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Unformatted text preview: oid of some frequency I!I, I in the
range 0 to 11'. W e shall now determine these frequencies. This goal is readily accomplished
by expressing t he frequency !I as in Eq. (8.14).
(a) The frequency 0.5.". is in the range (0 t o 11') so t hat i t cannot be reduced further.
(b) T he frequency 1.611' = 2.".  0.4."., a nd ! I, = 0.4.".. Therefore, a sinusoid of
frequency 1.611' can be expressed as a sinusoid of frequency I!I,I = 0.4."..
(e) 2.511' = 2.". + 0.5."., a nd !I, = 0.5.".. Therefore, a sinusoid of frequency 2.5.". c an be
'expressed as a sinusoid of frequency I!I,I = 0.5."..
(d) 5.6.". = 3(211')  0.4."., and ! I, = 0.411'. Therefore, a sinusoid of frequency 5.6.".
can be expressed as a sinusoid of frequency I!I, I = 0.4."..
(e) 34.116 = 5(2.".) + 2.7, and ! I, = 2.7. Therefore, a sinusoid of frequency 34.116
can be expressed as a sinusoid of frequency I!I,I = 2.7. • 8 .2 S ome Useful D iscretetime S ignal m odels 553 • T he f undamental r ange f requencies c an b e d etermined b y using a simple g raphical artifice as follows: m ark all t he f requencies o n a t ape u sing a linear scale, s tarting
w ith zero frequency. Now w ind t his t ape c ontinuously a round t he two poles, one
a t 10fi = 0 a nd t he o ther a t 10fi = ."., a s i llustrated i n Fig. 8.11. T he r educed
v alue o f a ny f requency m arked o n t he t ape is i ts p rojection o n t he h orizontal (10,1)
axis. F or i nstance, t he r educed f requency corresponding t o 0 = 1.6.". is 0.411' ( the
p rojection o f 1.6.". o n t he h orizontal O f a xis). Similarly, frequencies 2.5.".,5.611', a nd
34.116 c orrespond t o f requencies 0.5.".,0.4."., a nd 2.7 o n t he 10fi a xis.
{:, E xercise E 8.4
Show t hat the sinusoids of frequencies r l = (a) 211' (b) 3.". (c) 5.". (d) 3.2.". (e) 22.1327 (f)."..J..2
can be expressed as sinusoids of frequencies (a) 0 (b).". (c).". (d) O.S". (e) 3 (f)".  2, respectively.
\7. {:, E xercise E 8.5
Show that a discretetime sinusoid of frequency". + X can be expressed as a sinusoid with
frequency".  X (0:'0 x :'0 ".). This fact shows that a sinusoid with frequency above 11' by amount
x has the frequency identical to a sinusoid of frequency below". by the same amount x , and the
maximum rate o f oscillation occurs at r l = ".. As r l increases beyond "., the rate of oscillation
actually decreases. \7. o C omputer E xample C 8.4
I n the fundamental range of frequencies from  ". to ". find a sinusoid t hat is indistinguishable from the sinusoid cos (Efk). Verify by plotting these two sinusoids t hat they
are indeed identical.
T he sinusoid cos (Ef k) is identical to the sinusoid cos (Ef  2".) k = cos (  1 ~" k) =
cos (l~"k). W e may verify t hat these two sinusoids are identical.
k =15:15; k =k';
t k1=cos( 3 *pl*k/7);
t k2=cos( 11 * pi*k/7);
s tem(k,fk1,'x'),hold o n,
s tem(k,fk2),hold o ff JOlt .~~ 87r t~...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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