Convolution - Signal Processing of Random Physiological Signals - page 91

a

0 t

-u ( a - t )

FIGURE 10.12 : Combining the operations of folding and translating about both axes. The step
function folded about both axes and displaced by + a gives the unit step function expressed as

− u ( a − t )

Combining the operations of folding about the y -axis, then folding about the x -
axis, and translating the step function by a positive displacement of + a gives the results

shown in Fig. 10.12. The unit step function is then expressed as − u ( a − t ).

10.1.1 Real Translation Theorem
One can summarize the operation of translation along an axis by a well-known theorem
which states that any function f ( t ), delayed in its beginning to some time t = a , can be

represented as a time-shifted function as in (10.9).

f ( t − a ) μ ( t − a ) (10.9)

1

, − t ≥ a

where μ ( t − a ) =

a
Proof of the time shift theorem is based in LaPlace Transform of the two functions,
which is often referred to as the Real Translation Theorem or simply the Shifting Theorem
as shown in the resulting (10.10).

0 , − t <

e − as
s

L [ μ

( t

− a )] = (10.10)

where, e − as

.
Let us return to the visualization of convolution. The function f (

shifts the transform from t = 0to t =∞

) should not be
a problem since it is similar to the function f ( t ), except for the change of independent
variable from units of time t to units of delay

τ

τ . It is important to keep in mind that

once the variable is changed from time t to delay τ that the unit of delay τ becomes the

independent variable and that time t in the function f ( t − τ ) is the same as displacement

a in the function f ( a − t ).