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Dsp notation  (Page 3/3)

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Convolution

Convolution is given by and should be displayed as

f g τ ∞ ∞ f τ g t τ
The convolution operator is defined by a csymbol which can be used through the following CNXML/MathML code.

Mathml mark-up

f g

Circular convolution

Standard notation for circular convolution , or periodic convolution, uses a circled asterick, ⊛ , between two variables or functions. See the equation below as an example: y t ⊛ f t g t Although a csymbol for regular convolution exists, there is still not a propermathematical way to mark-up circular convolution. Until a csymbol or MathML tag is created, we advise using thefollowing code:

⊛ f g

Twiddle factor

The Twiddle factor should be displayed as

W N 2 N
The twiddle factor could be given a definitionURL, which in conjunction with, probably givenby .

Mathml mark-up

W N

Fourier transform

Although the concept of the Fourier transform is well-known, there are variations in the precise definition of the transform. Itis recommended that the following notation be used for the Fourier transform. Which particular transform is used(discrete Fourier transform, continuous-time Fourier series, discrete-time Fourier transform, etc) should be apparentfrom context (these all assume that x is the function to be transformed):

  • CTFT: X Ω
X Ω
  • DTFT: X ω
X ω
  • DFT: X n
X n

The Fourier operatorℱ[·]could be given a definitionURL, which in conjunction with csymbol would allow us to facilitate the coding of the mathematical expression.

Ctft

There are a few ways of describing the Fourier transform. The recommended notation for the transform is given as:

X Ω t ∞ ∞ x t Ω t

Laplace transform

The Laplace transform is defined as

ℒ x t X s t ∞ ∞ x t s t
where s σ Ω is the complex frequency. The unilateral Laplace transform, or one-side Laplace, is defined exactly the same but thelowlimit of the integral is 0 rather than ∞ .
The Laplace operatorℒ[·]could be given a definitionURL, which in conjunction with csymbol would allow us to facilitate the coding of the mathematical expression.

ℒ x t

Z-transform

The (unilateral) z-transform is given by

Z f n F z n 0 ∞ f n z n
where z σ ω is complex in general.
The z-transform operator Z[·]could be given a definitionURL, which in conjunction with csymbol would allow us to facilitate the coding of the mathematical expression.

Z f n

Complex numbers

As is common in electrical engineering, it is recommended that the complex number be used to denote 1 . Use to mark-up all instances of the imaginary number, . Use the following notation and code for operations dealing with complex numbers.

  • The modulus of a complex number z will be represented as z .
z
  • The argument of complex number z will be rendered as z using the arg tag in MathML. The standard notation in electrical engineeringoften denotes the argument simply as ∠ z , but this does not utilize the arg tag. See the codeblock below to see how both of these representations should beimplemented:
z ∠ z
  • The conjugate of a complex number z will be represented as z * . The default rendering for the conjugate tag in MathML is z . The first notation is more common to electrical engineers and can be rendered using thefollowing MathML code:
z *

The abs tag in MathML content can be used to display the modulus with the default rendering. However, the arg and the conjugate tags' default rendering is not the recommended rendering. The correct rendering will be implemented using an XSL transformation.The Connexions project has already implemented the use of j for the imaginary unit in its stylesheets.

Inner product

The inner product (or scalar product) of two vectors is given by

x 1 x 2 t ∞ ∞ x 1 t x 2 * t
The MathML code used to denote an inner product follows. Note that this code is for a more simplified expression, onlydealing with x and y rather than a subscripted variable as seen in the equation above.

x y
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Source:  OpenStax, Ece 454 and ece 554 supplemental reading. OpenStax CNX. Apr 02, 2012 Download for free at http://cnx.org/content/col11416/1.1
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