Convolution Theorem

6 Key excerpts on "Convolution Theorem"

• 

  eBook - PDF

  Lectures on the Fourier Transform and Its Applications

  • Brad G. Osgood(Author)
  • 2019(Publication Date)
  • American Mathematical Society

    (Publisher)

  • In words: Convolution in the time domain corresponds to multiplication in the frequency domain. The Convolution Theorem is indeed remarkable, but the game was rigged! We defined convolution to make it work. What we’ve just gone through is the same sort of thing we did when we found the formula for the Fourier coefficients for a periodic function. Or when we defined the inner product of two vectors by first asking for an algebraic condition that guaranteed orthogonality. That game was rigged, too. Remember the principle, going back to Archimedes: first suppose the problem is solved and see what the answer must be. The second step, assuming the first one works, is to turn that solution into a definition and then announce to the world that you have solved your original problem based on your brilliant definition. 2 Recall that when we studied Fourier series, convolution came up in the form ( g ∗ f )( t ) = 1 0 g ( t − x ) f ( x ) dx . In that setting, for the integral to make sense, i.e., to be able to evaluate g ( t − x ) at points outside the interval from 0 to 1, we had to assume that g was periodic. That’s not an issue in the present setting, where we assume that f ( t ) and g ( t ) are defined for all t so the factors in the integrand, g ( t − x ) f ( x ), are defined everywhere. 2 Mathematicians, in particular, are very good at presenting their results and writing their books in this way — do step one in secret and tell the world only step two. It’s extremely irritating. 162 3. Convolution Questions may be raised as to whether the integral converges, but at least the setup makes sense. Let’s see a quick application of our brilliant new discovery. As an exercise you can show (by hand, using the definition as an integral) that (Π ∗ Π)( x ) = Λ( x ) . Recall that Λ is the triangle function. Applying the Convolution Theorem, we find that F Λ( s ) = F (Π ∗ Π)( s ) = sinc s · sinc s = sinc 2 s , just like before. To ask again, was there any doubt? Remark on notation.

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  Practical Signals Theory with MATLAB Applications

  • Richard J. Tervo(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  The exact output from a given linear system at any specific time is a function of the current input signal, the response function, and the state of the system as determined by the history of previous input signals. In particular, the history of previous input signals is especially important in fully describing the output from a linear system. In general, any non-zero input signal has a lingering effect on the output, which must be accounted for as it combines with the same or other input signals. The overall interaction between a signal and the response function is described by the mathematical operation called convolution, and a signal is said to be convolved with the response function of a linear system to produce the output signal. To completely describe the interaction between signals and systems requires an understanding of the convolution relationship between a system response function and an input signal. Once this association is established, study and analysis of the system can take place in either the time domain or the frequency domain. 3.4 Convolution The introduction of the convolution integral follows naturally from the superposi- tion property of linear systems. When a signal is fed into a system, the effect of that input may persist for some time, even after the input disappears. For example, striking a bell with a hammer produces a ringing that only gradually fades away. If the bell is struck again before all the echoes have ceased, the new sound will be affected by the fact that the system was not at rest before the second input occurred. The overall response of a system to some input must therefore take into consider- ation the entire time frame before, during, and after the arrival of a specific input signal. Yet, because of the superposition property, the instantaneous output values are formed from the sum of the individual input responses at that time. This is the underlying principle leading to the convolution integral.

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  eBook - ePub

  Digital Signals Theory

  • Brian McFee(Author)
  • 2023(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 3 Convolution

  DOI: 10.1201/9781003264859-3

  This chapter introduces the fundamental concepts behind filtering and convolution. We'll stick to the basics in this chapter and leave the more detailed analysis for later chapters.

  Specifically, we'll cover:

  • How to build up convolution from simpler operations of delay, gain, and mixing
  • Formally defining convolution
  • The concept of impulse response
  • Different convolution modes
  • Algebraic properties of convolution (without proof)

  3.1 DELAY, GAIN, AND MIX

  The term convolution gets thrown around quite a bit in signal processing, and it can sound more complicated than it really is. In the simplest terms, convolution consists of three basic operations:

  1. delaying a signal by some fixed number of samples,
  2. applying a gain to the delayed signal (changing its amplitude),
  3. mixing (adding) the delayed and gained signal with the original signal.

  Before we get into the equations in full generality, let's work through a couple of simple examples.

  3.1.1 Example 1: delay and mix

  As a first example, let's consider the case where there is no gain applied to either the delayed or original signal, so we only have to worry about delay and mixing. If our input signal is x[n], and our delay is

  k > 0

  samples, the output of this process will be a new signal y[n] defined as

  As a general convention, we use

  x ⁢

  [ n ]

  to denote an “input” signal, and

  y ⁢

  [ n ]

  to denote an “output” signal, meaning the result of some computational process applied to

  x ⁢

  [ n ]

  .

  y ⁢

  [ n ]

  =

  x ⁢

  [ n ]

  +

  x ⁢

  [

  n - k

  ]

  .

  That is, the nth output sample

  y ⁢

  [ n ]

  is the sum of the input sample

  x ⁢

  [ n ]

  and the input sample k steps back in time

  x ⁢

  [

  n - k

  ]

  .

  Fig. 3.1 demonstrates this process for an input x[n] generated by a square wave, and a delay of

  k = 3

  samples.

  Figure 3.1

  Top: a signal x[n] and a delayed copy of the signal

  x [ n − 3 ]

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  eBook - PDF

  An Introduction to Analog and Digital Communications

  • Simon Haykin, Michael Moher(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  This property is known as the Convolution Theorem. Its use permits us to exchange a convolution operation in the time domain for a multiplication of two Fourier transforms, an operation that is ordinarily easier to manipulate. We have more to say on convolution later in the chapter when the issue of filtering is discussed. Using the shorthand notation for convolution, we may rewrite Eq. (2.51) in the sim- ple form (2.52) Note that Properties 11 and 12, described by Eqs. (2.49) and (2.51), respectively, are the dual of each other.  Drill Problem 2.5 Develop the detailed steps that show that the modulation and con- volution theorems are indeed the dual of each other.  PROPERTY 13 Correlation Theorem Let and Then, assuming that and are complex valued, (2.53) where is the complex conjugate of and is the time variable involved in defining the inverse Fourier transform of the product G 1 1f 2G 2 … 1f 2. t G 2 1f 2, G 2 … 1f 2 L q q g 1 1t2g 2 … 1t  t2 dt Δ G 1 1f 2G 2 … 1f 2 g 2 1t2 g 1 1t2 g 2 1t2 Δ G 2 1f 2. g 1 1t2 Δ G 1 1f 2 g 1 1t2  g 2 1t2 Δ G 1 1f 2G 2 1f 2 L q q g 1 1t2g 2 1t  t2 dt Δ G 1 1f 2G 2 1f 2 g 2 1t2 Δ G 2 1f 2. g 1 1t2 Δ G 1 1f 2 G 1 1f 2  G 2 1f 2  G 2 1f 2  G 1 1f 2  g 1 1t2g 2 1t2 Δ G 1 1f 2  G 2 1f 2 G 12 1f 2  G 1 1f 2  G 2 1f 2 G 2 1f 2. G 1 1f 2 G 12 1f 2 G 12 1f 2  L q q G 1 1l2G 2 1f  l2 dl 38 CHAPTER 2  FOURIER REPRESENTATION OF SIGNALS AND SYSTEMS To prove Eq. (2.53), we begin by reformulating the convolution integral with the roles of the time variables t and interchanged, in which case we may simply rewrite Eq. (2.51) as (2.54) As already pointed out in the statement of Property 13, the inverse Fourier transform of the product term has as its time variable; that is, is its kernel. With the formula of Eq. (2.54) at hand, Eq. (2.53) follows directly by combining reflection rule (special case of the dilation property) and conjugation rule.

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  Numerical Methods of Exploration Seismology

  With Algorithms in MATLAB®

  • Gary F. Margrave, Michael P. Lamoureux(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  (2.13), making the formulas dimensionally consistent. The limits of integration are formally written from −∞ to ∞ to accommodate all possible input signals. If a(τ) is causal, then the lower limit may be altered to zero. If a(τ) van- ishes for τ > τ max , then the upper limit may be set to τ max . In other words, the integration extends over all relevant values of τ . Convolution arises in many parts of physical theory other than the application of linear, stationary filters. In all cases, the interpretation given here of forming a superposition of scaled and time-shifted copies of a physical response (or impulse response) is appropriate. However, the choice of which function, a or b, to interpret as the impulse response is arbitrary. Since convolution is commutative, i.e., a • b = b • a, then either function can be interpreted as the impulse response, with the other being the input signal that provides the weighting. This means that all signals can be considered as filters and all filters as signals. In closing this section, it is appropriate to comment on the meaning of the word filtering in this context. In ordinary life, filtering is encountered in situations where something, for example water, is separated into desirable and undesirable parts. Usually, water is passed through a physical device called a filter that somehow traps and removes impu- rities, thereby separating the water into pure water and contaminants. It is probably not obvious at this point that convolution has a similar action on signals. The connection will 54 2 Signal Theory: Continuous become clear when the link between convolution and the Fourier transform is explicitly made later in this chapter. The Fourier transform is a tool that can decompose a signal into a continuous sum of sines and cosines at various frequencies. What distinguishes one signal from another are the amplitudes (i.e., strengths) and phases (i.e., starting values) of each sinusoid at these various frequencies.

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  eBook - PDF

  Quantitative Coherent Imaging

  Theory, Methods and Some Applications

  • J.M. Blackledge(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  This important operation is discussed in detail in part 3. Autoconvolution and autocorrelation Two other definitions which are important in the context of convolution and correlation are: Quantitative coherent imaging o o o o o o ^- J dkF(k) J dk'G(k')exp(ik'x')^ J exp[ix(k - k')dx -O O OO OO = h J dkF W J dk'G(k')exp(ik'x')6(k-k') — OO — OO o o = -?- / F(k)G(k)exp(ikx')dk 2π J — o o Hence, f(x)®g(x)*=*F{k)G(k) This theorem is sometimes referred to as the Fultung theorem from the German word 'Fultung' meaning 'folding'. The convo-lution theorem also holds in Fourier space. That is, the convo-lution of two complex spectra is equal to the Fourier transform of the product of these functions in real space, or f(x)g(x)^^F(k)»G(k) This result is sometimes referred to as the product theorem. The correlation theorem The correlation theorem follows from the Convolution Theorem and can be written in the form f(x)Qg(x)^F(-k)G(k) f and g real, f*(x)Qg(x)*=*F*(k)G(k) f and g complex. If / is a real function, then the real part of its Fourier transform F(k) is symmetric and the imaginary part of F(k) is asymmetric, i.e. F r (-k) = F r (k) and Fi(-k) = -Fi(k) In this case F(-k) = F r (-k) + iFi(-k) 30

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