I asked a question some days ago and it was very well answer. However, I have a new one. I don't understand why the phase of the discrete fourier transformation is not correct. It has the correct answer, but it also has other points that are not correct.

Here, the correct analytical answer:

FourierTransform[Exp[-t]*UnitStep[t], t, w, FourierParameters -> {1, -1}] Plot[Abs[-(I/(-I + 2*Pi*f))], {f, -5, 5}, PlotRange -> All] Plot[Arg[-(I/(-I + 2*Pi*f))], {f, -5, 5}, PlotRange -> All] 

Here, the numerical results I obtained:

signal[i_] := Exp[-(i*samplingPeriode)]*UnitStep[(i*samplingPeriode)] initialTime = -10; finalTime = 10; signalDuration = finalTime - initialTime; bandwidth = 50; nyquistRate = 2*bandwidth; nyquistSamplingPeriode = 1/nyquistRate; samplingFrequency = 20*nyquistRate; samplingPeriode = 1/samplingFrequency; numberOfSamples = signalDuration/samplingPeriode; yValuesSampledSignal = Table[signal[i], {i, initialTime/samplingPeriode, finalTime/samplingPeriode}]; xValuesSampledSignal = Table[i*samplingPeriode, {i, initialTime/samplingPeriode, finalTime/samplingPeriode}]; sampledSignal = Partition[Riffle[xValuesSampledSignal, yValuesSampledSignal], 2]; yValuesDiscreteFourierTransform = samplingPeriode* Fourier[yValuesSampledSignal, FourierParameters -> {1, -1}]; rotatedYValuesDiscreteFourierTransform = RotateRight[yValuesDiscreteFourierTransform, numberOfSamples/2]; xValuesDiscreteFourierTransform = Table[i*(1/signalDuration), {i, -(numberOfSamples/2), numberOfSamples/2}]; discreteFourierTransform = Partition[ Riffle[xValuesDiscreteFourierTransform, yValuesDiscreteFourierTransform], 2]; absYValuesDiscreteFourierTransform = Abs[rotatedYValuesDiscreteFourierTransform]; absDiscreteFourierTransform = Partition[ Riffle[xValuesDiscreteFourierTransform, absYValuesDiscreteFourierTransform], 2]; argYValuesDiscreteFourierTransform = Arg[rotatedYValuesDiscreteFourierTransform]; argDiscreteFourierTransform = Partition[ Riffle[xValuesDiscreteFourierTransform, argYValuesDiscreteFourierTransform], 2]; yValuesRecontructedSignal = (1/samplingPeriode)* InverseFourier[yValuesDiscreteFourierTransform, FourierParameters -> {1, -1}]; xValuesRecontructedSignal = xValuesSampledSignal; reconstructedSignal = Partition[ Riffle[xValuesRecontructedSignal, yValuesRecontructedSignal], 2]; ListLinePlot[sampledSignal, PlotRange -> All] ListPlot[Take[ absDiscreteFourierTransform, {numberOfSamples/2 - 100, numberOfSamples/2 + 100}], Filling -> Axis, PlotRange -> All] ListPlot[Take[ argDiscreteFourierTransform, {numberOfSamples/2 - 100, numberOfSamples/2 + 100}], Filling -> Axis, PlotRange -> All] ListLinePlot[reconstructedSignal, PlotRange -> All] 

I asked a question some days ago and it was very well answer. However, I have a new one. I don't understand why the phase of the discrete fourier transformation is not correct. It has the correct answer, but it also has other points that are not correct.

Here, the correct analytical answer:

FourierTransform[Exp[-t]*UnitStep[t], t, w, FourierParameters -> {1, -1}] Plot[Abs[-(I/(-I + 2*Pi*f))], {f, -5, 5}, PlotRange -> All] Plot[Arg[-(I/(-I + 2*Pi*f))], {f, -5, 5}, PlotRange -> All] 

Here, the numerical results I obtained:

signal[i_] := Exp[-(i*samplingPeriode)]*UnitStep[(i*samplingPeriode)] initialTime = -10; finalTime = 10; signalDuration = finalTime - initialTime; bandwidth = 50; nyquistRate = 2*bandwidth; nyquistSamplingPeriode = 1/nyquistRate; samplingFrequency = 2*nyquistRate; samplingPeriode = 1/samplingFrequency; numberOfSamples = signalDuration/samplingPeriode; yValuesSampledSignal = Table[signal[i], {i, initialTime/samplingPeriode, finalTime/samplingPeriode,samplingPeriode}]; xValuesSampledSignal = Table[i*samplingPeriode, {i, initialTime/samplingPeriode, finalTime/samplingPeriode,samplingPeriode}]; sampledSignal = Partition[Riffle[xValuesSampledSignal, yValuesSampledSignal], 2]; yValuesDiscreteFourierTransform = samplingPeriode* Fourier[yValuesSampledSignal, FourierParameters -> {1, -1}]; rotatedYValuesDiscreteFourierTransform = RotateRight[yValuesDiscreteFourierTransform, numberOfSamples/2]; xValuesDiscreteFourierTransform = Table[i*(1/signalDuration), {i, -(numberOfSamples/2), numberOfSamples/2}]; discreteFourierTransform = Partition[ Riffle[xValuesDiscreteFourierTransform, yValuesDiscreteFourierTransform], 2]; absYValuesDiscreteFourierTransform = Abs[rotatedYValuesDiscreteFourierTransform]; absDiscreteFourierTransform = Partition[ Riffle[xValuesDiscreteFourierTransform, absYValuesDiscreteFourierTransform], 2]; argYValuesDiscreteFourierTransform = Arg[rotatedYValuesDiscreteFourierTransform]; argDiscreteFourierTransform = Partition[ Riffle[xValuesDiscreteFourierTransform, argYValuesDiscreteFourierTransform], 2]; yValuesRecontructedSignal = (1/samplingPeriode)* InverseFourier[yValuesDiscreteFourierTransform, FourierParameters -> {1, -1}]; xValuesRecontructedSignal = xValuesSampledSignal; reconstructedSignal = Partition[ Riffle[xValuesRecontructedSignal, yValuesRecontructedSignal], 2]; ListLinePlot[sampledSignal, PlotRange -> All] ListPlot[Take[ absDiscreteFourierTransform, {numberOfSamples/2 - 100, numberOfSamples/2 + 100}], Filling -> Axis, PlotRange -> All] ListPlot[Take[ argDiscreteFourierTransform, {numberOfSamples/2 - 100, numberOfSamples/2 + 100}], Filling -> Axis, PlotRange -> All] ListLinePlot[reconstructedSignal, PlotRange -> All]