqabaGccaWG0baaleaacaWGUbGaeyypa0JaaGymaaqaaiabg6HiLcqdcqGHris5aaaa@5C4F@ Where the coefficients are given by a 0 = 1 T 0 ∫ − T 0 /2 T 0 /2 x(t)dt MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaaGimaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGubWaaSbaaSqaaiaaicdaaeqaaaaakmaapedabaGaamiEaiaacIcacaWG0bGaaiykaiaadsgacaWG0baaleaacqGHsisldaWcgaqaaiaadsfadaWgaaadbaGaaGimaaqabaaaleaacaaIYaaaaaqaamaalyaabaGaamivamaaBaaameaacaaIWaaabeaaaSqaaiaaikdaaaaaniabgUIiYdaaaa@48CE@ a n = 2 T 0 ∫ − T 0 /2 T 0 /2 x(t)cos⁡n Ω 0 tdt MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiaaikdaaeaacaWGubWaaSbaaSqaaiaaicdaaeqaaaaakmaapedabaGaamiEaiaacIcacaWG0bGaaiykaiGacogacaGGVbGaai4Caiaad6gacqqHPoWvdaWgaaWcbaGaaGimaaqabaGccaWG0bGaamizaiaadshaaSqaaiabgkHiTmaalyaabaGaamivamaaBaaameaacaaIWaaabeaaaSqaaiaaikdaaaaabaWaaSGbaeaacaWGubWaaSbaaWqaaiaaicdaaeqaaaWcbaGaaGOmaaaaa0Gaey4kIipaaaa@5045@ b n = 2 T 0 ∫ − T 0 /2 T 0 /2 x(t)sin⁡n Ω 0 tdt MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkgadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiaaikdaaeaacaWGubWaaSbaaSqaaiaaicdaaeqaaaaakmaapedabaGaamiEaiaacIcacaWG0bGaaiykaiGacohacaGGPbGaaiOBaiaad6gacqqHPoWvdaWgaaWcbaGaaGimaaqabaGccaWG0bGaamizaiaadshaaSqaaiabgkHiTmaalyaabaGaamivamaaBaaameaacaaIWaaabeaaaSqaaiaaikdaaaaabaWaaSGbaeaacaWGubWaaSbaaWqaaiaaicdaaeqaaaWcbaGaaGOmaaaaa0Gaey4kIipaaaa@504B@ In above integrals the limits were put as − T 0 /2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiabgkHiTmaalyaabaGaamivamaaBaaaleaacaaIWaaabeaaaOqaaiaaikdaaaaaaa@3965@ and T 0 /2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaalyaabaGaamivamaaBaaaleaacaaIWaaabeaaaOqaaiaaikdaaaaaaa@3878@ , but other limits can be used so long as the distance between them is the period T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaa@379C@ , e.g. 0 and T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaa@379C@ . The expansion components have following meaning : a 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaaGimaaqabaaaaa@37A9@ : The average of the signal (or DC component) a 1 cos⁡ Ω 0 + b 1 sin⁡ Ω 0 t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaaGymaaqabaGcciGGJbGaai4BaiaacohacqqHPoWvdaWgaaWcbaGaaGimaaqabaGccqGHRaWkcaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaci4CaiaacMgacaGGUbGaeuyQdC1aaSbaaSqaaiaaicdaaeqaaOGaamiDaaaa@460E@ : The fundamental component (remember the sum of two sinusoids of the same frequeny is a sinusoid at that frequency, see ), or the first harmonic. a 2 cos⁡ ω 0 t+ b 2 sin⁡ ω 0 t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaaGOmaaqabaGcciGGJbGaai4BaiaacohacqaHjpWDdaWgaaWcbaGaaGimaaqabaGccaWG0bGaey4kaSIaamOyamaaBaaaleaacaaIYaaabeaakiGacohacaGGPbGaaiOBaiabeM8a3naaBaaaleaacaaIWaaabeaakiaadshaaaa@4787@ : The second harmonic a 3 cos⁡ ω 0 t+ b 3 sin⁡ ω 0 t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaaG4maaqabaGcciGGJbGaai4BaiaacohacqaHjpWDdaWgaaWcbaGaaGimaaqabaGccaWG0bGaey4kaSIaamOyamaaBaaaleaacaaIZaaabeaakiGacohacaGGPbGaaiOBaiabeM8a3naaBaaaleaacaaIWaaabeaakiaadshaaaa@4789@ : The third harmonic ... Find the Fourier expansion for the symmetric square wave of . Solution We observe straightaway that the DC component is zero since the positive and regative parts of the signal are equal: a 0 =0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIWaaaaa@3973@ Of course when using a we will get the same result. Next, since the waveform is odd-symmetric (antisymmetric) (symmetric with respect to the origin), the cosine components are also zero : a n =0 all n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaamOBaaqabaGccqGH9aqpcaaIWaGaaGzbVlaaywW7caWGHbGaamiBaiaadYgacaaMf8UaamOBaaaa@4211@

(periodic square wave) It is left with the sine components given by b n = 4A T 0 ∫ 0 T 0 /2 sin⁡n Ω 0 tdt = 4A T 0 −1 n ω 0 [ cos⁡n Ω 0 t ] 0 T 0 /2 = −4A 2π 1 n [ 1−1 ]=0 , n even = −4A 2π 1 n [ −1−1 ]= 4A 2π 1 n , n old MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@A2CB@ These bn size 12{b rSub { size 8{n} } } {}coefficients can be put in a more concise form : Thus the Fourier expansion is x(t)= ∑ n=1 ∞ 4A π 1 (2n−1) sin⁡(2n−1) Ω 0 t , n=1, 2, 3,... = 4A π (sin⁡ Ω 0 t+ 1 3 sin⁡3 Ω 0 t+ 1 5 sin⁡5 Ω 0 t+...) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@8A3F@

(magnitude spectrum) is the plot of the normalized coefficients with respect to the normalized angular frequency. We know that the sum of two sinusoids of the same frequency is another sinusoid at that frequency, specifically acos⁡Ωt+bsin⁡Ωt= a 2 + b 2 cos⁡(Ωt+ tan⁡ −1 b a ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadggaciGGJbGaai4BaiaacohacqqHPoWvcaWG0bGaey4kaSIaamOyaiGacohacaGGPbGaaiOBaiabfM6axjaadshacqGH9aqpdaGcaaqaaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGIbWaaWbaaSqabeaacaaIYaaaaaqabaGcciGGJbGaai4BaiaacohacaGGOaGaeuyQdCLaamiDaiabgUcaRiGacshacaGGHbGaaiOBamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaalaaabaGaamOyaaqaaiaadggaaaGaaiykaaaa@5712@ Because of this, expansion can be changed to the form of amplitude and phase: x(t)= c 0 + ∑ n=1 ∞ c n cos⁡(n Ω 0 t+ Φ 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpcaWGJbWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSYaaabCaeaacaWGJbWaaSbaaSqaaiaad6gaaeqaaOGaci4yaiaac+gacaGGZbGaaiikaiaad6gacqqHPoWvdaWgaaWcbaGaaGimaaqabaGccaWG0bGaey4kaSIaeuOPdy0aaSbaaSqaaiaaicdaaeqaaOGaaiykaaWcbaGaamOBaiabg2da9iaaigdaaeaacqGHEisPa0GaeyyeIuoaaaa@5146@ Where c 0 = a 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadogadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaaicdaaeqaaaaa@3A87@ c n = a n 2 + b n 2 n=1, 2, 3,... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadogadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaGcaaqaaiaadggadaqhaaWcbaGaamOBaaqaaiaaikdaaaGccqGHRaWkcaWGIbWaa0baaSqaaiaad6gaaeaacaaIYaaaaaqabaGccaaMf8UaaGzbVlaad6gacqGH9aqpcaaIXaGaaiilaiaaysW7caaIYaGaaiilaiaaysW7caaIZaGaaiilaiaac6cacaGGUaGaaiOlaaaa@4E08@ Φ n = tan⁡ −1 − b n a n n=1, 2, 3,... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiabfA6agnaaBaaaleaacaWGUbaabeaakiabg2da9iGacshacaGGHbGaaiOBamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaalaaabaGaeyOeI0IaamOyamaaBaaaleaacaWGUbaabeaaaOqaaiaadggadaWgaaWcbaGaamOBaaqabaaaaOGaaGzbVlaaywW7caWGUbGaeyypa0JaaGymaiaacYcacaaMe8UaaGOmaiaacYcacaaMe8UaaG4maiaacYcacaGGUaGaaiOlaiaac6caaaa@51DB@ In this expansion we can recognize c0 size 12{c rSub { size 8{0} } } {} as the average component, c1cos(ω0t+Φ1) size 12{c rSub { size 8{1} } "cos" \( ω rSub { size 8{0} } t+Φ rSub { size 8{1} } \) } {} the fundamental component, and c2cos(2ω0t+Φ2) size 12{c rSub { size 8{2} } "cos" \( 2ω rSub { size 8{0} } t+Φ rSub { size 8{2} } \) } {} the second harmonic... The plot of the coefficients versus frequency is the magnitude spectrum, and the plot of the phase Φn size 12{Φ rSub { size 8{n} } } {}versus ferquency is the phase spectrum. Both spectra are discrete or line spectra. Find the Fourier expansion of the waveform in . Solution The coefficients are c 0 = a 0 c n = a n 2 + b n 2 = b n , n=1, 2, 3,... Φ n = tan⁡ −1 − b n a n =− 90 0 (=−π/2 ) , n=1, 2, 3,... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@8091@ The magnitude spectrum is as previously, the phase spectrum is given in . The expansion expression is

(phase spectrum) x(t)= ∑ n=1 ∞ 4A π 1 2n−1 cos⁡[ (2n−1) Ω 0 t+ 90 0 ] = ∑ n=1 ∞ 4A π 1 2n−1 cos⁡(2n−1) Ω 0 t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@755A@

Complex exponential expansion The Fourier expansion in the form of complex exponentials are more fundamental since it is more compact and is related directly to the Fourier transform. The expansion is x(t)= ∑ n=−∞ ∞ X n e jn Ω 0 t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpdaaeWbqaaiaadIfadaWgaaWcbaGaamOBaaqabaGccaWGLbWaaWbaaSqabeaacaWGQbGaamOBaiabfM6axnaaBaaameaacaaIWaaabeaaliaadshaaaaabaGaamOBaiabg2da9iabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHris5aaaa@4AA9@ The two symmetric components Xn size 12{X rSub { size 8{n} } } {} and X−n size 12{X rSub { size 8{ - n} } } {} always appear in pairs and the sum of each pair is a real signal. Relations between the complex exponential and trigonometric coefficients are X 0 = a 0 = c 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0Jaam4yamaaBaaaleaacaaIWaaabeaaaaa@3D5A@ X n = a n +j b n 2 = c n 2 e j Φ n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiaadggadaWgaaWcbaGaamOBaaqabaGccqGHRaWkcaWGQbGaamOyamaaBaaaleaacaWGUbaabeaaaOqaaiaaikdaaaGaeyypa0ZaaSaaaeaacaWGJbWaaSbaaSqaaiaad6gaaeqaaaGcbaGaaGOmaaaacaWGLbWaaWbaaSqabeaacaWGQbGaeuOPdy0aaSbaaWqaaiaad6gaaeqaaaaaaaa@4828@ X n = a n −j b n 2 = c n 2 e −j Φ n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiaadggadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGQbGaamOyamaaBaaaleaacaWGUbaabeaaaOqaaiaaikdaaaGaeyypa0ZaaSaaaeaacaWGJbWaaSbaaSqaaiaad6gaaeqaaaGcbaGaaGOmaaaacaWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaeuOPdy0aaSbaaWqaaiaad6gaaeqaaaaaaaa@4920@ The coefficients Xn size 12{X rSub { size 8{n} } } {} can be computed directly from X n = 1 T 0 ∫ 0 T 0 x(t) e −jn Ω 0 t dt MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGubWaaSbaaSqaaiaaicdaaeqaaaaakmaapedabaGaamiEaiaacIcacaWG0bGaaiykaaWcbaGaaGimaaqaaiaadsfadaWgaaadbaGaaGimaaqabaaaniabgUIiYdGccaWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaamOBaiabfM6axnaaBaaameaacaaIWaaabeaaliaadshaaaGccaWGKbGaamiDaaaa@4CC4@ The limits of the integral can be -T0/2 size 12{"-T" rSub { size 8{0} } /2 } {}and T0/2 size 12{T rSub { size 8{0} } /2 } {}instead of 0 and T as above. Since the coefficients Xn size 12{X rSub { size 8{n} } } {} are generally complex, we write X n =| X n | e j Φ n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaabdaqaaiaadIfadaWgaaWcbaGaamOBaaqabaaakiaawEa7caGLiWoacaWGLbWaaWbaaSqabeaacaWGQbGaeuOPdy0aaSbaaWqaaiaad6gaaeqaaaaaaaa@42B1@ The variation of ∣Xn∣ size 12{ lline "Xn" rline } {} is the magnitude spectrum, the variation of Φn size 12{Φ rSub { size 8{n} } } {} is the phase spectrum of the signal. For a real signal, the magnitude spectrum is even-symmetric (symmetric), and the phase spectrum is odd-symmetric (antisymmetric). Find the Fourier expansion of a uniform impulse sequence. Solution Let’s consider an uniform sequence of impulse Aδ(t) size 12{Aδ \( t \) } {} of interval (period) T0 size 12{T rSub { size 8{0} } } {} a: x(t)= ∑ k=−∞ +∞ Aδ(t−k T 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpdaaeWbqaaiaadgeacqaH0oazcaGGOaGaamiDaiabgkHiTiaadUgacaWGubWaaSbaaSqaaiaaicdaaeqaaOGaaiykaaWcbaGaam4Aaiabg2da9iabgkHiTiabg6HiLcqaaiabgUcaRiabg6HiLcqdcqGHris5aaaa@4B7E@ The expansion coefficients are given by X n = 1 T 0 ∫ − T 0 /2 T 0 /2 Aδ(t) e −jn Ω 0 t dt = 1 T 0 ∫ −ε +ε Aδ(t) e −jn Ω 0 t dt = A T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7F8E@

(signal and its spectrum) is the magnitude spectrum. From these coefficients we can synthesize the signal as x(t)= A T 0 ∑ n=−∞ +∞ e jn Ω 0 t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpdaWcaaqaaiaadgeaaeaacaWGubWaaSbaaSqaaiaaicdaaeqaaaaakmaaqahabaGaamyzamaaCaaaleqabaGaamOAaiaad6gacqqHPoWvdaWgaaadbaGaaGimaaqabaWccaWG0baaaaqaaiaad6gacqGH9aqpcqGHsislcqGHEisPaeaacqGHRaWkcqGHEisPa0GaeyyeIuoaaaa@4C24@

The sinx/x function Concerning in the Fourier analysis there is a special function we need to know, that is the sinx/x size 12{"sin"x/x} {}function, also called function, or sampling function . The variation on sinx/x size 12{"sin"x/x} {} with respect to x is shown in . It is an even-symmetric function with unit area, having a maximum of 1 at origin, and zero crossings at regular interval of π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiabec8aWbaa@379A@ . The distance between the origin and the first zero crossing is also π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiabec8aWbaa@379A@ . The function is oscillating and progressively decaying. The first minimum peaks have the value of -0.2178, and the next maximum peaks have the value of 0.1284.

Function sinx/x (or sincx, or Sa(x)) The zero crossing points are determined by sin⁡x x =0 ⇒ sin⁡x=0 ⇒ x=±nπ , n=1, 2, 3,... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaamaalaaabaGaci4CaiaacMgacaGGUbGaamiEaaqaaiaadIhaaaGaeyypa0JaaGimaiaaywW7cqGHshI3caaMf8Uaci4CaiaacMgacaGGUbGaamiEaiabg2da9iaaicdacaaMf8UaeyO0H4TaaGzbVlaadIhacqGH9aqpcqGHXcqScaWGUbGaeqiWdaNaaGjbVlaacYcacaaMf8UaaGzbVlaad6gacqGH9aqpcaaIXaGaaiilaiaaysW7caaIYaGaaiilaiaaysW7caaIZaGaaiilaiaac6cacaGGUaGaaiOlaaaa@646D@ And the extrema of the function occur at sin⁡x=±1 ⇒ x=±(2n+1) π 2 , n=1, 2, 3,... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiGacohacaGGPbGaaiOBaiaadIhacqGH9aqpcqGHXcqScaaIXaGaaGzbVlabgkDiElaaywW7caWG4bGaeyypa0JaeyySaeRaaiikaiaaikdacaWGUbGaey4kaSIaaGymaiaacMcadaWcaaqaaiabec8aWbqaaiaaikdaaaGaaGjbVlaacYcacaaMf8UaaGzbVlaad6gacqGH9aqpcaaIXaGaaiilaiaaysW7caaIYaGaaiilaiaaysW7caaIZaGaaiilaiaac6cacaGGUaGaaiOlaaaa@5EBF@ The first few peak values are x=±3π/2,x=±5π/2,... size 12{x= +- 3π/2,x= +- 5π/2, "." "." "." } {} Notice these are in middle of the zero crossings. However, due to the decaying 1/x these peak values do not occur exactly in the middle but a little bit earlier. Some authors plot the function sin⁡πsin⁡πx MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiGacohacaGGPbGaaiOBaiabec8aWjGacohacaGGPbGaaiOBaiabec8aWjaadIhaaaa@4004@ instead of sinx/x. (a) Find the Fourier expansion coefficients of the periodic square wave given in , and plot the amplitude spectrum for the case τ/T0=1/6 size 12{" "τ/T rSub { size 8{0} } =1/6} {}. (b) Repeat above question when the square wave is delayed so that the central pulse begins at t=0 Solution (a) The coefficients are X n = 1 T 0 ∫ − T 0 /2 T 0 /2 A e −jn Ω 0 t dt = A T 0 ∫ −τ/2 τ/2 e −jn Ω 0 t dt = A T 0 [ e −jn Ω 0 t −jn Ω 0 t ] −τ/2 τ/2 = A T 0 2sin⁡n Ω 0 τ/2 n Ω 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@90A9@ On replacing Ω0=2π/T0 size 12{ %OMEGA rSub { size 8{0} } =2π/T rSub { size 8{0} } } {} we have X n = Aτ T 0 sin n πτ / T 0 n πτ / T 0 size 12{X rSub { size 8{n} } = { {Aτ} over {T rSub { size 8{0} } } } { {"sin"n ital "πτ"/T rSub { size 8{0} } } over {n ital "πτ"/T rSub { size 8{0} } } } } {} which has the form of the sinx/x function with x=nπτ/T0 size 12{x=n ital "πτ"/T rSub { size 8{0} } } {}. The

(even - symmetric square wave) maximum occurs at orgin n = 0 and is X 0 = lim n→0 X n = A τ 1 T 0 = Aτ T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaaGimaaqabaGccqGH9aqpdaWfqaqaaiaadYgacaWGPbGaamyBaaWcbaGaamOBaiabgkziUkaaicdaaeqaaOGaamiwamaaBaaaleaacaWGUbaabeaakiabg2da9maalaaabaGaamyqaiabes8a0naaBaaaleaacaaIXaaabeaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaOGaeyypa0ZaaSaaaeaacaWGbbGaeqiXdqhabaGaamivamaaBaaaleaacaaIWaaabeaaaaaaaa@4D1F@

(amplitude spectrum for τ/ T 0 =1/6 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaamaalyaabaGaeqiXdqhabaGaamivamaaBaaaleaacaaIWaaabeaaaaGccqGH9aqpdaWcgaqaaiaaigdaaeaacaaI2aaaaaaa@3C20@ ) is the plot of the amplitude spectrum for the case of τ/T0 size 12{τ/T rSub { size 8{0} } } {}= 1/6. Notice the sinx/x envelope. (b) Now the square wave is delayed (shifted to the right) by τ/2 size 12{τ/2} {} and appears as in . The

(shifted square wave) expansion coefficients are X n = 1 T 0 ∫ 0 τ/2 A e −jn Ω 0 t dt= A T 0 [ 1 −jn Ω 0 e −jn Ω 0 t ] 0 τ = Aτ T 0 sin⁡nπτ/ T 0 nπτ/ T 0 e −jnπτ/ T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOabaeqabaGaamiwamaaBaaaleaacaWGUbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaOWaa8qmaeaacaWGbbGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiaad6gacqqHPoWvdaWgaaadbaGaaGimaaqabaWccaWG0baaaaqaaiaaicdaaeaacqaHepaDcaGGVaGaaGOmaaqdcqGHRiI8aOGaamizaiaadshacqGH9aqpdaWcaaqaaiaadgeaaeaacaWGubWaaSbaaSqaaiaaicdaaeqaaaaakmaadmaabaWaaSaaaeaacaaIXaaabaGaeyOeI0IaamOAaiaad6gacqqHPoWvdaWgaaWcbaGaaGimaaqabaaaaOGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiaad6gacqqHPoWvdaWgaaadbaGaaGimaaqabaWccaWG0baaaaGccaGLBbGaayzxaaWaa0baaSqaaiaaicdaaeaacqaHepaDaaaakeaaaeaacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7cqGH9aqpdaWcaaqaaiaadgeacqaHepaDaeaacaWGubWaaSbaaSqaaiaaicdaaeqaaaaakmaalaaabaGaci4CaiaacMgacaGGUbGaamOBaiabec8aWjabes8a0jaac+cacaWGubWaaSbaaSqaaiaaicdaaeqaaaGcbaGaamOBaiabec8aWjabes8a0jaac+cacaWGubWaaSbaaSqaaiaaicdaaeqaaaaakiaadwgadaahaaWcbeqaaiabgkHiTiaadQgacaWGUbGaeqiWdaNaeqiXdqNaai4laiaadsfadaWgaaadbaGaaGimaaqabaaaaaaaaa@8F7D@

(magnitude spectrum of) The result is the same as before but with a phase factor. Since ∣e−jnπτ/T0∣=1 size 12{ lline e rSup { size 8{ - ital "jn" ital "πτ"/T rSub { size 6{0} } } } rline =1} {} the magnitude variation is exactly the same as before. In we plot the magnitude spectrum rather than the amplitude spectrum as a matter of choice. Magnitude means absolute values (only positive) whereas amplitude means values which can be positive or regative. However the phase spectrum is different (see Example later)

Gibbs effect (Gibbs phenomenon)

Gibbs effect in Fourier expansion For a given periodic function x(t) we cannot, in general, expand it fully as, in theory, the number of coefficients is infinite. Thus we must drop the high harmonics. This is called truncation. When recombining (synthesizing) the signal from the truncated series we will not get the original signal. It is demonstrated that the Fourier expansion is optimal, meaning that the mean square error between the original signal and the reconstructed signal from the truncated series is smallest compared to other expansions of the same number of coefficients. For the Fourier expansion, of course the more the number of harnomics are taken into accounts the less the error. But an interesting fact is that there is always overshoot and undershoot at abrupt changes of the signal waveform even if the number of harmonics is very very large. This is the Gibbs effect, or Gibbs phenomenon. For a square wave the overshoot and undershoot is about 9% . In the original triangular waveform is compared to the reconstructed waveform from the first seven expansion coefficients. From this we can guess the ripples will still be clearly pronounced even dozens or more harmonics are take, into account, especially around the abrupt changes. Actually, the Gibbs phenomenon includes both the ripple and the overshoot, undershoot.