Singularity functions are discontinuous functions. A singularity is a point at which a function does not possess a derivative. In other words, a singularity function is discontinuous at its singular points. Hence a function that is described by polynomial in t is thus a singularity function. The commonly used singularity functions are:
At first, we take up the study of a unit-step function.
The continuous-time unit-step function is defined as:
Equation (2.1) defines what a unit-step function. The value of a unit-step function is one, for values of t > 0, and it is zero, for values of t < 0. It is undefined at t = 0. The unit-step function has a value between 0 and 1, at t = 0. The value of the unit-step function changes suddenly, at t = 0. Because of the step change in unit-step function at t = 0, the value of derivative of unit-step function is infinite at t = 0. In other words, the unit-step function is discontinuous at t = 0. It can be seen that the derivative of unit-step function is zero at all instants, except t = 0. The Laplace transform of the unit-step function is expressed by equation (2.2). It can be seen that the unit-step function has a pole at origin. A pole of a function is defined as a value at which the value of the function is infinite. That is if s = 0, the value of the Laplace transform of the unit-step function is infinite.
The unit step function u(t) is represented as shown in Fig. 9. The unit step function is used widely in network theory and control theory. It can be seen that the unit step function has a discontinuity at t = 0 and is continuous for all other values of t . One way to remember the unit step function is to know how it relates to its argument. The argument in equation (2.1) is t . When the argument is positive, the unit step function is positive and has unit value. It is zero when its argument is negative. In Fig. 9, the region where t > 0 is shown by the area, with the gray shade. In Fig. 9, the region where t < 0 is shown by the area, without a shade. The pole-zero plot of the unit-step function is also shown in Fig. 9. As stated earlier, the Laplace transform function of the unit-step function has a single pole at the origin, and the pole is marked by a ‘×’ in Fig. 9.
Reflection operation on the independent variable of the unit-step function
On knowing how to interpret a unit-step function based on its argument, it is easy to visualize how u(-t) would be. This function, u(-t) is illustrated in Fig. 10.
The value of u(-t) is unity for values of t < 0. When t < 0, t has a negative value, and then - t is positive. Hence the argument of u(-t) is positive when t < 0, and u(-t) is equal to unity. On the other hand, - t is negative when t > 0, and u(-t) is equal to zero for positive values of t.
Another example using the unit step function is shown in Fig. 11. This function is called the signum function and it is written as sgn(t).
The signum function is defined by equation (2.3). It can also be expressed as shown by equations (2.4) and (2.5). As expressed by equation (2.5), this function has a value of unity for t > 0, since u(t) equals unity in this range. When t < 0, - t is positive and u-(t) equals unity in this range. The plot in Fig. 11 shows how the signum function appears as a function of time. The signum function is often not used in network theory, but it is used in communication and control theory.
Shifting operation on the independent variable of the unit-step function
A shifted time function is displayed in Fig. 12a.
The shifted function can be expressed as shown below:
In a shifted unit step function, defined by equation (2.6), the step change occurs at t = t, whereas the step change occurs at t = 0 for the unit step function defined by equation (2.1). It can be seen that the shifted unit step function is obtained by shifting the unit step function to the right by t seconds. It is seen from equation (2.6) that when t > t, the argument of the shifted unit step function is positive and then the function has unit value. When the argument of the shifted unit step function is negative, the function has zero value. It can be seen that the argument of the shifted unit step function is negative for t < t.
The shifted function, displayed in Fig. 12a, can be reflected and this function is displayed by the sketch in Fig. 12b. We can express the shifted-reflected unit-step function as follows.
When t < t, ( t - t ) is negative, and hence we get equation (2.6b). We can express equation (2.6b) in another way, as shown by equation (2.6c).
Synthesis of a signal
It is possible to use singularity functions to generate or synthesize different signals. An example is shown below to show how a rectangular pulse signal can be visualized as the combination of two step functions.
A rectangular pulse function of unit amplitude, illustrated in Fig. 13, is obtained as the combination of a unit step function and a shifted step function with a magnitude of - 1.
When the two signals shown in Fig. 14 are added, we get the rectangular pulse shown in Fig. 13. From Fig. 14, we get that
A system may receive a single rectangular pulse, as shown in Fig. 13. If this pulse repeats itself after a fixed period, then the resulting signal is a square-wave periodic signal.
The ramp function is illustrated in Fig. 15. It can be defined as follows:
Equation (2.8) defines the ramp function. The ramp function has zero value in the range defined by t < 0. When t > 0, the ramp function increases linearly with time. The Laplace transform expression for the ramp function is displayed by equation (2.9). This function has two poles at the origin in s-domain.
The unit-step function and the ramp function are related. We can define the unit-step function, as the derivative of the ramp function, as shown by equation (2.10). Alternatively, we can state that the ramp function is the integral of the unit-step function, as shown by equation (2.11).
Equation (2.9) presents the Laplace transform of the ramp function. The Laplace transform of the unit-step function defined by equation (2.1) can be obtained by multiplying the Laplace transform of the ramp function by s. Since the unit step function is the derivative of the ramp function, it can be seen that multiplication by s in s-domain amounts to differentiation in time-domain. Conversely, division of an expression by s in s-domain amounts to integration of the corresponding time-function. Since the ramp function is the integral of the unit-step function, we can divide the Laplace transform expression of the unit-step function, shown by equation (2.2) by s, and obtain the Laplace transform expression of the ramp function, shown by equation (2.9). From equation (2.9), it can be seen that the ramp function has double poles at origin. In this case, the multiplicity of poles at origin is two.
Unlike the unit step function, the ramp function is continuous for all t, including t = 0. However in s-domain, this function has singular points at the origin, as shown by equation (2.9).
The effect of operations on the independent variable of the ramp function is shown by the sketches in Fig. 16. The plots of the shifted ramp function and the reflected ramp function are displayed.
It is possible to shift the ramp function and then reflect it, as shown in Fig. 17. The ramp function is a signal generated by some electronic circuits. With additional electronic circuitry, it is possible to generate saw-tooth waveform displayed in Fig. 18. Such a signal is used in a cathode-ray oscilloscope as the timing signal. Such a signal is used in a TV also for horizontal and vertical scanning.
The unit impulse function, designated d(t), is also called the Delta dirac function. Its use in network theory, control theory and signal theory is widespread and it is important because of its properties and the insight it offers about the network to which it is applied.
The impulse function is displayed, as shown in Fig. 19. The impulse function is related to the unit-step function, as expressed by equation (2.12). It is the integral of the impulse function. Alternatively, the unit impulse function is defined as the derivative of the unit step function, as expressed by equation (2.13). Given that the Laplace transform of unit step function is 1/s and that differentiation in time-domain corresponds to multiplication by s in s-domain, the Laplace transform of the unit impulse function becomes 1, as expressed by equation (2.14).
Integration in time-domain corresponds to dividing by s in the s-domain. Since the unit step function is the integral of the unit impulse function, equation (2.2) can be obtained by dividing the expression in equation (2.14) by s. It can also be seen from equation (2.14) that the impulse function has no pole or zero and this lends importance to the response of a network to an impulse input. The poles of a network's forced response to any input other than an impulse signal reflect the poles of the excitation signal. On the other hand, when the excitation signal is the impulse function, then the impulse response exhibits only the poles of the network and thus the impulse response gives more information about the network.
Even though the unit impulse function is defined as the derivative of the unit-step function as shown by equation (2.13), it needs to be mentioned that equation (2.13) does not conform to the normal definition of a function, since u(t) is discontinuous at t = 0 and hence it is not strictly differentiable at t = 0. However, a function , that is an approximation of the unit step function, illustrated in Fig. 20, can be used to justify equation (2.13).
Function uD(t) rises from zero to one in a short time interval of length D. As D tends to zero, function uD(t) tends to be equal to u(t). In practice, the unit-step function generated is similar to uD(t) with a relatively short time interval of length D, but for analysis it is idealized and it is treated to be u(t). The derivative of uD(t) is displayed in Fig. 21.
From Fig. 21, it can be seen that the unit impulse function can be defined as follows. Firstly the unit step function is defined as:
Equation (2.15) states that as D tends to zero, function uD(t) tends to be equal to u(t). Equation (2.16) expresses dD(t) as the derivative of uD(t). Equation (2.17) states that as D tends to zero, function dD(t) becomes equal to d(t).
Using the concept of a shifted step function, function dD(t) can be expressed by equation (2.18). Function dD(t) is displayed as a pulse function in Fig. 22.
The impulse function can be defined as shown by equation (21.9). We obtain equation (2.20) from Fig. 21.
The impulse function can be approximated in some other ways too. The ideal impulse function is represented by a spike at the origin as shown in Fig. 19. It has the following properties.
Even though the ideal impulse function is unrealizable in practice, a pulse function as shown in Fig. 22 is often realizable, where D is very small compared with the time constant of the network to which the approximate impulse function is applied.
Sifting/Sampling Property
Given a continuous time-function f(t), sampling property is defined as:
This property is also known as the sifting property. Since d(t - a) = 0 when t ≠ a, the above integral can be expressed to be:
Sampling property shows how a function can be sampled at a given instant.
Since the impulse function has value only at t = a, the value of f(t) when t ≠ a is not important, we get equation (2.23). Equation (2.23) shows we call this property as the sifting property.
Time Scaling Property
Time-scaling property of an impulse function is explained now. Given that a > 0, the time-scaling property is expressed by equations (2.24a) and (2.24b). We can express that d(at) approximates to dD(t) as D tends to zero. The graphical representation of time-scaling is presented in Fig. 23. It can be seen that the area shrinks to (1/a) due to time-scaling, when a >1. When a < 1, the area expands. We can use the time-scaling property and the sifting property as shown below.
When a function is sampled by a time-scaled and shifted impulse function, as shown by equation (2.25a), we can obtain the expression on the right-hand side as follows. By the use of sifting property, we obtain equation (2.25b). Hence we obtain equation (2.25c).
The impulse response is significant since it reveals the nature of the system. The poles of impulse response are the poles of the system. We can use convolution integral to obtain the response of a system to any input. To apply the convolution integral, we make use of the impulse response. The topic of convolution integral is presented on another page.
It is possible to synthesize waveforms using singularity functions. An example has already been presented in Figs. 13 and 14, wherein a pulse function has been generated as the sum of step functions. Some more examples are presented below.
WORKED EXAMPLE 1
Express f(t) in Fig. 24a as the sum of singularity functions.
Solution:
It can be seen from Fig. 24a that
Here f1(t) is a ramp function and f2(t) is a shifted ramp function. We can express function f(t) as follows:
This example shows how addition of a ramp signal and a shifted ramp signal leads to a different function.
WORKED EXAMPLE 2
A saw tooth pulse is shown in Fig. 25a. Express it as the sum of singularity functions.
Solution:
It can be seen from Fig. 25a that function f(t) can be described by equation (2.30).
This saw tooth pulse function can be visualized to be the sum of three singularity functions, as shown in Fig. 25. Equation (2.31) expresses function f(t) as the sum of three singularity functions. When t < 1, we have a ramp function representing function f(t). At t = 1, value of function f(t) changes from 1 to 0. By adding a negative shifted unit-step function to the ramp function, we can bring the value of function f(t) changes from 1 to 0. For t > 1, value of function f(t) is zero. If we had only the sum the ramp function and the negative shifted unit-step function, the resulting function would continue to increase for t > 1. By adding a negative shifted unit-ramp function to the sum of the ramp function and the negative shifted unit-step function, we get the single saw-tooth pulse.
WORKED EXAMPLE 3
A triangular pulse is shown in Fig. 26a. Express it as the sum of singularity functions.
Solution:
It can be seen from Fig. 26a that function f(t) can be described by equation (2.32), when 0 < t < 1.
When 1 < t <2, we need to form an equation for the line with a negative slope. Equation (2.33) is an equation that represents a line. The unknown constants can be evaluated from the values the line has at t = 1 and t = 2. Then we get equation (2.34).
Equation (2.35) describes the triangular pulse. It is easy to see that the triangular pulse is part of the ramp function when t < 1. When t > 1, the slope of the triangular function is - 1. The ramp function, that starts at t = 0, has a slope of one. We can add a negative shifted ramp function starts at t = 1 to the ramp function, that starts at t = 0. Let the slope of the shifted ramp function starts at t = 1 be -2. Then the resultant function obtained by adding the two ramp functions has a slope of -1 for 1 < t <2. For t > 2, the triangular pulse has zero value. Hence we can add a shifted unit ramp function that starts at t = 2 to the other tow ramp functions. Equation (2.36) expresses the triangular pulse as the sum of three ramp functions.
In Fig. 26, a graphical representation of how to obtain the triangular pulse is presented.
WORKED EXAMPLE 4
A pulse is shown in Fig. 27. Express it as the sum of singularity functions.
SOLUTION:
The solution is shown in Fig. 28 and it is self-explanatory.
The equation that defines the pulse in Fig. 27 is presented below.
This page has introduced and explained the three singularity functions. It has been shown how pulses can be generated using these functions. Next we take up the topic of complex frequency.