26
Practical MATLAB
® Applications for Engineers
or
f( )
[ ( )]
[ ( )]
t
real f t
j imag f t
ϭ
ϩ
, for the analog case
R.1.79 The complex conjugate sequence of ƒ(n) is denoted by ƒ*(n), where
ƒ*(n) = real [ƒ(n)] − j imag [ƒ(n)] = a(n) − jb(n)
R.1.80 Let f(n) be a complex sequence, where ƒ(n) = a(n) + jb(n). This sequence can further
be decomposed into
ƒ(n) = [a e (n) + a o (n)] + j[b e (n) + b o (n)]
where the subscripts e and o denote the even and odd parts, respectively, of the a(n)
and b(n) of f(n).
The same relation holds when n is replaced by t for the analog case.
R.1.81 A signal or sequence is periodic (with either period T or N) if the following relations hold
ƒ(t) = ƒ(t + kT) (analog)
or
ƒ(n) = ƒ(n + kN) (discrete)
for any k = 0, ±1, ±2, ±3, ….
When the signal or sequence does not satisfy the preceding relations, it is nonperiodic. A periodic signal is defi ned for all t (−∞, ∞). Periodic signals or sequences are
basically ideal concepts. Most practical signals are basically nonperiodic.
R.1.82 The energy E of a signal ƒ(t) or sequence ƒ(n) is defi ned by
E
f t dt
ϭ
Ͻ
ϩ
( )
2
ϱ
Ϫϱ
ϱ
∫
where |f(t)|
2
|= f(t) . f(t)*, for the continuous case, and
E
f n
n
n
ϭ
ϭ
ϩ
( )
2 ∆ Ͻ ϱ
Ϫϱ
ϱ
∑
for the discrete case.
R.1.83 A fi nite length sequence with fi nite magnitudes will always have fi nite energy or
an infi nite sequence with a fi nite number of samples may not have infi nite energy.
R.1.84 A signal ƒ(t) defi ned over the range t o ≤ t ≤ t 1 , with a fi nite number of maxima and
minima, is associated with a fi nite energy content E (in joules).
R.1.85 Let the energy of the signal f(t) exist and be fi nite, then the signal f(t) is referred to
as an energy signal.
R.1.86 The average power of a fi nite discrete sequence f(n) (or time-limited signal f(t)) is
defi ned by the following equations:
P
t t
f t dt
av
t
t
ϭ
1
0
2
0
1
Ϫ
( )
∫
(analog case)
CRC_47760_CH001.indd 26
CRC_47760_CH001.indd 26
7/25/2008 4:16:15 PM
7/25/2008 4:16:15 PM
Practical MATLAB
® Applications for Engineers
or
f( )
[ ( )]
[ ( )]
t
real f t
j imag f t
ϭ
ϩ
, for the analog case
R.1.79 The complex conjugate sequence of ƒ(n) is denoted by ƒ*(n), where
ƒ*(n) = real [ƒ(n)] − j imag [ƒ(n)] = a(n) − jb(n)
R.1.80 Let f(n) be a complex sequence, where ƒ(n) = a(n) + jb(n). This sequence can further
be decomposed into
ƒ(n) = [a e (n) + a o (n)] + j[b e (n) + b o (n)]
where the subscripts e and o denote the even and odd parts, respectively, of the a(n)
and b(n) of f(n).
The same relation holds when n is replaced by t for the analog case.
R.1.81 A signal or sequence is periodic (with either period T or N) if the following relations hold
ƒ(t) = ƒ(t + kT) (analog)
or
ƒ(n) = ƒ(n + kN) (discrete)
for any k = 0, ±1, ±2, ±3, ….
When the signal or sequence does not satisfy the preceding relations, it is nonperiodic. A periodic signal is defi ned for all t (−∞, ∞). Periodic signals or sequences are
basically ideal concepts. Most practical signals are basically nonperiodic.
R.1.82 The energy E of a signal ƒ(t) or sequence ƒ(n) is defi ned by
E
f t dt
ϭ
Ͻ
ϩ
( )
2
ϱ
Ϫϱ
ϱ
∫
where |f(t)|
2
|= f(t) . f(t)*, for the continuous case, and
E
f n
n
n
ϭ
ϭ
ϩ
( )
2 ∆ Ͻ ϱ
Ϫϱ
ϱ
∑
for the discrete case.
R.1.83 A fi nite length sequence with fi nite magnitudes will always have fi nite energy or
an infi nite sequence with a fi nite number of samples may not have infi nite energy.
R.1.84 A signal ƒ(t) defi ned over the range t o ≤ t ≤ t 1 , with a fi nite number of maxima and
minima, is associated with a fi nite energy content E (in joules).
R.1.85 Let the energy of the signal f(t) exist and be fi nite, then the signal f(t) is referred to
as an energy signal.
R.1.86 The average power of a fi nite discrete sequence f(n) (or time-limited signal f(t)) is
defi ned by the following equations:
P
t t
f t dt
av
t
t
ϭ
1
0
2
0
1
Ϫ
( )
∫
(analog case)
CRC_47760_CH001.indd 26
CRC_47760_CH001.indd 26
7/25/2008 4:16:15 PM
7/25/2008 4:16:15 PM