Digital Signal Processing Reference
In-Depth Information
The signal characteristics of evenness, oddness, and periodicity are defined
next. These three characteristics appear often in the study of signals and systems.
Models of common signals that appear in physical systems are then defined.
The signals considered are exponential signals and sinusoids whose amplitudes vary
exponentially. Singularity functions are defined; the unit step and the unit impulse
functions are emphasized. These two functions will prove to be very useful not only
in the time domain, but also in the frequency domain.
TABLE 2.4
Key Equations of Chapter Two
Equation Title
Equation Number
Equation
Independent-variable transformation
(2.6)
y(t) = x(at + b)
Signal-amplitude transformation
(2.8)
y(t) = Ax(t) + B
1
2
[x(t) + x(-t)]
Even part of a signal
(2.13)
x
e
(t) =
1
2
[x(t) - x(-t)]
Odd part of a signal
(2.14)
x
o
(t) =
Definition of periodicity
(2.15)
x(t) = x(t + T), T 7 0
Fundamental frequency in
hertz and radians/second
1
T
0
Hz, v
0
= 2pf
0
=
2p
T
0
rad/s
(2.16)
f
0
=
x(t) = Ce
at
Exponential function
(2.18)
e
ju
Euler's relation
(2.19)
= cos u + j sin u
e
ju
+ e
-ju
2
Cosine equation
(2.21)
cos u =
e
ju
- e
-ju
2j
Sine equation
(2.22)
sin u =
sin u
cos u
e
ju
= 1
∠u and arg e
ju
= tan
-1
Complex exponential in polar form
(2.23) and (2.24)
B
R
= u
1,
t 7 0
b
Unit step function
(2.32)
u(t) =
0,
t 6 0
Unit impulse function
(2.40)
d(t - t
0
)
= 0,
t Z t
0
;
q
d(t - t
0
)dt
= 1
L
- q
q
Sifting property of unit impulse function
(2.41)
f(t)d(t - t
0
)dt = f(t
0
)
L
- q
Multiplication property of unit
impulse function
(2.42)
f(t)d(t - t
0
) = f(t
0
)d(t - t
0
)
Test for time invariance
(2.73)
y(t)
t - t
0
= y(t)
x(t - t
0
)
Test for linearity
(2.77)
a
1
x
1
(t) + a
2
x
2
(t) : a
1
y
1
(t) + a
2
y
2
(t)
