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:
δt=0
δ
t
0
for t≠0t0
Delta function property II:
∫-∞∞δtdt=1
t
δ
t
1
The delta function has a useful property, namely the
sampling property.
xt=∫-∞∞xτδt-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 xtxt
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:
ut=1ift≥00otherwise
u
t
1
t0
0
Trigonometric functions
The
trigonometric functions are central
to signal processing and telecommunications. They are defined as follows, where
ΩΩ is the angular frequency.
Ω=2πF0
Ω
2
F0
, where
F0F0
is the
frequency of the signal.
Sine:
xt=sinΩt
x
t
Ω
t
Cosine:
xt=cosΩt
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.
ⅈ=-1
1
is the imaginary unit.
complex exponential:
xt=ⅇⅈΩt
x
t
Ω
t
Euler's relations
The
complex exponential function can be written
as a sum of its real and imaginary part.
xt=ⅇⅈΩt=cosΩt+ⅈsinΩt
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:
cosΩt=ⅇⅈΩt+ⅇ-ⅈΩt2
Ω
t
Ω
t
Ω
t
2
Euler relation 2:
sinΩt=ⅇⅈΩt-ⅇ-ⅈΩt2ⅈ
Ω
t
Ω
t
Ω
t
2
The importance of Euler's relations can hardly be stressed enough.