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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 .43 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 squarepulse 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 moothedout 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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 Spring '13
 Bayliss
 Signal Processing, The Land

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