2.3 Discrete-time signals

So far, we have treated what are known as analog signals and systems. Mathematically, analog signals are functions having continuous quantities as theirindependent variables, such as space and time. Discrete-time signals are functions defined on the integers; they are sequences. One ofthe fundamental results of signal theory will detail conditions under which an analog signal can be converted into a discrete-time one andretrieved without error . This result is important because discrete-time signals can be manipulated bysystems instantiated as computer programs. Subsequent modules describe how virtually all analog signal processing can beperformed with software.

As important as such results are, discrete-time signals are more general, encompassing signals derived fromanalog ones and signals that aren't. For example, the characters forming a text file form a sequence,which is also a discrete-time signal. We must deal with such symbolic valued signals and systems as well.

As with analog signals, we seek ways of decomposing real-valueddiscrete-time signals into simpler components. With this approach leading to a better understanding of signal structure,we can exploit that structure to represent information (create ways of representing information with signals) and to extractinformation (retrieve the information thus represented). For symbolic-valued signals, the approach is different: We develop acommon representation of all symbolic-valued signals so that we can embody the information they contain in a unified way. Froman information representation perspective, the most important issue becomes, for both real-valued and symbolic-valued signals,efficiency; What is the most parsimonious and compact way to represent information so that it can be extracted later.

Real- and complex-valued signals

A discrete-time signal is represented symbolically as $s(n)$ , where $n=\{\dots , -1, 0, 1, \dots \}$ . We usually draw discrete-time signals as stem plots toemphasize the fact they are functions defined only on the integers. We can delay a discrete-time signal by an integerjust as with analog ones. A delayed unit sample has the expression $\delta (n-m)$ , and equals one when $n=m$ .

Discrete-time cosine signal

Complex exponentials

The most important signal is, of course, the complex exponential sequence .

Sinusoids

Discrete-time sinusoids have the obvious form $s(n)=A\cos (2\pi fn+\phi )$ . As opposed to analog complex exponentials and sinusoids thatcan have their frequencies be any real value, frequencies of their discrete-time counterparts yield unique waveforms only when $f$ lies in the interval $\left(-\left(\frac{1}{2}\right) , \frac{1}{2}\right]$ . This property can be easily understood by noting that addingan integer to the frequency of the discrete-time complex exponential has no effect on the signal's value.

Unit sample

The second-most important discrete-time signal is the unit sample , which is defined to be

Unit sample

Examination of a discrete-time signal's plot, like that of the cosine signal shown in [link] , reveals that all signals consist of a sequence of delayed andscaled unit samples. Because the value of a sequence at each integer $m$ is denoted by $s(m)$ and the unit sample delayed to occur at $m$ is written $\delta (n-m)$ , we can decompose any signal as a sum of unit samples delayed to the appropriate location and scaled bythe signal value.

Discrete-time systems can act on discrete-time signals in wayssimilar to those found in analog signals and systems. Because of the role of software in discrete-time systems, many moredifferent systems can be envisioned and “constructed” with programs than can be with analog signals. In fact, a specialclass of analog signals can be converted into discrete-time signals, processed with software, and converted back into ananalog signal, all without the incursion of error. For such signals, systems can be easily produced in software, withequivalent analog realizations difficult, if not impossible, to design.

Symbolic-valued signals

Another interesting aspect of discrete-time signals is thattheir values do not need to be real numbers. We do have real-valued discrete-time signals like the sinusoid, but wealso have signals that denote the sequence of characters typed on the keyboard. Such characters certainly aren't realnumbers, and as a collection of possible signal values, they have little mathematical structure other than that they aremembers of a set. More formally, each element of the symbolic-valued signal $s(n)$ takes on one of the values $\{a_1, \dots , a_{\mathrm{K}}\}$ which comprise the alphabet $A$ . This technical terminology does not mean we restrict symbols to being members of the Englishor Greek alphabet. They could represent keyboard characters, bytes (8-bit quantities), integers that convey dailytemperature. Whether controlled by software or not, discrete-time systems are ultimately constructed from digitalcircuits, which consist entirely of analog circuit elements. Furthermore, the transmission andreception of discrete-time signals, like e-mail, is accomplished with analog signals and systems. Understandinghow discrete-time and analog signals and systems intertwine is perhaps the main goal of this course.