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Discrete time signals

Module by: Anders Gjendemsjø. E-mail the author

Summary: 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 identified by an index: ...x(n-1),x(n), x(n+1),...

Example 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 0 0.5 , x⁢1=2.4 x 1 2.4 and so on.

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

Example 2

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]

Elementary signals & relations

The unit sample

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

unit sample:

δ⁢n={1  if  n=00  otherwise   δ n 1 n0 0
The unit sample function is very useful in that it can be seen as the elementary constituent in any discrete signal. Let x⁢nxn be a sequence. Then we can express x⁢nxn as follows (using the unit sample definition and the delay operation)
x⁢n=∑k=−∞∞x⁢k⁢δ⁢n−k x n k x k δ n k
(1)

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 Figure 1 for plots.

unit step:

u⁢n={1  if  n≥00  otherwise   u n 1 n0 0
Figure 1: Two unit step functions.
(a) Unit step function, no delay.
Figure 1(a) (unit_step_no_delay.png)
(b) Unit step function, delayed by 5.
Figure 1(b) (unit_step_delay_5.png)

Trigonometric functions

The discrete trigonometric functions are defined as follows. nn is the sequence index and ωω is the angular frequency. ω=2⁢π⁢f ω 2 f , where f is the digital frequency.

Discrete sine:

x⁢n=sinω⁢n x n ω n

Discrete cosine:

x⁢n=cosω⁢n x n ω n
Figure 2: A discrete sine with digital frequency 1/20.
Figure 2 (sine_discrete.png)

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=−1 1 is the imaginary unit.

complex exponential:

x⁢n=ei⁢ω⁢n x n ω n

Euler's relations

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

x⁢n=ei⁢ω⁢n=cosω⁢n+i⁢sinω⁢n x n ω n ω n ω n
(2)
By complex conjugating Equation 2 and add / subtract the result with Equation 2 we obtain Euler's relations.

Euler relation 1:

cosω⁢n=ei⁢ω⁢n+e−(i⁢ω⁢n)2 ω n ω n ω n 2

Euler relation 2:

sinω⁢n=ei⁢ω⁢n−e−(i⁢ω⁢n)2⁢i ω n ω n ω n 2
The importance of Euler's relations can hardly be stressed enough.

Matlab files

unit_step_discrete.m

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