Sample spaces

To model systems that yield uncertain or random measurements, we let ? denote the set of all possible distinct, indecomposable measurements that could be observed. The set ? is called the sample space. Here are some examples corresponding to the applications discussed at the beginning of the chapter.

Signal processing. In a radar system, the voltage of a noise waveform at time t can be viewed as possibly being any real number. The first step in modeling such a noise voltage is to consider the sample space consisting of all real numbers, i.e., ? = ( ? ?, ?).

Computer memories. Suppose we store an n-bit word consisting of all 0s at a particular location. When we read it back, we may not get all 0s. In fact, any n-bit word may be read out if the memory location is faulty. The set of all possible n-bit words can be modeled by the sample space

Optical communication systems. Since the output of a photodetector is a random number of photoelectrons. The logical sample space here is the nonnegative integers,

Notice that we include 0 to account for the possibility that no photoelectrons are observed.

Wireless communication systems. Noncoherent receivers measure the energy of the incoming waveform. Since energy is a nonnegative quantity, we model it with the sample space consisting of the nonnegative...