Convolution - Signal Processing of Random Physiological Signals - page 106

If a function is convolved with the derivative of a delta function, the result is the
derivative of the original function as shown by (10.28).

δ ( t ) z ( t )

z ( t ) = (10.28)

where the prime ( ) denotes d/dt . The first derivative of ( t ) is called a doublet .

The convolution of a step function with the derivative of a delta function will yield
the delta function (10.29).

δ

δ ( t )

μ ( t ) = δ ( t ) (10.29)

It should be noted that the delta function is zero at all other values of time except at the in-
stant of time, t , where the delta function exists. This relation is expressed mathematically
as δ ( t − t 0 ) = 0 for t = t 0 or as (10.30).

t 2

δ

( t

− t 0 ) ∂ t = 0 (10.30)

t 1

For t 1 t 2

Other properties of convolution are of interest because of their use in checking
validity of a particular computation. For example, in the convolution of two functions,
as in (10.31), the area of a convolution product is equal to the product of the areas of the
functions. This means that the area under f 3 ( t ) is equal to the area under the function,
f 1 ( t ), times the area under the function, f 2 ( t ).

< t 0 <

∞

f 3 ( t ) = f 1 ( τ ) f 2 ( t − τ ) d τ (10.31)

−∞

Keep in mind that the area is computed by integrating over the interval, −∞ < t < ∞ .

Proof of this property follows in (10.32) through (10.36).

⎡ ⎤

∞ ∞ ∞

⎣ ⎦ dt

f 3 ( t ) dt = f 1 ( τ ) f 2 ( t − τ ) d τ (10.32)

−∞ −∞ −∞

⎡
⎣

⎤
⎦ d

∞ ∞

[ Area under f 3 ( t )]

= τ − τ τ (10.33)

f 1 ( ) f 2 ( t ) dt

−∞ −∞