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Important discrete time signals

The signals and relations presented in this module are quite similar to those in the Analog signals module. So do compare and find similarities and differences!

Sequences

Generally a time discrete signal is a sequence of real or complex numbers. Each component in the sequence is identifiedby an index: ...x(n-1),x(n), x(n+1),...

[x(n)] = [0.5 2.4 3.2 4.5]is a sequence. Using the index to identify a component we have x 0 0.5 , x 1 2.4 and so on.

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Manipulating sequences

  • Addition

    Add individually each component with similar index
  • Multiplication by a constant

    Multiply every component by the constant
  • Multiplication of sequences

    Multiply each component individually
  • Delay

    A delay by k implies that we shift the sequence by k. For this to make sense the sequence has to be of infinite length.

Given the sequences [x(n)] = [0.5 2.4 3.2 4.5]and [y(n)]= [0.0 2.2 7.2 5.5].

a)Addition. [z(n)]=[x(n)]+[y(n)]=[0.5 4.6 10.4 10.0]

b)Multiplication by a constant c=2. [w(n)]= 2 *[x(n)]= [1.0 4.8 6.4 9.0]

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Elementary signals&Relations

The unit sample

The unit sample is a signal which is zero everywhere except when its argument is zero, thenit is equal to 1. Mathematically

n 1 n 0 0
The unit sample function is very useful in that it can be seen as the elementary constituent in any discrete signal.Let x n be a sequence. Then we can express x n as follows (using the unit sample definition and the delay operation)
x n k x k n k

The unit step

The unit step function is equal to zero when its index is negative and equal to one for non-negative indexes,see for plots.

u n 1 n 0 0
Unit step function, no delay.
Unit step function, delayed by 5.
Two unit step functions.

Trigonometric functions

The discrete trigonometric functions are defined as follows. n is the sequence index and is the angular frequency. 2 f , where f is the digital frequency.

x n n
x n n
A discrete sine with digital frequency 1/20.

The complex exponential function

The complex exponential function is central to signal processing and some call it the most important signal. Remember that it is a sequence and that 1 is the imaginary unit.

x n n

Euler's relations

The complex exponential function can be written as a sum of its real and imaginary part.

x n n n n
By complex conjugating and add / subtract the result with we obtain Euler's relations.
n n n 2
n n n 2
The importance of Euler's relations can hardly be stressed enough.

Matlab files

unit_step_discrete.m

Take a look at

  • Introduction
  • Analog signals
  • Discrete vs Analog signals
  • Frequency definitions and periodicity
  • Energy&Power
  • Exercises
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Questions & Answers

Complementary angles
Idrissa Reply
Commplementary angles
Idrissa Reply
Complementary angles
Idrissa
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Sherica
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Sherica
Complementary angles
Idrissa
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Sherica
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
y=10×
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Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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is it 3×y ?
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J, combine like terms 7x-4y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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AMJAD
preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
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AMJAD
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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Prasenjit
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Damian
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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Source:  OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
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