# 1.1 Discrete time signals

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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.

## Manipulating sequences

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].

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

## 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

$(n)=\begin{cases}1 & \text{if n=0}\\ 0 & \text{otherwise}\end{cases}$
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)
$x(n)=\sum_{k=()}$ 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.

$u(n)=\begin{cases}1 & \text{if n\ge 0}\\ 0 & \text{otherwise}\end{cases}$

## Trigonometric functions

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

$x(n)=\sin (n)$
$x(n)=\cos (n)$

## 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 $i=\sqrt{-1}$ is the imaginary unit.

$x(n)=e^{in}$

## Euler's relations

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

$x(n)=e^{in}=\cos (n)+i\sin (n)$
By complex conjugating and add / subtract the result with we obtain Euler's relations.
$\cos (n)=\frac{e^{in}+e^{-(in)}}{2}$
$\sin (n)=\frac{e^{in}-e^{-(in)}}{2i}$
The importance of Euler's relations can hardly be stressed enough.

## Matlab files

Take a look at

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