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THE CONTINUOUS - TIME FOURIER SERIES (CTFS)

Module by: Nguyen Huu Phuong

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 Figure 1, can be expanded into sinsoidal components having frequencies which are the multiples of the fundamental frequency of the waveform.
Figure 1: A periodic waveform of period T 0 T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaa@37A4@
Let’s begin with the time signal x(t)x(t) size 12{x \( t \) } {}, periodic at period T0 (sec) or angular frequency Ω0=2π/T0Ω0=2π/T0 size 12{ %OMEGA rSub { size 8{0} } =2π/T rSub { size 8{0} } } {} (rad/sec) or frequency F 0 =1/ T 0 F 0 =1/ T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaaGimaaqabaGccqGH9aqpdaWcgaqaaiaaigdaaeaacaWGubWaaSbaaSqaaiaaicdaaeqaaaaaaaa@3B2E@ (Hz) Figure 1. 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 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@ (1)
Where the coefficients are given by
a 0 = 1 T 0 ∫ − T 0 /2 T 0 /2 x(t)dt 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@ (2)
a n = 2 T 0 ∫ − T 0 /2 T 0 /2 x(t)cos⁡n Ω 0 tdt 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@ (3)
b n = 2 T 0 ∫ − T 0 /2 T 0 /2 x(t)sin⁡n Ω 0 tdt 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@ (4)
In above integrals the limits were put as − T 0 /2 − T 0 /2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiabgkHiTmaalyaabaGaamivamaaBaaaleaacaaIWaaabeaaaOqaaiaaikdaaaaaaa@3965@ and T 0 /2 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 T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaa@379C@ , e.g. 0 and T 0 T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaa@379C@ .
The expansion components have following meaning :
  • a 0 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 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 Equation 5), or the first harmonic.
  • a 2 cos⁡ ω 0 t+ b 2 sin⁡ ω 0 t 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 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
  • ...
Example 1 
Find the Fourier expansion for the symmetric square wave of Figure 2.
Solution
We observe straightaway that the DC component is zero since the positive and regative parts of the signal are equal:
a 0 =0 a 0 =0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIWaaaaa@3973@
Of course when using Equation 2a 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 a n =0  all n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaamOBaaqabaGccqGH9aqpcaaIWaGaaGzbVlaaywW7caWGHbGaamiBaiaadYgacaaMf8UaamOBaaaa@4211@
Figure 2: Example 1 (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 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 bnbn 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+...) 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=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOabaeqabaGaamiEaiaacIcacaWG0bGaaiykaiabg2da9maaqahabaWaaSaaaeaacaaI0aGaamyqaaqaaiabec8aWbaaaSqaaiaad6gacqGH9aqpcaaIXaaabaGaeyOhIukaniabggHiLdGcdaWcaaqaaiaaigdaaeaacaGGOaGaaGOmaiaad6gacqGHsislcaaIXaGaaiykaaaaciGGZbGaaiyAaiaac6gacaGGOaGaaGOmaiaad6gacqGHsislcaaIXaGaaiykaiabfM6axnaaBaaaleaacaaIWaaabeaakiaadshacaaMe8UaaiilaiaaywW7caaMf8UaamOBaiabg2da9iaaigdacaGGSaGaaGjbVlaaikdacaGGSaGaaGjbVlaaiodacaGGSaGaaiOlaiaac6cacaGGUaaabaaabaGaaGzbVlaaywW7cqGH9aqpdaWcaaqaaiaaisdacaWGbbaabaGaeqiWdahaaiaacIcaciGGZbGaaiyAaiaac6gacqqHPoWvdaWgaaWcbaGaaGimaaqabaGccaWG0bGaey4kaSYaaSaaaeaacaaIXaaabaGaaG4maaaaciGGZbGaaiyAaiaac6gacaaIZaGaeuyQdC1aaSbaaSqaaiaaicdaaeqaaOGaamiDaiabgUcaRmaalaaabaGaaGymaaqaaiaaiwdaaaGaci4CaiaacMgacaGGUbGaaGynaiabfM6axnaaBaaaleaacaaIWaaabeaakiaadshacqGHRaWkcaGGUaGaaiOlaiaac6cacaGGPaaaaaa@8A3F@
Figure 3: Example 1 (magnitude spectrum)
Figure 3 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 ) 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@ (5)
Because of this, expansion Equation 1 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 ) 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@ (6)
Where
c 0 = a 0 c 0 = a 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadogadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaaicdaaeqaaaaa@3A87@ (7)
c n = a n 2 + b n 2   n=1, 2, 3,... c n = a n 2 + b n 2   n=1, 2, 3,... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadogadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaGcaaqaaiaadggadaqhaaWcbaGaamOBaaqaaiaaikdaaaGccqGHRaWkcaWGIbWaa0baaSqaaiaad6gaaeaacaaIYaaaaaqabaGccaaMf8UaaGzbVlaad6gacqGH9aqpcaaIXaGaaiilaiaaysW7caaIYaGaaiilaiaaysW7caaIZaGaaiilaiaac6cacaGGUaGaaiOlaaaa@4E08@ (8)
Φ n = tan⁡ −1 − b n a n   n=1, 2, 3,... Φ n = tan⁡ −1 − b n a n   n=1, 2, 3,... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiabfA6agnaaBaaaleaacaWGUbaabeaakiabg2da9iGacshacaGGHbGaaiOBamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaalaaabaGaeyOeI0IaamOyamaaBaaaleaacaWGUbaabeaaaOqaaiaadggadaWgaaWcbaGaamOBaaqabaaaaOGaaGzbVlaaywW7caWGUbGaeyypa0JaaGymaiaacYcacaaMe8UaaGOmaiaacYcacaaMe8UaaG4maiaacYcacaGGUaGaaiOlaiaac6caaaa@51DB@ (9)
In this expansion we can recognize c0c0 size 12{c rSub { size 8{0} } } {} as the average component, c1cos(ω0t+Φ1)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)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 ΦnΦn size 12{Φ rSub { size 8{n} } } {}versus ferquency is the phase spectrum. Both spectra are discrete or line spectra.
Example 2 
Find the Fourier expansion of the waveform in Example 1.
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,... 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 Figure 4.
The expansion expression is
Figure 4: Example 2 (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 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 x(t)= ∑ n=−∞ ∞ X n e jn Ω 0 t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpdaaeWbqaaiaadIfadaWgaaWcbaGaamOBaaqabaGccaWGLbWaaWbaaSqabeaacaWGQbGaamOBaiabfM6axnaaBaaameaacaaIWaaabeaaliaadshaaaaabaGaamOBaiabg2da9iabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHris5aaaa@4AA9@ (10)
The two symmetric components XnXn size 12{X rSub { size 8{n} } } {} and X−nX−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 X 0 = a 0 = c 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0Jaam4yamaaBaaaleaacaaIWaaabeaaaaa@3D5A@ (11)
X n = a n +j b n 2 = c n 2 e j Φ n 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@ (12)
X n = a n −j b n 2 = c n 2 e −j Φ n 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@ (13)
The coefficients XnXn 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 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@ (14)
The limits of the integral can be -T0/2-T0/2 size 12{"-T" rSub { size 8{0} } /2 } {}and T0/2T0/2 size 12{T rSub { size 8{0} } /2 } {}instead of 0 and T as above.
Since the coefficients XnXn size 12{X rSub { size 8{n} } } {} are generally complex, we write
X n =| X n | e j Φ n X n =| X n | e j Φ n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaabdaqaaiaadIfadaWgaaWcbaGaamOBaaqabaaakiaawEa7caGLiWoacaWGLbWaaWbaaSqabeaacaWGQbGaeuOPdy0aaSbaaWqaaiaad6gaaeqaaaaaaaa@42B1@ (15)
The variation of ∣Xn∣∣Xn∣ size 12{ lline "Xn" rline } {} is the magnitude spectrum, the variation of ΦnΦn 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).
Example 3 
Find the Fourier expansion of a uniform impulse sequence.
Solution
Let’s consider an uniform sequence of impulse Aδ(t)Aδ(t) size 12{Aδ \( t \) } {} of interval (period) T0T0 size 12{T rSub { size 8{0} } } {} Figure 5a:
x(t)= ∑ k=−∞ +∞ Aδ(t−k T 0 ) 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 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@
Figure 5: Example 4 (signal and its spectrum)
Figure 6 is the magnitude spectrum. From these coefficients we can synthesize the signal as
x(t)= A T 0 ∑ n=−∞ +∞ e jn Ω 0 t 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/xsinx/x size 12{"sin"x/x} {}function, also called function, or sampling function . The variation on sinx/xsinx/x size 12{"sin"x/x} {} with respect to x is shown in Figure 6. It is an even-symmetric function with unit area, having a maximum of 1 at origin, and zero crossings at regular interval of π 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 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.
Figure 6: 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,... 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,... 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,...x=±3π/2,x=±5π/2,... size 12{x= +- 3π/2,x= +- 5π/2, "." "." "." } {} Notice these are in middle of the zero crossings. However, du