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

For example jt cos nwot dt to2 ao 365a recall

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Unformatted text preview: g the Fundamental Frequency and Period We have seen t hat every periodic signal can be expressed as a s um of sinusoids of a fundamental frequency wo a nd its harmonics. O ne may ask w hether a sum of sinusoids of a ny frequencies represents a periodic signal. I f so, how does one determine t he p eriod? Consider t he following three functions: f I(t) = 2 + 7 cos (~t h (t) = 2 cos (2t h (t) = 3 sin + Ih) + 3 cos (~t + Ih) + 5 cos ( tt + ( 3) + BIl + 5 sin (71"t + ( 2) (3V2t+B) + 7 cos (6V2t + 4» Recall t hat every frequency in a periodic signal is an integral multiple of t he f undamental frequency woo Therefore, t he r atio of any two frequencies is of t he form m in where m a nd n are integers. This means t hat t he ratio of any two frequencies is a r ational n umber. W hen t he r atio of two frequencies is a r ational number, they are said to be h armonically related. T he l argest positive number of which all the frequencies are integral multiples is t he f undamental frequency. T he frequencies in t he s pectrum of f I (t) a re ~, ~, a nd t (we do n ot consider dc). T he r atios of t he su~cessive frequencies are ~ a~d ~, respectively. Because b oth t hese numbers are ratlOnal, all t he t hree frequencies in t he s pectrum are harmonically related a nd t he signal i t (t) is periodic. T he largest 3 .4-3 T he Role o f Amplitude and Phase Spectra in W ave Shaping T he trigonometric Fourier series of a signal f (t) shows explicitly t he sinusoidal components of f (t). We c an synthesize f (t) by adding the sinusoids in the s pectrum of f (t). To synthesize t he square-pulse periodic signal f (t) of Fig. 3.8a, we a dd successive harmonics in its s pectrum s tep by s tep a nd observe t he s imilarity of t he resulting signal t o f (t). T he Fourier series for this function as found in Example 3.4 is 1 f (t) = 2 2 + - (cos t 71" 1 1 1 - cos 3t + - cos 5t - - cos 7t 3 5 7 + ... ) We s tart t he synthesis with only t he first t erm in t he series (n = 0), a c onstant ~ (dc); this is a gross approximation o fthe s quare wave, as shown in Fig. 3 .lla. I n the next s tep we a dd t he dc (n = 0) a nd t he first harmonic (fundamental), which results in a signal shown in Fig. 3 .llb. Observe t hat t he synthesized signal somewhat resembles f (t). I t is a s moothed-out version of f (t). T he s harp corners in f (t) are not reproduced in this signal because s harp corners indicate rapid changes a nd t heir reproduction requires rapidly varying ( that is, higher frequency) components, which are excluded. Figure 3.11c shows t he s um of dc, first, a nd t hird harmonics (even harmonics are absent). As we increase t he n umber of harmonics progressively, as i llustrated in Figs. 3.11d (sum up t o t he fifth harmonic) and 3.11e (sum up t o *' t The l argest n umber o f which %;-, .. " ~ are integral multiples is t he r atio of t he G CF ( greatest common factor) of t he n umerators s et ( ai, a2, " ', a m) t o t he L CM ( least common m ultiple) o f t he d enominator s et ( bl' b2, " ', bm ). For instance, for t he...
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