Impulse sampler

A real sampler, as shown in Fig. 18.1, is closed for a finite period of time. This time of closure is usually small compared with the sampling period. Therefore the real sampler can be closely approximated by an impulse sampler. An impulse sampler is a device that converts a continuous input signal into a sequence of impulses or delta functions. Remember, these are impulses, not pulses. The height of each of these impulses is infinite. The width of each is zero. The area of the impulse or the strength of the impulse is equal to the magnitude of the input function at the sampling instant. 

The impulse sampler is, of course, a mathematical fiction an impulse sampler is not physically realizable. But the behavior of a real sampler and hold circuit is practically identical to that of the idealized impulse sampler and hold circuit. The impulse sampler is used in the analysis of sampled-data systems and in the design of sampled-data controllers because it greatly simplifies these calculations. 

Thus the sequence of impulses/ ) that comes out of an impulse sampler can be expressed ... 

Equation (18.4) expresses the sequence of impulses that comes out of an impulse sampler in the time domain. Equation (18.5) gives the sequence in the Laplace domain. Substituting io) for s gives the impulse sequence in the frequency domain. 

Suppose we sample this/, with an impulse sampler. The sequence of impulses/ coming out of the impulse sampler will be, according to Eq. (18.11),... 

Sequences of impulses, such as the output of an impulse sampler, can be z-transformed. For a specified sampling period 7, the z transformation of an impulse-sampled signal/,, is defined by the equation... 

Passing the step function through an impulse sampler gives/(f, = where... 

If/( is a unit impulse, putting it through an impulse sampler should give an /, , that is still just a unit impulse 5( . But Eq. (18.4) says that... 

Consider an arbitrary functionThe original function), ) before the time delay is assumed to be zero for time less than zero. Running the delayed function through an impulse sampler and z-transforming give... 

A sequence of impulses m,, comes out of the impulse sampler on the input of the process. Each of these impulses will produce a response from the process. 

Consider the sampled-data system shown in Fig. 18.11a in the Laplace domain. The input enters through an impulse sampler. The continuous output of the process is... 

When goes through the impulse sampler it becomes which can be... 

Thus the overall transfer function of the process can be expressed as a product of the two individual pulse transfer functions if there is an impulse sampler between the elements. 

With an impulse sampler between the elements, the overall system transfer function is, from Eq. (18.71),... 

Suppose we add the fictitious deadtime element and impulse sampler sketched with dashed lines in Fig. 18.13. Then by letting the dead time vary continuously between 0 and 1, we could obtain a description of x, at any time in between the sampling periods. 

Figure 27.4 From the response of a real sampler to the response of an ideal impulse sampler.

From the last equation we find an expression for the sequence of impulses coming out of the impulse sampler in the Laplace domain (i.e., the 5-domain) ... 

Impulse sampler, ideal (see Sampler, ideal impulse)... 

We can extend eq. (27.1) to apply for any time. Thus the sequence of impulses y t) coming out of an impulse sampler is expressed by the following equation in the time domain ... 

Define the ideal impulse sampler. How does it differ from a real sampler Draw two sketches indicating the outputs of an ideal impulse and a real sampler. 

Develop two mathematical expressions describing the output of an ideal impulse sampler one in the time domain and the other in the Laplace domain. 

In Section 27.1 we developed the Laplace transform of a sequence of impulses produced by an ideal impulse sampler. Thus if y(t) is the continuous signal entering the sampler and y (i) is the sequence of impulses produced by the sampler, the Laplace transform of y (t) is given by eq. (27.2a) ... 

Passing the step function through an impulse sampler gives = Kun(t)I t)-, where /(,) is the sequenee of unit impulses defined in Eq. (14.3). Using the definition of z transformation [Eq. (14.22)J gives... 

We are now ready to use the concepts of impulse-sampled functions, pulse transfer functions, and holds to study the dynamics of sampled-data systems. Consider the sampled-data system shown in Fig. 14.9u in the Laplace domain. The input enters through an impulse sampler. The continuous output of the process T(5) is... 

When goes through the impulse sampler it becomes Y y which can be expressed [see Eq. (14.69)] as... 

---