SIGNALS COMMONLY ENCOUNTERED IN SIGNAL PROCESSING APPLICATIONS

1. INTRODUCTION

In signal processing applications, we come across several types of signals, which can be defined mathematically. The most commonly encountered signals are:

·         Delta or impulse function

·         Step function

·         Ramp function

·         Parabolic and exponential functions

·         Periodic functions

These functions are usually applied as inputs to various systems, and we evaluate the performance of the systems based on these inputs.

2. THE DELTA (IMPULSE) FUNCTION

The ideal delta (impulse) function is defined as a function that has infinite amplitude and zero duration. Such a function can exist only in theory, but not in practice. This is because every practical signal is an energy function that requires a finite period (however small this may be) for its existence; it can not exist for a period of zero duration. Hence we conclude that the practical delta function is one that has an extremely high amplitude and existence of an extremely short duration of time.

2.1 DELTA FUNCTION IN CONTINUOUS TIME DOMAIN

Delta function is defined in the continuous time-domain (CTD) mode using the function

Equation (1) says that the area under the curve δ(t) integrated between the infinite limits is unity. As stated before, delta function exists only theoretically in its ideal form. In practice, there are several mathematical functions that can be approximated (within limits) to be the delta function.

            Consider Fig. 1, which shows a rectangular pulse of width τ (tau) and height 1/τ. Now, when τ → 0, 1/τ → ¥. The figure can thus be approximated as a delta function in the limits shown. In mathematical form, the delta function can be approximated in the form

2.2 DELTA FUNCTION IN DISCRETE TIME DOMAIN

The delta function, also called as Dirac’s delta function, named after its originator P. A. M. Dirac, is defined in discrete time-domain (DTD) mode as

                                                            δ( (n) = 1, n = 0

                                                                    = 0, elsewhere  (2)                                          

Figure 2 shows the representation of the unit delta function.  As shown in the figure, the unit delta function exists only at time n = 0, and has no value at other places in the graph.

Since the delta function has zero (or very short) duration, we call this as an impulse function also, since it acts like a sudden shock. It may be noted that a karate chop very nearly approximates a delta function. The impulse input function is usually used to test the ability of a given system to withstand sudden shocks of extremely large amplitudes, and which can occur at any instant of time in the system. It is quite common that the impulse may occur at any instant of time. The following examples illustrate this idea.

Example 1:   Plot the following impulse functions:

                        (a) δ(n -1)   

                        (b) δ(n +2)

Solution:

(a) To obtain the position of the delayed delta function, we write

                                   

                                    δ(n - 1) = δ(0) = 1, at n = 1

                                                             = 0, elsewhere                   (1)                            

       

From (1), we find that δ(n - 1) is a delta function shifted to the right by a one unit of time.

(b)  By a similar argument, we find that

                           δ (n + 2) = δ(0) = 1, at n = -2  (2)

Equation (2) reveals that δ(n + 2) is delta function shifted to the left by 2 units of time. The functions δ(n - 1) and δ(n + 2) are plotted, as shown in Fig. 3. 

  

3. STEP FUNCTION

We define unit step function in the continuous-time domain mode domain as

                                  u(t) = 1,  0 £ t £ ¥

                                                                 = 0,  elsewhere    (3)                                                                 

           

In the discrete-time domain mode, unit step function becomes

                                                          u(n) = 1,  0 £  n £ ¥

                                                                   =0, elsewhere  (4)                                                      

Figure 4 shows the unit step function in the continuous time-domain mode and Fig. 5 shows the unit step function in the discrete time-domain mode. It may be noted that:

      ·         In a step function, the transition of the waveforms from 0 to 1 occurs in zero time.

·         The step function can assume any desired amplitude. However, when the amplitude   is unity, we call it as unit step function.

4. UNIT RAMP FUNCTION

We define the unit ramp function in the continuous-time domain mode as:

                                 r(t) = t,     0 £ t £ ¥

                                                               = 0, elsewhere           (5)

           

and the unit ramp function in the discrete-time domain mode as:

                                    r(n) = n,  0 £ n £ ¥

                                                                   = 0,  elsewhere     (6)                                               

Figure 6 shows the continuous time-domain version of the unit ramp function, and Fig. 7 shows its discrete time-domain version. As shown in the figures, the ramp function is a waveform whose amplitude is proportional to time.

5. PARABOLIC FUNCTIONS

We define a parabolic waveform in continuous time-domain mode as:

                       

                                    p(t) = t², - ¥ £ t £ ¥

                                            = 0, elsewhere (7)

                                                         

In the discrete time-domain mode, we define it as

                                                             p(n) = n²,  - ¥ £ n £ ¥

                                                                            = 0,  elsewhere   (8)    

Figure 8 shows the parabolic waveform in the continuous time-domain mode. The corresponding discrete version is shown in Fig. 9.

We can see that the parabolic waveform can be obtained by integrating the ramp waveform (in the continuous time-domain mode).  In turn, the ramp waveform can be obtained by differentiating the parabolic waveform.

            Similarly, the ramp waveform (in the continuous time-domain mode) can be obtained by integrating the step waveform and in turn, the step waveform can be obtained by differentiating ramp waveform.

      Finally, we find that the delta function may be obtained by differentiating the step waveform and in turn, the step waveform can be obtained by integrating the delta function.

In general functions of the type f(t) = a^t, where a is a constant,  are known as exponential functions. Usually, for exponential functions, a is chosen as e, the base of natural logarithm.

6. PERIODIC FUNCTIONS

So far, we have discussed waveforms that are non-periodic. Now, let us discuss about a few typical periodic waveforms. Sinusoidal waveforms are considered to be the most fundamental of all the periodic waveforms. They are mathematically expressed as

                                                      y(t) = V_max sin ωt  (9)

in the continuous time-domain mode, where V_max = amplitude and ω = angular frequency. In the discrete time-domain mode, it will take the form 

                                                        y(n) = V_max sin nω  (10)

Other periodic waveforms in use are the square, triangular, and sweep waveforms. These waveforms can be derived from the waveforms we have already discussed. Hence, they will not be described here, now. Rather, they will be described as and when need arises.