Analog signal processing
Encyclopedia
Analog signal processing is any signal processing
conducted on analog signal
s by analog means. "Analog" indicates something that is mathematically represented as a set of continuous values. This differs from "digital" which uses a series of discrete quantities to represent signal. Analog values are typically represented as a voltage
, electric current
, or electric charge
around components in the electronic devices. An error or noise affecting such physical quantities will result in a corresponding error in the signals represented by such physical quantities.
Examples of analog signal processing include crossover filters in loudspeakers, "bass", "treble" and "volume" controls on stereos, and "tint" controls on TVs. Common analog processing elements include capacitors, resistors, inductors and transistors.
as H(s), where s is a complex number
in the form of s=a+ib, or s=a+jb in electrical engineering terms (electrical engineers use j because current is represented by the variable i). Input signals are usually called x(t) or X(s) and output signals are usually called y(t) or Y(s).
is the basic concept in signal processing that states an input signal can be combined with the system's function to find the output signal. It is the integral of the product of two waveforms after one has reversed and shifted; the symbol for convolution is *.
That is the convolution integral and is used to find the convolution of a signal and a system; typically a = -∞ and b = +∞.
Consider two waveforms f and g. By calculating the convolution, we determine how much a reversed function g must be shifted along the x-axis to become identical to function f. The convolution function essentially reverses and slides function g along the axis, and calculates the integral of their (f and the reversed and shifted g) product for each possible amount of sliding. When the functions match, the value of (f*g) is maximized. This occurs because when positive areas (peaks) or negative areas (troughs) are multiplied, they contribute to the integral.
is a function that transforms a signal or system in the time domain into the frequency domain, but it only works for certain ones. The constraint on which systems or signals can be transformed by the Fourier Transform is that:
This is the Fourier transform integral:
Most of the time the Fourier transform integral isn't used to determine the transform. Usually a table of transform pairs is used to find the Fourier transform of a signal or system. The inverse Fourier transform is used to go from frequency domain to time domain:
Each signal or system that can be transformed has a unique Fourier transform; there is only one time signal and one frequency signal that goes together.
. It allows a transform of any system or signal because it is a transform into the complex plane instead of just the jω line like the Fourier transform. The major difference is that the Laplace transform has a region of convergence for which the transform is valid. This implies that a signal in frequency may have more than one signal in time; the correct time signal for the transform is determined by the region of convergence. If the region of convergence includes the jω axis, jω can be substituted into the Laplace transform for s and it's the same as the Fourier transform. The Laplace transform is:
and the inverse Laplace transform, if all the singularities of X(s) are in the left half of the complex plane, is:
s are plots of magnitude vs. frequency and phase vs. frequency for a system. The magnitude axis is in Decibel
(dB). The phase axis is in either degrees or radians. The frequency axes are in a logarithmic scale
. These are useful because for sinusoidal inputs, the output is the input multiplied by the value of the magnitude plot at the frequency and shifted by the value of the phase plot at the frequency.
shows either the phase shift or magnitude of a signal at each frequency that it exists at. These can be found by taking the Fourier transform of a time signal and are plotted similarly to a bode plot.
are the building block of analog signal processing. All real world signals can be represented as an infinite sum of sinusoidal functions via a Fourier series
. A sinusoidal function can be represented in terms of an exponential by the application of Euler's Formula
.
) is defined as a signal that has an infinite magnitude and an infinitesimally narrow width with an area under it of one, centered at zero. An impulse can be represented as an infinite sum of sinusoids that includes all possible frequencies. It is not, in reality, possible to generate such a signal, but it can be sufficiently approximated with a large amplitude, narrow pulse, to produce the theoretical impulse response in a network to a high degree of accuracy. The symbol for an impulse is δ(t). If an impulse is used as an input to a system, the output is known as the impulse response. The impulse response defines the system because all possible frequencies are represented in the input.
, is a signal that has a magnitude of zero before zero and a magnitude of one after zero. The symbol for a unit step is u(t). If a step is used as the input to a system, the output is called the step response. The step response shows how a system responds to a sudden input, similar to turning on a switch. The period before the output stabilizes is called the transient part of a signal. The step response can be multiplied with other signals to show how the system responds when an input is suddenly turned on.
The unit step function is related to the Dirac delta function by;
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
conducted on analog signal
Analog signal
An analog or analogue signal is any continuous signal for which the time varying feature of the signal is a representation of some other time varying quantity, i.e., analogous to another time varying signal. It differs from a digital signal in terms of small fluctuations in the signal which are...
s by analog means. "Analog" indicates something that is mathematically represented as a set of continuous values. This differs from "digital" which uses a series of discrete quantities to represent signal. Analog values are typically represented as a voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
, electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
, or electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
around components in the electronic devices. An error or noise affecting such physical quantities will result in a corresponding error in the signals represented by such physical quantities.
Examples of analog signal processing include crossover filters in loudspeakers, "bass", "treble" and "volume" controls on stereos, and "tint" controls on TVs. Common analog processing elements include capacitors, resistors, inductors and transistors.
Tools used in analog signal processing
A system's behavior can be mathematically modeled and is represented in the time domain as h(t) and in the frequency domainFrequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
as H(s), where s is a complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
in the form of s=a+ib, or s=a+jb in electrical engineering terms (electrical engineers use j because current is represented by the variable i). Input signals are usually called x(t) or X(s) and output signals are usually called y(t) or Y(s).
Convolution
ConvolutionConvolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
is the basic concept in signal processing that states an input signal can be combined with the system's function to find the output signal. It is the integral of the product of two waveforms after one has reversed and shifted; the symbol for convolution is *.
That is the convolution integral and is used to find the convolution of a signal and a system; typically a = -∞ and b = +∞.
Consider two waveforms f and g. By calculating the convolution, we determine how much a reversed function g must be shifted along the x-axis to become identical to function f. The convolution function essentially reverses and slides function g along the axis, and calculates the integral of their (f and the reversed and shifted g) product for each possible amount of sliding. When the functions match, the value of (f*g) is maximized. This occurs because when positive areas (peaks) or negative areas (troughs) are multiplied, they contribute to the integral.
Fourier transform
The Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
is a function that transforms a signal or system in the time domain into the frequency domain, but it only works for certain ones. The constraint on which systems or signals can be transformed by the Fourier Transform is that:
This is the Fourier transform integral:
Most of the time the Fourier transform integral isn't used to determine the transform. Usually a table of transform pairs is used to find the Fourier transform of a signal or system. The inverse Fourier transform is used to go from frequency domain to time domain:
Each signal or system that can be transformed has a unique Fourier transform; there is only one time signal and one frequency signal that goes together.
Laplace transform
The Laplace transform is a generalized Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
. It allows a transform of any system or signal because it is a transform into the complex plane instead of just the jω line like the Fourier transform. The major difference is that the Laplace transform has a region of convergence for which the transform is valid. This implies that a signal in frequency may have more than one signal in time; the correct time signal for the transform is determined by the region of convergence. If the region of convergence includes the jω axis, jω can be substituted into the Laplace transform for s and it's the same as the Fourier transform. The Laplace transform is:
and the inverse Laplace transform, if all the singularities of X(s) are in the left half of the complex plane, is:
Bode plots
Bode plotBode plot
A Bode plot is a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency axis, to show the system's frequency response...
s are plots of magnitude vs. frequency and phase vs. frequency for a system. The magnitude axis is in Decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
(dB). The phase axis is in either degrees or radians. The frequency axes are in a logarithmic scale
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...
. These are useful because for sinusoidal inputs, the output is the input multiplied by the value of the magnitude plot at the frequency and shifted by the value of the phase plot at the frequency.
Time domain
This is the domain that most people are familiar with. A plot in the time domain shows the amplitude of the signal with respect to time.Frequency domain
A plot in the frequency domainFrequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
shows either the phase shift or magnitude of a signal at each frequency that it exists at. These can be found by taking the Fourier transform of a time signal and are plotted similarly to a bode plot.
Signals
While any signal can be used in analog signal processing, there are many types of signals that are used very frequently.Sinusoids
SinusoidsSine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...
are the building block of analog signal processing. All real world signals can be represented as an infinite sum of sinusoidal functions via a Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
. A sinusoidal function can be represented in terms of an exponential by the application of Euler's Formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
.
Impulse
An impulse (Dirac delta functionDirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
) is defined as a signal that has an infinite magnitude and an infinitesimally narrow width with an area under it of one, centered at zero. An impulse can be represented as an infinite sum of sinusoids that includes all possible frequencies. It is not, in reality, possible to generate such a signal, but it can be sufficiently approximated with a large amplitude, narrow pulse, to produce the theoretical impulse response in a network to a high degree of accuracy. The symbol for an impulse is δ(t). If an impulse is used as an input to a system, the output is known as the impulse response. The impulse response defines the system because all possible frequencies are represented in the input.
Step
A unit step function, also called the Heaviside step functionHeaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....
, is a signal that has a magnitude of zero before zero and a magnitude of one after zero. The symbol for a unit step is u(t). If a step is used as the input to a system, the output is called the step response. The step response shows how a system responds to a sudden input, similar to turning on a switch. The period before the output stabilizes is called the transient part of a signal. The step response can be multiplied with other signals to show how the system responds when an input is suddenly turned on.
The unit step function is related to the Dirac delta function by;