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Signals and Systems

Complex Envelope Implementation

The complex envelope is a baseband analytic signal. It is obtained by suppressing the negative (or positive) frequencies of a real bandpass signal, and transferring the remaining spectral content to baseband. Due to the complex notation and the spectral efficiency of the resulting signal, the complex envelope is very useful in both linear system simulation […]

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Signals and Systems

Analytic Signal and Complex Envelope

The analytic signal is a complex representation of a real signal, obtained by suppressing the negative frequency components of the original signal. From the analytic signal is derived the equivalent baseband signal, also an analytic representation known as complex envelope. Both representations are used in signal processing techniques due to their main advantages: spectral efficiency […]

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Signals and Systems

The Mathematics of Linear Distortion

The mathematics of linear distortion only applies to linear and time invariant systems [1] [2]. Therefore, these systems and their translation to the frequency domain, where the mathematical analysis is simplified, are briefly summarized. Then it is discussed how the presented theory can be applied to real transmission media and/or electronic components. Finally, the mathematics […]

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Signals and Systems

Hilbert Transform

The Hilbert transform is a linear operation applied to real signals. In practical terms, the Hilbert transform translates into a phase shift of -90º at the positive frequencies (and +90º at the negative frequencies) that make up the signal. The relevance of the Hilbert transform in telecommunication engineering is due to its contribution in obtaining […]

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Signals and Systems

Spatial Fourier Transform vs Temporal

The Fourier transform makes it possible to obtain the components that make up a signal in the transformed domain. In the most common case, the Fourier transform of temporal signals provides the frequencies (ω) that make up the signal. Equivalently, the spatial Fourier transform, applied on a signal defined in space, provides its components typically […]

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Signals and Systems

Signal Propagation in Communications

In communications, signal propagation theory studies the movement of signals from the transmitter to the receiver through a transmission medium. Depending on the characteristics of the signal and the medium, different situations may arise. In general, the cases are classified according to the type of distortion they produce in the transmitted signal. This text is […]

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Signals and Systems

Phase Velocity: Waves and Signals

Phase velocity typically denotes the speed with which a sinusoidal wave or tone moves in the medium in which it propagates. The term is coined because the velocity at which the tone moves is equivalent to the speed at which any one of its phases moves, whether it is a phase that produces a zero, […]

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Signals and Systems

Group Velocity: Mathematical Proof

The term group velocity defines the rate at which the envelope of a signal propagates in a transmission medium. This signal transmitted over the medium is obtained when a modulating signal modulates a central frequency, typically called the carrier [1]. As a result, the modulating signal and the envelope are equivalent. While the envelope propagates […]

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Signals and Systems

Waves in Electronics and Communications

This article reviews the basic theory of waves and wave motion focusing on a practical application: electronics and communications. In addition, images and graphs of moving waves are provided to facilitate the understanding of the concepts explained. The text is organized as follows. First, waves are defined from a general and also mathematical point of […]

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Signals and Systems

Frequency Constant Phase Shift and Distortion

Transmission media and electronic components can have multiple effects on the signals propagating through them [1] [2]. To avoid phase distortion, it is necessary that the phase of the system, the phase shift introduced by the medium, is linear with frequency. This case, whose effect consists of a time delay of the signal, is often […]