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Unformatted text preview: soid cos Ok is periodic only if O/27r is
a r ational number. Second, discretetime sinusoids whose frequencies 0 differ by
a n integral multiple of 27r a re identical. Consequently, a discretetime sinusoid of
a ny frequency 0 is identical t o some discretetime sinusoid whose frequency lies in
t he interval  7r t o 7r (called t he f undamental frequency range). Further, because
c os (  Ok + {I) = cos (Ok  0), a sinusoid of a frequency in t he range from  7r t o 0 can
b e expressed as a sinusoid of frequency in the range 0 t o 7r. T hus, a discretetime
sinusoid of any frequency can be expressed as a sinusoid of frequency in t he range 0
t o 7r. T hus, in practice, a discretetime sinusoid frequency is a t most 7r. T he highest
r ate o f oscillation in a discretetime sinusoid occurs when its frequency is 7r. In a
given time, a sinusoid of frequency other t han 7r will have a fewer number of cycles
( or oscillations) t han t he sinusoid o f frequency 7r. T his peculiarity of nonuniqueness
o f waveforms in discretetime sinusoids of different frequencies has a far reaching
consequences in signal processing by discretetime systems.
O ne useful measure of t he size of a discretetime signal is i ts energy defined by
t he s um L:k If[k]i2, if it is finite. I f t he signal energy is infinite, t he p roper measure
i s its power, if it exists. T he signal power is t he t ime average of its energy (averaged
o ver t he entire time interval from k =  00 t o 00). For periodic signals, t he time
averaging need be performed only over one period in view o f t he periodic repetition
o f t he signal. Signal power is also equal to t he m ean squared value o f t he signal
(averaged over t he entire time interval from k =  00 t o 00).
Sampling a continuoustime sinusoid cos (wt + 0) a t uniform intervals of T
s econds results in a discretetime sinusoid cos (Ok +0), where 0 = wT. A continuous 569 Problems time sinusoid of frequency F Hz must be sampled a t a r ate no less t han 2 F Hz.
Otherwise, t he r esulting sinusoid is aliased; t hat is, it appears as a sampled version
of a sinusoid of lower frequency.
Discretetime signals classification is identical t o t hat of continuoustime signals, discussed in chapter l .
A signal j [k] delayed by m t ime units (rightshifted) is given by f [k  m]. O n
t he o ther hand, f [k] advanced (leftshifted) by m t ime units is given by f [k +m]. A
signal f[k], when time inverted, is given by f [k]. T hese operations are t he s ame as
those for t he continuoustime case. T he case of time scaling, however, is somewhat
different because of t he discrete n ature o f variable k. Unlike t he c ontinuoustime
case, where time compression results in t he s ame d ata a t a higher speed, time
compression in t he discretetime case eliminates p art of t he d ata. Consequently,
this operation is called decimation o r downsampling. T ime expansion operation of
discretetime signals results in t ime e xpanding t he signal, t hus c...
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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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