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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