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Discrete signal

Description

A discrete (digital) signal is the discretized variant of a continuous(analog) signal. It describes the value of the signal at specific intervals. Typically used in signal processing. It is a Discrete function.

\[\htmlClass{sdt-0000000041}{x}[\htmlClass{sdt-0000000117}{n}] = \htmlClass{sdt-0000000041}{x}(\htmlClass{sdt-0000000117}{n}\htmlClass{sdt-0000000048}{T_{s}}) = \htmlClass{sdt-0000000121}{a} \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000116}{\omega} \htmlClass{sdt-0000000117}{n} \htmlClass{sdt-0000000048}{T_{s}} + \htmlClass{sdt-0000000123}{\phi}) = \htmlClass{sdt-0000000121}{a} \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000016}{\hat \omega} \htmlClass{sdt-0000000117}{n} + \htmlClass{sdt-0000000123}{\phi})\]

Symbols Used:

This is the symbol for normalized radian frequency. It is measured in radians (\( \htmlClass{udt-0000000005}{rad} \))
\( T_{s} \)

This symbol represents sampling period, the amount of time between samples taken in an analog-to-digital converter.
\( x \)

This symbol represents a function that represents a signal.
\( \omega \)

This symbol represents radian frequency, the speed of rotation. It is measured in radians per second.
\( n \)

This symbol represents any given whole number, \( n \in \htmlClass{sdt-0000000014}{\mathbb{W}}\).
\( a \)

This is a symbol for any generic constant. It can hold any numerical value
\( \phi \)

This symbol means the same as Angle, which uses \(\htmlClass{sdt-0000000024}{\theta}\). It is a secondary symbol to use that represents an angle, when a different angle is already using \(\htmlClass{sdt-0000000024}{\theta}\).
\( \cos \)

This is the symbol for cosine, a trigonometric function that calculates the ratio of the adjacent side to the hypotenuse of a right-angled triangle.

Derivation

Imagine you have some analog signal \(\htmlClass{sdt-0000000041}{x}(\htmlClass{sdt-0000000118}{t})\). We define \(\htmlClass{sdt-0000000118}{t} = 0\) to be the moment the converter starts taking samples. Your analog-to-digital converter samples at a rate of \(\htmlClass{sdt-0000000055}{f_{s}}\).

  1. The value of \(\htmlClass{sdt-0000000041}{x}[\htmlClass{sdt-0000000117}{n}]\) at \(n = 0\) will then be equal to \(x(0)\).
  2. It follows that the value of \(\htmlClass{sdt-0000000061}{x}\) at \(n = 1\) will be the value of \(\htmlClass{sdt-0000000041}{x}(\htmlClass{sdt-0000000118}{t})\), when \(\htmlClass{sdt-0000000118}{t}\) is how long it has been since the last sample. It will therefore be,
    \[\htmlClass{sdt-0000000041}{x}[1] = \htmlClass{sdt-0000000041}{x}(\htmlClass{sdt-0000000048}{T_{s}})\]
  3. Using the same logic, we can say that...
    \[\htmlClass{sdt-0000000041}{x}[2] = \htmlClass{sdt-0000000041}{x}(2 \cdot \htmlClass{sdt-0000000048}{T_{s}})\]
  4. We can now generalize the pattern, and say that...
    \[\htmlClass{sdt-0000000041}{x}[\htmlClass{sdt-0000000117}{n}] = \htmlClass{sdt-0000000041}{x}(\htmlClass{sdt-0000000117}{n} \htmlClass{sdt-0000000048}{T_{s}})\]
    as required.

Example

Coming soon...