Unformatted text preview: r). T he a rea under this product is C(t2) = A2,
giving us another point on the curve crt) a t t = t2 (Figure 2.7i). This procedure can
be repeated for all values of t, from - 00 t o 0 0. T he r esult will be a curve describing
crt) for all time t. Note t hat when t ::; - 3, f (r) and g(t - r) do not overlap (see
Fig. 2.7h); therefore, crt) = 0 for t ::; - 3.
Summary of the Graphical Procedure (c) T he p rocedure for graphical convolution can be summarized as follows: (d) 1. Keep t he function f (r) fixed.
(e) : --- t) _ _ / (r;) g ( I-·t) r 1 =11 > 0 2. Visualize the function g(r) as a rigid wire frame, and rotate (or invert) this
frame about t he vertical axis (r = 0) to obtain g( - r).
3. Shift t he inverted frame along the r axis by to seconds. T he shifted frame now
represents 9 (to - r).
4. T he a rea under the product of f (r) a nd g(to - r) ( the shifted frame) is c(to),
t he value of t he convolution a t t = to·
5. R epeat this procedure, shifting t he frame by different values (positive and
negative) t o o btain crt) for all values of t. ( f) 0 2 2+11 r;- ..... t2"": V g ( I-r;)
1 = 12 / (r;) . L. 1 <0 (g) ~. 0 2 +12 2 r;- 1,-- /(r;) (h) -1 2 +1, r;- 0 c ( I) (i)
I, -3 12 0 II 1- F ig. 2 .7 Graphical explanation of the convolution operation. Convolution: its bark is worse than its bite!
T he graphical procedure discussed here appears very complicated and discouraging a t first reading. Indeed, some people claim t hat convolution has driven many
electrical engineering undergraduates t o c ontemplate theology either for salvation
or as a n a lternative career ( I EEE Spectrum, March 1991, p.60). Actually, t he b ark 130 2 T ime-Domain A nalysis of Continuous-Time S ystems 2.4 o f c onvolution is worse t han i ts bite! I n g raphical convolution, we need t o d etermine t he a rea u nder t he p roduct f (r)g(t - r) for all values of t from - 00 t o 0 0.
However, a m athematical d escription o f f (r)g(t - r) is generally valid over a r ange
o f t . T herefore, r epeating t he p rocedure for every value o f t a mounts t o r epeating
i t o nly a few t imes for different ranges o f t .
W e c an also use t he c ommutative p roperty o f convolution t o o ur a dvantage
by c omputing f (t) * g (t) or g (t) * f (t), w hichever is simpler. As a rule of t humb,
c onvolution c omputations a re simplified i f we choose t o i nvert t he s impler o f t he
t wo functions. F or e xample, if t he m athematical d escription o f g (t) is simpler t han
t hat o f f (t), t hen f (t) * g (t) will b e e asier t o c ompute t han g (t) * f (t). I n c ontrast,
i f t he m athematical d escription o f f (t) is s impler, t he reverse will b e t rue.
W e shall d emonstrate g raphical convolution w ith t he following examples. L et
us s tart by r eworking E xample 2.4 using t his g raphical method. h (t) ! (t) t(a) o t(b) (c) • E xample 2 .6
Determine graphically y(t) = f (t) * h(t) for f (t) = e -tu(t) and h(t) = e - 2t u(t).
Figures 2.8a a.nd 2.8b show f (t) and h(t) respectively, and Fig. 2.8c shows f (r) and
h( - r) as functions of r . The function h( t - r) is now o...
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