1.1 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),...

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.

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

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.

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.

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.

Euler's relations

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

Matlab files

unit_step_discrete.m

Take a look at

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