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For any result o r relationship between j (t) a nd F (w), t here exists a dual result or
relationship, obtained by interchanging t he roles of j (t) a nd F (w) in t he original
result (along with some minor modifications arising because of t he factor 27r a nd
a sign change). For example, t he timeshifting property, to be proved later, s tates
t hat i f j (t) { =? F (w), t hen 27r8(w) 7 F ig. 4.16 A near symmetry between the direct and the inverse Fourier transforms. a >O + j w)n+1 1 8 (t) 1
i1t F(co ) e jrot dro a >O 27r8(w  wa)
9 cos wat 7r[8(w  wa) + 8(w + wa)l 10 sin wat j7r[8(w 11 u (t) 7r8(w)+i;; 12 sgn t .1 13 c oswatu(t) I[8(w  wo) 14 s inwatu(t) f;[8(w  wo)  8(w 15 e  at s in wat u (t) Wo a >O 16 e  at cos wat u (t) a +iw a>O 17 rect 18 * sinc(Wt) 19 ~(*) + wa)  8(w  wa)l jw + 8(w + wo)l + wl~w2
+ wo)l + wr.?w2 Symmetry o f Direct and Inverse Transform Operations:
TimeFrequency Duality j (t  to) { =? F (w)e jwto (4.30a) T he d ual of this property (the frequencyshifting property) states t hat (*)
r eet t Of t he two differences, t he former c an b e e liminated b y c hange of variable from w t o F (in hertz).
In this case
w = 27rF
a nd Ui'v) Therefore, t he d irect a nd t he inverse t ransforms a re given by 20 L
n =oo a nd 00 00 21 8 (tnT) Wo L
n =oo 8(w  nwo) 2
w a= T" T his leaves only one significant difference, t hat o f s ign change in t he e xponential index. Otherwise
t he t wo operations a re s ymmetrical. 254 4 C ontinuousTime Signal Analysis: T he Fourier Transform 4.3 Some P roperties o f t he Fourier Transform
f (t) f (t)e iwot F(w  wo) { => F (oo) 1 (4.30b) Observe t he r ole reversal of t ime a nd frequency in these two equations (with t he
minor difference o f t he sign change in t he e xponential index). T he value o f t his
principle lies i n t he fact t hat whenever we derive any result, we can be sure that
i t h as a dual. T his possibility c an give valuable insights a bout m any unsuspected
properties o r r esults in signal processing.
T he p roperties o f t he F ourier transform are useful not only in deriving t he d irect
a nd inverse t ransforms o f m any functions, b ut also in obtaining several valuable
results in s ignal processing. T he r eader should n ot fail t o observe t he e verpresent
duality in this discussion. We begin w ith t he s ymmetry property, which is o ne o f
t he c onsequences o f t he d uality principle discussed. 4 .32 0 t T  ' t 2 0 0 Ca) f (t) 4n Symmetry Property 00 T Cb) T his p roperty s tates t hat if f (t) { => F ig. 4 .17 F(w) t hen F(t) { => 27rf(w) (4.31) Proof: According t o Eq. (4.8b) 1 00 f (t) =  1 27r Hence 2 7rf(t) = I: Symmetry property of the Fourier transform. (with the minor adjustment of the factor 211"). This result appears a s pair 18 in Table 4.1
(with 7 /2 == w ).
As an interesting exercise, the reader should generate the dual of every pair in Table
4.1 by applying the symmetry property.
• t:. E xercise E 4.5...
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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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