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, for 0

1, for 0

tu t

t

<= ≥

which have the zero amplitude for all 0t < and the amplitude of 1 for all 0t ≥ , and its plot is shown in Figure 1-10

t

( )u t

1

0

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 for 1, 2, 3,

1 for 0,1, 2,3, 4,

nu n

n

= − − −= =

and its plot is shown in Figure 1-11.

n

( )u n

1•••••

1 2 31−2−3−4− 0

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 0t ≥ and zero for all 0t < as

( )0, for 0

, for 0

tAu t

A t

<= ≥

and its plot is shown in Figure 1-12.

SIGNALS 3

t

( )Au t

A

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 0t = ), 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 0t ≥ and 0 for all

0t < is given by:

( ) ( )( )

0, for 0

, for 0

tf t u t

f t t

<= ≥

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

0t−∞ < < will be forced to zero, and the signal will be turned on at 0t = . The plot of ( ) ( )f t u t is shown in Figure 1.13.

4 Fundamental of signal processing

( )f t

t

t

( ) ( )f t u t

SIGNALS 5

Figure 1-13:

The anti-causality property of unit step function

( )f t

t

t

( ) ( )f t u t−

0

0

6 Fundamental of signal processing

The non-causal signal

The time-shifting operation

The time shift to the right with 0t of unit step function sets a signal to “turn on” at time 0t rather than 0t = .

( ) 00

0

1, for0, for

t tu t t

t t≥

− = <

t

( )0u t t−

1

0t

Figure 1-14: The continuous-time unit step function time shifted to the right by 0t

The time shift to the left with 0t of unit step function sets a signal to “turn on” earlier than at time 0t = at the time 0t− .

( ) 00

0

1, for0, for

t tu t t

t t≥ −

+ = < −

SIGNALS 7

t

( )0u t t+

1

0t− 0

Figure 1-15: The continuous-time unit step function time shifted to the left by 0t

The time reversal

( ) ( ) 00 0

0

1, for( )

0, fort t

u t t u t tt t≤

− = − − = >

t

( )0u t t+

1

0t

Figure 1-16: The continuous-time unit step function time shifted to the right by 0t and time reflected

EXAMPLE 1:

Sketch the generic step function ( )0Au t t− .

SOLUTION:

8 Fundamental of signal processing

The unit step function ( )u t is scaled by scalar factor of A and time shifted by 0t to the right

( ) 00

0

0 t tAu t t

A t t

<− = ≥

t

( )u t

A

0t

Figure 1-17: The continuous-time generic step function.

The symmetrical unit rectangular pulse

The symmetrical unit rectangular pulse can be constructed by two unit step function ( )u t shifted 1 2 to the left and 1 2 to the right.

( ) 1 12 2

t u t u t = + − −

∏

Also alternative presentation is

( )112

0 otherwise

tt

≤=

∏

SIGNALS 9

t

( )t∏

1

12

12

−

Figure 1-17: The continuous-time unit pulse function

Note that the symmetrical unit rectangular pulse is an even function.

EXAMPLE 2:

Write an analytical expression to describe the waveform shown in Figure 1.8.

t

( )f t

2

53

Figure 1-18: The continuous-time pulse function time shifted to the right

SOLUTION:

10 Fundamental of signal processing

t

( )2 3u t −

2

53

t3

( )2 5u t− −

t5+ =

( )f t

35

2

2−

2

t

( )f t

2−

Figure 1-19: The continuous-time pulse function time shifted to the right

( ) ( ) ( )2 3 5f t u t u t= − − −

The signum or sign function

( )1 0

sgn 0 0

1 0

t

t t

t

>= =− <

( ) ( ) ( )sgn t u t u t= − −

SIGNALS 11

t

( )u t

1

1−

Figure 1-20: The continuous-time pulse function time shifted to the right

The unit ramp function

The continuous-time unit ramp function denoted as ( )r t is the integral of the unit step function

( ) ( )t

r t u dτ τ−∞

= ∫

and can be defined as:

( ) ( )0 0

0

tr t tu t

t t

<= = ≥

Alternatively, note that the step function is the derivative of unit ramp function

( ) ( )r tu t

t∂

=∂

12 Fundamental of signal processing

t

( )r t

1 2

12

Figure 1-21: The continuous-time ramp function

The time –shifted to the right unit ramp function

t

( )0r t t−

0t

Figure 1-22: The continuous-time ramp function time shifted to the right by 0t

The time-shifted ramp function having slope of m is denoted by ( )0r mt t− , and mathematically defined by:

SIGNALS 13

( )0

0

000

tmt t for tmr mt ttfor tm

− ≥− = <

and it is plotted in Figure 1-23.

t

( )0r mt t−

0t−

0tm

Figure 1-23: Ramp function with slope of m and time-shifted by 0t .

0

0

0

0

0

y mt tt y t

ty tm

= −= = −

= =

14 Fundamental of signal processing

t

( )S t

2

3

Figure 1-24: The continuous-time saw tooth function

The signal between the interval of 0 2t≤ ≤ is a line between two points

( )1 1 10, 0P t s= = and ( )2 2 22, 3P t s= = given by equation:

( ) ( ) ( )2 11

2 1

3 0 30 for 0 22 0 2

s sS t t t t t tt t− −

= − = − = ≤ ≤− −

( ) 0 otherwiseS t =

( ) ( ) ( )3 22

S t t u t u t= − −

t

( )tΛ

1−

1

1

SIGNALS 15

Figure 1-25: The continuous-time unit triangle function

( )1 for 1 1

0 for 1t t

tt

− − < <Λ = >

EXAMPLE:

t

( )f t

1−

2

1

1P

2P

3P

Figure 1-26:

( )1 1 1, 0P x y= − =

( )2 2 20, 2P x y= =

( )2 11 1

2 1

y yy y x xx x−

− = −−

16 Fundamental of signal processing

( ) ( )2 00 ( 1)0 1

y x−− = − −

− −

( )2 1y x= +

( ) ( )2 1f t t= +

( ) ( ) ( ) ( ) ( )2 1 1 4 2 1 1t u t tu t t u t+ + − + − −

t

( )f t

5

6

97

Figure 1-27:

( ) ( ) ( ) ( ) ( ) ( )6 6 65 5 2 7 7 9 92 2 2t u t t u t t u t − − − − − + − −

SIGNALS 17

The unit Impulse function (Dirac distribution or delta function)

The analytical expression for the unit impulse function is denoted as ( )tδ . Where

( )1 0

0 0

tt

tδ

== ≠

( )( ) 0

0 0

t dt tt

t

δδ

∞

−∞

== ≠

∫

( )unbounded 0

0 0

tt

tδ

== ≠

The impulse function is abstraction of the pulse with an infinitely large amplitude and infinitesimally small pulse width. The unit impulse can be visualized as a pulse with amplitude of 1 ε and width of ε , or as a triangle

( ) 1t dtδ∞

−∞=∫

The unit impulse function ( )tδ is not bounded at 0t =

t

( )tδ

1ε

2ε

2ε

−t

0ε →

1ε

εε− t0ε →

18 Fundamental of signal processing

Figure 1.30: The unit impulse function

( ) ( )0, for 01, for 0

t tu t d

tδ τ τ

−∞

<= = ≥∫

( ) utt

δ∂

=∂

Sampling or sifting properties of a function by an unit impulse function

( ) ( ) ( ) ( ) ( )0 0x t t dt x t dt xδ δ∞ ∞

−∞ −∞= =∫ ∫

( ) ( ) ( )0 0x t t t dt x tδ∞

−∞− =∫

The unit sample sequence (unit impulse sequence) The analytical expression for the unit impulse function is denoted as ( )nδ . Where

( )1, 0

0, 0

nn

nδ

== ≠

SIGNALS 19

n

( )nδ

Figure 1.31: The unit impulse function

properties of the unit impulse function

The sampling (sifting) property

( ) ( ) ( )0 0f t t t dt f tδ+∞

−∞− =∫

( ) ( ) ( ) ( ) ( ) ( ) ( )0 0

0 00 0 0 0 0

t t

t tf t t t dt f t t t dt f t t t dt f tδ δ δ+ +

− −

+∞

−∞− = − = − =∫ ∫ ∫

The time scaling property

( ) ( )1mt tm

δ δ=

( ) ( ) ( )1dmt dt dm mτ

δ δ τ δ τ τ+∞ +∞ +∞

−∞ −∞ −∞= =∫ ∫ ∫

20 Fundamental of signal processing

The real exponential function

The exponential signals are mathematically denoted

The sinusoidal signal

( ) ( )sinS t A t tω ϕ= + −∞ < < ∞

Where A is the amplitude or peak value,ω the angular frequency in radian per second ( )secrad , and ϕ , the phase in radian. The frequency f in Hertz ( )cycle second is

2f ω π= and 1f T= . The sinusoid is periodic with period of 2π ω .

( ) ( )sin 2S t A ft tπ ϕ= + −∞ < < ∞

( ) 2sin tS t A tTπ

ϕ = + −∞ < < ∞

t

( )sin A tω ϕ+

A

T

Sometimes the amplitude is considered as peak-to-peak value that is twice of peak value. The amplitude of a sine wave signal is given as of its root-mean-square (rms) value which is peak value divided by 2 .

SIGNALS 21

2peak

rms

VV =

For example peak value of home power supply in this country is 120 2 volts and rms value of 120 volts.

Frequency f is given in per second ( )1s or Hertz ( )Hz and period T is given in second

( )s .

The phase ϕ is with respect to an arbitrary time reference.

Sinusoidal signal can be

Signals can be represented as the sum of sinusoids. For example square wave signal

A

A−

t

( )f t

( ) 0 0 04 1 1sin sin 3 sin 5 ...

3 5Af t t t tω ω ωπ

= + + +

Where A is the amplitude of the square wave and 0ω is called the fundamental frequency 0

2Tπ

ω = , where T is period of the square wave.

22 Fundamental of signal processing

A

A−

t

( )f t