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By Robert Gray

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This example is different from the spinning wheel in that the sample space is discrete instead of continuous and that the probabilities of events are defined by sums instead of integrals, as one should expect when doing discrete math. 9) hold in this case as well (since sums behave like integrals), which in turn implies that the simple properties (a)–(d) also hold. A Single Coin Flip as Signal Processing The coin flip example can also be derived in a very different way that provides our first example of signal processing.

For example, the infinite two-sided space for a given A is At = { all waveforms {x(t); t ∈ (−∞, ∞)}; x(t) ∈ A, all t}, t∈ with a similar definition for one-sided spaces and for time functions on a finite time interval. Note that we indexed sequences (discrete time signals) using subscripts, as in xn , and we indexed waveforms (continuous time signals) using parentheses, as in x(t). In fact, the notations are interchangeable; we could denote waveforms as {x(t); t ∈ } or as {xt ; t ∈ }. The notation using subscripts for sequences and parentheses for waveforms is the most common, and we will usually stick to it.

1, 0, 1, . }; ai ∈ Ai } , i∈Z and the one-sided space Ai = { all sequences {ai ; i = 0, 1, . }; ai ∈ Ai } . i∈Z+ ∞ ∞ These two spaces are also denoted by i=−∞ Ai or ×∞ i=−∞ Ai and i=0 Ai or ×∞ A , respectively. i i=0 The two spaces under discussion are often called sequence spaces. Even if the original space A is discrete, the sequence space constructed from A will be continuous. For example, suppose that Ai = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} for all integers i. Then ×∞ i=0 Ai is the space of all semiinfinite (one-sided) decimal sequences, which is the same as the space of all real numbers in the CHAPTER 2.