Signal Processing and Linear Systems-B.P.Lathi copy

# O n t he o ther hand fig 810b which shows cos nk has

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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 iscrete-time 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 discrete-time 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.

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