26 Practical MATLAB® Applications for Engineers
or
f( )[ ( )] [ ( )] t, for the analog casereal f t j imag f t
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) = [ae(n) + ao(n)] + j[be(n) + bo(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 rela-
tions hold
ƒ(t) = ƒ(t + kT) (a nalog)
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 nonpe-
riodic. 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
Eftdt
()
2
∫where |f(t)|^2 |= f(t). f(t)*, for the continuous case, and
Efnn
n
()
2
∆
∑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 to ≤ 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
tt
av ft dt
tt
1
0201
∫ () (analog case)