Singularity Functions
1
Singularity functions
1-2-1 The unit-step function The continuous-time unit-step function
The continuous-time unit-step function is denoted as u ( t ) and is defined mathematically by: 0, u (t ) = 1, for t < 0 for t ≥ 0
which have the zero amplitude for all t < 0 and the amplitude of 1 for all t ≥ 0 , and its plot is shown in Figure 1-10
u (t ) 1
0
t
2
Fundamental of signal processing Figure 1-10: The continuous-time unit step function
The discrete-time unit-step function
The discrete-time unit-step function is denoted as u [ n ] , and is defined mathematically by: 0 u [ n] = 1 and its plot is shown in Figure 1-11. for n = −1, −2, −3, for n = 0,1, 2,3, 4,
u (n) 1
• • • • • −4 −3 −2 −1 0 1
2
3
n
Figure 1-11: The discrete-time unit step function
The amplitude scaling
If A , is an arbitrary nonzero real number, than Au ( t ) is step function with amplitude of A for all t ≥ 0 and zero for all t < 0 as 0, Au ( t ) = A, and its plot is shown in Figure 1-12. for t < 0 for t ≥ 0
SIGNALS
3
Au ( t ) A t
0
Figure 1-12: The continuous-time generic step function with amplitude of A .
The causality property of unit step function
The signal f ( t ) defined over time domain of −∞ ≤ t ≤ +∞ , starts at t = −∞ . If there is a desire that the signal be in causal form (starts at t = 0 ), it can be described as f ( t ) u ( t ) . The product f ( t ) u ( t ) of any signal f ( t ) is equal to f ( t ) for all t ≥ 0 and 0 for all t < 0 is given by:
0, f (t ) u (t ) = f (t ) ,
for t < 0 for t ≥ 0
Note that the signal f ( t ) exist over −∞ < t < ∞ and by multiplying the function f ( t ) by unit-step function u ( t ) , any nonzero value of f ( t ) in the time interval of −∞ < t < 0 will be forced to zero, and the signal will be turned on at t = 0 . The plot of f ( t ) u ( t ) is shown in Figure 1.13.
4
Fundamental of signal processing
f (t )
t
f ( t ) u (t )
t
Singularity functions
1-2-1 The unit-step function The continuous-time unit-step function
The continuous-time unit-step function is denoted as u ( t ) and is defined mathematically by: 0, u (t ) = 1, for t < 0 for t ≥ 0
which have the zero amplitude for all t < 0 and the amplitude of 1 for all t ≥ 0 , and its plot is shown in Figure 1-10
u (t ) 1
0
t
2
Fundamental of signal processing Figure 1-10: The continuous-time unit step function
The discrete-time unit-step function
The discrete-time unit-step function is denoted as u [ n ] , and is defined mathematically by: 0 u [ n] = 1 and its plot is shown in Figure 1-11. for n = −1, −2, −3, for n = 0,1, 2,3, 4,
u (n) 1
• • • • • −4 −3 −2 −1 0 1
2
3
n
Figure 1-11: The discrete-time unit step function
The amplitude scaling
If A , is an arbitrary nonzero real number, than Au ( t ) is step function with amplitude of A for all t ≥ 0 and zero for all t < 0 as 0, Au ( t ) = A, and its plot is shown in Figure 1-12. for t < 0 for t ≥ 0
SIGNALS
3
Au ( t ) A t
0
Figure 1-12: The continuous-time generic step function with amplitude of A .
The causality property of unit step function
The signal f ( t ) defined over time domain of −∞ ≤ t ≤ +∞ , starts at t = −∞ . If there is a desire that the signal be in causal form (starts at t = 0 ), it can be described as f ( t ) u ( t ) . The product f ( t ) u ( t ) of any signal f ( t ) is equal to f ( t ) for all t ≥ 0 and 0 for all t < 0 is given by:
0, f (t ) u (t ) = f (t ) ,
for t < 0 for t ≥ 0
Note that the signal f ( t ) exist over −∞ < t < ∞ and by multiplying the function f ( t ) by unit-step function u ( t ) , any nonzero value of f ( t ) in the time interval of −∞ < t < 0 will be forced to zero, and the signal will be turned on at t = 0 . The plot of f ( t ) u ( t ) is shown in Figure 1.13.
4
Fundamental of signal processing
f (t )
t
f ( t ) u (t )
t