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),... [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 AdditionAdd individually each component with similar index Multiplication by a constantMultiply every component by the constant Multiplication of sequencesMultiply each component individually DelayA 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]

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 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 xn be a sequence. Then we can express xn as follows (using the unit sample definition and the delay operation) x n k x k δ 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 n0 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 ω is the angular frequency. ω 2 f , where f is the digital frequency. x n ω n x n ω 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 ω n

Euler's relations The complex exponential function can be written as a sum of its real and imaginary part. x n ω n ω n ω n By complex conjugating and add / subtract the result with we obtain Euler's relations. ω n ω n ω n 2 ω n ω n ω 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 ?