Figure 1 shows the evolution from Fourier series to Fourier transform.

In Equation we replace Ω0Ω0 size 12{ %OMEGA rSub { size 8{0} } } {} by 2πF02πF0 size 12{2πF rSub { size 8{0} } } {}, and write X(nF0)X(nF0) size 12{X \( ital "nF" rSub { size 8{0} } \) } {} for XnXn size 12{X rSub { size 8{n} } } {}, then

The expansion coefficients are given by

Periodic signals have line (discrete) spectrum.

Now we replace the analysis equation X(nF0)X(nF0) size 12{X \( ital "nF" rSub { size 8{0} } \) } {} for x(t) in the synthesis equation:

x ( t ) = ∑ n = − ∞ ∞ 1 T 0 ∫ − T 0 / 2 T 0 / 2 x ( t ) e − j2π nF 0 t dt e j2π nF 0 t x ( t ) = ∑ n = − ∞ ∞ 1 T 0 ∫ − T 0 / 2 T 0 / 2 x ( t ) e − j2π nF 0 t dt e j2π nF 0 t size 12{x \( t \) = Sum cSub {n= - infinity } cSup { infinity } { left [ { {1} over {T rSub { size 8{0} } } } Int rSub { - T rSub { size 6{0} } /2} rSup {T rSub { size 6{0} } /2} {x \( t \) e rSup { size 8{ - j2π ital "nF" rSub { size 6{0} } t} } ital "dt"} right ] size 12{e rSup {j2π ital "nF" rSub { size 6{0} } t} }} } {}

Let T0→∞T0→∞ size 12{T rSub { size 8{0} } rightarrow infinity } {} to push all periods on both sides of the central period of x(t) to infinity, thus transforming the periodic signal into an aperiodic one. On the other hand when the period T0→∞T0→∞ size 12{T rSub { size 8{0} } rightarrow infinity } {}, 1/T0→dt1/T0→dt size 12{1/T rSub { size 8{0} } rightarrow ital "dt"} {} (an infinitesimal quantity), nF0→FnF0→F size 12{ ital "nF" rSub { size 8{0} } rightarrow F} {}(continuous frquency) and the limits ±T0→±∞±T0→±∞ size 12{ +- T rSub { size 8{0} } rightarrow +- infinity } {}, the discrete spectrum becomes continuous. Thus as T0→∞T0→∞ size 12{T rSub { size 8{0} } rightarrow infinity } {},

x ( t ) = ∑ n = − ∞ ∞ 1 T 0 ∫ T 0 / 2 T 0 / 2 x ( t ) e − j2π nf 0 dt x ( t ) = ∑ n = − ∞ ∞ 1 T 0 ∫ T 0 / 2 T 0 / 2 x ( t ) e − j2π nf 0 dt size 12{x \( t \) = Sum cSub {n= - infinity } cSup { infinity } { left [ { {1} over {T rSub { size 8{0} } } } Int rSub {T rSub { size 6{0} } /2} rSup {T rSub { size 6{0} } /2} {x \( t \) e rSup { size 8{ - j2π ital "nf" rSub { size 6{0} } ital "dt"} } } right ]} } {}

The term in brackets is, by definition, the Fourier transform (or the Fourier integral) X(F)X(F) size 12{X \( F \) } {} of x(t)x(t) size 12{x \( t \) } {}.

Thus

The inverse transform is then

x(t)x(t) size 12{x \( t \) } {}and X(F)X(F) size 12{X \( F \) } {} form a continuous-time Fourier transform (CTFT) pair:

x ( t ) ← → X ( F ) x ( t ) ← → X ( F ) size 12{x \( t \) leftarrow rightarrow X \( F \) } {}

The Fourier transform X(F)X(F) size 12{X \( F \) } {} is, in general, complex and we write

where ∣X(f)∣∣X(f)∣ size 12{ lline X \( f \) rline } {} is the magnitude spectrum, and Φ(F)Φ(F) size 12{Φ \( F \) } {} the phase spectrum:

It can be shown that if the signal x(t)x(t) size 12{x \( t \) } {} is real, then XR(F)XR(F) size 12{X rSub { size 8{R} } \( F \) } {}is an even-symmetric (symmetric) and XI(F)XI(F) size 12{X rSub { size 8{I} } \( F \) } {} an old-synmetric (antisymmetric), thus the magnitude spectrum ∣X(F)∣∣X(F)∣ size 12{ lline X \( F \) rline } {} is an even-symmetric and the phase spectrum is an odd-symmetric, i.e.

(a) The given pulse is plotted in Figure 2. It can be denoted as

x(t)=Ap[ − τ 2 , τ 2 ] x(t)=Ap[ − τ 2 , τ 2 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpcaWGbbGaamiCamaadmaabaGaeyOeI0YaaSaaaeaacqaHepaDaeaacaaIYaaaaiaacYcadaWcaaqaaiabes8a0bqaaiaaikdaaaaacaGLBbGaayzxaaaaaa@44A6@

The CTFT Fourier transform is

X(F)= ∫ −∞ ∞ x(t) e −j2πFt df = ∫ −τ/2 τ/2 A e −j2πFt df           =A [ e −j2πFt −j2πFt ] −τ/2 τ/2 =Aτ sin⁡πFτ πFτ X(F)= ∫ −∞ ∞ x(t) e −j2πFt df = ∫ −τ/2 τ/2 A e −j2πFt df           =A [ e −j2πFt −j2πFt ] −τ/2 τ/2 =Aτ sin⁡πFτ πFτ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@97EC@

Notice that the result has the form of sinx/x function (section). The magnitude spectrum is

| X(F) |=| Aτ sin⁡πFτ πFτ |=Aτ| sin⁡πFτ πFτ | | X(F) |=| Aτ sin⁡πFτ πFτ |=Aτ| sin⁡πFτ πFτ | MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaamiwaiaacIcacaWGgbGaaiykaaGaay5bSlaawIa7aiabg2da9maaemaabaGaamyqaiabes8a0naalaaabaGaci4CaiaacMgacaGGUbGaeqiWdaNaamOraiabes8a0bqaaiabec8aWjaadAeacqaHepaDaaaacaGLhWUaayjcSdGaeyypa0Jaamyqaiabes8a0naaemaabaWaaSaaaeaaciGGZbGaaiyAaiaac6gacqaHapaCcaWGgbGaeqiXdqhabaGaeqiWdaNaamOraiabes8a0baaaiaawEa7caGLiWoaaaa@6072@

For the amplitude spectrum we leave its negative part as is (i.e. we do not take the alsolute value) (Figure 3).

As X(F) is a real function, its phase is zero at all frequency. But in Fourier analysis the phase spectrum is interpreted differently. That is for a real function, the phase spectrum is zero for positive value and is π π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@37A2@ for negative value. Besides, to ensure the phase spectrum is an odd-symmetric function, the phase is understood to be + π π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@37A2@ for positive frequency and - π π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@37A2@ for negative frequency (Figure 3).

(b) In order to find the transform of δ(t) δ(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaacIcacaWG0bGaaiykaaaa@39DC@ we consider the amplitude A as 1/ τ τ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0baa@37AA@ , hence the spectrum of the rectangular pulse of amplitude 1/ τ τ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0baa@37AA@ , width τ τ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0baa@37AA@ , is

X(F)= 1 τ τ sin⁡πFτ πFτ = sin⁡πFτ πFτ X(F)= 1 τ τ sin⁡πFτ πFτ = sin⁡πFτ πFτ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaGGOaGaamOraiaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaacqaHepaDaaGaeqiXdq3aaSaaaeaaciGGZbGaaiyAaiaac6gacqaHapaCcaWGgbGaeqiXdqhabaGaeqiWdaNaamOraiabes8a0baacqGH9aqpdaWcaaqaaiGacohacaGGPbGaaiOBaiabec8aWjaadAeacqaHepaDaeaacqaHapaCcaWGgbGaeqiXdqhaaaaa@564B@

Now let τ→0τ→0 size 12{τ rightarrow 0} {} then X(F)→1X(F)→1 size 12{X \( F \) rightarrow 1} {} which is the transform of δ(t)δ(t) size 12{δ \( t \) } {}:

δ ( t ) ↔ 1 δ ( t ) ↔ 1 size 12{δ \( t \) ↔1} {}

The amplitude spectrum is 1 at all frequencies and the phase spectrum is zero. From the amplitude spectrum of the rectangular pulse of Figure 3, as τ→0 τ→0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0jabgkziUkaaicdaaaa@3A51@ the points 1/ τ τ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0baa@37AA@ and -1/ τ τ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0baa@37AA@ go to ±∞ ±∞ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgglaXkabg6HiLcaa@3944@ and the central lobe extends to ±∞ ±∞ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgglaXkabg6HiLcaa@3944@ the amplitude spectrum of δ(t) δ(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaacIcacaWG0bGaaiykaaaa@39DC@ is 1 for all frequencies.

(c) The delayed pulse is denoted as

x(t− t 0 )=Ap[ t 0 − τ 2 ,  t 0 + τ 2 ] x(t− t 0 )=Ap[ t 0 − τ 2 ,  t 0 + τ 2 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiabgkHiTiaadshadaWgaaWcbaGaaGimaaqabaGccaGGPaGaeyypa0JaamyqaiaadchadaWadaqaaiaadshadaWgaaWcbaGaaGimaaqabaGccqGHsisldaWcaaqaaiabes8a0bqaaiaaikdaaaGaaiilaiaaysW7caWG0bWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSYaaSaaaeaacqaHepaDaeaacaaIYaaaaaGaay5waiaaw2faaaaa@4DBD@

Its CTFT is

X(F)= ∫ t 0 −τ/2 t 0 +τ/2 A e −j2πFt dt =A [ e −j2πFt −j2πFt ] t 0 −τ/2 t 0 +τ/2           =Aτ sin⁡πFτ πFτ e −j2πF t 0 X(F)= ∫ t 0 −τ/2 t 0 +τ/2 A e −j2πFt dt =A [ e −j2πFt −j2πFt ] t 0 −τ/2 t 0 +τ/2           =Aτ sin⁡πFτ πFτ e −j2πF t 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@9630@

Notice the appearance of the phase factor e−j2πFt0e−j2πFt0 size 12{e rSup { size 8{ - j2π ital "Ft" rSub { size 6{0} } } } } {}. The magnitude response is exactly as in (a). However the appearance of the phase factor makes the phase spectrum quite different. It is understood as

Φ(F)=∠( sin⁡πFτ πFτ −2πF t 0 ) Φ(F)=∠( sin⁡πFτ πFτ −2πF t 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfA6agjaacIcacaWGgbGaaiykaiabg2da9iabgcIiqlaacIcadaWcaaqaaiGacohacaGGPbGaaiOBaiabec8aWjaadAeacqaHepaDaeaacqaHapaCcaWGgbGaeqiXdqhaaiabgkHiTiaaikdacqaHapaCcaWGgbGaamiDamaaBaaaleaacaaIWaaabeaakiaacMcaaaa@4F1C@

Where ∠ ∠ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgcIiqdaa@3783@ denotes the phase (or argument). The phase spectrum is detailed as follows: