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Analog signals

Module by: Anders Gjendemsjø. E-mail the author

Summary: Analog signals

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

Delta function property I:

δ⁢t=0 δ t 0 for t≠0t0

Delta function property II:

∫−∞∞δ⁢tdt=1 t δ t 1
The delta function has a useful property, namely the sampling property.
x⁢t=∫−∞∞x⁢τ⁢δ⁢t−τdτ x t τ x τ δ t τ
(1)
At this stage this may seem not particulary useful, so for now just convince yourself that the above relation holds.

(We assume that x⁢txt is "well behaved" at t=τtτ, 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 Figure 1 for plots.

unit step:

u⁢t={1  if  t≥00  otherwise   u t 1 t0 0
Figure 1: Two unit step functions.
(a) Unit step function, no delay.
Figure 1(a) (unit_step_no_delay_analog.png)
(b) Unit step function, delayed by 5.
Figure 1(b) (unit_step_delay_5_analog.png)

Trigonometric functions

The trigonometric functions are central to signal processing and telecommunications. They are defined as follows, where ΩΩ is the angular frequency. Ω=2⁢π⁢F0 Ω 2 F0 , where F0F0 is the frequency of the signal.

Sine:

x⁢t=sinΩ⁢t x t Ω t

Cosine:

x⁢t=cosΩ⁢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. j=−1 1 is the imaginary unit.

complex exponential:

x⁢t=ej⁢Ω⁢t 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=ej⁢Ω⁢t=cosΩ⁢t+j⁢sinΩ⁢t x t Ω t Ω t Ω t
(2)
By complex conjugating Equation 2 and add / subtract the result with Equation 2 we obtain Euler's relations.

Euler relation 1:

cosΩ⁢t=ej⁢Ω⁢t+e−(j⁢Ω⁢t)2 Ω t Ω t Ω t 2

Euler relation 2:

sinΩ⁢t=ej⁢Ω⁢t−e−(j⁢Ω⁢t)2⁢j Ω t Ω t Ω t 2
The importance of Euler's relations can hardly be stressed enough.

Matlab file

unit_step_analog.m

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