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

# In practice t he s ampling interval t c annot be zero

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Unformatted text preview: soid cos Ok is periodic only if O/27r is a r ational number. Second, discrete-time sinusoids whose frequencies 0 differ by a n integral multiple of 27r a re identical. Consequently, a discrete-time sinusoid of a ny frequency 0 is identical t o some discrete-time 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 discrete-time 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 discrete-time sinusoid frequency is a t most 7r. T he highest r ate o f oscillation in a discrete-time 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 discrete-time sinusoids of different frequencies has a far reaching consequences in signal processing by discrete-time systems. O ne useful measure of t he size of a discrete-time 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 continuous-time sinusoid cos (wt + 0) a t uniform intervals of T s econds results in a discrete-time 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. Discrete-time signals classification is identical t o t hat of continuous-time signals, discussed in chapter l . A signal j [k] delayed by m t ime units (right-shifted) is given by f [k - m]. O n t he o ther hand, f [k] advanced (left-shifted) 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 continuous-time 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 ontinuous-time case, where time compression results in t he s ame d ata a t a higher speed, time compression in t he discrete-time case eliminates p art of t he d ata. Consequently, this operation is called decimation o r downsampling. T ime expansion operation of discrete-time 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.

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