The Scientist and Engineer's Guide to
Digital Signal Processing
By Steven W. Smith, Ph.D.

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In the 1960s it was predicted that artificial intelligence would revolutionize the way humans interact with computers and other machines. It was believed that by the end of the century we would have robots cleaning our houses, computers driving our cars, and voice interfaces controlling the storage and retrieval of information. This hasn't happened; these abstract tasks are far more complicated than expected, and very difficult to carry out with the step-by-step logic provided by digital computers.

However, the last forty years have shown that computers are extremely capable in two broad areas, (1) data manipulation, such as word processing and database management, and (2) mathematical calculation, used in science, engineering, and Digital Signal Processing. All microprocessors can perform both tasks; however, it is difficult (expensive) to make a device that is optimized for both. There are technical tradeoffs in the hardware design, such as the size of the instruction set and how interrupts are handled. Even

more important, there are marketing issues involved: development and manufacturing cost, competitive position, product lifetime, and so on. As a broad generalization, these factors have made traditional microprocessors, such as the Pentium®, primarily directed at data manipulation. Similarly, DSPs are designed to perform the mathematical calculations needed in Digital Signal Processing.

Figure 28-1 lists the most important differences between these two categories. Data manipulation involves storing and sorting information. For instance, consider a word processing program. The basic task is to store the information (typed in by the operator), organize the information (cut and paste, spell checking, page layout, etc.), and then retrieve the information (such as saving the document on a floppy disk or printing it with a laser printer). These tasks are accomplished by moving data from one location to another, and testing for inequalities (A=B, A<B, etc.). As an example, imagine sorting a list of words into alphabetical order. Each word is represented by an 8 bit number, the ASCII value of the first letter in the word. Alphabetizing involved rearranging the order of the words until the ASCII values continually increase from the beginning to the end of the list. This can be accomplished by repeating two steps over-and-over until the alphabetization is complete. First, test two adjacent entries for being in alphabetical order (IF A>B THEN ...). Second, if the two entries are not in alphabetical order, switch them so that they are (A⇄B). When this two step process is repeated many times on all adjacent pairs, the list will eventually become alphabetized.

As another example, consider how a document is printed from a word processor. The computer continually tests the input device (mouse or keyboard) for the binary code that indicates "print the document." When this code is detected, the program moves the data from the computer's memory to the printer. Here we have the same two basic operations: moving data and inequality testing. While mathematics is occasionally used in this type of application, it is infrequent and does not significantly affect the overall execution speed.

In comparison, the execution speed of most DSP algorithms is limited almost completely by the number of multiplications and additions required. For example, Fig. 28-2 shows the implementation of an FIR digital filter, the most common DSP technique. Using the standard notation, the input signal is referred to by x[ ], while the output signal is denoted by y[ ]. Our task is to calculate the sample at location n in the output signal, i.e., y[n]. An FIR filter performs this calculation by multiplying appropriate samples from the input signal by a group of coefficients, denoted by: a₀, a₁, a₂, a₃, …, and then adding the products. In equation form, y[n] is found by:

This is simply saying that the input signal has been convolved with a filter kernel (i.e., an impulse response) consisting of: a₀, a₁, a₂, a₃, …. Depending on the application, there may only be a few coefficients in the filter kernel, or many thousands. While there is some data transfer and inequality evaluation in this algorithm, such as to keep track of the intermediate results and control the loops, the math operations dominate the execution time.

In addition to preforming mathematical calculations very rapidly, DSPs must also have a predictable execution time. Suppose you launch your desktop computer on some task, say, converting a word-processing document from one form to another. It doesn't matter if the processing takes ten milliseconds or ten seconds; you simply wait for the action to be completed before you give the computer its next assignment.

In comparison, most DSPs are used in applications where the processing is continuous, not having a defined start or end. For instance, consider an engineer designing a DSP system for an audio signal, such as a hearing aid. If the digital signal is being received at 20,000 samples per second, the DSP must be able to maintain a sustained throughput of 20,000 samples per second. However, there are important reasons not to make it any faster than necessary. As the speed increases, so does the cost, the power consumption, the design difficulty, and so on. This makes an accurate knowledge of the execution time critical for selecting the proper device, as well as the algorithms that can be applied.