Image processing (convolution)

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Image processing (convolution)

calculation steps for convolution: (Dynamic Demo)
    • The h (n) is folded around the longitudinal axis to H (-N);

    • H (n-m) is shifted to H (-M);

    • Multiplies all the corresponding entries of X (m) and H (n-m) and adds the discrete convolution result y (n).

Description

Make m′=n-m, do variable substitution, then convolution formula becomes

Therefore, the position of X (M) and H (n-m) is swapped (that is, the linear time-invariant system with input of X (n), the unit impulse response to H (n) and the linear time invariant system with the input h (n) and the unit impulse response to X (n) have the same output.

Discrete convolution is also referred to as "linear convolution" or "direct convolution" to differentiate other types of convolution.

The stability and causality of the system

Linear and time invariant two constraints define a class of systems that can be used for convolution and representation. Stability and causality are also important limitations.

stability System: for each bounded input produces a bounded output system for a stable system.

When and only then, the linear time-invariant system is stable.

Causal system:The output y (n) of the system depends only on the input at this time and at that time, depending on X (n), X (n-1), X (n-2) ....

non-causal system: the output y (n) of the system depends on the future input x (n+1), X (n+2), ....

Description

Many important networks, such as ideal low-pass filters, are non-causal and non-realized systems. But digital signal processing is often non-real-time, even real-time processing, but also allow a large delay, then for a certain output Y (n), there is a large number of "future" input x (n+1), X (n+2) ... Records can be called in memory, so it can be very close to implementing these non-causal systems, that is, a causal system with a very large delay approximation of the non-causal system, which is the digital system more than the analog system to obtain close to the ideal characteristics of the reason.

The necessary and sufficient conditions of causality system: H (N) ≡0,n〈0.

a stable causal system:A system that satisfies both stability and causality. The unit impulse response of this system is both unilateral and absolutely integrable, i.e.

This kind of stability and causality system is both achievable and stable, which is the most important system.

Http://zlgc.seu.edu.cn/jpkc2/ipkc/signal/new/course/one/1_3_2.htm

This paper discusses the method of generating the output image by using the small neighborhood of pixels in the input image, which is called filtering (filtering) in signal processing. Among them, the most commonly used is the linear filter : The output pixel is the weighted sum of the input neighborhood pixels.

1. Related operators (Correlation Operator)

Definition: That is, where H is called the correlation nucleus (Kernel).

Steps:

1) Slide the core so that its center is on the (i,j) pixel of the input image g

2) pixel value of the output image (I,j) by using the summation of the above formula

3) fully manipulate until all the pixel values of the output image are calculated

Cases:

A = [1 8 h = [8 1 6
23 5 7 14 16 3 5 7
4 6 13 20 22 4 9 2]
10 12 19) 21 3
11 18 25) 2 9]

Computes the (2,4) element of the output image =

Matlab function: IMFilter (A,H)

2. Convolution operator (convolution)

Definition:,, where

Steps:

1) Rotate the nucleus around the center 180 degrees

2) Slide the core so that its center is on the (i,j) pixel of the input image g

3) pixel value of the output image (I,j) by using the summation of the above formula

4) fully manipulate until all the pixel values of the output image are calculated

Example: Calculating the (2,4) element of the output image =

matlab function: Matlab function: IMFilter (a,h, ' conv ')% IMFilter is related operator by default, so it is necessary to pass the parameter ' conv ' When doing convolution calculation

3. Marginal effect

When filtering the edges of the image, a portion of the nucleus is located outside the edge of the image.

Common strategies include the following:

1) Use constant padding: IMFilter is filled by default with 0, which causes the image edges to be processed to be black.

2) Copy edge pixel: I3 = IMFilter (i,h, ' replicate ');

4. Common Filter

The Fspecial function can generate the kernel of several well-defined filter related operators.

Example: Unsharp Masking filter

?
12345 I = imread(‘moon.tif‘);h = fspecial(‘unsharp‘);I2 = imfilter(I,h);imshow(I), title(‘Original Image‘)figure, imshow(I2), title(‘Filtered Image‘)

Correlation and convolution

http://www.baisi.net/viewthread.php?tid=911430

The

Correlation and convolution are two completely different concepts, which should be particularly noted, although they have similar places in mathematical calculations. To understand these two concepts, we should start with their physical background, for the reasons of space I only talk about the continuous signal situation for members to discuss. The research of the
signal is the system's processing object, and the system is the carrier of the signal. How do you study signal passing through the system? Simply say two aspects: first, the signal decomposition into a plurality of simple signals and (often using Fourier transform), the system response is divided into 0 input and 0 State response solution. What does it look like when a communication signal is passed through a channel? is to seek the solution of the signal through the system. The solution of system response is divided into time domain and frequency domain, and time domain solution is divided into classical and modern times, and the above mentioned is the modern time domain solution. The reason is that there is convolution, it is applied in the 0 state response, its mathematical principle is to take advantage of the integration of integral. For the 0 State response, the requirement contains a non-homogeneous equation with an initial condition of 0 for the excitation function, and the natural idea is to decompose the complex excitation signal into a simple time signal (note that the sine signal is not simple enough and the Fourier transform is only the first step). Continuous-time system analysis, mathematical processing is attributed to the establishment and solution of differential equations. For the sake of popularity below is not so rigorous, the differential equation solution from the derivative of the function to obtain the original function, the method is many (high number of lessons) of the essential differential inverse to seek integral. Integration process We know that the use of small rectangular summation instead of curved trapezoidal, small rectangle is the rectangular pulse, pulse width toward zero, the rectangular pulse into the Δ function, summed into the integral (and the limit). This is why the system gives the solution of the Δ function, the impulse response H (t), the reason why the excitation signal e decomposition into a simple time signal is decomposed into a series of Δ functions and. The Δ function is zero when t≠0, and how to represent an impulse function for E at t=t0? is e (t0) δ (t-t0), why is that so? T-t0≠0 are zero, only in the T=t0 value of 1 (in order to understand that, Δ function definition is not so), multiply E (t0), is not e in the value of T=t0, Sum is E, in turn, E is also expressed as Δ function and. E through the system, E (t0) constant, δ (t-t0) solution is H (t-t0), finally get E (t0) H (t-t0), of course, this is a point, also required and. Note that the sum here is integral and, because it is continuous, think of the discrete sequence. To whom do you sum? Just asked for the T0, all, is to let t0 change, so there is
∫e (t0) H (t-t0) dt0=∫e (Tau) H (t-τ) dτ
Make sure of the integral variable AH (it comes from let t0 change)!!! This is the key to not being confused with the relevant formula. With convolution integral and computer and numerical integration calculation, time domain analysis is still indispensable. Otherwise missing the above three factors any one, hehe I, consequences? It's like having an FFT.

2. As the processing signal through the system to deal with such integral ∫e (tau) H (t-τ) dτ, can solve the problem of the integral, it also solves the problem of the signal through the system. So, for the sake of research, the predecessors had a convolution integral name for the purpose of the study, referred to as convolution. Also due to the simple study of convolution (without regard to the physical background, common methods or thinking of mathematics), the above-written two functions X (t), Y (t) convolution R (t) =∫x (Tau) y (t-τ) dτ, and then there are many properties and algorithms. But back in the signal processing you have to be able to explain the physical meaning of it, knowing that X (t), Y (t) represent E (t), H (t) and not be confused by external things.

3. Correlation was first used to describe the concept of the relationship between random variables in probability theory, such as correlation coefficients. In fact, the signal is generally a random process, in order to achieve signal detection, identification and extraction, often to understand the similarity of two signals, or a signal after a delay after the similarity of its own. But the correlation coefficient is defective, because the molecule is the inner product of two signals, such as Sinx and COSX, from the waveform is only different phase, and the correlation coefficient is 0 (because of the sine and Yu Yingzheng intersection), so the introduction of the correlation function, the original two functions directly into a function and the delay of another function for the inner product. Deterministic signals also have the same concept, the relevant formula and convolution formula is very similar, and can use convolution expression, so some people feel that the two concepts are also related, in fact, there is no link between the concept. Due to the delay of the first function and the second function of the correlation function as the inner product, the correlation function does not satisfy the interchange, and the convolution can. The delay varies with different results, so the correlation coefficient is the number, and the correlation function is the function that is the delay function. Formula is
R (Tau) =∫x (t) y (t+τ) DT
Integral is the calculation of the inner product, so is the original function of the integral, the result is the function of delay tau, so the convolution formula is very similar, but the physical meaning of each quantity is different, must be clear.
Because the computational problem is purely mathematical problem, do not consider the physical background, so by two formulas, note the symbol of the variable, by integral transformation, to become consistent, there is X (t) and Y (t) cross-correlation function R (t) =x (-T) *y (t), * is convolution.

A brief discussion on "convolution"

In functional analysis, convolution is a mathematical operator that generates a third function through two functions f and G , characterizing the accumulation of the overlapping portions of the function F and the G that have been flipped and moved. If a function that participates in convolution is regarded as an indicator function of interval, convolution can also be regarded as the generalization of "sliding average".

First, Brief introduction

Convolution is an important operation in analytical mathematics. Set: f(x),g(x) is the two integrable functions on the R1, as integral:

It can be proved that, regarding almost all, the above integrals are present. Thus, with the different values of x , this integral defines a new function h (x), called the convolution of the function F and G , as h (x) = (f*g) (x). Easy to verify,(f * g) (x) = (G * f) (x), and (f * g) (x) is still an integrable function. That is to say, the convolution is substituted for multiplication, L1 (R1) 1 space is an algebra, even a BA-khz algebra.

Convolution has a close relationship with Fourier transform. Using a little property, that is, the product of the Fourier transform of two functions equals the Fourier transform after convolution, the processing of many problems in Fourier analysis can be simplified.

The function f*g obtained by convolution is generally smoother than F and G . Especially when G is a smooth function with compactly supported sets,F is a local integrable, their convolution f * g is also a smooth function. By using this property, for any integrable function f, it is possible to construct a smooth function column fsapproximation to F , which is called the smoothing or regularization of functions.

The concept of convolution can also be extended to series, measure and generalized functions.

Second, the definition

The convolution of the function F and g is a function of the amount of translation, which is the integral of one of the functions that is flipped and translated and multiplied with the other function.

The interval of integration depends on the domain of f and G .

Three, fast convolution algorithm

When it is a finite length of n , it takes about N2 operations. Some fast algorithms can be used to reduce the complexity of O (n log n).

The most common fast convolution algorithm is the use of fast Fourier transforms by circular convolution. It can also be borrowed by other practices that do not include FFT, such as number-theoretic conversions.

Four, convolution theorem

Convolution theorem indicates that the Fourier transform of function convolution is the product of function Fourier transform. That is, the convolution in one domain is equivalent to the product in another domain, for example, the convolution in the time domain corresponds to the product in the frequency domain.

This theorem is also established for variations of various Fourier transforms, such as Laplace transform, bilateral Laplace transform, Z-transform, Mellin transform, and Hartley transform. The Fourier transform defined on the locally compact abelian group can also be generalized to the harmonic analysis.

Convolution theorem can be used to simplify the computation of convolution. For the sequence of length n , according to the definition of convolution, we need to do 2n -1 pairs of bit multiplication, the computational complexity is, and the Fourier transform is used to transform the sequence into the frequency domain, only a set of bit multiplication is needed, and the fast algorithm of Fourier transform is used. The total computational complexity is. This result can be applied in the fast multiplication calculation.

V. Application

Convolution has many applications in engineering and mathematics:

    • In statistics, the weighted sliding average is a convolution.
    • In probability theory, the probability density function of two statistical independent variables x and y is the convolution of the probability density function of x and Y.
    • In acoustics, an echo can be represented by a convolution of the source sound with a function reflecting various reflection effects.
    • In electronic engineering and signal processing, the output of any linear system can be obtained by convolution the input signal with the system function (System impulse response).
    • In physics, any linear system (conforming to the superposition principle) has a convolution.

Convolution

In functional analysis, the volume (roll ) , the rotation , or the fold is a mathematical operator that generates a third function by two functions F and G , and the function F accumulate with overlapping portions of G that have been flipped and moved. If a function that participates in convolution is regarded as an indicator function of interval, convolution can also be regarded as the generalization of "sliding average".

Simple Introduction

Convolution is an important operation in analytical mathematics. Set: F(x),g(x) is the two integrable functions on which the integrals are made:

It can be proved that, regarding almost all, the above integrals are present. Thus, with the different values of x , this integral defines a new function h(x), called the convolution of the function F and G , which is recorded as h (x) = (f * g) (x). Easy to verify, (f * g) (x) = (g * f) (x), and (F * g) (x) is still an integrable function. That is to say, the convolution is substituted for multiplication, and theL1 (R1) space is an algebra, even a BA-khz algebra.

Convolution has a close relationship with Fourier transform. Using a little property, that is, the product of the Fourier transform of two functions equals the Fourier transform after convolution, the processing of many problems in Fourier analysis can be simplified.

The function f * g obtained by convolution is generally smoother than F and G . Especially when G is a smooth function with compactly supported sets,f is a local integrable, their convolution f * g is also a smooth function. With this property, for any integrable function f, it is simple to construct a column of F sthat is approximated to F, This method is called smoothing or regularization of functions.

The concept of convolution can also be extended to series, measure and generalized functions.

Defined

The convolution of the function F and g is a function of the amount of translation, which is the integral of one of the functions that is flipped and translated and multiplied with the other function.

The interval of integration depends on the domain of f and G .

For a function defined in a discrete domain, the convolution is defined as

Fast convolution algorithm

When it is a finite length of n , it takes about n2 operations. Some fast algorithms can be used to reduce the complexity of O(nln n).

The most common fast convolution algorithm is the use of a fast Fourier transform with circular folds. It can also be done by other methods that do not include FFT, such as number theory conversions.

Convolution of multivariate functions

Depending on the definition of flipping, panning, and integrals, you can also define integrals on a multivariate function similarly:

Properties

Various convolution operators satisfy the following properties:

Exchange Law
Binding law
Distribution Law
Number Multiplication Binding law

Where a is any real number (or complex number).

Differential theorem

where Df represents the differential of F , if in the discrete domain refers to the difference operator, including forward differential and back differential two kinds:

    • Forward differential:
    • Back-to-differential:
Convolution theorem

convolution theorem indicates that the Fourier transform of function convolution is the product of function Fourier transform. That is, the convolution in one domain is equivalent to the product in another domain, for example, the convolution in the time domain corresponds to the product in the frequency domain.

which represents the Fourier transform of F .

This theorem is also established for variations of various Fourier transforms, such as Laplace transform, bilateral Laplace transform, Z-transform, Mellin transform, and Hartley transform (see Mellin inversion theorem). The Fourier transform defined on the locally compact abelian group can also be generalized to the harmonic analysis.

Convolution theorem can be used to simplify the computation of convolution. For sequences of length n, the calculation of the convolution is done in terms of 2n − 1 pairs of bit multiplication, the computational complexity is, and the Fourier transform is used to transform the sequence into the frequency domain, only a set of bit multiplication is needed. With the fast algorithm of Fourier transform, the total computational complexity is. This result can be applied in the fast multiplication calculation.

Convolution on a group

If G is a group with a m measure (for example, a locally compact topological group under the HAL measure in the Hausdorff space), for M-Lebesgue integrable real or complex functions f on G and g, you can define their convolution:

For the convolution defined on these groups, it is also possible to give properties such as convolution theorem, but this requires the representation theory of these groups and the Peter-Outer theorem of harmonic analysis.

Application

Convolution has many applications in engineering and mathematics:

    • In statistics, the weighted sliding average is a convolution.
    • In probability theory, the probability density function of two statistical independent variables x and y is the convolution of the probability density function of x and Y.
    • In acoustics, an echo can be represented by a convolution of the source sound with a function reflecting various reflection effects.
    • In electronic engineering and signal processing, the output of any linear system can be obtained by convolution the input signal with the system function (System impulse response).
    • In physics, any linear system (conforming to the superposition principle) has a convolution.
See
    • Anti-convolution
    • Fourier transform
External links
    • PlanetMath on the convolution of the information.
    • Visual convolution Java Applet
From "Http://zh.wikipedia.org/w/index.php?title= convolution &oldid=18610515"

Convolution convolution

Definition: an infinite integral operation for two functions in mathematics. For functions F1 (T) and F2 (t), their convolution is expressed as: "" is a convolution operation symbol.
Applied disciplines: Electric power (primary discipline); General theory (level two)

In functional analysis, convolution (convolution), convolution, or convolution (English: convolution) is a mathematical operator that generates a third function through two functions f and G, with the accumulation of a function f with the overlapping portion of a flip and pan and G. If a function that participates in convolution is regarded as an indicator function of interval, convolution can also be regarded as the generalization of "sliding average".

Simple introduction to the basic connotation

Definition of convolution

Convolution is an important operation in analytical mathematics. Set F( x), g( x) is the two integrable functions on the R1, making integrals (as shown on the right): it can be proved that almost all real numbers x, the above points are present. This way, with xValue, this integral defines a new function h (x), called a function FAnd gThe convolution, recorded as h (x) = (f*g) (x)。 Easy to verify, (f * g) (x) = (G * f) (x)And (f * g) (x)is still an integrable function.   That is to say, the convolution is substituted for multiplication, L1 (R1) 1 space is an algebra, even a BA-khz algebra. Convolution has a close relationship with Fourier transform.   Using a little property, that is, the product of the Fourier transform of two functions equals the Fourier transform after convolution, the processing of many problems in Fourier analysis can be simplified. The functions obtained by convolution f*gGenerally than FAnd gare smooth. Especially when gAs a smooth function with compact set, FWhen they are locally integrable, their convolution F * gis also a smooth function. Using this property, for any integrable function F, you can simply construct a column that approximates the FThe Smooth function column FS, this method is called smoothing or regularization of functions. The concept of convolution can also be extended to series, measure and generalized functions. Defining functions FAnd gConvolution, which is the integral of one of the functions that flips and pans the product of another function, is a function of the amount of translation. The integration interval depends on FAnd gThe domain of the definition.   The convolution of the function f and g can be defined as: Z (t) =f (t) *g (t) =∫f (m) g (t-m) DM. For functions defined in discrete domains, convolution is defined as a fast convolution algorithm when it is finite length N, it takes about n^2Second operation. With some fast algorithms can be reduced to O ( NLog N) complexity. The most common fast convolution algorithm uses the fast Fourier transform by using the circular folding product. It can also be done by other methods that do not include FFT, such as number theory conversions. Multivariate function convolution according to the definition of flip, shift, integral, can also be similar to define the integral of the multivariate function: properties various convolution operators satisfy the following properties: The law of commutative law and the Law of distributive law of the Union aIs any real number (or complex number). Differential theorem where D FSaid FDifferential, if it is a difference operator in a discrete domain, including forward differential and back difference: Forward differential: Back differential: convolution theorem convolution theoremIt is pointed out that the Fourier transform of function convolution is the product of function Fourier transform.   That is, the convolution in one domain is equivalent to the product in another domain, for example, the convolution in the time domain corresponds to the product in the frequency domain. which represents FFourier transform. This theorem is also established for variations of various Fourier transforms, such as Laplace transform, bilateral Laplace transform, Z-transform, Mellin transform, and Hartley transform (see Mellin inversion theorem).   The Fourier transform defined on the locally compact abelian group can also be generalized to the harmonic analysis. Convolution theorem can be used to simplify the computation of convolution. For a length of NSequence, which is calculated according to the definition of convolution, requires doing 2 N-1 pairs of bit multiplication, and its computational complexity is, and by using Fourier transform to transform the sequence to the frequency domain, only a set of bit multiplication, using the fast algorithm of Fourier transform, the total computational complexity is. This result can be applied in the fast multiplication calculation. The convolution on the group if GThere is a certain mA group of measures (for example, a locally compact topological group under the Harr measure on a Hausdorff space), for GOn m-Lebesgue a real or complex function that can be accumulated FAnd g, they can be defined as convolution: for the convolution defined on these groups, it is also possible to give properties such as convolution theorem, but this requires the representation theory of these groups and the Peter-weyl theorem of harmonic analysis. The application of convolution has many applications in engineering and mathematics: Statistics , the weighted sliding average is a convolution. probability theory , two statistical independent variables x and y probability density function is the convolution of the probability density function of x and Y. In acoustics, an echo can be represented by a convolution of the source sound with a function reflecting various reflection effects. In electronic engineering and signal processing, the output of any linear system can be obtained by convolution the input signal with the system function (System impulse response).   In physics, any linear system (conforming to the superposition principle) has a convolution. Introduce a practical example of probabilistic applications. Assuming that the arrival rate of the demand time is possion (λ) distribution, the distribution function of the size of the demand is D (.), the distribution function of the demand for unit time is F (x):

Convolution applications (1 photos) where D (k) (x) is a K-order convolution. Convolution is a linear operation, and the common mask operation in image processing is convolution, which is widely used in image filtering.   Castlman's book is very detailed about the convolution. Gaussian transform is to use Gaussian function to convolution the image. The Gaussian operator can be obtained directly from the discrete Gaussian function: for (i=0; i<n; i++) {for (j=0; j<n; J + +) {G[i*n+j]=exp (-((I-(N-1)/2) ^2+ (J (N-1)/2) ^2)/   (2*delta^2));   Sum + = G[i*n+j]; }} and dividing by sum to get the normalized operator n is the size of the filter, the Delta option first, before referring to the convolution, we must mention the background of the convolution.   Convolution is in the signal and linear system based on or in the background, it is meaningless to separate the convolution from this background, in addition to the so-called pleated formula on the mathematical meaning and integral (or summation, discrete case). Signal and linear systems, the discussion is the change in the signal after a linear system (that is, the input and output and the so-called system, the mathematical relationship between the three).   The meaning of the so-called linear system is that this so-called system, which brings the output signal to the mathematical relationship between the input signal is a linear operation relationship.   So, in fact, we have to design the so-called system transfer function according to the signal form we need to deal with, then the transfer function and the input signal of this system are the so-called convolution relation in the mathematical form. One of the most important cases of convolution is the convolution theorem in signal and linear systems or digital signal processing. By using this theorem, the convolution operation in time domain or space domain can be equivalent to the multiplication operation of frequency domain, and the fast algorithm such as FFT is used to realize effective calculation and save the operation cost.
Extended reading:
  • http://zh.wikipedia.org/wiki/convolution

  • Http://zhidao.baidu.com/question/19032593.html

linear time Invariant systemLinear time invariant (LTI) It consists of two definitions for the concept linear system of the linear system and the discrete time system and the Nonlinear System: (1) The linear characteristics of the system that satisfies the superposition principle are defined according to whether the input and output relations of the system are linear. Even if to two excitation X1 (n) and X2 (n), with T[AX1 (n) +bx2 (n)]=at[x1 (n)]+bt[x2 (n)], the formula A, B is any constant. The nonlinear system is not satisfied with the above relationship. (2) A system consisting of a linear element and a separate power supply is defined according to the component characteristics of the constituent system. [1] constant system time-invariant system: is the system parameters do not change with the time, that is, regardless of the input signal action time successively, the output signal response shape are the same, only from the occurrence of different time. The mathematical representation of T[x (n)]=y[n] is t[x (n-n0)]=y[n-n0], which indicates that the sequence x (n) is shifted first and then the transformation is equivalent to the transformation before it is shifted. Linear time Invariant system: it can be expressed by unit impulse response, which satisfies the superposition principle and has the time invariant characteristic.   The unit impulse response is the system output when the input is a unit pulse sequence, generally expressed as H (n), i.e. h (n) =t[δ (n). Any input sequence x (n) corresponding y (n) =t[x (n)]=t[δ (n-k)]; Because the system is linear, the upper formula can be written as Y (n) =t[δ (n-k)], and because the system is time-invariant, there is t[δ (n-k)]=h (n-k), resulting in Y (n) =h ( N-K) =x (n) *h (n); This formula is called discrete convolution, denoted by "*". The homogeneous property of linear time-invariant systems if the excitation f (t) produces a response of Y (t), then the response generated by the excitation AF (t) is ay (t), which is the homogeneous nature.   Where a is an arbitrary constant. F (t) system Y (t), Af (t) system ay (t) superposition if the response of the excitation F1 (t) and F2 (t) is Y1 (t), Y2 (t), then the excitation F1 (t) +f2 (t) is generated should be Y1 (t) +y2 (t), which is called the superposition property. If the response of linear excitation F1 (t) and F2 (t) is Y1 (t), Y2 (t), then the response of the excitation a1f1 (t) +a2f2 (t) is a1y1 (t) +a2y2 (t), which is called linear. If the excitation f (t) produces a response of Y (t), then the response generated by the excitation F (t-t0) is Y (t-t0), which is called invariance, also known as constancy or delay. It shows that when the excitation f (t) delay time T0, its response y (T) also delay time t0, and the waveform is unchanged. Differential stressExcitation f (t) produces a response of Y (t), then the excitation f ' (t) produces a response that is Y ' (t), for this nature is differential. Integral if the response generated by the excitation F (t) is Y (t), then the response of the excitation F (t) is the integral of Y (t). This property is called integral.
Resources
  • "Signals and Systems" (second edition) Lu Youxin Zhang Mingyu Electronics Publishing House

Convolution theorem

convolution theorem indicates that the Fourier transform of function convolution is the product of function Fourier transform. That is, the convolution in one domain corresponds to the product in another domain, for example, the convolution in the time domain corresponds to the product in the frequency domain.

which represents the Fourier transform of F .

By Fourier inverse transformation, can also be written

Note that the above is only the correct transformation of the definition of a particular form, the transformation may be formalized by other means, so that the above-mentioned relationship has other constant factors.

This theorem is also established for variations of various Fourier transforms, such as Laplace transform, bilateral Laplace transform, Z-transform, Mellin transform, and Hartley transform (see Mellin inversion theorem). The Fourier transform defined on the locally compact abelian group can also be generalized to the harmonic analysis.

Convolution theorem can be used to simplify the computation of convolution. For sequences of length n, the calculation of the convolution is done in terms of 2n − 1 pairs of bit multiplication, the computational complexity is, and the Fourier transform is used to transform the sequence into the frequency domain, only a set of bit multiplication is needed. With the fast algorithm of Fourier transform, the total computational complexity is. This result can be applied in the fast multiplication calculation.

Prove
.

Substituting y = zx; dy = dz

Related items
    • Convolution

Impuls response function method for impulse response functions


Image:


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Maichong xiangying Hanshu FANGFA
Impulse response Function Method (volume name: Automatic control and system engineering)
Impuls Response Function method

A Nonparametric model identification method for estimating the impulse response function of a linear system. For deterministic linear systems without random noise, when the input signal is a pulse function δ (t), the output response H (t) of the system is called the impulse response function. for any input u (t), the output y (t) of the linear system is expressed as the convolution of the impulse response function and the input, that is, if the system is physically achievable, then the output is 0 before the input starts, that is, when τ<0 h (tau) = 0, here tau is the integral variable. For discrete systems, the impulse response function is an infinite weight sequence, and the output of the system is the convolution of the input sequence UT and the weight sequence HT:. The impulse response function of the system is a very important non-parametric model. The method of identifying impulse response function is divided into direct method, correlation method and indirect method. ① Direct method: The waveform is more ideal pulse signal input system, the response of the time domain record the output response of the system, can be response curve or discrete value. ② Correlation method: By the famous Wiener-Hough equation: If the input signal U (t) autocorrelation function R (t) is a pulse function kδ (t), then the impulse response function ignores a constant factor meaning equal to the input and output of the cross-correlation function, namely h (t) = (1/k) Ruy (t). When using the correlation method to identify the impulse response of the system, the common pseudo-random signal is used as the input signal, and the cross-correlation function Ruy (t) can be obtained by the correlation instrument or digital computer, because the autocorrelation function R (t) of pseudo-random signal is approximate to a pulse function, so h (t) = (1/k) Ruy This is a more general approach. It is also possible to enter an approximate white noise signal with a wide enough bandwidth to obtain an approximate representation of H (t). ③ Indirect method: The Power spectrum analysis method can be used to estimate the frequency response function H (ω), and then use the Fourier inverse transform to transform it to the time domain, so then get the impulse response H (t).

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