The signals signals and relations presented in this module are quite similar to those in the Discrete time signals module. So do compare and find similarities and differences!

Manipulating signals Mathematical operations on analog signals are unambiguous. We require that the signals are defined over the same time interval when using operations such as addition, multiplication, division and so on.

Elementary signals & relations

The (Dirac) delta function The delta function is a peculiar function that has zero duration, infinite height, but still unit area! Mathematically we have the following two properties δ t 0 for t0 t δ t 1 The delta function has a useful property, namely the sampling property. x t τ x τ δ t τ At this stage this may seem not particulary useful, so for now just convince yourself that the above relation holds. (We assume that xt is "well behaved" at tτ, that is continuous and finite.)

The unit step function The unit step function is equal to zero when its argument is negative and equal to one for non-negative arguments, see for plots. u t 1 t0 0

Unit step function, no delay. Unit step function, delayed by 5. Two unit step functions.

Trigonometric functions The trigonometric functions are central to signal processing and telecommunications. They are defined as follows, where Ω is the angular frequency. Ω 2 F0 , where F0 is the frequency of the signal. x t Ω t x t Ω t See also Frequency definitions & periodicity.

The complex exponential function The complex exponential function is central to signal processing and some call it the most important signal. 1 is the imaginary unit. x t Ω t

Euler's relations The complex exponential function can be written as a sum of its real and imaginary part. x t Ω t Ω t Ω t By complex conjugating and add / subtract the result with we obtain Euler's relations. Ω t Ω t Ω t 2 Ω t Ω t Ω t 2 The importance of Euler's relations can hardly be stressed enough.

Matlab file unit_step_analog.m

Take a look at Introduction Discrete time signals Discrete vs Analog signals Frequency definitions and periodicity Energy & Power Exercises ?