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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 of all possible cases of linear distortion are summarized in a table, and each case is explained individually.

1. Context: Linear Time Invariant (LTI) Systems

This section briefly reviews linear time invariant (LTI) systems. LTI systems are the basic fundamental concept on which the mathematical analysis of linear distortion is articulated [1].

Generically, the following notation is used throughout the text for any system: \footnotesize x_i(t) represents a temporary input signal and \footnotesize y_i(t) represents the associated output signal. Where i is an integer that allows to distinguish the different additive contributions of the input and output signals.

1.1 Linearity

A system is linear if and only if it satisfies the superposition principle. Being \footnotesize a_i a real number, and given an input \footnotesize a_1x_1(t) + a_2x_2(t) , the output would be \footnotesize a_1y_1(t) + a_2y_2(t) . Graphically:

Linear system meets superposition
Linear System meets Superposition

1.2 Time Invariance

If the system does not change its properties over time, it is said to be time invariant. Using the time origin as a reference, an input \footnotesize x(t) produces an output \footnotesize y(t) . If the system is time invariant any delay \footnotesize \tau in the input signal, \footnotesize x(t -\tau) , produces the same delay in the output, \footnotesize y(t -\tau) .

1.3 Impulse Response

LTI systems are defined by an impulse response, h(t), which relates the input and output using the convolution operator, \footnotesize (*) . Mathematically [1]:

\begin{equation} y(t) = x(t) * h(t) = \int_{-\infty}^\infty x(\tau)h(t-\tau)\,d\tau \end{equation}

A detailed explanation of the impulse response and how it is obtained is beyond the scope of this text. However, it is worth noting that it is clear from equation (1) that the convolution operator is a linear operator.

1.4 The Frequency Domain

1.4.1 Relevance

All signals passing through a transmission medium or an electronic component consist of a sum of tones. In fact, more precisely, they are determined by a spectral density in the frequency domain [1]. In line with the above, the Fourier transform makes it possible to decompose and represent signals and LTI systems in the frequency domain. In this way, many concepts related to signal processing can be simplified, including the mathematics of linear distortion, as shown below.

It should be noted that in a linear system all the frequency components appearing in the output were already present in the input signal. Otherwise, when new frequencies appear in the output signal, we find a nonlinear system. Obviously, nonlinear behavior cannot be analyzed with the LTI system theory explained in this article.

1.4.2 The Fourier Transform

A detailed explanation of the Fourier transform is beyond the scope of this text. Therefore, we will focus on the concepts minimally necessary to understand the mathematics of linear distortion. Mathematically, the Fourier transform of a generic function defined in the time domain, x(t), produces its corresponding function in the frequency domain, X(f), by the following equation [1]:

\begin{equation} X(f) = \mathcal{F}(x(t))= \int_{-\infty}^\infty x(t)e^{-j2 \pi f t}\,dt \end{equation}

Given an LTI system, and using equation (2), the Fourier transforms of the input signal x(t), the impulse response h(t), and the output signal y(t) can be obtained. Specifically: X(f), H(f) and Y(f) respectively.

1.4.3 Temporal Convolution = Spectral Product

The key concept that simplifies frequency domain analysis, including the mathematics of linear distortion, is the following. The convolution operator in the time domain becomes multiplication in the frequency domain. Mathematically it follows that:

\begin{equation} Y(f) = X(f) ·H(f) \end{equation}

This is why the function H(f) is called the transfer function of the LTI system. The following graph illustrates this concept and the temporal and spectral relationship.

Time - Frequency relationship. Simplifies signal processing, for  example the mathematics of linear distortion.
Time Domain vs. Frequency Domain for LTI Systems

2. Application to Transmission Media and Electronic Components

In general, signal transmission media are not pure LTI systems. There are always weather conditions, at least temperature, that can affect the impulse response of the system. The same is true for electronic media and components. There are always voltage and/or current and/or power and/or temperature values that can modify the behavior of the circuit elements. In addition, in any case, there is the additive thermal noise that appears at the output [1].

Therefore, when studying transmission media and electronic stages as if they were LTI systems, the analysis is only valid under certain operating conditions. Typically, within an operating range (voltages, currents, powers, temperature, noise….) in which system variations are negligible.

3. The Mathematics of Linear Distortion

The basic model with which the mathematics of linear distortion will be analyzed is explained below. Subsequently, a table is shown that summarizes and synthesizes in a schematic way all the possibilities and casuistry. Finally, the cases in the table are explained in more detail.

3.1 Mathematical Model

Fourier transforms, even those of real signals, can give rise to complex functions. Therefore the transfer function of the LTI system, H(f), can be represented in terms of its modulus, \footnotesize |H(f)| , and phase, \footnotesize \phase{H(f)} . Mathematically:

\begin{equation} H(f) = |H(f)|e^{j\phase{H(f)}} \end{equation}

For simplicity, without losing generality, this expression and all the equivalent ones can be reduced as follows:

\begin{equation} H(f) = |H(f)|\phase{H(f)} \end{equation}

Consequently, substituting equation (4) into equation (3), the modulus and phase at the output of the LTI system is calculated as follows:

\begin{equation} |Y(f)| = |X(f)| ·|H(f)| \end{equation}

\begin{equation} \phase{Y(f)} = \phase{X(f)} + \phase{H(f)} \end{equation}

3.2 Summary Table

According to the separation of H(f) in modulus and phase, the following table summarizes all cases without distortion and with amplitude or phase distortion. Without loss of generality, in all cases it should be assumed that X(f) and H(f) have spectral content in the same bandwidth.

ConditionH(f)EffectY(f)y(t)ProofExample
\footnotesize \phase{H(f)}=0 \footnotesize |H(f)|=K
\footnotesize K > 1
Gain
g = K
\footnotesize g·X(f) \footnotesize g·x(t) LinkLink
\footnotesize |H(f)|=K
\footnotesize K = 1
All pass \footnotesize X(f) \footnotesize x(t) LinkLink
\footnotesize |H(f)|=K
\footnotesize K < 1
Attenuation
\footnotesize l=\cfrac{1}{K}
\footnotesize \cfrac{X(f)}{l} \footnotesize \cfrac{x(t)}{l} LinkLink
\footnotesize |H(f)|\not=K Amplitude
Distortion
\footnotesize \not\propto X(f) \footnotesize \not\propto x(t) LinkLink
\footnotesize |H(f)|=1 \footnotesize \phase{H(f)}=-2{\pi}fd Delay
d seconds
\footnotesize X(f)·e^{-j2{\pi}fd} \footnotesize x(t-d) LinkLink
\footnotesize \phase{H(f)}\not\propto-2{\pi}fd Phase
Distortion
\footnotesize \not=X(f)·e^{-j2{\pi}fd} \footnotesize \not=x(t-d) LinkLink
Synthesis of possibilities without and with linear distortion, associated mathematics and examples. X(f) and H(f) are assumed to have spectral content in the same bandwidth. K is a constant.

3.3 Explanation

3.3.1 The Mathematics of Linear Amplitude Distortion

First, the table independently analyzes the effect of the modulus of H(f) on the amplitude of the frequency components of the input X(f). For this purpose it is considered that H(f) does not produce any phase shift.

When |H(f)| is a constant K, the frequency components of the input are affected uniformly. Depending on the value of K, there may be cases with amplitude gain (K > 1), amplitude attenuation (K < 1), or no effect (K=1), which is called an all-pass filter. In other words, the output of the system is a replica, although it may be attenuated or amplified, of the input signal. Therefore there is no amplitude distortion.

When |H(f)| is not a constant, there will be at least two frequency components of the input signal X(f) that will undergo different amplitude changes. Consequently, the output signal Y(f) will change its aspect with respect to the input. Therefore, amplitude distortion will have occurred.

Examples of all of the above cases can be found in this link.

3.3.2 The Mathematics of Linear Phase Distortion

Secondly, the table independently analyzes the effect of the phase shift introduced by H(f) on the phase of the frequency components of the input X(f). For this purpose it is considered that |H(f)|=1, so that it has no impact on the amplitude of these components.

When the phase of H(f) is linear with frequency and proportional to a value d, all input components suffer the same time delay, d seconds. In other words the output of the system is a delayed replica of the input signal. Therefore, no phase distortion occurs.

On the contrary, when the phase of H(f) is not linear with frequency, at least two components of the input suffer different time delay. In other words the output of the system is not a replica of the input signal. Therefore, phase distortion has occurred. There is an exception to the previous statement. The application of frequency independent constant phase in bandpass signals does not imply that there is distortion in the demodulated signal [1].

Examples of all of the above cases can be found in this link.

3.3.3 Global Effects

Given an arbitrary transfer function H(f), any combination of the cases explained in section 3.31 and section 3.3.2 may occur. Consequently, there will be cases without distortion and cases with amplitude and/or phase distortion.

Additionally, there will be cases in which signal inversion occurs (multiplication by -1, equivalent to a phase shift of an odd multiple of 180º). These cases do not change the discussion, since the inversion does not add distortion. An inverted signal is perfectly recoverable by reversing the inversion.


Bibliography
[1] Communication Systems, A. Bruce Carlson.
[2] Signals and Systems, A. V. Openheim.


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