The Fourier series expansion of a signal reveals its frequency structure we need. But unfortunately, the Fourier series expansion only applies to periodic signals while in reality signals are mostly aperiodic, and the Fourier transform was developed for the latter.

The Fourier transform pair

The evolution from Fourier series to Fourier transform Fig.3.10 shows the evolution from Fourier series to Fourier transform. In Equation (3.6) we replace Ω0 size 12{ %OMEGA rSub { size 8{0} } } {} by 2πF0 size 12{2πF rSub { size 8{0} } } {}, and write X(nF0) size 12{X \( ital "nF" rSub { size 8{0} } \) } {} for Xn size 12{X rSub { size 8{n} } } {}, then x(t)= ∑ n=−∞ ∞ X(n F 0 ) e j2π F 0 t dt MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpdaaeWbqaaiaadIfacaGGOaGaamOBaiaadAeadaWgaaWcbaGaaGimaaqabaGccaGGPaGaamyzamaaCaaaleqabaGaamOAaiaaikdacqaHapaCcaWGgbWaaSbaaWqaaiaaicdaaeqaaSGaamiDaaaakiaadsgacaWG0baaleaacaWGUbGaeyypa0JaeyOeI0IaeyOhIukabaGaeyOhIukaniabggHiLdaaaa@5049@ The expansion coefficients are given by X(nF0)=1T0∫−T0/2T0/2x(t)e−j2πnF0tdt size 12{X \( ital "nF" rSub { size 8{0} } \) = { {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"} } {} Periodic signals have line (discrete) spectrum. Now we replace the analysis equation 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 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→∞ 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→∞ size 12{T rSub { size 8{0} } rightarrow infinity } {}, 1/T0→dt size 12{1/T rSub { size 8{0} } rightarrow ital "dt"} {} (an infinitesimal quantity), nF0→F size 12{ ital "nF" rSub { size 8{0} } rightarrow F} {}(continuous frquency) and the limits ±T0→±∞ size 12{ +- T rSub { size 8{0} } rightarrow +- infinity } {}, the discrete spectrum becomes continuous. Thus as 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 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) size 12{X \( F \) } {} of x(t) size 12{x \( t \) } {}. Thus X(F)=∫−∞∞x(t)e−j2πFtdt size 12{X \( F \) = Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {x \( t \) e rSup { size 8{ - j2π ital "Ft"} } ital "dt"} } {} The inverse transform is then x(t)=∑n=−∞∞dFX(F)ej2πFt⇒∫∞∞X(F)ej2πFtdF size 12{x \( t \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } { ital "dFX" \( F \) e rSup { size 8{j2π ital "Ft"} } } drarrow Int rSub { size 8{ infinity } } rSup { size 8{ infinity } } {X \( F \) e rSup { size 8{j2π ital "Ft"} } ital "dF"} } {} x(t) size 12{x \( t \) } {}and X(F) size 12{X \( F \) } {} form a continuous-time Fourier transform (CTFT) pair: x ( t ) ← → X ( F ) size 12{x \( t \) leftarrow rightarrow X \( F \) } {}

Magnitude and phase spectra The Fourier transform X(F) size 12{X \( F \) } {} is, in general, complex and we write X(F)=XR(F)+jXI(F)=∣X(F)∣ejΦ(F) size 12{X \( F \) =X rSub { size 8{R} } \( F \) + ital "jX" rSub { size 8{I} } \( F \) = lline X \( F \) rline e rSup { size 8{jΦ \( F \) } } } {} where ∣X(f)∣ size 12{ lline X \( f \) rline } {} is the magnitude spectrum, and Φ(F) size 12{Φ \( F \) } {} the phase spectrum: ∣X(F)∣=XR2(F)+XI2(F) size 12{ lline X \( F \) rline = sqrt {X rSub { size 8{R} } rSup { size 8{2} } \( F \) +X rSub { size 8{I} } rSup { size 8{2} } \( F \) } } {} Φ(F)=tan−1XI(F)XR(F) size 12{Φ \( F \) ="tan" rSup { size 8{ - 1} } { {X rSub { size 8{I} } \( F \) } over {X rSub { size 8{R} } \( F \) } } } {} It can be shown that if the signal x(t) size 12{x \( t \) } {} is real, then XR(F) size 12{X rSub { size 8{R} } \( F \) } {}is an even-symmetric (symmetric) and XI(F) size 12{X rSub { size 8{I} } \( F \) } {} an old-synmetric (antisymmetric), thus the magnitude spectrum ∣X(F)∣ size 12{ lline X \( F \) rline } {} is an even-symmetric and the phase spectrum is an odd-symmetric, i.e. | X(F) |=| X(−F) | and Φ(F)=−Φ(−F) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaamiwaiaacIcacaWGgbGaaiykaaGaay5bSlaawIa7aiabg2da9maaemaabaGaamiwaiaacIcacqGHsislcaWGgbGaaiykaaGaay5bSlaawIa7aiaaywW7caWGHbGaamOBaiaadsgacaaMf8UaeuOPdyKaaiikaiaadAeacaGGPaGaeyypa0JaeyOeI0IaeuOPdyKaaiikaiabgkHiTiaadAeacaGGPaaaaa@5418@ (a) Find the Fourier transform, the magnitude and phase spectrum of an even-symmetric rectangular pulse of amplitude A and width τ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0baa@37AA@ . (b) Apply above result to find the transform of the unit impulse τ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0baa@37AA@ (t) (delta Dirac function).(c) Repeat the question when the above pulse is delayed by t 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshadaWgaaWcbaGaaGimaaqabaaaaa@37C4@ . Solution (a) The given pulse is plotted in Fig.3.11a. It can be denoted as x(t)=Ap[ − τ 2 , τ 2 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpcaWGbbGaamiCamaadmaabaGaeyOeI0YaaSaaaeaacqaHepaDaeaacaaIYaaaaiaacYcadaWcaaqaaiabes8a0bqaaiaaikdaaaaacaGLBbGaayzxaaaaaa@44A6@

Example 3.2.1 (the pulse and its delay) 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 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 3.1.3). The magnitude spectrum is | 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) (Fig.3.11b).

Example 3.2.1 (magnitude and phase spectra) 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 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 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@37A2@ for positive frequency and - π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@37A2@ for negative frequency (Fig.3.11b). (b) In order to find the transform of δ(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaacIcacaWG0bGaaiykaaaa@39DC@ we consider the amplitude A as 1/ τ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0baa@37AA@ , hence the spectrum of the rectangular pulse of amplitude 1/ τ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0baa@37AA@ , width 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 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaGGOaGaamOraiaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaacqaHepaDaaGaeqiXdq3aaSaaaeaaciGGZbGaaiyAaiaac6gacqaHapaCcaWGgbGaeqiXdqhabaGaeqiWdaNaamOraiabes8a0baacqGH9aqpdaWcaaqaaiGacohacaGGPbGaaiOBaiabec8aWjaadAeacqaHepaDaeaacqaHapaCcaWGgbGaeqiXdqhaaaaa@564B@ Now let τ→0 size 12{τ rightarrow 0} {} then X(F)→1 size 12{X \( F \) rightarrow 1} {} which is the transform of δ(t) size 12{δ \( t \) } {}: 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 Fig.3.11b, as τ→0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0jabgkziUkaaicdaaaa@3A51@ the points 1/ τ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0baa@37AA@ and -1/ 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) 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 ] 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 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@9630@ Notice the appearance of the phase factor e−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 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfA6agjaacIcacaWGgbGaaiykaiabg2da9iabgcIiqlaacIcadaWcaaqaaiGacohacaGGPbGaaiOBaiabec8aWjaadAeacqaHepaDaeaacqaHapaCcaWGgbGaeqiXdqhaaiabgkHiTiaaikdacqaHapaCcaWGgbGaamiDamaaBaaaleaacaaIWaaabeaakiaacMcaaaa@4F1C@

Example 3.2.1 (phase spectrum of the delay pulse) 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: Φ(F)=−2πF t 0 , sin⁡πFτ πFτ >0 and F>0 or F<0 =−2πF t 0 +π , sin⁡πFτ πFτ <0 and F>0 =−2πF t 0 −π , sin⁡πFτ πFτ <0 and F<0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@BACD@ The phase spectrum is depicted in Fig.3.11c.

Some properties of the CTFT The continuous-time Fourier transform has many properties which help us to find the Fourier transform faster than just using the definition. In the following only some common properties are mentioned. (a) Linearity a1x1(t)+a2x2(t)↔a1X1(F)+a2X2(F) size 12{a rSub { size 8{1} } x rSub { size 8{1} } \( t \) +a rSub { size 8{2} } x rSub { size 8{2} } \( t \) ↔a rSub { size 8{1} } X rSub { size 8{1} } \( F \) +a rSub { size 8{2} } X rSub { size 8{2} } \( F \) } {} a 1 , a 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaaGymaaqabaGccaGGSaGaaGjbVlaadggadaWgaaWcbaGaaGOmaaqabaaaaa@3BC7@ are constants (b) Time-shift x(t−t0)↔X(f)e−j2πFt0 size 12{x \( t - t rSub { size 8{0} } \) matrix { {} # ↔ {} # {} } X \( f \) e rSup { size 8{ - j2π ital "Ft" rSub { size 6{0} } } } } {} When the signal is delayed by t0 size 12{t rSub { size 8{0} } } {} its transform is phase-shfted by 2πFt0 size 12{2π ital "Ft" rSub { size 8{0} } } {}. (c) Frequency shift (also called modulation theorem) x(t)ej2πF0t↔X(F−F0) size 12{x \( t \) e rSup { size 8{j2πF rSub { size 6{0} } t} } matrix { matrix { {} # ↔{} } {} # {} } X \( F - F rSub {0} size 12{ \) }} {} When the signal is phase-shifted by +2πF t 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgUcaRiaaikdacqaHapaCcaWGgbGaamiDamaaBaaaleaacaaIWaaabeaaaaa@3BEA@ its transform is frequency-shifted by F 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaaGimaaqabaaaaa@3796@ . Find the magnitude spectrum of the amplitude modulation (AM) signal x(t)=m(t)cos⁡2π F c t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpcaWGTbGaaiikaiaadshacaGGPaGaci4yaiaac+gacaGGZbGaaGOmaiabec8aWjaadAeadaWgaaWcbaGaam4yaaqabaGccaWG0baaaa@45AC@ where the spectrum of the low frequency signal representing information is known, and cos⁡2π F c t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacogacaGGVbGaai4CaiaaikdacqaHapaCcaWGgbWaaSbaaSqaaiaadogaaeqaaOGaamiDaaaa@3E13@ is the carrier (its frequency F c MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaam4yaaqabaaaaa@37C4@ is much higher than the frquencies of m(t)). The above amplitude modulation is called DSB-SC (double sideband with suppressed carrier).

Example 3.2.2 (spectrum of m(t) and x(t)) Solution The magnitude spectrum | M(f) | MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaamytaiaacIcacaWGMbGaaiykaaGaay5bSlaawIa7aaaa@3C1D@ of the modulating information m(t) is assumed to be as in Fig.3.12 where F M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaamytaaqabaaaaa@37AE@ is its maximum frequency. F M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaamytaaqabaaaaa@37AE@ is much smaller than the carrier frequency F c MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaam4yaaqabaaaaa@37C4@ . Let’s express the cosine in terms of complex exponentials x(t)=m(t)cos⁡2π F c t= 1 2 m(t) e j2π F c t + 1 2 m(t) e −j2π F c t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpcaWGTbGaaiikaiaadshacaGGPaGaci4yaiaac+gacaGGZbGaaGOmaiabec8aWjaadAeadaWgaaWcbaGaam4yaaqabaGccaWG0bGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGTbGaaiikaiaadshacaGGPaGaamyzamaaCaaaleqabaGaamOAaiaaikdacqaHapaCcaWGgbWaaSbaaWqaaiaadogaaeqaaSGaamiDaaaakiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGaamyBaiaacIcacaWG0bGaaiykaiaadwgadaahaaWcbeqaaiabgkHiTiaadQgacaaIYaGaeqiWdaNaamOramaaBaaameaacaWGJbaabeaaliaadshaaaaaaa@60E7@ If M(F) is the spectrum of m(t) then applying the frequency shift property will give the spectrum of the AM signal as X ( F ) = 1 2 M ( F − F c ) + 1 2 M ( F + F c ) size 12{X \( F \) = { {1} over {2} } M \( F - F rSub { size 8{c} } \) + { {1} over {2} } M \( F+F rSub { size 8{c} } \) } {} Thus the spectrum of x(t) is the spectrum of m(t) shifted to frquency -F C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaam4qaaqabaaaaa@37A4@ and F C MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaam4qaaqabaaaaa@37A4@ and with amplitude equals to half that of m(t) (Fig.3.12) (d) Time convolution (also called convolution theorem ) Convolution of two signals x 1 (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiDaiaacMcaaaa@3A25@ and x 2 (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiDaiaacMcaaaa@3A25@ , denoted x 1 (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiDaiaacMcaaaa@3A25@  x 2 (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiDaiaacMcaaaa@3A25@ , is defined as x1(t)∗x2(t)=∫−∞∞x1(t')x2(t−t')dt' size 12{x rSub { size 8{1} } \( t \) * x rSub { size 8{2} } \( t \) = Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {x rSub { size 8{1} } \( t' \) x rSub { size 8{2} } \( t - t' \) ital "dt"'} } {} It can be demonstrated that x1(t)∗x2(t)↔X1(F)X2(F) size 12{x rSub { size 8{1} } \( t \) * x rSub { size 8{2} } \( t \) ↔X rSub { size 8{1} } \( F \) X rSub { size 8{2} } \( F \) } {} which means convolution in time domain conesponds to normal multiplication in transform domain. Relating to the time convolution there is this useful relation x(t)∗δ(t)=x(t) size 12{x \( t \) * δ \( t \) =x \( t \) } {} i.e time convolution a signal x(n) with the unit impulse δ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKbaa@378A@ (t) is just that signal x(n). The converse of the time convolution is the frequency convolution stated as x1(t)x2(t)↔X1(t)∗X2(t) size 12{x rSub { size 8{1} } \( t \) x rSub { size 8{2} } \( t \) ↔X rSub { size 8{1} } \( t \) * X rSub { size 8{2} } \( t \) } {} (e) Parseval’s theorem This theorem equates the energy in time domain to that in frequency domain: E=∫−∞∞∣x(t)∣2dt=∫−∞∞∣X(F)∣2dF size 12{E= Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } { lline x \( t \) rline rSup { size 8{2} } ital "dt"= Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } { lline X \( F \) rline rSup { size 8{2} } ital "dF"} } } {}

CTFT of basic signals In this subsection common Fourier transform pairs are mentioned with illustrated figures but without proofs. (a) Narrow pulse δt size 12{δt} {} This is a pulse of amplitude 1 and of infinitesimal width: The transform is δt↔δt size 12{δt↔δt} {}

Narrow pulse δt MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaadshaaaa@3883@ (b) Unit impulse δ(t) size 12{δ \( t \) } {} This is the delta Dirac function, not the narrow pulse mentioned above: The transform is δ ( t ) ↔ 1 size 12{δ \( t \) ↔1} {}

Unit impulse δ(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaacIcacaWG0bGaaiykaaaa@39DC@ (c) A constant The transform is A↔Aδ(−F)=Aδ(F) size 12{A↔Aδ \( - F \) =Aδ \( F \) } {}

A constant (d) Causal exponential This is the function x(t)= e −at , t≥0 =0 , t<0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiEaiaacIcacaWG0bGaaiykaiabg2da9iaadwgadaahaaWcbeqaaiabgkHiTiaadggacaWG0baaaOGaaGzbVlaacYcacaaMf8UaaGzbVlaadshacqGHLjYScaaIWaaabaaabaGaaGzbVlaaysW7caaMe8UaaGjbVlabg2da9iaaicdacaaMf8UaaGzbVlaacYcacaaMf8UaaGzbVlaadshacqGH8aapcaaIWaaaaaa@5896@ The transform is X(F)= 1 a+j2πF MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaGGOaGaamOraiaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGHbGaey4kaSIaamOAaiaaikdacqaHapaCcaWGgbaaaaaa@40B2@ hence the magnitude spectrum | X(F) |= 1 | a+j2πF | = 1 a 2 + (2πF) 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaamiwaiaacIcacaWGgbGaaiykaaGaay5bSlaawIa7aiabg2da9maalaaabaGaaGymaaqaamaaemaabaGaamyyaiabgUcaRiaadQgacaaIYaGaeqiWdaNaamOraaGaay5bSlaawIa7aaaacqGH9aqpdaWcaaqaaiaaigdaaeaadaGcaaqaaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaGGOaGaaGOmaiabec8aWjaadAeacaGGPaWaaWbaaSqabeaacaaIYaaaaaqabaaaaaaa@5118@

Causal exponential (e) Unit step The transform and magnitude spectrum are respectively X(F)= 1 2 δ(F)+ 1 2 1 jπF MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaGGOaGaamOraiaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiabes7aKjaacIcacaWGgbGaaiykaiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaWaaSaaaeaacaaIXaaabaGaamOAaiabec8aWjaadAeaaaaaaa@45E7@ | X(F) |= 1 2 δ 2 (F)+ ( 1 πF ) 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaamiwaiaacIcacaWGgbGaaiykaaGaay5bSlaawIa7aiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaOaaaeaacqaH0oazdaahaaWcbeqaaiaaikdaaaGccaGGOaGaamOraiaacMcacqGHRaWkcaGGOaWaaSaaaeaacaaIXaaabaGaeqiWdaNaamOraaaacaGGPaWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@49D8@

Unit step (f) Cosine and sine The cosine Acos⁡2π F 0 t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeaciGGJbGaai4BaiaacohacaaIYaGaeqiWdaNaamOramaaBaaaleaacaaIWaaabeaakiaadshaaaa@3EAB@ has Fourier transform X(F)= A 2 δ(F− F 0 )+ A 2 (F+ F 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaGGOaGaamOraiaacMcacqGH9aqpdaWcaaqaaiaadgeaaeaacaaIYaaaaiabes7aKjaacIcacaWGgbGaeyOeI0IaamOramaaBaaaleaacaaIWaaabeaakiaacMcacqGHRaWkdaWcaaqaaiaadgeaaeaacaaIYaaaaiaacIcacaWGgbGaey4kaSIaamOramaaBaaaleaacaaIWaaabeaakiaacMcaaaa@4924@ If the cosine is written as AcosΩ0t the transform is X(Ω)=πδ(Ω−Ω0)+πδ(Ω+Ω0) size 12{X \( %OMEGA \) = ital "πδ" \( %OMEGA - %OMEGA rSub { size 8{0} } \) + ital "πδ" \( %OMEGA + %OMEGA rSub { size 8{0} } \) } {} Similarly, the transform of the sine sin2F0t and the sine sinΩ0t size 12{"sin" %OMEGA rSub { size 8{0} } t} {}are respectively X(f)=jA2δ(F−F0)+jA2(F−F0) size 12{X \( f \) =j { {A} over {2} } δ \( F - F rSub { size 8{0} } \) +j { {A} over {2} } \( F - F rSub { size 8{0} } \) } {} x(Ω)=jπδ(Ω−Ω0)+jπδ(Ω−Ω0) size 12{x \( %OMEGA \) =j ital "πδ" \( %OMEGA - %OMEGA rSub { size 8{0} } \) +j ital "πδ" \( %OMEGA - %OMEGA rSub { size 8{0} } \) } {}

Consine function

The CTFT for systems Analog (discrete-time) systems are characterized by their impulse (impulsive) responses, similarly to digital (discrete-time) systems. In the time domain the output signal y(t) of the system is the convolution of the input signal x(t) with the system impulse response h(t): x ( t ) = x ( t ) ∗ h ( t ) size 12{x \( t \) =x \( t \) * h \( t \) } {} By the convolution theorem, the above equation is transformed into the Fourier domain as Y(F)=X(F)H(F) size 12{Y \( F \) =X \( F \) H \( F \) } {} where H(F), the Fourier transform of the impulse response h(t), is the frequency characterization of the system and is called frequency response. From above we write H(F)=Y(F)X(F) size 12{H \( F \) = { {Y \( F \) } over {X \( F \) } } } {} Now the frequency response can be interpreted as the ratio of the Fourier transform of the output signal to the transform of the input signal. The frequency response is, in general, complex and we write H(F)=∣H(F)∣ejΦ(F) size 12{H \( F \) = lline H \( F \) rline e rSup { size 8{jΦ \( F \) } } } {} where ∣H(F)∣ size 12{ lline H \( F \) rline } {} is the magnitude response (spectrum) and Φ(F) is the phase response (spectrum). For example for ideal systems the output is y(t)=Gx(t− t 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhacaGGOaGaamiDaiaacMcacqGH9aqpcaWGhbGaamiEaiaacIcacaWG0bGaeyOeI0IaamiDamaaBaaaleaacaaIWaaabeaakiaacMcaaaa@412C@ where G is a scale factor (gain or attenuation), and t0 size 12{t rSub { size 8{0} } } {} is a time delay. The above equation is Fourier transformed to Y(F)=GX(F) e −2πF t 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMfacaGGOaGaamOraiaacMcacqGH9aqpcaWGhbGaamiwaiaacIcacaWGgbGaaiykaiaadwgadaahaaWcbeqaaiabgkHiTiaaikdacqaHapaCcaWGgbGaamiDamaaBaaameaacaaIWaaabeaaaaaaaa@44E2@ then H(F)=Y(F)X(F)=Ge−2πFt0 size 12{H \( F \) = { {Y \( F \) } over {X \( F \) } } = ital "Ge" rSup { size 8{ - 2π ital "Ft" rSub { size 6{0} } } } } {}

Ideal systems Thus for ideal systems, the magnitude response | H(F) | MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaamisaiaacIcacaWGgbGaaiykaaGaay5bSlaawIa7aaaa@3BF8@ is constant, independent of frequency, and the phase response Φ(F)=−2πF t 0 =−2π t 0 F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfA6agjaacIcacaWGgbGaaiykaiabg2da9iabgkHiTiaaikdacqaHapaCcaWGgbGaamiDamaaBaaaleaacaaIWaaabeaakiabg2da9iabgkHiTiaaikdacqaHapaCcaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaamOraaaa@47C3@ is proportional to frequency F (Fig.3.19), Such systems are called linear phase. For real systems, the magnitude response is even-symmetric (symmetric), and the phase response is odd-symmetric (antisymmetric) as for real signals (Equation (3.16)).