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Unformatted text preview: n between t he convolution a nd t he correlation of f (t) a nd g(t) [given by Eq. (3.30)]. Note t hat g( 7 - t) is t he g(7) time-shifted
by t . T hus, 1/J f g(t) is equal t o t he a rea under t he p roduct of t he f pulse a nd 9 pulse
time-shifted b y t ( without time-inversion). In convolution also we follow the same
procedure, e xcept t hat t he 9 pulse is t ime-inverted before it is shifted by t . T his
observation suggests t hat 1/Jfg(t) is equal t o f (t) * g (-t) [the convolution of j (t)
w ith time-inverted g(t)l, t hat is, 1/Jfg(t) = f (t) * g (-t) (3.31) This can be formally proved as follows. Letting g( - t) = w (t), f (t) * g( - t) = f (t) * w(t) = [ : f(7)W(t - 7) d7 = [ : f (7)g(7 - t) d7 = 1/Jfg(t) To reiterate, 1/Jfg(t) is equal to t he a rea under t he p roduct of t he f pulse a nd 9
pulse time-shifted by t ( without time-inversion), a nd is given by t he convolution of
f (t) w ithg(-t).
E xercise E 3.4
Show t hat tPfg(t), t he c orrelation function of J (t) a nd get) in Fig. 2.11 is given b y c (t) in
Fig. 2.12. \ l
. F ig. 3 .6 3 .3 Signal representation by Orthogonal Signal S et In this section we show a way of representing a signal as a sum of orthogonal
signals. Here again we c an benefit from t he insight gained from a similar problem in
vectors. We know t hat a vector can be represented as a sum of orthogonal vectors,
which form t he c oordinate system of a vector space. T he problem in signals is
analogous, a nd t he results for signals are parallel to those for vectors. So, let us
review t he case of vector representation. 3.3-1 Orthogonal Vector Space Let us investigate a three-dimensional Cartesian vector space described by three
mutually orthogonal vectors X l, X 2, a nd X 3, as illustrated in Fig. 3.6. First, we
shall seek t o a pproximate a three-dimensional vector f in t erms of two mutually
orthogonal vectors X l a nd X 2: f':, E xercise E 3.5
Show t hat t Pfg(t), t he c orrelation function of J (t) a nd get) in Fig. 2.12 is given b y c(t) i n
F ig.2.11. \ l Representation of a vector in three-dimensional space. f::e C IXI + C2X2 T he e rror e in this approximation is f':, or
f= Autocorrelation Function C orrelation of a signal with itself is called t he a utocorrelation. T he autocorrelation function 1/Jf(t) of a signal f (t) is defined as
In C hapter 4, we shall show t hat t he a utocorrelation function provides valuable
spectral information about t he signal. q XI + C2X2 + e As in t he earlier geometrical argument, we see from Fig 3.6 t hat t he l ength of e is
minimum when e is p erpendicular to t he X I-X2 plane, a nd q XI a nd C 2X2 a re the
projections (components) of f o n X l a nd X 2, respectively. Therefore, t he c onstants
C I a nd C2 are given by formula (3.6). Observe t hat t he e rror vector is o rthogonal t o
b oth t he vectors X l a nd X 2.
Now, let us determine t he ' best' a pproximation t o f in terms of all three mutually orthogonal vectors X l, X 2, a nd X 3:
(3.33) 3 Signal Representation by Orthogonal Sets 184 Figure 3.6 shows t hat a unique choice of C l...
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