Introduction

• that the real and the imaginary part of the spectrum of a causal signal are connected by the Hilbert-Transform.
• that the real and the imaginary part of an analytic signal are connected by the Hilbert-Transform.
• how the even and the odd part of a real signal are transformed into the frequency domain.

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Select a real and causal signal v(t).

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Here the applet can be finished or the help site can be activated. The scrollbar changes the parameter for generating the selected causal signal v(t).

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Implemented signals p = Parameter t0 = reference value for the time domain, reference value for the frequency domain is w0 = p/t0

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The causal signal is split up in an even and an odd part. The even one is shown in the upper left, the odd one in the upper right. The Fourier transform of both parts is shown in the lower area. The even time signal is transformed to the real part P(jw) of the spectrum V(jw), the odd time signal to the complex part jQ(jw) (see textbook at page 219 for the symmetrical relations of the Fourier-Transformation). As you know from the theory of causal signals, P(jw) is the Hilbert-Transform of Q(jw).

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A real and even signal v(t) can be chosen here.

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Here the applet can be finished or the help site can be activated. The scrollbar changes a parameter for generating the selected signal.

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Implemented signals p = Parameter t0 = reference value for the time domain, reference value for the frequency domain is w0 = p/t0

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With v(t) an analytic signal is generated: vanalyt(t) = v(t) + j·H{v(t)} The real and even signal v(t) is transformed to the even part of the real and right-sided spectrum V(jw) of vanalyt(t). Veven(jw) is shown in the upper left. The odd part Vodd(jw) is the Fourier transform of jH{v(t)} and is shown in the upper right.

Introduction

• what modulation means both in the time and the frequency domain.
• how a single-sideband signal is generated.
• a possibility for the recovery of the input signal.
• which effects can be expected on the output signal depending on the channel properties.

Applet functions

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Here a band limited signal can be chosen. Two scrollbars respectively change the width and the position of the signal.

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In the lower area the selected input signal and its Fourier transform are drawn.

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By pushing a certain signal button the time signal and its Fourier transform appear in a new window.

(the real part of signals is drawn in red, the imaginary one in blue)

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Select the sideband which should be transmitted.

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This button opens a new window for selecting the channel.

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Channel window The following options are possible: 0dB/3dB attentuation , no/linear phase RC Low Pass Characteristic with the following transfer function (see textbook at page 158f.): (wm = modulation frequency) The selected channel transfer function is plotted in the upper right, its modulus in the lower left and its phase in the lower right.

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The reset button closes all open windows.

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Here you can finish the applet or activate the help site.

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There is also a sound signal implemented. By selecting this, additional sound buttons will be activated to play the respective sound signal.

Introduction

• how to generate an analytic signal from a real bandpass signal.
• that sampling of a complex bandpass signal only requires half the sampling frequency.
• how the original signal can be extracted from the sampled complex signal.

Applet functions

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Select a real bandpass signal.

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In the lower area the selected input signal and its Fourier tranform are shown.

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By pushing a certain signal button the time signal and its Fourier transform appear in a new window.

(the real part of signals is drawn in red, the imaginary one in blue)

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The sampling rate ws is constant. Here you can change the band width of the input signal, e.g. sampling at Nyquist rate means band width of the analytic signal w2 - w1 equal sampling rate ws.

Sampling at Nyquist rate (the lower band and its periodical repetitions are drawn in orange (light-blue) for real (imaginary) signals)

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The reset button closes all open windows.

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Here you can finish the applet or activate the help site.

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There is also a sound signal implemented. By selecting this additional sound buttons will be activated to play the respective sound signal.