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Unformatted text preview: tement c ertainly does not apply t o continuous-time sinusoids.
This r esult means t hat discrete-time sinusoids of frequencies separated by integral multiples of 211' are identical. T he most dramatic consequence of this fact is
t hat a d iscrete-time sinusoid cos ( nk + 8) has a unique waveform only for t he values
o f n over a r ange of 211'. We may select this range to be 0 t o 211', or 11' t o 311', or even
-11' t o 11'. T he i mportant t hing is t hat t he r ange must be of width 211'. A sinusoid
of any frequency outside this interval is identical to a sinusoid of frequency within
this range of w idth 211'. We shall select this range -11' t o 11' a nd call i t t he f undamental r ange o f f requencies. T hus, a sinusoid of any frequency n is identical
t o some sinusoid of frequency nf in t he f undamental range -11' t o 11'. Consider, for
example, sinusoids of frequencies n = 8.711' a nd 9.611'. We c an a dd o r s ubtract any
integral multiple of 211' from these frequencies and the sinusoids will still remain
unchanged. T o reduce these frequencies t o t he f undamental range (-11' t o 11'), we
need t o s ubtract 4 x 211' = 811' from 8.711' a nd s ubtract 5 x 211' = 1011' from 9.611', to
yield frequencies 0.711' a nd -0.411', respectively. Thus
cos ( 8.7d
cos ( 9.6d + 8) =
+ 8) = cos (0.71I'k + 0)
+ 8) cos ( -O.4d (8.11) We shall now show t hat t he r ange of frequencies t hat c an be distinguished can
be further reduced from (-11', 11') t o (0, 11'). According t o Eq. (8.6), cos ( -nk + 8) =
cos ( nk - 8). I n other words, t he frequencies in t he r ange (0 t o -11') c an be expressed
as frequencies in t he r ange (0 t o 11') w ith opposite phase. For example, t he second
sinusoid in Eq. (8.11) can be expressed as
cos (9.611'k + 8) = cos ( -O.4d + 8) = cos ( O.4d - 8) (8.13) T his result shows t hat a sinusoid of any frequency n c an always be expressed as a
sinusoid of a frequency Infl, where I nfllies in t he r ange 0 t o 11'. Note, however, a
possible sign change in t he p hases of t he two sinusoids. In o ther words a discretetime sinusoid of any frequency, no m atter how high, is identical in ever~ r espect t o
a sinusoid within t he f undamental frequency range, such as -11' t o 11'. I n contrast, a
discrete-time sinusoid of any frequency, no m atter how high, can be expressed, with
a possible sign change in phase, as a sinusoid of frequency in t he r ange (0, 1 I')j t hat
is, w ithin h alf t he f undamental f requency r ange.
A s ystematic procedure t o reduce t he frequency of a sinusoid cos ( nk + 8) is t o
express n a st Infl ::s: 11', a nd m a n integer (8.14) This procedure is always possible. T he reduced frequency o fthe sinusoid cos ( nk+8)
is t hen 1 0fl.
t Equation (8.14) c an also b e e xpressed as fl., = fl.lmodulo 2 ". 552 8 D iscrete-time S ignals a nd S ystems E xample B .1
Consider sinusoids of frequencies !I equal to (a) 0.511' (b) 1.611' (c) 2.511' (d) 5.611' (e)
34.116. Each o f these sinusoids is equivalent to a sinus...
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