Tài liệu Signals And Systems P2 pdf

Signals and Systems with MATLAB Applications, Second Edition 1-19

Orchard Publications

Summary

int(d) % Integrate the delta function

ans =

Heaviside(t-a)*k

1.8 Summary

• The unit step function that is defined as

• The unit step function offers a convenient method of describing the sudden application of a volt￾age or current source.

• The unit ramp function, denoted as , is defined as

• The unit impulse or delta function, denoted as , is the derivative of the unit step . It is also

defined as

and

• The sampling property of the delta function states that

or, when ,

• The sifting property of the delta function states that

• The sampling property of the doublet function states that

u0( )t

u0( )t

0 t0 <

⎩1 t0 >

⎨

⎧ =

u1( )t

u1( )t u0()τ τ d –∞

t

∫ =

δ( )t u0( )t

δ τ()τd –∞

t

∫ u0 = ( )t

δ( )t = 0 for all t 0 ≠

f( )δ t ( ) t a – = f a( )δ( )t

a 0 =

f( )δ t ( )t = f 0( )δ( )t

f t( )δ( ) t – α dt –∞

∞

∫ = f( ) α

δ'( )t

f( )δ t '( ) t a – = f a( )δ'( ) t a – – f '( )δ a ( ) t a –

Chapter 1 Elementary Signals

1-20 Signals and Systems with MATLAB Applications, Second Edition

Orchard Publications

1.9 Exercises

1. Evaluate the following functions:

a.

b.

c.

d.

e.

f.

2.

a. Express the voltage waveform shown in Figure 1.24, as a sum of unit step functions for

the time interval .

b. Using the result of part (a), compute the derivative of , and sketch its waveform.

Figure 1.24. Waveform for Exercise 2

sin t tδ π

6 – -- ⎝ ⎠ ⎛ ⎞

cos t 2tδ π

4 – -- ⎝ ⎠ ⎛ ⎞

t

2 δ t π

2 – -- ⎝ ⎠ ⎛ ⎞ cos

tan t 2tδ π

8 – -- ⎝ ⎠ ⎛ ⎞

t

2

e

–t

δ( ) t 2 – dt –∞

∞

∫

t

2 δ

1 t π

2 – -- ⎝ ⎠ ⎛ ⎞ sin

v t( )

0t7 < < s

v t( )

−10

−20

10

20

1 23 4 5 6 7

0

v t( )

t s( )

e

–2t

v t( ) ( ) V